The Radon function is calculated from an image matrix such as f(x, y) in specific directions. The Radon function accounts for linear integrals over different paths or certain paths (from different sources/beams) in a specific direction [9,26–32]. This function takes multiple parallel-beam projections of the input image from different angles by rotating the source around the center of the image.

In Fig. 3, a single image at a specified rotation angle of the Radon transform is illustrated. Besides, Fig. 4 indicates the geometry of the Radon Transform.

The computation of projections can be down from any angle. Generally, the Radon transform (Rθ(x′)) of the function f(x, y) is the linear integral of it, in parallel to y’-axis [34–36] (see Eqs. (1) to (4)): (1)Rθ(x′)=∫−∞+∞f(ρ1x,y,θ,ρ2x,y,θ)dy (2)ρ1x,y,θ=x′cos⁡(θ)−y′sin⁡(θ) (3)ρ2x,y,θ=x′sin⁡(θ)+y′cos⁡(θ) (4)[x′y′]=[cos⁡(θ)sin⁡(θ)−sin⁡(θ)cos⁡(θ)][xy]where the pair (x’, y’) is the new place of the (x, y) after rotating with the angle theta in a two-dimensional Cartesian coordinate system.

This section conceptualizes our new skew detection approach so-called Radon CLF (Radon Curve Length Fitness Function). Usually, the local region in a document image has a consistent orientation and frequency [38]. So, it can be modeled as a surface wave characterized entirely by the dominant orientation and frequency pattern. This approximation model is practical enough for our purpose of evaluating the performance of the Radon Transform for the skew estimation problem. According to Eq. (5), a local region of the image can be modeled as a surface wave [39]: (5)I(x, y)=Acos(2πf(x cos(θ)+y sin(θ)))where f is the frequency, the theta is the dominant orientation, and A is the amplitude of the cosine function. A is the intensity adjustment parameter of the synthesized image I(x, y). An example image and its projection by the Radon Transform are shown in Fig. 5. As it can be seen in the projection function in Fig. 5b, providing that it has been projected in the actual orientation, which is parallel to the local orientation of the input image, it can be approximately treated as a semi-sinusoidal plane wave. Besides, the noisy version of the image in Fig. 5a and its Radon projection are depicted in Fig. 6. Although, the noise power is utterly high (the Gaussian noise with the standard deviation (σ) about 20), the comparison between Figs. 5b, and 6b shows that the semi-sinusoidal structure of the Radon projection is still satisfied with a few distortions. Figs. 7 and 8 are depicted with a new angle. Now, two new Radon transform maps are compared between a noise-free directional image and its noisy version with σ = 9. With comparing two Figs. 7a and 8a in the image signal space, it can be viewed that the noise effect is quite eye-catching. On the other side, when we make comparison between Figs. 7b and 8b in the Radon space, it indicates that the presence of noise does not impact the projected pattern in this space, considerably.

These comparisons show that the proposed skew detection algorithm can abundantly tolerate the additive noise more efficiently in the Radon space rather than the signal space, providing that an appropriate feature extraction method is available for the Radon space. This observed phenomenon should be evaluated further in the following sections.

In Fig. 9, f(x) is shown as an example of an one-dimensional continuous function. The Arc length L of a function such as y = f(x) between a and b (from the point (a, f (a)) to the point (b, f (b))) can be derived using Eq. (6): (6)L=∫ab1+(dydx)2 dxwhere dydx denotes the first derivative of the function f(x). Supposing that C would be a curve in Euclidean (or, generally, a metric) space X = Rⁿ, so C is a continuous function of an image where f: [a, b] → X of the interval [a, b] into X. From a partition a = x₀ < … < x_n−1 < x_n = b of the interval [a, b], there is a finite collection of points f(x₀), f(x₁),…, f(x_n−1), and f(x_n), which can be used to calculate the length of the line segment connecting the two points. According to Eq. (7), the arc length L of C is then defined to be L(C): (7)L(C)=∑i=0n−1d(f(xi),f(xi+1))where the supermom is calculated of all possible sections of [a, b] and n is unbounded. This definition of the arc length does not require that C be defined by a differentiable function. Generally, the notion of differentiability is not defined in a metric space.

