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<front>
<journal-meta>
<journal-id journal-id-type="pmc">CMC</journal-id>
<journal-id journal-id-type="nlm-ta">CMC</journal-id>
<journal-id journal-id-type="publisher-id">CMC</journal-id>
<journal-title-group>
<journal-title>Computers, Materials &#x0026; Continua</journal-title>
</journal-title-group>
<issn pub-type="epub">1546-2226</issn>
<issn pub-type="ppub">1546-2218</issn>
<publisher>
<publisher-name>Tech Science Press</publisher-name>
<publisher-loc>USA</publisher-loc>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">17527</article-id>
<article-id pub-id-type="doi">10.32604/cmc.2021.017527</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Article</subject>
</subj-group>
</article-categories>
<title-group>
<article-title>Research on Face Anti-Spoofing Algorithm Based on Image Fusion</article-title>
<alt-title alt-title-type="left-running-head">Research on Face Anti-Spoofing Algorithm Based on Image Fusion</alt-title>
<alt-title alt-title-type="right-running-head">Research on Face Anti-Spoofing Algorithm Based on Image Fusion</alt-title>
</title-group>
<contrib-group content-type="authors">
<contrib id="author-1" contrib-type="author">
<name name-style="western">
<surname>Yu</surname>
<given-names>Pingping</given-names>
</name>
<xref ref-type="aff" rid="aff-1">1</xref>
</contrib>
<contrib id="author-2" contrib-type="author">
<name name-style="western">
<surname>Wang</surname>
<given-names>Jiayu</given-names>
</name>
<xref ref-type="aff" rid="aff-1">1</xref></contrib>
<contrib id="author-3" contrib-type="author" corresp="yes">
<name name-style="western">
<surname>Cao</surname>
<given-names>Ning</given-names>
</name>
<xref ref-type="aff" rid="aff-2">2</xref></contrib>
<contrib id="author-4" contrib-type="author">
<name name-style="western">
<surname>Dintera</surname>
<given-names>Heiner</given-names>
</name>
<xref ref-type="aff" rid="aff-3">3</xref></contrib>
<aff id="aff-1"><label>1</label><institution>School of Information Science and Engineering, Hebei University of Science and Technology</institution>, <addr-line>Shijiazhuang, 050000</addr-line>, <country>China</country></aff>
<aff id="aff-2"><label>2</label><institution>School of Internet of Things and Software Technology, Wuxi Vocational College of Science and Technology</institution>, <addr-line>Wuxi, 214028</addr-line>, <country>China</country></aff>
<aff id="aff-3"><label>3</label><institution>German-Russian Institute of Advanced Technologies</institution>, <addr-line>Karan, 420126</addr-line>, <country>Russia</country></aff>
</contrib-group>
<author-notes><corresp id="cor1">&#x002A;Corresponding Author: Ning Cao. Email: <email>ning.cao2008@hotmail.com</email></corresp></author-notes>
<pub-date pub-type="epub" date-type="pub" iso-8601-date="2021-03-17"><day>17</day><month>03</month><year>2021</year>
</pub-date>
<volume>68</volume>
<issue>3</issue>
<fpage>3861</fpage>
<lpage>3876</lpage>
<history>
<date date-type="received"><day>01</day><month>02</month><year>2021</year></date>
<date date-type="accepted"><day>05</day><month>03</month><year>2021</year></date>
</history>
<permissions>
<copyright-statement>&#x00A9; 2021 Yu et al.</copyright-statement>
<copyright-year>2021</copyright-year>
<copyright-holder>Yu et al.</copyright-holder>
<license xlink:href="https://creativecommons.org/licenses/by/4.0/">
<license-p>This work is licensed under a <ext-link ext-link-type="uri" xlink:type="simple" xlink:href="https://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution 4.0 International License</ext-link>, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.</license-p>
</license>
</permissions>
<self-uri content-type="pdf" xlink:href="TSP_CMC_17527.pdf"></self-uri>
<abstract>
<p>Along with the rapid development of biometric authentication technology, face recognition has been commercially used in many industries in recent years. However, it cannot be ignored that face recognition-based authentication techniques can be easily spoofed using various types of attacks such photographs, videos or forged 3D masks. In order to solve this problem, this work proposed a face anti-fraud algorithm based on the fusion of thermal infrared images and visible light images. The normal temperature distribution of the human face is stable and characteristic, and the important physiological information of the human body can be observed by the infrared thermal images. Therefore, based on the thermal infrared image, the pixel value of the pulse sensitive area of the human face is collected, and the human heart rate signal is detected to distinguish between real faces and spoofing faces. In order to better obtain the texture features of the face, an image fusion algorithm based on DTCWT and the improved Roberts algorithm is proposed. Firstly, DTCWT is used to decompose the thermal infrared image and visible light image of the face to obtain high-and low-frequency subbands. Then, the method based on region energy and the improved Roberts algorithm are then used to fuse the coefficients of the high- and low-frequency subbands. Finally, the DTCWT inverse transform is used to obtain the fused image containing the facial texture features. Face recognition is carried out on the fused image to realize identity authentication. Experimental results show that this algorithm can effectively resist attacks from photos, videos or masks. Compared with the use of visible light images alone for face recognition, this algorithm has higher recognition accuracy and better robustness.</p>
</abstract>
<kwd-group kwd-group-type="author">
<kwd>Anti-spoofing</kwd>
<kwd>infrared thermal images</kwd>
<kwd>image fusion</kwd>
<kwd>heart rate detection</kwd>
</kwd-group>
</article-meta>
</front>
<body>
<sec id="s1">
<label>1</label>
<title>Introduction</title>
<p>With the development of biometrics, face recognition plays a pivotal role in applications such as identity recognition systems, criminal justice database systems, and public surveillance systems. The subsequent face spoofing attacks have also increased sharply. Attackers often use photos, videos, 3D modeling, masks, and other methods to imitate real human faces and obtain system access authorization for illegal intrusion and face recognition. This poses a serious threat to the security of the face recognition system. It is a necessary research topic to accurately judge the authenticity of human faces and identify human facial information to resist these complex and diverse deception attacks [<xref ref-type="bibr" rid="ref-1">1</xref>].</p>
<p>Face anti-spoofing detection systems are mainly divided into three categories: systems based on specific equipment, systems based on human-computer interaction, and systems based on pure algorithms.</p>
<p>There are relatively many methods based on human-computer interaction to prevent spoofing attacks. For example, Alsufyani et al. [<xref ref-type="bibr" rid="ref-2">2</xref>] used the random movement of infrared light to track the relative movement trajectory of the human eye. Singh et al. [<xref ref-type="bibr" rid="ref-3">3</xref>] and Pan et al. [<xref ref-type="bibr" rid="ref-4">4</xref>] proposed to detect the user&#x2019;s blinking and lip movements to resist people face spoofing attack; Tirunagari et al. [<xref ref-type="bibr" rid="ref-5">5</xref>] used dynamic correlation models to preprocess the video to extract texture features. The disadvantage of this type of detection method which seeks user cooperation is that it takes too long to detect the user&#x2019;s specified action, and it needs the user to request coordinated action. This will affect the user&#x2019;s experience.</p>
<p>Relying on pure algorithms for rapid detection and resolution through user videos or images is also a research hotspot. Wen et al. [<xref ref-type="bibr" rid="ref-6">6</xref>] proposed an algorithm combining image deformation analysis features; Pinto et al. [<xref ref-type="bibr" rid="ref-7">7</xref>] proposed a method of visual frequency analysis to detect video Face attacks; M&#x00E4;&#x00E4;tt&#x00E4; et al. [<xref ref-type="bibr" rid="ref-8">8</xref>] used LBP features to complete the detection of spoofing attacks; Alhassan et al. [<xref ref-type="bibr" rid="ref-9">9</xref>] combined DMD, LBP, and SVM to perform a liveness test score. Li et al. [<xref ref-type="bibr" rid="ref-10">10</xref>] proposed a face recognition algorithm based on LBP-EHMM; Wild et al. [<xref ref-type="bibr" rid="ref-11">11</xref>] proposed a detection algorithm based on bagging strategy; Pinto et al. [<xref ref-type="bibr" rid="ref-12">12</xref>] proposed a face activity detection method based on visual rhythm analysis. This type of texture feature-based detection method is based on gray-scale image extraction. The extracted features are not comprehensive enough, which affects the final detection result and has limited accuracy. Lee et al. [<xref ref-type="bibr" rid="ref-13">13</xref>] identify real faces from photos by analyzing data. Zhang et al. [<xref ref-type="bibr" rid="ref-14">14</xref>] used an adaptive ellipse fitting method to roughly determine the face area. Then, the study performed AdaBoost-based classification according to face template matching and face skin color distribution statistics, and finally detected facial occlusion. Xia et al. [<xref ref-type="bibr" rid="ref-15">15</xref>] proposed face occlusion detection based on a convolutional neural network. The network model was trained through a large number of occlusion samples. The image to be detected was input to the network, and the result of detecting whether the left and right eyes, nose and mouth were occluded was directly outputted. Kim et al. [<xref ref-type="bibr" rid="ref-16">16</xref>] proposed a face activity detection method for face spoofing attacks on mobile phones. According to the difference in the diffusion speed of reflected light from fake photos and live images, a real-time live detection based on the diffusion speed of reflected light from a single image was proposed. They used the following method: i) introduce the total variation flow to obtain the diffusion speed; ii) use the different diffusion speeds of the reflected light from the active skin and the fake face to distinguish whether there is activity; iii) use the LSP code to extract the speed feature vector on the reflected light diffusion speed distribution map, and iv) use the SVM classifier to determine whether the image comes from a living human face.</p>
