Scale space Radon transform

IET Image ProcessingVolume 15, Issue 9 p. 2097-2111 ORIGINAL RESEARCH PAPEROpen Access Scale space Radon transform Djemel Ziou, Corresponding Author Djemel Ziou [email protected] Département d'informatique, Université de Sherbrooke, Québec, Canada Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China Correspondence Djemel Ziou, Département d'informatique, Université de Sherbrooke, Québec, QC J1K 2R1, Canada. Email: [email protected]Search for more papers by this authorNafaa Nacereddine, Nafaa Nacereddine Research Center in Industrial Technologies, Algiers, AlgeriaSearch for more papers by this authorAicha Baya Goumeidane, Aicha Baya Goumeidane Research Center in Industrial Technologies, Algiers, AlgeriaSearch for more papers by this author Djemel Ziou, Corresponding Author Djemel Ziou [email protected] Département d'informatique, Université de Sherbrooke, Québec, Canada Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China Correspondence Djemel Ziou, Département d'informatique, Université de Sherbrooke, Québec, QC J1K 2R1, Canada. Email: [email protected]Search for more papers by this authorNafaa Nacereddine, Nafaa Nacereddine Research Center in Industrial Technologies, Algiers, AlgeriaSearch for more papers by this authorAicha Baya Goumeidane, Aicha Baya Goumeidane Research Center in Industrial Technologies, Algiers, AlgeriaSearch for more papers by this author First published: 18 March 2021 https://doi.org/10.1049/ipr2.12180Citations: 2 [Correction added on 21-April-2021, after first online publication: corrections has been made in the tables 1, 2, 3, 5, and 6] AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract An extension of Radon transform by using a measure function capturing the user need is proposed. The new transform, called scale space Radon transform, is devoted to the case where the embedded shape in the image is not filiform. A case study is brought on a straight line and an ellipse where the SSRT behaviour in the scale space and in the presence of noise is deeply analyzed. In order to show the effectiveness of the proposed transform, the experiments have been carried out, first, on linear and elliptical structures generated synthetically subjected to strong altering conditions such blur and noise and then on structures images issued from real-world applications such as road traffic, satellite imagery and weld X-ray imaging. Comparisons in terms of detection accuracy and computational time with well-known transforms and recent work dedicated to this purpose are conducted, where the proposed transform shows an outstanding performance in detecting the above-mentioned structures and targeting accurately their spatial locations even in low-quality images. 1 INTRODUCTION Radon transform has been widely used for the detection of embedded thin structures in images and other signals [1, 2]. It has been also used to define other transforms such as ridgelet and curvelet transforms [3, 4]. Even if it has been introduced more than one century ago, its properties are still studied [5-8]. Radon transform can be interpreted as a matching between an embedded thin structure in an image and the Dirac distribution ( δ) of an implicit parametric shape. If it exists, the parametric model maximizing the matching is the desired one. Although the use of Dirac is motivated by strong arguments, there are some practical limitations. Indeed, the embedded shape is often represented in an image by elongated areas of more than one pixel width. The road in satellite image or a lane marking in a road traffic image are examples. In such cases, the estimated model using the Radon transform can be different from the user expectation, and hence is inexact. Consequently, a trick allowing the control of the parametric shape position is needed. This issue, which seems not be tackled before in the case of parametric shapes, is the scope of this paper. The idea is to express the desired parametric model through a kernel defined on a parametric shape instead of being limited to the Dirac distribution. The proposed transform, which we call scale space Radon transform (SSRT), is a matching of the kernel and an embedded shape in an image. More formally, for an implicit shape S ( X , μ ) and a kernel g ( S ( X , μ ) , α ) , where α is a scale parameter and μ a parameter vector, we