In the case of the discreet Radon function, Rθ(x), this article defines the curve length of the function, Lθ,Rθ, as Eq. (8): (8)Lθ,Rθ=∑xΔRθ(x)

Thus, the estimated skew of an input image, θ^, can be obtained the using proposed Eq. (9): (9)θ^=Argmax(Lθ,Rθ)

If the orientation is the same as the local skew, the projection function will result in a semi-sinusoidal plane wave. If not, the resultant pattern can be an erratic, non-sinusoidal signal with a smaller amplitude. This fact is shown in Fig. 10. Fig. 10 makes a comparison between Radon projection patterns on the actual orientation at θ = 70 (the red line with error = 0) and some incorrect orientations, such as 71 (error = + 1^◦), and 60 (error = −10^◦). This study proposes the curve length of the projected Radon pattern as a fitness function for skew estimation, and call it the Radon CLF algorithm.

It will be shown that in the correct orientation, such as θ = 70 in Fig. 10, the length of the curve would be well over the curve lengths of other incorrect orientations.

Performance metrics are a vital part of every algorithm analysis. To evaluate the performance of our proposed method, we have used the Mean Squared Error (MSE) (for both the skew estimator and image comparison), the Mean Absolute Error (MAE) (for the skew estimator), Accuracy (for the skew estimator), Peak Signal-to-Noise Ratio (PSNR) (for the images comparison), and Structural Similarity Measure (SSIM) (for the images comparison) along with the Computational Time.

In statistics, the MSE is considered the Mean Squared Deviation (MSD) of an estimator. It can measure the average of the square of the errors or the average squared difference between the actual/ground-truth value and the estimated value ([40]). (10)MSE=1n∑i=1n(θi−θ^i)2

In Eq. (10), θi is the true value of the angle for the ith sample, and θ^i is the estimated angle for it. The θi−θ^i  is the error signal for the ith sample, and should be calculated for all n available samples.

In comparison to the MSE, the Mean Absolute Error or MAE is the absolute average of the difference between the ground-truth and the predicted value [40] (see Eq. (11)). (11)MAE=1n∑i=1n|θi−θ^i|

In addition to MSE and MAE, the Accuracy is defined using Eqs. (12) to (14): (12)|T|=card(T) ↔ T={{∀Ii∈DB|θ−θ^=0}}where |T| is used for the cardinal of the set T, and it means the total number of error-free predictions (true predictions) in the whole dataset (Card (DB)). Similarly, the |F| is used for the cardinal of the set F which means the total number of false predictions (predictions with error) in the entire dataset. (13)|F|=card(F) ↔ F={{∀Ii∈DB|θ−θ^≠0}} (14)Accuracy=|T||T|+|F|

The 1D MSE formula can be extended to the 2D space ([41,42]): MSE^2D. In this study, it is needed to compare two images as two matrices. Eq. (15) represents the MSE^2D for two images: (15)MSE2D=1m×n∑i=0n−1∑j=0m−1(I(i,j)−I^(i,j)2where I(x, y) is an m×n original image while the I^(x,y) would be whether a noisy image or a disturbing version of the original image. The MSE^2D measures differences between two images and shows the quality degradation. The zero MSE^2D means that the two images are the identical (the perfect similarity).

The PSNR or Peak Signal-to-Noise Ratio also represents a measure of the image error. Both the PSNR, and MSE^2D are usually used to measure an image’s quality after its variation ([41,42]). The PSNR can be calculated directly found on the MSE^2D ([40]), according to Eq. (16): (16)PSNR=10×log10(2552MSE2D)

In addition to PSNR and MSE^2D, Structural Similarity Measure or SSIM have been deployed in this paper using Eq. (17) ([42]). In a 2D space, the MSE^2D will calculate distance as the mean of the square error between each corresponding pixel for the two target images. In contrast, the SSIM tries to do the opposite, and looks for similarities within pixels ([42–44]). To remedy some of the issues associated with MSE for image processing, SSIM have used. In Eq. (17), μ_x and μ_y are the average of x and y while σ _x and σ _y are the standard deviation of them, respectively. Similarly, μ_x² and μ_y² denote the variances and σ _xy is the covariance of x and y. The c₁ and c₂ are two adjustable constant parameters. (17)SSIM=(2μxμy+c1)(2σxy+c2)(μx2+μy2+c1)(σx2+σy2+c2)

It is worth mentioning that the SSIM can vary between −1 and 1; where SSIM = 1 indicates the perfect similarity ([40,43,44]).