<p>Bao et al. [<xref ref-type="bibr" rid="ref-17">17</xref>] proposed a face anti-spoofing detection algorithm that used fusion color texture features using the difference in color features and detailed texture features between real faces and spoofing attack images. The algorithm mainly used infrared at night and lacked spectrum collection color information. Li et al. [<xref ref-type="bibr" rid="ref-18">18</xref>] proposed a face anti-spoofing method based on P-CNN and ELM to detect 2D spoofing attacks. Combining traditional digital images, Zhang et al. [<xref ref-type="bibr" rid="ref-19">19</xref>] proposed a forensic algorithm for face photos and video spoofing attacks based on the recursive elimination of color texture Markov features and support vector machine features. However, the algorithm could not detect whether the face uses 3D means such as silicone masks. Relying on pure algorithms for anti-spoofing detection of faces, the complexity of the algorithm is relatively high. There are certain restrictions on the detection environment and imaging CMOS cameras, which leads to an increase in the algorithm complexity of the entire face recognition process.</p>
<p>Seeking user cooperation and relying on pure algorithms are not ideal detection methods. With the popularity of infrared cameras, depth cameras and other equipments and the reduction of costs, anti-spoofing algorithms based on specific equipment have become the mainstream. Sun et al. [<xref ref-type="bibr" rid="ref-20">20</xref>] fused the human eye features of color and infrared to detect whether the driver is tired; Wang et al. [<xref ref-type="bibr" rid="ref-21">21</xref>] proposed a three-dimensional face recognition method with elastic matching of the radial curve of the face, but the user experience was lacking.</p>
<fig id="fig-1">
<label>Figure 1</label>
<caption>
<title>General flow chart of the face anti-spoofing system</title>
</caption><graphic mimetype="image" mime-subtype="png" xlink:href="fig-1.png"/>
</fig>
<p>This paper proposes a face anti-fraud algorithm based on the fusion of thermal infrared images and visible light images. By detecting the pulse sensitive area in the infrared thermal image, the grayscale value signal of the image is statistically analyzed, and the heart rate waveform is calculated to distinguish real and fake faces. For faces, we use the method based on dual-tree complex wavelet transform (DTCWT) and improved Roberts operator to fuse the images, and finally identify the identity. The overall flow chart is shown in <xref ref-type="fig" rid="fig-1">Fig. 1</xref>.</p>
<p>The rest of this article is arranged as follows. The second part introduces real and fake face detection method based on life information analysis. The third part introduces the image fusion based on DTCWT and the improved Roberts algorithm. The fourth part introduces face recognition and gives the results of this method.</p>
</sec>
<sec id="s2">
<label>2</label>
<title>Real and Fake Face Detection Method Based on Life Information Analysis</title>
<p>The difference between a real face and a deceptive face is that a real face has some vital information, such as capillaries and pulse. The surface of the human body can radiate infrared thermal energy, and the energy is mainly concentrated in the infrared band with a wavelength of 9.312<inline-formula id="ieqn-1"><!--<alternatives><inline-graphic xlink:href="ieqn-1.png"/>--><!--<tex-math id="tex-ieqn-1"><![CDATA[$\sim$]]></tex-math>--><mml:math id="mml-ieqn-1"><mml:mo>~</mml:mo></mml:math><!--</alternatives>--></inline-formula>9.464 <inline-formula id="ieqn-2"><!--<alternatives><inline-graphic xlink:href="ieqn-2.png"/>--><!--<tex-math id="tex-ieqn-2"><![CDATA[$\mu$]]></tex-math>--><mml:math id="mml-ieqn-2"><mml:mi>&#x03BC;</mml:mi></mml:math><!--</alternatives>--></inline-formula>m [<xref ref-type="bibr" rid="ref-22">22</xref>]. The normal temperature distribution of the human face is stable and characteristic, and the important physiological information of the human body can be observed by using the infrared thermal image. Since there are a large number of capillaries in the face, the blood in the blood vessels changes with the beating of the heart. When the heart contracts, the blood increases and the heat radiation energy increases.When the heart relaxes, the blood decreases and the heat radiation energy decreases. Therefore, the pixel value of the infrared thermal image reflected on the infrared thermal imager will fluctuate with the heart&#x2019;s beating, while the thermal infrared image of the spoofing face does not have this phenomenon.</p>
<sec id="s2_1">
<label>2.1</label>
<title>Heart Rate Signal Detection</title>
<p>Before signal processing, we perform grayscale value processing on the facial thermal image video taken by the infrared thermal imager, and the conversion formula is</p>
<p><disp-formula id="eqn-1">
<label>(1)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-1.png"/>-->
<!--<tex-math id="tex-eqn-1"><![CDATA[$$\begin{equation}
G_{d}=0.299R+0.587G+0.114B
 \label{eqn-1}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-1" display="block"><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>299</mml:mn><mml:mi>R</mml:mi><mml:mo>+</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>587</mml:mn><mml:mi>G</mml:mi><mml:mo>+</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>114</mml:mn><mml:mi>B</mml:mi></mml:math><!--</alternatives>--></disp-formula></p>
<p>where <italic>G<sub>d</sub></italic> is the grayscale value after conversion, and <italic>R</italic>, <inline-formula id="ieqn-3"><!--<alternatives><inline-graphic xlink:href="ieqn-3.png"/>--><!--<tex-math id="tex-ieqn-3"><![CDATA[$\mathrm{G}$]]></tex-math>--><mml:math id="mml-ieqn-3"><mml:mstyle mathvariant="normal"><mml:mi>G</mml:mi></mml:mstyle></mml:math><!--</alternatives>--></inline-formula> and <italic>B</italic> are the red, green, and blue component values of the pixel before conversion.</p>
<p>We perform face detection on the input test video and select a region of interest (ROI) to reduce the interference of light changes caused by other factors. Since the forehead part of the gray-scale thermal image of the human face is most sensitive to pulse beats, the forehead part of the grayscale thermal image is selected as ROI for processing in this article. It is mainly divided into two steps:</p>
<p>1) Set ROI template for grayscale thermal image.</p>
<p>In the first few frames of the gray-scale image sequence, we select a relatively clear facial image. We set the distance between the center points of the pupils of the two eyes to 4<italic>d</italic>. A <inline-formula id="ieqn-4"><!--<alternatives><inline-graphic xlink:href="ieqn-4.png"/>--><!--<tex-math id="tex-ieqn-4"><![CDATA[$3d \times 3d/2$]]></tex-math>--><mml:math id="mml-ieqn-4"><mml:mn>3</mml:mn><mml:mi>d</mml:mi><mml:mo>&#x00D7;</mml:mo><mml:mn>3</mml:mn><mml:mi>d</mml:mi><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:math><!--</alternatives>--></inline-formula> rectangle (<italic>r</italic><sub>1</sub>) in the center of the forehead is selected <italic>d</italic> above the straight line from the pupils, as shown in <xref ref-type="fig" rid="fig-2">Fig. 2</xref>.</p>
<fig id="fig-2">
<label>Figure 2</label>
<caption>
<title>ROI template and rectangular area 
 
</title>
</caption><graphic mimetype="image" mime-subtype="png" xlink:href="fig-2.png"/>
</fig>
<p>2) After the ROI template is obtained, extract the sensitive areas of each frame of the gray image sequence.</p>
<p>Due to the large noise interference of individual frames, it is impossible to ensure that the <italic>r</italic><sub>1</sub> region information of each frame image is complete and effective. Therefore, the sensitive area of each frame of the image needs to be selected according to the template <italic>r</italic><sub>1</sub>. In each frame of the gray image sequence, a rectangular block <italic>R<sub>n</sub></italic> of size <inline-formula id="ieqn-5"><!--<alternatives><inline-graphic xlink:href="ieqn-5.png"/>--><!--<tex-math id="tex-ieqn-5"><![CDATA[$5d \times 3d$]]></tex-math>--><mml:math id="mml-ieqn-5"><mml:mn>5</mml:mn><mml:mi>d</mml:mi><mml:mo>&#x00D7;</mml:mo><mml:mn>3</mml:mn><mml:mi>d</mml:mi></mml:math><!--</alternatives>--></inline-formula> containing the ROI template <italic>r</italic><sub>1</sub> in the middle is found, as shown in the dotted area in <xref ref-type="fig" rid="fig-2">Fig. 2</xref>.</p>
<p>The normalized cross-correlation function is used in the <italic>R<sub>n</sub></italic> area to examine the matching of each <inline-formula id="ieqn-6"><!--<alternatives><inline-graphic xlink:href="ieqn-6.png"/>--><!--<tex-math id="tex-ieqn-6"><![CDATA[$3d \times 3d/2$]]></tex-math>--><mml:math id="mml-ieqn-6"><mml:mn>3</mml:mn><mml:mi>d</mml:mi><mml:mo>&#x00D7;</mml:mo><mml:mn>3</mml:mn><mml:mi>d</mml:mi><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:math><!--</alternatives>--></inline-formula> candidate block <italic>r<sub>n</sub></italic> with the ROI template <italic>r</italic><sub>1</sub> to ensure that each frame image can select the most accurate sensitive area of grayscale value. The normalized cross-correlation function formula is obtained as follows</p>
<p><disp-formula id="eqn-2">
<label>(2)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-2.png"/>-->
<!--<tex-math id="tex-eqn-2"><![CDATA[$$\begin{equation}
\gamma \left(u,\,v\right)=\frac{\sum_{x,\,y} \left[f \left(x,\,y\right)-\overline{f}_{u,\,v}\right]
 \left[t \left(x-u,\,y-v\right)-\overline{t}\right]}{ \left\{\sum_{x,\,y} \left[f \left(x,\,y\right)-\overline{f}_{u,\,v}\right]^{2}\cdot \sum_{x,\,y} \left[t \left(x-u,\,y-v\right)-\overline{t}\right]^{2}\right\}^{1/2}}
 \label{eqn-2}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-2" display="block"><mml:mi>&#x03B3;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>v</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mover accent="false"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mo accent="true">&#x00AF;</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>-</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi><mml:mo>-</mml:mo><mml:mi>v</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mover accent="false"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo accent="true">&#x00AF;</mml:mo></mml:mover></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mover accent="false"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mo accent="true">&#x00AF;</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>-</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi><mml:mo>-</mml:mo><mml:mi>v</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mover accent="false"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo accent="true">&#x00AF;</mml:mo></mml:mover></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:math><!--</alternatives>--></disp-formula></p>