define the SSRT by F ( μ , α ) = ∫ D I ( X ) g ( S ( X , μ ) , α ) d X . (1)The kernel g ( ) must be chosen such that F ( μ , α ) satisfies the following requirements: (1) has at least one maximum with regards to the parameter vector μ; (2) has a nice behaviour in scale space; (3) is robust to noise; and (4) can be computed efficiently. The second requirement means that if the integral exists, then the displacement of the maxima in scale space is smooth. It follows that, the Radon transform is a particular case of the SSRT; lim α → 0 g ( S ( x , μ ) , α ) = δ ( S ( x , μ ) ) . This restriction has three desirable consequences that are: the shape detection is reduced to a maxima detection in the Radon space, Radon transform can be used as a basis when studying the properties of the SSRT, and the user need can be expressed through the scale parameter. The third other requirements do not need additional explanations. In this paper, we will use the Gaussian function as a kernel and study the behaviour of SSRT maximum in the scale space for the straight line and the ellipse. Through the work presented in this paper, we will show that the Gaussian kernel fulfils the four requirements mentioned before. Before we detail mathematically the proposed transform, let us give, in the following, an overview on some recent state-of-the-art dealing with linear structures detection. The authors in [9] propose a statistical method based on the Hough transform (HT) for line-segment detection. Their method focuses on the voting cells around a peak. They start with analyzing the voting values distribution in each column around a peak in the Hough space. With the aim at detecting a final peak accurately and robustly, they compute the normal angle and distance of the peak by using fitting and interpolation techniques, respectively. In another work of the same authors [10], a minimum entropy-based Hough transform method has been proposed, where the voting entropies around a peak in Hough space are analyzed to determine the line-segment parameters. Despite the improvements brought by these methods compared to the standard Hough Transform in terms of accuracy for the thin line detection, the problem with this HT-based method as with RT-based one [11] resides in the thick lines such as the traffic lane and the road images used in their experiments. The latter are detected by the diagonals (instead by the medians) which does not agree with the reality. Indeed, the best manner to detect a thick line is to determine its median and thickness values, in other words, its centerline and width. Miao et al. [12] presents an automatic approach for 2D road centerline extraction from classified satellite images that integrates tensor voting, principal curves, and the geodesic method. This method consists of three main steps. Tensor voting is first used to extract feature points from the classified image. The extracted feature points are then projected onto the principal curves using the subspace constrained mean shift (SCMS) method [13]. Finally, the feature points are linked by the geodesic method to create the central line. The experimental results indicate that, in contrast to SCMS, the proposed method can achieve unbiased centerlines with higher computational efficiency. The main disadvantage of this method is its inability to work on the raw image since it uses only the classified road map and, in addition, it consists in a step-by-step method which makes it dependent on the output quality of each processed step. The estimation of the line width is important for many linear feature detection applications such as road network extraction from remotely sensed imagery for the purpose of updating a digital road database. Different road widths often indicate different road classes [14]. In order to extract accurate centerlines, the authors in [14] proposed the use of Radon transform to extract centerlines from classified road imagery. A mean filter is proposed to locate the true peak in the Radon image and a profile analysis technique is used to further refine the line parameters. This method shows a good behaviour when dealing with perfect straight road segments even if experiments on real-world applications remain insufficient. However, it is not suitable for irregular or occluded linear shape structures. Recently, Polisano et