When the skew angle is detected by an algorithm, the next step would be Skew Correction. Technically, it is just a simple rotation procedure for a 2D image. So far, different methods such as contour-oriented projection, direct/indirect based method, and others, have been introduced to correct skewed images. In our simulation, the rotation of an input image is done through the Affine Transformation (AT) using Eqs. (18) and (19) ([45]): (18)x∗=cos⁡(θ^)x+sin⁡(θ^)y (19)y∗=−sin⁡(θ^)x+cos⁡(θ^)y

Here, the (x, y) is the coordinates of a pixel in the skewed input image, θ^ denotes the detected angle, and (x*, y*) is a new place of the pixel.

One of the most important things related to any algorithm is its robustness against noise effects. This section, hankers for drawing a broad picture of the noise tolerance of our model. Fig. 13 represents an input directional image with its noisy version (An additive Gaussian noise with σ = 10). The Error image, which is defined as a differential image between two images, also is depicted in this experiment. The Error energy shows that the degrading effect of the additive Gaussian noise is so high. To make an appropriate compassion to Fig. 13, the experiment is repeated; but this time, the image space is replaced by the Radon space.

In Fig. 14, the Radon projection maps are depicted for the input image, its noisy version, and the Error image in the Radon space. When the noisy image in the signal space in Fig. 13 is compared with the noisy image in the Radon space in Fig. 14, it shows that in the Radon space, the information related to the orientation can still be extracted from the noisy image. In contrast, its corresponding noisy image in the signal space has almost no helpful information about the actual direction. It indeed means that the noise effect in the Radon space is much lower, and an appropriate estimator can spot a dominant orientation even with a high power noise.

We compare the MSE^2D, PSNR, and SSIM for the input image and the noisy image in both image signal space and Radon space to perform a more thorough analysis. Fig. 15 shows the MSE^2D for two different spaces in various noise powers. In our implementations, the σ was increased from 0 to 65.

The figure indicates that the MSE^2D−SS (the MSE in the signal space) is growing up exponentially with rising noise power. On the opposite side, MSE^2D−RS (the MSE in the Radon space) is well below the corresponding values in the signal space with little variations of the MSE between σ = 0 to σ = 65.

In addition to the MSE, the PSNR is also illustrated in Fig. 16 for two different spaces. The achieved results show that there is a big gap between the PSNR in the signal space (PSNR^SS) (the red line) and the PSNR in the Radon space (PSNR^RS) (the blue line). Indeed, the PSNR^SS is dominated by the PSNR^RS. This indicates the Radon space keeps up the image quality rather than the signal space. Then, in order to remedy some of the issues associated with MSE and PSNR, SSIM is deployed. As mentioned, the SSIM value can vary between −1 and 1, where 1 indicates the perfect similarity. Fig. 17 compares the SSIM between the signal space and the Radon space for different values of the σ. The results unveil that SSIM^RS (SSIM in the Radon space) decreased gradually from 1 to about 0.8 with increasing of the σ while SSIM^SS (SSIM in the signal space) was diving sharply from 1 to almost 0 during similar noise conditions. This indicates the perfect similarity is more achievable in the Radon space rather than signal space, providing that an appropriate feature extraction procedure is available. It is worth mentioning that all inputs were normalized using Eq. (20) before computing the Error Image, MSE^2D, PSNR, and SSIM. (20)xs=x−xminxmax−xmin

In Eq. (20), x_S is the scaled version (the normalized version), and x is the input image (either in the signal space or the Radon space). Alongside, x_min and x_max are the minimum and maximum of the x, respectively.

In image processing problems where feature detection is the only need, mapping of an original image from image space to corresponding feature space via a useful transform, with subsequent processing in lower dimension feature space, would be an appropriate. Radon domain, when properly executed, can lead to minimum entropy or maximum sparseness. High-resolution Radon Transform methods can efficiently remove random or correlated noise, improve signal clarity, by utilizing the move-out or curvature of the signal of interest. This article has deployed 2D Mean Square Error (MSE^2D), Peak Signal-to-Noise Ratio (PSNR), and Structural Similarity Measure (SSIM) to evaluate the accuracy of the proposed feature extraction algorithm. There are noticeable gaps between Red & Blue lines, corresponding to Image & Radon spaces for all three evaluation metrics. This indicates the proposed feature detection method is strong enough to benefit the Radon space potential characteristics, such as getting the least entropy or the most sparseness.