<p>Among them, <inline-formula id="ieqn-7"><!--<alternatives><inline-graphic xlink:href="ieqn-7.png"/>--><!--<tex-math id="tex-ieqn-7"><![CDATA[$\mu$]]></tex-math>--><mml:math id="mml-ieqn-7"><mml:mi>&#x03BC;</mml:mi></mml:math><!--</alternatives>--></inline-formula> refers to the correlation interval on the abscissa axis of the grayscale image, <inline-formula id="ieqn-8"><!--<alternatives><inline-graphic xlink:href="ieqn-8.png"/>--><!--<tex-math id="tex-ieqn-8"><![CDATA[$\upsilon$]]></tex-math>--><mml:math id="mml-ieqn-8"><mml:mi>&#x03C5;</mml:mi></mml:math><!--</alternatives>--></inline-formula> refers to the correlation interval on the ordinate axis, <inline-formula id="ieqn-9"><!--<alternatives><inline-graphic xlink:href="ieqn-9.png"/>--><!--<tex-math id="tex-ieqn-9"><![CDATA[$f(x, y)$]]></tex-math>--><mml:math id="mml-ieqn-9"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> represents the candidate block <italic>r<sub>n</sub></italic>, <inline-formula id="ieqn-10"><!--<alternatives><inline-graphic xlink:href="ieqn-10.png"/>--><!--<tex-math id="tex-ieqn-10"><![CDATA[$t(x, y)$]]></tex-math>--><mml:math id="mml-ieqn-10"><mml:mi>t</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> represents the pixel value of the ROI template <italic>r</italic><sub>1</sub>, <inline-formula id="ieqn-11"><!--<alternatives><inline-graphic xlink:href="ieqn-11.png"/>--><!--<tex-math id="tex-ieqn-11"><![CDATA[$\overline{f}_{\mu , \upsilon }$]]></tex-math>--><mml:math id="mml-ieqn-11"><mml:msub><mml:mrow><mml:mover accent="false" class="mml-overline"><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mo accent="true">&#x00AF;</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>&#x03BC;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x03C5;</mml:mi></mml:mrow></mml:msub></mml:math><!--</alternatives>--></inline-formula> and <inline-formula id="ieqn-12"><!--<alternatives><inline-graphic xlink:href="ieqn-12.png"/>--><!--<tex-math id="tex-ieqn-12"><![CDATA[$\overline{t}$]]></tex-math>--><mml:math id="mml-ieqn-12"><mml:mover accent="false" class="mml-overline"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo accent="true">&#x00AF;</mml:mo></mml:mover></mml:math><!--</alternatives>--></inline-formula> are the mean values of the pixel values of the candidate block <italic>r<sub>n</sub></italic> and the ROI template <italic>r</italic><sub>1</sub>, respectively.</p>
<p>The obtained normalized cross-correlation coefficients of each candidate block <italic>r<sub>n</sub></italic> are selected, and the block <italic>r<sub>n</sub></italic> with the largest absolute value and exceeding the specified threshold is used as the sensitive area of the frame. The grayscale value of the sensitive area of each frame of the gray image sequence is averaged to obtain <italic>g<sub>n</sub></italic>. These average values are arranged in chronological order to get the grayscale value waveform, which is the reflection of the heart rate signal. A set of grayscale value waveforms is shown in <xref ref-type="fig" rid="fig-3">Fig. 3</xref>.</p>
<fig id="fig-3">
<label>Figure 3</label>
<caption>
<title>Grayscale value waveform of real faces 
 
</title>
</caption><graphic mimetype="image" mime-subtype="png" xlink:href="fig-3.png"/>
</fig>
<p>With the acquisition of a set of heart rates, the average value is calculated and stored in the array. Finally, the calculated variance is compared with the threshold to get the result, which is used to judge whether the given test video is a spoofing attack. Then, we enter the face recognition process on the premise that it is a real face.</p>
</sec>
<sec id="s2_2">
<label>2.2</label>
<title>Anti-Spoofing Attack Method</title>
<p>The experiment uses an infrared thermal imager of model G100EX, the temperature measurement range is &#x2212;40<inline-formula id="ieqn-13"><!--<alternatives><inline-graphic xlink:href="ieqn-13.png"/>--><!--<tex-math id="tex-ieqn-13"><![CDATA[$^{\circ}$]]></tex-math>--><mml:math id="mml-ieqn-13"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula>C to 1500<inline-formula id="ieqn-14"><!--<alternatives><inline-graphic xlink:href="ieqn-14.png"/>--><!--<tex-math id="tex-ieqn-14"><![CDATA[$^{\circ}$]]></tex-math>--><mml:math id="mml-ieqn-14"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula>C, the temperature resolution is 0.04 at 30<inline-formula id="ieqn-15"><!--<alternatives><inline-graphic xlink:href="ieqn-15.png"/>--><!--<tex-math id="tex-ieqn-15"><![CDATA[$^{\circ}$]]></tex-math>--><mml:math id="mml-ieqn-15"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula>C, the pixel is <inline-formula id="ieqn-16"><!--<alternatives><inline-graphic xlink:href="ieqn-16.png"/>--><!--<tex-math id="tex-ieqn-16"><![CDATA[$320~\mathrm{(H)}\times 240$]]></tex-math>--><mml:math id="mml-ieqn-16"><mml:mn>320</mml:mn><mml:mspace width=".3em" /><mml:mstyle mathvariant="normal"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mstyle><mml:mo>&#x00D7;</mml:mo><mml:mn>240</mml:mn></mml:math><!--</alternatives>--></inline-formula> (V), and the response wavelength is <inline-formula id="ieqn-17"><!--<alternatives><inline-graphic xlink:href="ieqn-17.png"/>--><!--<tex-math id="tex-ieqn-17"><![CDATA[$8{\sim}14~\mu \mathrm{m}$]]></tex-math>--><mml:math id="mml-ieqn-17"><mml:mn>8</mml:mn><mml:mo>~</mml:mo><mml:mn>14</mml:mn><mml:mspace width=".3em" /><mml:mi>&#x03BC;</mml:mi><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle></mml:math><!--</alternatives>--></inline-formula>.</p>
<p>In a room with a temperature of 22<inline-formula id="ieqn-18"><!--<alternatives><inline-graphic xlink:href="ieqn-18.png"/>--><!--<tex-math id="tex-ieqn-18"><![CDATA[$^{\circ}$]]></tex-math>--><mml:math id="mml-ieqn-18"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula>C, thermal infrared video collection was performed on the experimenter himself, the paper photos of the experimenter, the electronic video of the experimenter, and the mask-wearing experimenter. Considering that the attacker may simulate the mask to the temperature of the human body to attack, the temperature of the mask is increased to approximately 37 degrees for shooting. The collection methods and the presentation results are shown in <xref ref-type="fig" rid="fig-4">Fig. 4</xref>.</p>
<fig id="fig-4">
<label>Figure 4</label>
<caption>
<title>Collected experimental data. ((a) A group of real face images; (b) a group of printed face images; (c) a group of electronic videos; (d) a group of faceswearing masks)</title>
</caption><graphic mimetype="image" mime-subtype="png" xlink:href="fig-4.png"/>
</fig>
<p>According to the abovementioned algorithm, the grayscale value waveform acquisition is performed for the three kinds of deception attacks, and the results are shown in <xref ref-type="fig" rid="fig-5">Fig. 5</xref>.</p>
<fig id="fig-5">
<label>Figure 5</label>
<caption>
<title>The grayscale value waveforms of various attacks. (a) Grayscale value waveform of electronic photo attack; (b) grayscale value waveform of electronic video attack; (c) grayscale value waveform of mask attack</title>
</caption><graphic mimetype="image" mime-subtype="png" xlink:href="fig-5.png"/>
</fig>
<p>From the waveform of <xref ref-type="fig" rid="fig-3">Fig. 3</xref>, it can be seen that the grayscale value of the real face is between 80 and 120, and the fluctuation range is large. The grayscale value of the photo attack and the video attack is quite different from the grayscale value of the real face. It can be clearly distinguished. The grayscale value obtained after wearing the mask is slightly different from the grayscale value of the real face, which is difficult to distinguish directly, but the waveforms of the three deception attacks are relatively stable compared with the waveforms of the real face. Therefore, we use the standard deviation to further calculate the stability of the waveform to determine whether it is a mask attack. The standard deviation formula is obtained as follows:</p>
<p><disp-formula id="eqn-3">
<label>(3)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-3.png"/>-->
<!--<tex-math id="tex-eqn-3"><![CDATA[$$\begin{equation}
\sigma =\sqrt{\frac{1}{N}\sum_{i=1}^{N}({g_{i}}-\mu )^{2}}
 \label{eqn-3}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-3" display="block"><mml:mi>&#x03C3;</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mstyle displaystyle='true'><mml:mstyle displaystyle='true'><mml:munderover><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo lspace='0pt' rspace='0pt'>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:munderover></mml:mstyle></mml:mstyle><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>&#x03BC;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt></mml:math><!--</alternatives>--></disp-formula></p>
<p>where N is the number of frames of the image, <italic>g<sub>i</sub></italic> is the grayscale value of the sensitive area of each frame, and <inline-formula id="ieqn-19"><!--<alternatives><inline-graphic xlink:href="ieqn-19.png"/>--><!--<tex-math id="tex-ieqn-19"><![CDATA[$\mu$]]></tex-math>--><mml:math id="mml-ieqn-19"><mml:mi>&#x03BC;</mml:mi></mml:math><!--</alternatives>--></inline-formula> is the average value of the grayscale value of the sensitive area of the N frames.</p>
<p>The results are shown in <xref ref-type="table" rid="table-1">Tab. 1</xref>. The standard deviation of real faces is generally above 6, and the standard deviation of photos, videos and masked faces is below 4. Therefore, the threshold for distinguishing true and false faces can be set between 4 and Between 6.</p>
<table-wrap id="table-1">
<label>Table 1</label>
<caption>
<title>The average and standard deviation of grayscale values of real and fake faces</title>
</caption>
<table>
<colgroup>
<col/>
<col/>
<col/>
<col/>
<col/>
</colgroup>
<thead>
<tr>
<th>Data group</th>
<th>Average grayscale value of the photo face</th>
<th>Average grayscale value of the video face</th>
<th>Average grayscale value of the mask face</th>
<th>Average grayscale value of the real face</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>21.191</td>
<td>22.196</td>
<td>113.214</td>
<td>94.691</td>
</tr>
<tr>
<td>2</td>
<td>20.865</td>
<td>23.865</td>