al. [15] present a new formulation for the problem of recovering lines in degraded images using the framework of atomic norm minimization. They have used a primal-dual splitting algorithm to solve the convex optimization problem. Then, they have applied it successfully to the deblurring of images and the recovering lines parameters by the Prony method [16], where they show in experiments the robustness of their method for strong blur and strong noise level. However, despite the robustness of this method to strong blur and noise presented in their experiments, the outstanding performance claimed by the authors remain to be discussed as will be shown later in experiments. Unlike all the works cited before, the proposed method in this paper which consists in an original parameterized integral transform as mentioned above, acts on the raw input data with an unsupervised manner in a unique step and provides all the detected shape parameters, in an elegant way, without any further handling. The next section is devoted to the study of the SSRT of the straight line as well as its behaviour in scale space. The SSRT of the ellipse is studied in Section 3. Section 4 is devoted to experiments where SSRT is applied and compared with other works for the detection of linear and elliptical structures generated synthetically or presented in images of real-world applications. Section 6 draws the main conclusions and points out the further work. 2 SCALE SPACE RADON TRANSFORM FOR STRAIGHT LINE Let us consider a binary image I ( x , y ) whose values are one within a box and null outside; I ( x , y ) = 1 if x ∈ [ − a , a ] and y ∈ [ − b , b ] . The geometric configuration of this image is shown in Figure 1. The Radon transform R T along a straight line parameterized by ( ρ , θ ) , is given by: R T ( ρ , θ ) = ∫ − a a ∫ − b b I ( x , y ) δ ( x cos θ + y sin θ − ρ ) d x d y . (2) FIGURE 1Open in figure viewerPowerPoint An image I ( x , y ) = 1 if x ∈ [ − a , a ] and y ∈ [ − b , b ] , and 0 otherwise Let us set L 1 = 2 ( a cos θ + b sin θ − ρ ) / sin 2 θ , L 2 = 2 ( a cos θ − b sin θ + ρ ) / sin 2 θ , and β = arctan ( b / a ) . The evaluation of this integral gives: for 0 ≤ θ ≤ π 2 − β R T ( ρ , θ ) = 2 b cos θ if 0 ≤ ρ ≤ a cos θ − b sin θ L 1 if a cos θ − b sin θ ≤ ρ ≤ a cos θ + b sin θ , for π 2 − β ≤ θ ≤ π 2 R T ( ρ , θ ) = 2 a sin θ if 0 ≤ ρ ≤ − a cos θ + b sin θ L 1 if − a cos θ + b sin θ ≤ ρ ≤ a cos θ + b sin θ , for π 2 ≤ θ ≤ π 2 + β R T ( ρ , θ ) = 2 a sin θ if 0 ≤ ρ ≤ a cos θ + b sin θ L 2 if a cos θ + b sin θ ≤ ρ ≤ − a cos θ + b sin θ , for π 2 + β ≤ θ ≤ π R T ( ρ , θ ) = − 2 b cos θ if 0 ≤ ρ ≤ − a cos θ − b sin θ L 2 if − a cos θ − b sin θ ≤ ρ ≤ − a cos θ + b sin θ . The function R T ( ) is the length of the segment within the box. Its maximum is reached when ρ = 0 and θ = π / 2 ± β , corresponding to the two diagonals of the box ( − a , − b ) , ( a , b ) and ( − a , b ) , ( a , − b ) . Recall that instead of the Dirac, we propose the use of Gaussian along the line. The SSRT for the configuration in Figure 1 is given by F ( ρ , θ , σ ) = 1 2 π σ ∫ − a a ∫ − b b I ( x , y ) e − ( x cos θ + y sin θ − ρ ) 2 / 2 σ 2 d x d y . (3) Step-by-step manipulations of Equation (3), as shown in Appendix A, leads to: F ( ρ , θ , σ ) = 1 sin 2 θ σ 2 π e − d 1 2 / 2 σ 2 − e − d 2 2 / 2 σ 2 − e − d 3 2 / 2 σ 2 + e − d 4 2 / 2 σ 2 + d 1 erf d 1 2 σ − d 2 erf d 2 2 σ − d 3 erf d 3 2 σ , + d 4 erf d 4 2 σ , (4)where d 1 = a cos θ + b sin θ − ρ , d 2 = a cos θ − b sin θ − ρ , d 3 = a cos θ − b sin θ + ρ , d 4 = a cos θ + b sin θ + ρ . Now, we will study the position of the global maximum (if any) of the F ( ) in the scale space for the rectangular structure in Figure 1, starting by the case where σ → 0 , and increasing its value to follow the movement of this maxima as a function of σ. When σ → 0 , the function F ( ) is nothing but the Radon transform; that is, F ( ) = R T ( ) . A formal proof is given in Appendix B. Let us now deal with the case where σ > 0 . We set the first derivative of F ( ) wrt ρ to zero and we plot it as an implicit function in Figure 2a. We can see that ρ = 0 , whatever are the values of σ and θ. This critical configuration ( ρ = 0 ) is a maximum as it is displayed in Figure 3a, and also shown analytically in the Appendix C. This is expected because for fixed values of σ > 0 and θ, the maximal directional