In a new scenario, the aim is to evaluate the noise power impact on the fitness function variations. In Fig. 18, several fitness functions are drawn for noisy images with σ = 0 (no noise) to σ = 11 with θ = 80. In almost all experiments, the fitness function has a global minimum of about θ = 80 (the perfect estimation). The second extremum is also highlighted in this figure. When the noise power increases sharply, a local minima may change and even be converted to a fake global minimum. This phenomenon will be investigated in the succeeding scenario. In Fig. 19, the skew and frequency of the synthetic input image are upgraded. In almost all curves for noisy images with σ = 0 (no noise) to σ = 9, the fitness function has a global minimum of about θ = 30. This means the estimator is doing magnificently even with a relatively high noise power. This indicates that the proposed approach tolerates noise effects marvelously. Now, the noise power increases, manifestly. Fig. 20, shows that when σ rises from 10 to 45, there is no longer a regular pattern. Besides, for the very high noise powers such as σ = 30, 40, and 45 the real extremums were flipped, and replaced by other false local minimums. As a result, in presence of the very high power additive Gaussian noise, the error can be grown sharply.

To reduce the run-time, the image size can be reduced by a scale factor α. This step can potentially speed up the processing time but it may lead to a reduction in the accuracy. As a result, in this section, the aim is to analyze the proposed re-scaling procedure effects on not only the run-time but also some critical algorithm’s performance measures such as the Accuracy, MSE, and MAE. For this purpose, a 65 × 65 synthetic image with θ = 70 is considered in Fig. 21. In this new experiment, the scaling factor (α) is selected from the set [1.0, 0.9, 0.7, 0.5], and α = 1 means there is no re-scaling procedure. Fig. 21 demonstrates that the global minimum of the fitness function has no drift due to the re-scaling procedure. Therefore, the estimator can detect the skew accurately for even α = 0.5 for a very tiny input image with the original size of 65 × 65. Then, the experiment is extended for a 600 × 600 synthetic image with θ = 70 at several scale factors, such as 1.0, 0.9, 0.7, 0.5, 0.4, 0.3, 0.15, and also 0.1. Similarly, the results are very satisfying even for α = 0.1 for the recent example (See Fig. 22).

To have more discussions, the outcomes of some new experiments are reported in both Tables 1, and 2. In these tables, α is the scale factor, Image size^* is the size of the scaled image, θ is the dominant orientation of an input image, θ^ represents the estimated skew, Error =θ−θ^, while Time shows the run-time in the second. In Table 1, for all scale factors that range between 0.1, and 1, the estimation error would be precisely zero, while the computation time decreases from 3.16612 s for α = 1 (Image size^* = 600 × 600) to just 0.08803 s for α = 0.1 (Image size^* = 60 × 60). This means the proposed Radon CLF is not only an accurate algorithm but also can reduce the run-time blatantly. In Table 2, the error would be zero except for α = 0.3, with a remarkably tiny image including only 20 rows and 20 columns of pixels. In this table, the estimation error is still zero for any Image size^* greater than 20 × 20. This implies that any more reduction of the input size can increase the probability of the error. Furthermore, Table 2 denotes that with re-scaling the input image from its original size of 65 × 65 to 26 × 26, the run-time falls from 0.09996 s to 0.04103, and at the same time, there is still no error.

Table 3 shows the achieved results due to running the Radon CLF algorithm on about 5696 images in the DSI5000 dataset. The dataset has been divided into two parts named DSI5000-p1 and DSI5000-p2. Each part has an equal number of samples, around 2848 images. According to Table 3, the DSI5000-p1 includes images with a lower size (65 × 65). In comparison with the DSI5000-p2, the DSI5000-p1 has a lower run-time together with a lower accuracy.

In addition to DSI5000, many real-world scanned image documents were also included in our simulations. Both handwriting, and printed samples, at various resolutions, were gathered from different Persian/Arabic & English multilingual resources such as books, booklets, letters, and newspapers. Fig. 23 shows some samples and the result of the Radon CLF algorithm. Our method can accurately detect the skew in real photos, according to experimental results.

Finally, Table 4 draws a comparison between the proposed approach and other available algorithms. Experimental results show that our algorithm is capable of skew compensating for large documents far faster than well-known existing methods, with a run-time of about 0.048 s and an Accuracy of 99.87% for DSI5000 dataset.