<td>113.799</td>
<td>85.682</td>
</tr>
<tr>
<td>3</td>
<td>20.356</td>
<td>24.356</td>
<td>113.612</td>
<td>93.584</td>
</tr>
<tr>
<td>4</td>
<td>20.034</td>
<td>24.031</td>
<td>113.147</td>
<td>96.250</td>
</tr>
<tr>
<td>5</td>
<td>20.168</td>
<td>24.168</td>
<td>112.368</td>
<td>100.358</td>
</tr>
<tr>
<td>6</td>
<td>21.999</td>
<td>24.686</td>
<td>112.687</td>
<td>101.632</td>
</tr>
<tr>
<td><inline-formula id="ieqn-20"><!--<alternatives><inline-graphic xlink:href="ieqn-20.png"/>--><!--<tex-math id="tex-ieqn-20"><![CDATA[$\vdots$]]></tex-math>--><mml:math id="mml-ieqn-20"><mml:mo>&#x22EE;</mml:mo></mml:math><!--</alternatives>--></inline-formula></td>
<td><inline-formula id="ieqn-21"><!--<alternatives><inline-graphic xlink:href="ieqn-21.png"/>--><!--<tex-math id="tex-ieqn-21"><![CDATA[$\vdots$]]></tex-math>--><mml:math id="mml-ieqn-21"><mml:mo>&#x22EE;</mml:mo></mml:math><!--</alternatives>--></inline-formula></td>
<td><inline-formula id="ieqn-22"><!--<alternatives><inline-graphic xlink:href="ieqn-22.png"/>--><!--<tex-math id="tex-ieqn-22"><![CDATA[$\vdots$]]></tex-math>--><mml:math id="mml-ieqn-22"><mml:mo>&#x22EE;</mml:mo></mml:math><!--</alternatives>--></inline-formula></td>
<td><inline-formula id="ieqn-23"><!--<alternatives><inline-graphic xlink:href="ieqn-23.png"/>--><!--<tex-math id="tex-ieqn-23"><![CDATA[$\vdots$]]></tex-math>--><mml:math id="mml-ieqn-23"><mml:mo>&#x22EE;</mml:mo></mml:math><!--</alternatives>--></inline-formula></td>
<td><inline-formula id="ieqn-24"><!--<alternatives><inline-graphic xlink:href="ieqn-24.png"/>--><!--<tex-math id="tex-ieqn-24"><![CDATA[$\vdots$]]></tex-math>--><mml:math id="mml-ieqn-24"><mml:mo>&#x22EE;</mml:mo></mml:math><!--</alternatives>--></inline-formula></td>
</tr>
<tr>
<td>Standard deviation</td>
<td>0.9258</td>
<td>1.5353</td>
<td>3.1228</td>
<td>6.6534</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>A face detection database is constructed which contains 30 real faces, 30 photo faces, 30 electronic video faces and 30 mask faces. The method proposed in this paper is verified by experiments on this database. The results are shown in <xref ref-type="table" rid="table-2">Tab. 2</xref>, which shows that the method proposed in this paper can effectively solve the problem of real face detection.</p>
<table-wrap id="table-2">
<label>Table 2</label>
<caption>
<title>Spoofing attack test results</title>
</caption>
<table>
<colgroup>
<col/>
<col/>
<col/>
<col/>
<col/>
</colgroup>
<thead>
<tr>
<th>Experimental data</th>
<th>Number of real faces</th>
<th>Number of spoofing attacks</th>
<th>Number of correct identification</th>
<th>The detection accuracy of real face and deception attack</th>
</tr>
</thead>
<tbody>
<tr>
<td><bold>120</bold></td>
<td>30</td>
<td>90</td>
<td>117</td>
<td>97.50%</td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
</sec>
<sec id="s3">
<label>3</label>
<title>Image Fusion Based on DTCWT and Improved Roberts Algorithm</title>
<p>The infrared thermal image contains the temperature information of the human body surface, which can be used to distinguish true and false faces through heart rate detection, but the lack of detailed information such as contour texture makes it impossible to recognize the identity of real faces. Visible light images contain rich detailed information, but they have low anti-interference ability under the influence of light, and important face information is often lost during face recognition. The thermal infrared image is not affected by light, and has good anti-interference ability and camouflage recognition ability. The fusion of the thermal infrared image and the visible light image can not only retain the rich detailed information in the visible light image and the temperature information in the infrared thermal image but also make up for the lack of light interference characteristic information [<xref ref-type="bibr" rid="ref-23">23</xref>]. Therefore, this paper uses the dual-tree complex wavelet transform to decompose the visible light image and the infrared thermal image, and obtain the high-frequency and low-frequency subband components of the same size as the source image. The low-frequency subband uses the method based on regional energy to fuse, and the edge enhancement method based on the improved Roberts operator is used for fusion of the high-frequency sub-band. Finally the dual-tree complex wavelet inverse transform is used to obtain the final fused image. The block diagram of the fusion algorithm is shown in the <xref ref-type="fig" rid="fig-6">Fig. 6</xref>.</p>
<fig id="fig-6">
<label>Figure 6</label>
<caption>
<title>Block diagram of the fusion algorithm 
 
</title>
</caption><graphic mimetype="image" mime-subtype="png" xlink:href="fig-6.png"/>
</fig>
<sec id="s3_1">
<label>3.1</label>
<title>Improved Roberts Algorithm</title>
<sec id="s3_1_1">
<label>3.1.1</label>
<title>Improved Roberts Operator</title>
<p>Roberts operator uses the difference between adjacent pixels in the diagonal direction (45<inline-formula id="ieqn-25"><!--<alternatives><inline-graphic xlink:href="ieqn-25.png"/>--><!--<tex-math id="tex-ieqn-25"><![CDATA[$^{\circ}$]]></tex-math>--><mml:math id="mml-ieqn-25"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula>, 135<inline-formula id="ieqn-26"><!--<alternatives><inline-graphic xlink:href="ieqn-26.png"/>--><!--<tex-math id="tex-ieqn-26"><![CDATA[$^{\circ}$]]></tex-math>--><mml:math id="mml-ieqn-26"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula> direction) in the <inline-formula id="ieqn-27"><!--<alternatives><inline-graphic xlink:href="ieqn-27.png"/>--><!--<tex-math id="tex-ieqn-27"><![CDATA[$2 \times 2$]]></tex-math>--><mml:math id="mml-ieqn-27"><mml:mn>2</mml:mn><mml:mo>&#x00D7;</mml:mo><mml:mn>2</mml:mn></mml:math><!--</alternatives>--></inline-formula> area to approximate the gradient amplitude for edge detection, and the magnitude of the gradient <inline-formula id="ieqn-28"><!--<alternatives><inline-graphic xlink:href="ieqn-28.png"/>--><!--<tex-math id="tex-ieqn-28"><![CDATA[$\mathrm{R} \left(x, y\right)$]]></tex-math>--><mml:math id="mml-ieqn-28"><mml:mstyle mathvariant="normal"><mml:mi>R</mml:mi></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> of a certain point <inline-formula id="ieqn-29"><!--<alternatives><inline-graphic xlink:href="ieqn-29.png"/>--><!--<tex-math id="tex-ieqn-29"><![CDATA[$f \left(x, y\right)$]]></tex-math>--><mml:math id="mml-ieqn-29"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> on the image is defined as follows:</p>
<p><disp-formula id="eqn-4">
<label>(4)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-4.png"/>-->
<!--<tex-math id="tex-eqn-4"><![CDATA[$$\begin{equation}
\mathrm{R} \left(x,\,y\right)=\sqrt{ \left[f \left(x+1,\,y+1\right)-f(x,\,y)\right]^{2}+
 \left[f \left(x+1,\,y\right)-f(x,\,y+1)\right]^{2}}
 \label{eqn-4}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-4" display="block"><mml:mstyle mathvariant="normal"><mml:mi>R</mml:mi></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt></mml:math><!--</alternatives>--></disp-formula></p>
<p>where we elect the threshold <italic>t</italic> and, when <inline-formula id="ieqn-30"><!--<alternatives><inline-graphic xlink:href="ieqn-30.png"/>--><!--<tex-math id="tex-ieqn-30"><![CDATA[$\mathrm{R} \left(x, y\right) > t$]]></tex-math>--><mml:math id="mml-ieqn-30"><mml:mstyle mathvariant="normal"><mml:mi>R</mml:mi></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x003E;</mml:mo><mml:mi>t</mml:mi></mml:math><!--</alternatives>--></inline-formula>, the pixel point <inline-formula id="ieqn-31"><!--<alternatives><inline-graphic xlink:href="ieqn-31.png"/>--><!--<tex-math id="tex-ieqn-31"><![CDATA[$f \left(x, y\right)$]]></tex-math>--><mml:math id="mml-ieqn-31"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> is determined to be an edge point. The traditional Roberts operator only calculates the information of 4 pixels in the diagonal direction, ignoring the pixel information in the vertical and horizontal directions. It is easy to cause missing edge pixels, and the threshold needs to be set manually, which has limitations [<xref ref-type="bibr" rid="ref-24">24</xref>].</p>
<p>Given the shortcomings of the traditional Roberts algorithm, this article considers adding vertical and horizontal direction information on the basis of the traditional Roberts operator (as shown in the <xref ref-type="fig" rid="fig-7">Fig. 7</xref>). Use the template in four directions of 0<inline-formula id="ieqn-32"><!--<alternatives><inline-graphic xlink:href="ieqn-32.png"/>--><!--<tex-math id="tex-ieqn-32"><![CDATA[$^{\circ}$]]></tex-math>--><mml:math id="mml-ieqn-32"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula>, 45<inline-formula id="ieqn-33"><!--<alternatives><inline-graphic xlink:href="ieqn-33.png"/>--><!--<tex-math id="tex-ieqn-33"><![CDATA[$^{\circ}$]]></tex-math>--><mml:math id="mml-ieqn-33"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula>, 90<inline-formula id="ieqn-34"><!--<alternatives><inline-graphic xlink:href="ieqn-34.png"/>--><!--<tex-math id="tex-ieqn-34"><![CDATA[$^{\circ}$]]></tex-math>--><mml:math id="mml-ieqn-34"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula>, 135<inline-formula id="ieqn-35"><!--<alternatives><inline-graphic xlink:href="ieqn-35.png"/>--><!--<tex-math id="tex-ieqn-35"><![CDATA[$^{\circ}$]]></tex-math>--><mml:math id="mml-ieqn-35"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula> in the <inline-formula id="ieqn-36"><!--<alternatives><inline-graphic xlink:href="ieqn-36.png"/>--><!--<tex-math id="tex-ieqn-36"><![CDATA[$3 \times 3$]]></tex-math>--><mml:math id="mml-ieqn-36"><mml:mn>3</mml:mn><mml:mo>&#x00D7;</mml:mo><mml:mn>3</mml:mn></mml:math><!--</alternatives>--></inline-formula> field. The template performs convolution operations on pixels.</p>
<fig id="fig-7">
<label>Figure 7</label>
<caption>
<title>Calculation improvement of the gradient amplitude of Roberts operator 
 
</title>
</caption><graphic mimetype="image" mime-subtype="png" xlink:href="fig-7.png"/>
</fig>
<p>The difference in the four directions is obtained as follows:</p>
<p><disp-formula id="eqn-5">
<label>(5)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-5.png"/>-->
<!--<tex-math id="tex-eqn-5"><![CDATA[$$\begin{equation}
 \left\{\begin{array}{l}
f_{0}=f \left(i,\,j+1\right)-f \left(i,\,j-1\right) \\[8pt] f_{45}=f \left(i-1,\,j+1\right)-f \left(i+1,\,j-1\right) \\[8pt] f_{90}=f \left(i-1,\,j\right)-f \left(i+1,\,j\right) \\[8pt] f_{135}=f \left(i-1,\,j-1\right)-f \left(i+1,\,j+1\right) \end{array}\right.
 \label{eqn-5}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-5" display="block"><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtable equalrows="false" columnlines="" equalcolumns="false"><mml:mtr><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>j</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>45</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>j</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>90</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>j</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>j</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>135</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>j</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo></mml:mo></mml:mrow></mml:math><!--</alternatives>--></disp-formula></p>
<p>Their corresponding convolution operators are obtained as follows:</p>
<p><disp-formula id="eqn-6">
<label>(6)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-6.png"/>-->
<!--<tex-math id="tex-eqn-6"><![CDATA[$$\begin{align}
& f_{0}\colon \left[\begin{array}{l@{\quad}l@{\quad}l}
0 & 0 & 0 \\[5pt]
1 & 0 & -1 \\[5pt]
0 & 0 & 0 \end{array}\right]\quad f_{45}\colon \left[\begin{array}{l@{\quad}l@{\quad}l}
0 & 0 & -1 \\[5pt]
0 & 0 & 0 \\[5pt]
1 & 0 & 0 \end{array}\right] \nonumber \\
& f_{90}\colon \left[\begin{array}{l@{\quad}l@{\quad}l}
0 & 1 & 0 \\[5pt]
0 & 0 & 0 \\[5pt]
0 & -1 & 0 \end{array}\right]\quad f_{135}\colon \left[\begin{array}{l@{\quad}l@{\quad}l}-1 & 0 & 0 \\[5pt]
0 & 0 & 0 \\[5pt] 0 & 0 & 1 \end{array}\right]
 \label{eqn-6}
\end{align}$$]]></tex-math>-->
<mml:math id="mml-eqn-6" display="block"><mml:mtable columnalign="left" columnspacing="1pt"><mml:mtr><mml:mtd></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mtable equalrows="false" columnlines="none none" equalcolumns="false"><mml:mtr><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mn>1</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn></mml:mtd></mml:mtr> </mml:mtable></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mspace width="1em"/><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>45</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mtable equalrows="false" columnlines="none none" equalcolumns="false"><mml:mtr><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mn>1</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn></mml:mtd></mml:mtr> </mml:mtable></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>90</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mtable equalrows="false" columnlines="none none" equalcolumns="false"><mml:mtr><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>1</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn></mml:mtd></mml:mtr> </mml:mtable></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mspace width="1em"/><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>135</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mtable equalrows="false" columnlines="none none" equalcolumns="false"><mml:mtr><mml:mtd columnalign="left"><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>0</mml:mn><mml:mspace width="1em"/></mml:mtd><mml:mtd columnalign="left"><mml:mn>1</mml:mn></mml:mtd></mml:mtr> </mml:mtable></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math><!--</alternatives>--></disp-formula></p>
<p>The improved Roberts operator considers the neighborhood information of pixels in 8 directions, which makes the edge extraction information more complete.</p>
</sec>
<sec id="s3_1_2">
<label>3.1.2</label>
<title>Median Filter Denoising</title>
<p>Although the improved Roberts operator can effectively extract the edge information, the noise generated under the interference of the complex environment in the infrared image and the visible light image will affect the gradient amplitude of pixel value, resulting in the extraction of the false edge formed by the noise. Therefore, we need to denoise the image. Median filtering can protect the edges of the signal from being blurred while filtering out noise. The algorithm is relatively simple and efficient. The two-dimensional median filter expression is obtained as follows:</p>
<p><disp-formula id="eqn-7">
<label>(7)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-7.png"/>-->
<!--<tex-math id="tex-eqn-7"><![CDATA[$$\begin{equation}
g \left(x,\,y\right)=med\{f \left(x-k,\,y-l\right),\,(k,\,l\in W)\}
 \label{eqn-7}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-7" display="block"><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>m</mml:mi><mml:mi>e</mml:mi><mml:mi>d</mml:mi><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi><mml:mo>-</mml:mo><mml:mi>l</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>l</mml:mi><mml:mo>&#x2208;</mml:mo><mml:mi>W</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math><!--</alternatives>--></disp-formula></p>
<p>Among them, <inline-formula id="ieqn-37"><!--<alternatives><inline-graphic xlink:href="ieqn-37.png"/>--><!--<tex-math id="tex-ieqn-37"><![CDATA[$f \left(x, y\right)$]]></tex-math>--><mml:math id="mml-ieqn-37"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> is the initial image, <inline-formula id="ieqn-38"><!--<alternatives><inline-graphic xlink:href="ieqn-38.png"/>--><!--<tex-math id="tex-ieqn-38"><![CDATA[$g \left(x, y\right)$]]></tex-math>--><mml:math id="mml-ieqn-38"><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> is the filtered image, and <italic>W</italic> is the two-dimensional template. A template of 3 * 3 area is used here.</p>
</sec>
<sec id="s3_1_3">
<label>3.1.3</label>
<title>Threshold Segmentation Based on Otus</title>
<p>Based on the image denoising process, the improved Roberts operator is used for edge extraction. It is necessary to set the threshold <inline-formula id="ieqn-39"><!--<alternatives><inline-graphic xlink:href="ieqn-39.png"/>--><!--<tex-math id="tex-ieqn-39"><![CDATA[$\mathrm{t}$]]></tex-math>--><mml:math id="mml-ieqn-39"><mml:mstyle mathvariant="normal"><mml:mi>t</mml:mi></mml:mstyle></mml:math><!--</alternatives>--></inline-formula>, and determine the edge point when the pixel point <inline-formula id="ieqn-40"><!--<alternatives><inline-graphic xlink:href="ieqn-40.png"/>--><!--<tex-math id="tex-ieqn-40"><![CDATA[$ \left(x, y\right)$]]></tex-math>--><mml:math id="mml-ieqn-40"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> is greater than <inline-formula id="ieqn-41"><!--<alternatives><inline-graphic xlink:href="ieqn-41.png"/>--><!--<tex-math id="tex-ieqn-41"><![CDATA[$\mathrm{t}$]]></tex-math>--><mml:math id="mml-ieqn-41"><mml:mstyle mathvariant="normal"><mml:mi>t</mml:mi></mml:mstyle></mml:math><!--</alternatives>--></inline-formula>. The selection of the <inline-formula id="ieqn-42"><!--<alternatives><inline-graphic xlink:href="ieqn-42.png"/>--><!--<tex-math id="tex-ieqn-42"><![CDATA[$\mathrm{t}$]]></tex-math>--><mml:math id="mml-ieqn-42"><mml:mstyle mathvariant="normal"><mml:mi>t</mml:mi></mml:mstyle></mml:math><!--</alternatives>--></inline-formula> value is particularly important. The efficiency of the threshold setting is low, and the adaptive ability is poor. So this paper adopts the method of maximum between-class variance (Otus) for threshold segmentation.</p>
<p>Suppose an image with a gray level of <italic>L</italic>, the range of <italic>L</italic> is <inline-formula id="ieqn-43"><!--<alternatives><inline-graphic xlink:href="ieqn-43.png"/>--><!--<tex-math id="tex-ieqn-43"><![CDATA[$ \left(0, 1, \ldots , L-1\right)$]]></tex-math>--><mml:math id="mml-ieqn-43"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mi>L</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula>, use <italic>n<sub>i</sub></italic> to represent the number of pixels with a gray level of <italic>n</italic>, and <italic>N</italic> to represent the total number of pixels, then, we obtain the following expression:</p>
<p><disp-formula id="eqn-8">
<label>(8)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-8.png"/>-->
<!--<tex-math id="tex-eqn-8"><![CDATA[$$\begin{equation}
N=n_{0}+n_{1}+\ldots +n_{L}=\sum_{i=0}^{L-1}n_{i}
 \label{eqn-8}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-8" display="block"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle='true'><mml:mstyle displaystyle='true'><mml:munderover><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo lspace='0pt' rspace='0pt'>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>L</mml:mi><mml:mo lspace='0pt' rspace='0pt'>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover></mml:mstyle></mml:mstyle><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math><!--</alternatives>--></disp-formula></p>
<p>Let <italic>p</italic>(<italic>i</italic>) be the probability that a pixel with gray level <italic>i</italic> appears:</p>
<p><disp-formula id="eqn-9">
<label>(9)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-9.png"/>-->
<!--<tex-math id="tex-eqn-9"><![CDATA[$$\begin{equation}
p \left(i\right)=\frac{n_{i}}{N}
 \label{eqn-9}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-9" display="block"><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac></mml:math><!--</alternatives>--></disp-formula></p>
<p>Set the initial threshold <inline-formula id="ieqn-44"><!--<alternatives><inline-graphic xlink:href="ieqn-44.png"/>--><!--<tex-math id="tex-ieqn-44"><![CDATA[$\mathrm{t}$]]></tex-math>--><mml:math id="mml-ieqn-44"><mml:mstyle mathvariant="normal"><mml:mi>t</mml:mi></mml:mstyle></mml:math><!--</alternatives>--></inline-formula> to divide the image into two parts A and B, where the grayscale range of A is <inline-formula id="ieqn-45"><!--<alternatives><inline-graphic xlink:href="ieqn-45.png"/>--><!--<tex-math id="tex-ieqn-45"><![CDATA[$ \left(0, 1, \ldots , t\right)$]]></tex-math>--><mml:math id="mml-ieqn-45"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula>, and the grayscale range of B is <inline-formula id="ieqn-46"><!--<alternatives><inline-graphic xlink:href="ieqn-46.png"/>--><!--<tex-math id="tex-ieqn-46"><![CDATA[$ \left(t+1, t+2, \ldots , L-1\right)$]]></tex-math>--><mml:math id="mml-ieqn-46"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mi>L</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula>, then the probability of A and B is obtained as follows:</p>
<p><disp-formula id="eqn-10">
<label>(10)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-10.png"/>-->
<!--<tex-math id="tex-eqn-10"><![CDATA[$$\begin{equation}
P_{A} \left(t\right)=\sum_{i=0}^{t}p_{i},\quad P_{b} \left(t\right)=\sum_{i=t+1}^{L-1}p_{i}
 \label{eqn-10}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-10" display="block"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle='true'><mml:mstyle displaystyle='true'><mml:munderover><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo lspace='0pt' rspace='0pt'>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:munderover></mml:mstyle></mml:mstyle><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle='true'><mml:mstyle displaystyle='true'><mml:munderover><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo lspace='0pt' rspace='0pt'>=</mml:mo><mml:mi>t</mml:mi><mml:mo lspace='0pt' rspace='0pt'>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>L</mml:mi><mml:mo lspace='0pt' rspace='0pt'>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover></mml:mstyle></mml:mstyle><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math><!--</alternatives>--></disp-formula></p>
<p>The gray average value of the two parts A and B is obtained as follows:</p>
<p><disp-formula id="eqn-11">
<label>(11)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-11.png"/>-->
<!--<tex-math id="tex-eqn-11"><![CDATA[$$\begin{equation}
\mu _{A} \left(t\right)=\frac{\sum_{i=0}^{t}ip_{i}}{P_{A} \left(t\right)},\quad \mu _{A} \left(t\right)=\frac{\sum_{i=t+1}^{L-1}ip_{i}}{P_{B} \left(t\right)}
 \label{eqn-11}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-11" display="block"><mml:msub><mml:mrow><mml:mi>&#x03BC;</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mstyle displaystyle='true'><mml:mstyle displaystyle='true'><mml:msubsup><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo lspace='0pt' rspace='0pt'>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mstyle></mml:mstyle><mml:mi>i</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:msub><mml:mrow><mml:mi>&#x03BC;</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mstyle displaystyle='true'><mml:mstyle displaystyle='true'><mml:msubsup><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo lspace='0pt' rspace='0pt'>=</mml:mo><mml:mi>t</mml:mi><mml:mo lspace='0pt' rspace='0pt'>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>L</mml:mi><mml:mo lspace='0pt' rspace='0pt'>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mstyle></mml:mstyle><mml:mi>i</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:math><!--</alternatives>--></disp-formula></p>
<p>Then, the between-class variance in the two parts A and B is obtained as follows:</p>
<p><disp-formula id="eqn-12">
<label>(12)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-12.png"/>-->
<!--<tex-math id="tex-eqn-12"><![CDATA[$$\begin{equation}
d \left(t\right)=P_{A}P_{B}({\mu _{A}}-{\mu _{B}})^{2}
 \label{eqn-12}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-12" display="block"><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>&#x03BC;</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03BC;</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math><!--</alternatives>--></disp-formula></p>
<p>When the between-class variance <inline-formula id="ieqn-47"><!--<alternatives><inline-graphic xlink:href="ieqn-47.png"/>--><!--<tex-math id="tex-ieqn-47"><![CDATA[$d \left(t\right)$]]></tex-math>--><mml:math id="mml-ieqn-47"><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> is the largest, <inline-formula id="ieqn-48"><!--<alternatives><inline-graphic xlink:href="ieqn-48.png"/>--><!--<tex-math id="tex-ieqn-48"><![CDATA[$\mathrm{t}$]]></tex-math>--><mml:math id="mml-ieqn-48"><mml:mstyle mathvariant="normal"><mml:mi>t</mml:mi></mml:mstyle></mml:math><!--</alternatives>--></inline-formula> is the optimal threshold.</p>
</sec>
</sec>
<sec id="s3_2">
<label>3.2</label>
<title>Fusion Strategy</title>
<sec id="s3_2_1">
<label>3.2.1</label>
<title>Dual-Tree Complex Wavelet Transform</title>
<p>The dual-tree complex wavelet transform (DTCWT) is composed of two parallel real wavelet transforms, using different low-pass and high-pass filters, each group of decomposition and reconstruction processes is carried out separately, and there is no interaction between data [<xref ref-type="bibr" rid="ref-25">25</xref>].</p>
<p>If <inline-formula id="ieqn-49"><!--<alternatives><inline-graphic xlink:href="ieqn-49.png"/>--><!--<tex-math id="tex-ieqn-49"><![CDATA[$f \left(t\right)$]]></tex-math>--><mml:math id="mml-ieqn-49"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> is the image input signal, <inline-formula id="ieqn-50"><!--<alternatives><inline-graphic xlink:href="ieqn-50.png"/>--><!--<tex-math id="tex-ieqn-50"><![CDATA[$s^{r} \left(t\right)$]]></tex-math>--><mml:math id="mml-ieqn-50"><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> and <inline-formula id="ieqn-51"><!--<alternatives><inline-graphic xlink:href="ieqn-51.png"/>--><!--<tex-math id="tex-ieqn-51"><![CDATA[$s^{l} \left(t\right)$]]></tex-math>--><mml:math id="mml-ieqn-51"><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> are the wavelet functions of the real and imaginary parts, respectively, and <inline-formula id="ieqn-52"><!--<alternatives><inline-graphic xlink:href="ieqn-52.png"/>--><!--<tex-math id="tex-ieqn-52"><![CDATA[$h^{r} \left(n\right)$]]></tex-math>--><mml:math id="mml-ieqn-52"><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> and <inline-formula id="ieqn-53"><!--<alternatives><inline-graphic xlink:href="ieqn-53.png"/>--><!--<tex-math id="tex-ieqn-53"><![CDATA[$h^{l} \left(n\right)$]]></tex-math>--><mml:math id="mml-ieqn-53"><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> are the real scaling function of the part and the imaginary part, then the wavelet coefficient <inline-formula id="ieqn-54"><!--<alternatives><inline-graphic xlink:href="ieqn-54.png"/>--><!--<tex-math id="tex-ieqn-54"><![CDATA[$W_{j}^{r} \left(k\right)$]]></tex-math>--><mml:math id="mml-ieqn-54"><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> and the scaling coefficient <inline-formula id="ieqn-55"><!--<alternatives><inline-graphic xlink:href="ieqn-55.png"/>--><!--<tex-math id="tex-ieqn-55"><![CDATA[$C_{j}^{r} \left(k\right)$]]></tex-math>--><mml:math id="mml-ieqn-55"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> of the real part transformation are obtained as follows:</p>
<p><disp-formula id="eqn-13">
<label>(13)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-13.png"/>-->
<!--<tex-math id="tex-eqn-13"><![CDATA[$$\begin{equation} D_{j}^{r} \left(k\right)=2^{\frac{j}{2}}\int_{-\infty }^{ \infty }f \left(t\right)s^{r} \left(2^{j}t-k\right)dt,\quad j=1,\,2,\,\ldots ,\,J
 \label{eqn-13} \end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-13" display="block"><mml:mrow></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mstyle displaystyle='true'><mml:msubsup><mml:mrow><mml:mo>&#x222B; </mml:mo></mml:mrow><mml:mrow><mml:mo lspace='0pt' rspace='0pt'>-</mml:mo><mml:mi>&#x221E;</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x221E;</mml:mi></mml:mrow></mml:msubsup></mml:mstyle><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>J</mml:mi></mml:mrow><mml:mrow></mml:mrow></mml:math>
<!--</alternatives>--></disp-formula></p>
<p><disp-formula id="eqn-14">
<label>(14)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-14.png"/>-->
<!--<tex-math id="tex-eqn-14"><![CDATA[$$\begin{equation} C_{j}^{r} \left(k\right)=2^{\frac{J}{2}}\int_{-\infty }^{\infty }f \left(t\right)h^{r} \left(2^{j}t-k\right)dt
 \label{eqn-14}\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-14" display="block"><mml:mrow></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mstyle displaystyle='true'><mml:msubsup><mml:mrow><mml:mo>&#x222B; </mml:mo></mml:mrow><mml:mrow><mml:mo lspace='0pt' rspace='0pt'>-</mml:mo><mml:mi>&#x221E;</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x221E;</mml:mi></mml:mrow></mml:msubsup></mml:mstyle><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mrow></mml:mrow></mml:math>
<!--</alternatives>--></disp-formula></p>
<p><inline-formula id="ieqn-56"><!--<alternatives><inline-graphic xlink:href="ieqn-56.png"/>--><!--<tex-math id="tex-ieqn-56"><![CDATA[$\mathrm{J}$]]></tex-math>--><mml:math id="mml-ieqn-56"><mml:mstyle mathvariant="normal"><mml:mi>J</mml:mi></mml:mstyle></mml:math><!--</alternatives>--></inline-formula> represents the maximum number of decomposition layers. Similarly, the wavelet coefficient <inline-formula id="ieqn-57"><!--<alternatives><inline-graphic xlink:href="ieqn-57.png"/>--><!--<tex-math id="tex-ieqn-57"><![CDATA[$W_{j}^{l} \left(k\right)$]]></tex-math>--><mml:math id="mml-ieqn-57"><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> and the scale coefficient <inline-formula id="ieqn-58"><!--<alternatives><inline-graphic xlink:href="ieqn-58.png"/>--><!--<tex-math id="tex-ieqn-58"><![CDATA[$C_{j}^{l} \left(k\right)$]]></tex-math>--><mml:math id="mml-ieqn-58"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> of the imaginary part are obtained as follows:</p>
<p><disp-formula id="eqn-15">
<label>(15)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-15.png"/>-->
<!--<tex-math id="tex-eqn-15"><![CDATA[$$\begin{equation}D_{j}^{l} \left(k\right)=2^{\frac{j}{2}}\int_{-\infty}^{\infty}f \left(t\right)s^{l} \left(2^{j}t-k\right)dt,\quad j=1,\,2,\,\ldots ,\,J
 \label{eqn-15} \end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-15" display="block"><mml:mrow></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mstyle displaystyle='true'><mml:msubsup><mml:mrow><mml:mo>&#x222B; </mml:mo></mml:mrow><mml:mrow><mml:mo lspace='0pt' rspace='0pt'>-</mml:mo><mml:mi>&#x221E;</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x221E;</mml:mi></mml:mrow></mml:msubsup></mml:mstyle><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mspace width="1em"/><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>J</mml:mi></mml:mrow><mml:mrow></mml:mrow></mml:math>
<!--</alternatives>--></disp-formula></p>
<p><disp-formula id="eqn-16">
<label>(16)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-16.png"/>-->
<!--<tex-math id="tex-eqn-16"><![CDATA[$$\begin{equation} C_{j}^{l} \left(k\right)=2^{\frac{J}{2}}\int_{-\infty}^{\infty}f \left(t\right)h^{l} \left(2^{j}t-k\right)dt
 \label{eqn-16}\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-16" display="block"><mml:mrow></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mstyle displaystyle='true'><mml:msubsup><mml:mrow><mml:mo>&#x222B; </mml:mo></mml:mrow><mml:mrow><mml:mo lspace='0pt' rspace='0pt'>-</mml:mo><mml:mi>&#x221E;</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x221E;</mml:mi></mml:mrow></mml:msubsup></mml:mstyle><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mrow></mml:mrow></mml:math>
<!--</alternatives>--></disp-formula></p>
<p>The final DTCWT output complete wavelet coefficient <inline-formula id="ieqn-59"><!--<alternatives><inline-graphic xlink:href="ieqn-59.png"/>--><!--<tex-math id="tex-ieqn-59"><![CDATA[$W_{j} \left(k\right)$]]></tex-math>--><mml:math id="mml-ieqn-59"><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> and scale function <inline-formula id="ieqn-60"><!--<alternatives><inline-graphic xlink:href="ieqn-60.png"/>--><!--<tex-math id="tex-ieqn-60"><![CDATA[$C_{j} \left(k\right)$]]></tex-math>--><mml:math id="mml-ieqn-60"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> are obtained as follows:</p>
<p><disp-formula id="eqn-17">
<label>(17)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-17.png"/>-->
<!--<tex-math id="tex-eqn-17"><![CDATA[$$\begin{equation}D_{j} \left(k\right)=D_{j}^{r} \left(k\right)+iD_{j}^{l} \left(k\right)
 \label{eqn-17} \end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-17" display="block"><mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:msubsup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow></mml:mrow></mml:math>
<!--</alternatives>--></disp-formula></p>
<p><disp-formula id="eqn-18">
<label>(18)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-18.png"/>-->
<!--<tex-math id="tex-eqn-18"><![CDATA[$$\begin{equation}C_{j} \left(k\right)=C_{j}^{r} \left(k\right)+iC_{j}^{l} \left(k\right)
 \label{eqn-18}\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-18" display="block"><mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow></mml:mrow></mml:math>
<!--</alternatives>--></disp-formula></p>
<p>The wavelet coefficients and scale coefficients obtained by the above decomposition obtained as follows:</p>
<p><disp-formula id="eqn-19">
<label>(19)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-19.png"/>-->
<!--<tex-math id="tex-eqn-19"><![CDATA[$$\begin{equation}D_{j} \left(t\right)=2^{\frac{i}{2}}\lambda _{i}\sum_{n\in Z} \left[D_{j}^{r} \left(n\right)s^{r} \left(2^{j}t-n\right)+D_{j}^{l}
 \left(n\right)s^{i} \left(2^{j}t-n\right)\right]
 \label{eqn-19} \end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-19" display="block"><mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>&#x03BB;</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:munder><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo lspace='0pt' rspace='0pt'>&#x2208;</mml:mo><mml:mi>Z</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mrow></mml:mrow></mml:math>
<!--</alternatives>--></disp-formula></p>
<p><disp-formula id="eqn-20">
<label>(20)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-20.png"/>-->
<!--<tex-math id="tex-eqn-20"><![CDATA[$$\begin{equation}C_{j} \left(t\right)=2^{\frac{i}{2}}\lambda _{L+1}\sum_{n\in Z} \left[C_{j}^{r} \left(n\right)s^{r} \left(2^{j}t-n\right)+C_{j}^{l}
 \left(n\right)s^{i} \left(2^{j}t-n\right)\right]
 \label{eqn-20}\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-20" display="block"><mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>&#x03BB;</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi><mml:mo lspace='0pt' rspace='0pt'>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:munder><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo lspace='0pt' rspace='0pt'>&#x2208;</mml:mo><mml:mi>Z</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mrow></mml:mrow></mml:math>
<!--</alternatives>--></disp-formula></p>
<p>where <inline-formula id="ieqn-61"><!--<alternatives><inline-graphic xlink:href="ieqn-61.png"/>--><!--<tex-math id="tex-ieqn-61"><![CDATA[$\lambda _{i}$]]></tex-math>--><mml:math id="mml-ieqn-61"><mml:msub><mml:mrow><mml:mi>&#x03BB;</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math><!--</alternatives>--></inline-formula> is the scale selection coefficient, the value range is 0 or 1, and the reconstructed signal <inline-formula id="ieqn-62"><!--<alternatives><inline-graphic xlink:href="ieqn-62.png"/>--><!--<tex-math id="tex-ieqn-62"><![CDATA[$f^{*} \left(t\right)$]]></tex-math>--><mml:math id="mml-ieqn-62"><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> is obtained as follows:</p>
<p><disp-formula id="eqn-21">
<label>(21)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-21.png"/>-->
<!--<tex-math id="tex-eqn-21"><![CDATA[$$\begin{equation}
f^{*} \left(t\right)=\sum_{j=1}^{L}W_{j} \left(t\right)+C_{j} \left(t\right)
 \label{eqn-21}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-21" display="block"><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle='true'><mml:mstyle displaystyle='true'><mml:munderover><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo lspace='0pt' rspace='0pt'>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:munderover></mml:mstyle></mml:mstyle><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></disp-formula></p>
</sec>
<sec id="s3_2_2">
<label>3.2.2</label>
<title>Low-Frequency Subband Fusion Strategy</title>
<p>The low-frequency subband part of the image represents the energy distribution of most of the background of the image. In this paper, the weighting method based on regional energy is used to determine the fusion coefficient of the low-frequency subband. The specific fusion steps are obtained as follows.</p>
<p><bold>Step 1:</bold> Calculate the regional energy of the low-frequency subband coefficients after DTCWT decomposition.</p>
<p><disp-formula id="eqn-22">
<label>(22)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-22.png"/>-->
<!--<tex-math id="tex-eqn-22"><![CDATA[$$\begin{equation}
E_{F} \left(x,\,y\right)=\frac{1}{(2s+1)^{2}}\sum_{i=-s}^{s}\sum_{j=-s}^{s}L(x+i,\,y+j)^{2}
 \label{eqn-22}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-22" display="block"><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>F</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mstyle displaystyle='true'><mml:mstyle displaystyle='true'><mml:munderover><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo lspace='0pt' rspace='0pt'>=</mml:mo><mml:mo lspace='0pt' rspace='0pt'>-</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:munderover></mml:mstyle></mml:mstyle><mml:mstyle displaystyle='true'><mml:mstyle displaystyle='true'><mml:munderover><mml:mrow><mml:mo>&#x2211;</mml:mo> </mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo lspace='0pt' rspace='0pt'>=</mml:mo><mml:mo lspace='0pt' rspace='0pt'>-</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:munderover></mml:mstyle></mml:mstyle><mml:mi>L</mml:mi><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi><mml:mo>+</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math><!--</alternatives>--></disp-formula></p>
<p>where <inline-formula id="ieqn-63"><!--<alternatives><inline-graphic xlink:href="ieqn-63.png"/>--><!--<tex-math id="tex-ieqn-63"><![CDATA[$E_{F}(x, y)$]]></tex-math>--><mml:math id="mml-ieqn-63"><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>F</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> represents the average energy in the image <italic>F<sub>l</sub></italic> within the neighborhood window of <inline-formula id="ieqn-64"><!--<alternatives><inline-graphic xlink:href="ieqn-64.png"/>--><!--<tex-math id="tex-ieqn-64"><![CDATA[$(2s+1)\times (2s+1)$]]></tex-math>--><mml:math id="mml-ieqn-64"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x00D7;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> centered on the point <inline-formula id="ieqn-65"><!--<alternatives><inline-graphic xlink:href="ieqn-65.png"/>--><!--<tex-math id="tex-ieqn-65"><![CDATA[$(x, y)$]]></tex-math>--><mml:math id="mml-ieqn-65"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula>; <italic>s</italic> usually takes 1, 2, 3; <inline-formula id="ieqn-66"><!--<alternatives><inline-graphic xlink:href="ieqn-66.png"/>--><!--<tex-math id="tex-ieqn-66"><![CDATA[$\mathrm{L}(x+i, y+j)$]]></tex-math>--><mml:math id="mml-ieqn-66"><mml:mstyle mathvariant="normal"><mml:mi>L</mml:mi></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>+</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> represents the low-pass subband coefficients after image decomposition.</p>
<p><bold>Step 2:</bold> Calculate the weight.</p>
<p><disp-formula id="eqn-23">
<label>(23)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-23.png"/>-->
<!--<tex-math id="tex-eqn-23"><![CDATA[$$\begin{equation}
\omega =\frac{E_{A} \left(x,\,y\right)}{E_{A} \left(x,\,y\right)+E_{B} \left(x,\,y\right)}
 \label{eqn-23}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-23" display="block"><mml:mi>&#x03C9;</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:math><!--</alternatives>--></disp-formula></p>
<p><bold>Step 3:</bold> Calculate the low-frequency subband fusion coefficient:</p>
<p><disp-formula id="eqn-24">
<label>(24)</label>
<!--<alternatives><graphic mimetype="image" mime-subtype="png" xlink:href="eqn-24.png"/>-->
<!--<tex-math id="tex-eqn-24"><![CDATA[$$\begin{equation}
f_{L}^{F} \left(x,\,y\right)=\omega f_{L}^{A} \left(x,\,y\right)+(1-\omega )f_{L}^{B} \left(x,\,y\right)
 \label{eqn-24}
\end{equation}$$]]></tex-math>-->
<mml:math id="mml-eqn-24" display="block"><mml:msubsup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>F</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:msubsup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>&#x03C9;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msubsup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.3em"/><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></disp-formula></p>
</sec>
<sec id="s3_2_3">
<label>3.2.3</label>
<title>High-Frequency Subband Fusion Strategy</title>
<p>The high-frequency subband of the image reflect most of the details of edges, textures, contours, etc. The traditional fusion rule of taking the absolute value is susceptible to noise and the fusion effect is low. Moreover, due to the light and other factors, a part of the edge information of the visible light image may be lost, resulting in the loss of local information. Therefore, this paper proposes an edge-enhanced fusion rule based on the improved Roberts operator. The specific fusion algorithm steps are as follows.</p>
<p><bold>Step 1:</bold> Perform median filter processing on the high-frequency subband image to remove noise.</p>
<fig id="fig-8">
<label>Figure 8</label>
<caption>
<title>Fused image of real human faces</title>
</caption><graphic mimetype="image" mime-subtype="png" xlink:href="fig-8.png"/>
</fig>
<p><bold>Step 2:</bold> Perform edge information extraction on the high-frequency subband image after denoising. The edge information is extracted according to the above mentioned improved Roberts algorithm. Perform convolution operations based on templates in the directions of 0<inline-formula id="ieqn-67"><!--<alternatives><inline-graphic xlink:href="ieqn-67.png"/>--><!--<tex-math id="tex-ieqn-67"><![CDATA[$^\circ$]]></tex-math>--><mml:math id="mml-ieqn-67"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula>, 45<inline-formula id="ieqn-68"><!--<alternatives><inline-graphic xlink:href="ieqn-68.png"/>--><!--<tex-math id="tex-ieqn-68"><![CDATA[$^\circ$]]></tex-math>--><mml:math id="mml-ieqn-68"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula>, 90<inline-formula id="ieqn-69"><!--<alternatives><inline-graphic xlink:href="ieqn-69.png"/>--><!--<tex-math id="tex-ieqn-69"><![CDATA[$^\circ$]]></tex-math>--><mml:math id="mml-ieqn-69"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula>, 135<inline-formula id="ieqn-70"><!--<alternatives><inline-graphic xlink:href="ieqn-70.png"/>--><!--<tex-math id="tex-ieqn-70"><![CDATA[$^\circ$]]></tex-math>--><mml:math id="mml-ieqn-70"><mml:msup><mml:mrow></mml:mrow><mml:mrow><mml:mo>&#x2218;</mml:mo></mml:mrow></mml:msup></mml:math><!--</alternatives>--></inline-formula>, etc. to obtain the gradient value of each pixel. Get <inline-formula id="ieqn-71"><!--<alternatives><inline-graphic xlink:href="ieqn-71.png"/>--><!--<tex-math id="tex-ieqn-71"><![CDATA[$R_{A} \left(x, y\right)$]]></tex-math>--><mml:math id="mml-ieqn-71"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> and <inline-formula id="ieqn-72"><!--<alternatives><inline-graphic xlink:href="ieqn-72.png"/>--><!--<tex-math id="tex-ieqn-72"><![CDATA[$R_{B} \left(x, y\right)$]]></tex-math>--><mml:math id="mml-ieqn-72"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula>.</p>
<p><bold>Step 3:</bold> Use the Otus threshold segmentation method to obtain the best thresholds <italic>t<sub>A</sub></italic> and <italic>t<sub>B</sub></italic>.</p>
<p><bold>Step 4:</bold> Obtain the high-frequency subband fusion coefficient <inline-formula id="ieqn-73"><!--<alternatives><inline-graphic xlink:href="ieqn-73.png"/>--><!--<tex-math id="tex-ieqn-73"><![CDATA[$f_{H}^{F} \left(x, y\right)$]]></tex-math>--><mml:math id="mml-ieqn-73"><mml:msubsup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>F</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> as follows:</p>
<fig id="eqn-25">
<graphic mimetype="image" mime-subtype="png" xlink:href="inline01.png"/>
</fig>
<p>The low-frequency subband fusion coefficient <inline-formula id="ieqn-74"><!--<alternatives><inline-graphic xlink:href="ieqn-74.png"/>--><!--<tex-math id="tex-ieqn-74"><![CDATA[$f_{L}^{F}(x, y)$]]></tex-math>--><mml:math id="mml-ieqn-74"><mml:msubsup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>F</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> and the high-frequency subband fusion coefficient <inline-formula id="ieqn-75"><!--<alternatives><inline-graphic xlink:href="ieqn-75.png"/>--><!--<tex-math id="tex-ieqn-75"><![CDATA[$f_{H}^{F}(x, y)$]]></tex-math>--><mml:math id="mml-ieqn-75"><mml:msubsup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>F</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><!--</alternatives>--></inline-formula> are inversely transformed by DTCWT to obtain the final fused image <italic>F</italic>.</p>
</sec>
<sec id="s3_2_4">
<label>3.2.4</label>
<title>Image Fusion Experiment</title>
<p>The real face that has been successfully verified is fused using the above algorithm, and the resulting fused image is shown in <xref ref-type="fig" rid="fig-8">Fig. 8</xref>. Face recognition is carried out on the fused image, and the identity information of the tested person is verified to realize the face anti-spoofing.</p>
<table-wrap id="table-3">
<label>Table 3</label>
<caption>
<title>Comparison of face recognition results between visible light image and fusion image</title>
</caption>
<table>
<colgroup>
<col/>
<col/>
<col/>
<col/>
<col/>
<col/>
</colgroup>
<thead>
<tr>
<th>Recognition methods</th>
<th>LBP (%)</th>
<th>HOG (%)</th>
<th><inline-formula id="ieqn-76"><!--<alternatives><inline-graphic xlink:href="ieqn-76.png"/>--><!--<tex-math id="tex-ieqn-76"><![CDATA[$\text{L}\text{B}\text{P}+ \text{H}\text{O}\text{G}$]]></tex-math>--><mml:math id="mml-ieqn-76"><mml:mstyle class="text"><mml:mtext>L</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>B</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>P</mml:mtext></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle class="text"><mml:mtext>H</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>O</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>G</mml:mtext></mml:mstyle></mml:math><!--</alternatives>--></inline-formula> (%)</th>
<th>MB-LBP (%)</th>
<th><inline-formula id="ieqn-77"><!--<alternatives><inline-graphic xlink:href="ieqn-77.png"/>--><!--<tex-math id="tex-ieqn-77"><![CDATA[$\text{L}\text{B}\text{P}+ \text{MB-LBP}$]]></tex-math>--><mml:math id="mml-ieqn-77"><mml:mstyle class="text"><mml:mtext>L</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>B</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>P</mml:mtext></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle class="text"><mml:mtext>MB-LBP</mml:mtext></mml:mstyle></mml:math><!--</alternatives>--></inline-formula> (%)</th>
</tr>
</thead>
<tbody>
<tr>
<td>Recognition rate of the visible light image under normal environment</td>
<td>88.5</td>
<td>90.3</td>
<td>93.3</td>
<td>92.6</td>
<td>94.4</td>
</tr>
<tr>
<td>Recognition rate of the fusion image under normal environment</td>
<td>89.9</td>
<td>91.4</td>
<td>93.7</td>
<td>92.5</td>
<td>94.6</td>
</tr>
<tr>
<td>Recognition rate of the visible light image with light changes</td>
<td>83.3%</td>
<td>84.8</td>
<td>86.7</td>
<td>86.1</td>
<td>90.1</td>
</tr>
<tr>
<td>Recognition rate of the fusion image with illumination transformation</td>
<td>88.4</td>
<td>90.9</td>
<td>92.6</td>
<td>91.3</td>
<td>93.8</td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
</sec>
</sec>
<sec id="s4">
<label>4</label>
<title>Test Results and Analysis</title>
<p>In order to verify the advantages of fusion images in face recognition, this paper collects human face information from 100 people. The visible image and thermal infrared image of human faces are collected under normal light conditions and large light changing conditions. Fusion images are obtained by using the above method, and the visible light image and the fusion image database of the human face are established. The visible light image and the fusion image under different lighting conditions are respectively applied to different face recognition n algorithms (LBP [<xref ref-type="bibr" rid="ref-26">26</xref>], HOG [<xref ref-type="bibr" rid="ref-26">26</xref>], <inline-formula id="ieqn-78"><!--<alternatives><inline-graphic xlink:href="ieqn-78.png"/>--><!--<tex-math id="tex-ieqn-78"><![CDATA[$\text{L}\text{B}\text{P}+ \text{H}\text{O}\text{G}$]]></tex-math>--><mml:math id="mml-ieqn-78"><mml:mstyle class="text"><mml:mtext>L</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>B</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>P</mml:mtext></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle class="text"><mml:mtext>H</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>O</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>G</mml:mtext></mml:mstyle></mml:math><!--</alternatives>--></inline-formula> [<xref ref-type="bibr" rid="ref-26">26</xref>], MB-LBP [<xref ref-type="bibr" rid="ref-27">27</xref>], <inline-formula id="ieqn-79"><!--<alternatives><inline-graphic xlink:href="ieqn-79.png"/>--><!--<tex-math id="tex-ieqn-79"><![CDATA[$\text{L}\text{B}\text{P}+ \text{MB-LBP}$]]></tex-math>--><mml:math id="mml-ieqn-79"><mml:mstyle class="text"><mml:mtext>L</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>B</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>P</mml:mtext></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle class="text"><mml:mtext>MB-LBP</mml:mtext></mml:mstyle></mml:math><!--</alternatives>--></inline-formula> [<xref ref-type="bibr" rid="ref-27">27</xref>]) to obtain the face recognition rate, as shown in <xref ref-type="table" rid="table-3">Tab. 3</xref>. From the results, it can be seen that there is not much difference between the recognition rate of visible image and fused image in the same face recognition algorithm under normal illumination; but in the case of large changes in light, the recognition rate of fusion image is significantly higher than that of visible image in the same face recognition algorithm. Therefore, face recognition based on fusion image has strong robustness to illumination changes.</p>
</sec>
<sec id="s5">
<label>5</label>
<title>Conclusion</title>
<p>This paper proposes an algorithm to resist facial spoofing attacks. Using thermal infrared images, the pixel values of real faces and fake faces of legitimate users are collected, and heart rate signals are detected to distinguish true and false faces. An image fusion algorithm based on DTCWT is proposed to decompose the visible light image and thermal infrared image of real human face. The obtained high-frequency sub-band uses the method based on regional energy for coefficient fusion, and the low-frequency subband uses the improved Roberts algorithm for coefficient fusion. Then use the DTCWT inverse transform to obtain a fusion image containing facial texture features. Different face recognition algorithms are used to verify the recognition rate visible light images and fusion images. The results show that the face recognition algorithm based on fusion images has a higher recognition rate. It can be seen that the algorithm proposed in this paper can effectively reduce the impact of illumination changes on face recognition results in practical application scenes. Combined with heart rate signal detection can effectively distinguish the real faces and spoofing attack face, so as to improve the security of face anti- spoofing system.</p>
</sec>
</body>
<back>
<fn-group><fn fn-type="other"><p><bold>Funding Statement:</bold> This research was funded by the Hebei Science and Technology Support Program Project (Grant No. 19273703D), and the Hebei Higher Education Science and Technology Research Project (Grant No. ZD2020318).</p></fn>
<fn fn-type="conflict"><p><bold>Conflicts of Interest:</bold> The authors declare that they have no conflicts of interest to report regarding the present study.</p></fn></fn-group>
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