Gaussian volume is within the structure when the line parameterized by ρ and θ crosses the center of the structure (the axes origin in case of Figure 1). In what follows, we consider only the configuration where ρ = 0 . The maximum in θ depends on σ. In Figure 2b, we set the first derivative of F ( ) wrt θ to zero and we plot the solution on F ( ) . For each small value of σ there are three solutions. Two are maxima (in red) and the one in the middle is a minimum (in blue). When σ increases, the two maxima move and attract each other in the scale space and merge at some specific value of σ related to b as we will see later. This induces a directional Gaussians (the kernel) movement as a function of σ in the spacial domain, as illustrated in Figure 4. The movement is a continuous deformation of critical points. When a maximum disappears at a given scale, it will not appear at a higher scale. This desirable property, known as the nice property of a Gaussian in scale space, makes easier the choice of the scale to ensure the existence of only one global maximum and to detect it. In the scale space, Figure 2b, as σ increases, the fusion of critical points occurs at some specific value of σ (optimal σ) and the optimal θ converges to the value θ = π / 2 . FIGURE 2Open in figure viewerPowerPoint (a) ρ evolution as function of σ and θ for the SSRT of the structure of Figure 1. (b) The SSRT F ( ) (the grid) evolution as a function of σ and θ with highlighted scale space FIGURE 3Open in figure viewerPowerPoint (a) Evolution of F ( ) as a function of θ and ρ for a = 20 , b = 5 and σ = 3 . (b) SNR of the SSRT for a straight line in case of the structure in Figure 1 FIGURE 4Open in figure viewerPowerPoint Directional Gaussians (the kernel) movement as σ changes in the spatial domain for σ = 1.3 , 0.8, and 0.4 from left to right, respectively Let us now compute the optimal σ maximizing F ( ) , that is, σ o p t = argmax σ F ( 0 , θ , σ ) and it is unique. This corresponds to the point ( σ , π / 2 ) where max σ F σ ( 0 , π / 2 , σ ) starts to decrease. Here, we take a decreasing rate of 0.001% which is expressed as ( lim σ → 0 F ( 0 , π 2 , σ ) − F ( 0 , π 2 , σ o p t ) ) / lim σ → 0 F ( 0 , π 2 , σ ) = 0.00001 , that is, ( 2 a − 2 a erf ( b / ( σ o p t 2 ) ) ) / ( 2 a ) = 0.00001 . Finally, the optimal value of σ to detect a 2 b -thickness line is obtained as σ o p t = b / ( 2 erf − 1 ( 0.99999 ) ) = 0.226 × b . The length of the linear structure 2 a can be deduced from F ( ) . Indeed, F ( ρ o p t , θ o p t , σ o p t ) = 2 a erf ( b / 0.226 2 b ) ≃ 2 a . It is worth pointing out that 99.99% of the area under the Gaussian is contained in the interval [- 4 σ 4 σ ], so this calculus is in conformity with the Gaussian function characteristics. In the case of noisy data, the effects of noise on Radon transform are reduced [6]. Let examine the signal-to-noise ratio (SNR) of the SSRT in case of Figure 1. Here the variance of the SSRT output in response to white centred noise is σ n 2 2 π σ 2 ∫ − a a ∫ − b b e − 2 ( x cos θ + y sin θ − ρ ) 2 / 2 σ 2 d x d y . The SNR is given by S N R F ( ρ , θ ) = 1 2 π σ ∫ − a a ∫ − b b I ( x , y ) e − ( x cos θ + y sin θ − ρ ) 2 / 2 σ 2 d x d y σ n 2 π σ ∫ − a a ∫ − b b e − 2 ( x cos θ + y sin θ − ρ ) 2 / 2 σ 2 d x d y . (5) Figure 3b shows that, for SSRT, the SNR increases with σ. It is greater than the one of Radon transform ( σ close to 0). Identically, it increases with θ reaching a maximum at θ = π / 2 . In this case, the expression of SNR is given as S N R F ( 0 , π / 2 ) = [ 2 π 1 4 a σ erf b σ 2 ] / [ σ n erf ( b / σ ) ] . Its plotting in function of σ is illustrated in Figure 3b for θ = π / 2 , a = 20 and b = 5 . Note that the SSRT's SNR is proportional to 1 / σ n . So, the detection of thick lines in noisy images is improved by using the SSRT. It is worth to recall that all these computations are done for the linear structure of Figure 1 to simplify the explanation of the SSRT behaviour. 3 SCALE SPACE RADON TRANSFORM FOR ELLIPSE Let us consider an image I ( x , y ) that may contain an area bordered by two concentric ellipses [17]. The image support is considered infinite so that we do not have to be concerned about its size. The SSRT of an ellipse is then given by E ( a , b , σ ) = 1 2 π σ ∫ 0 + ∞ ∫ 0 + ∞ I ( x , y ) e − C ( x , y , a , b , u , v , ψ ) 2 / 2 σ 2 d x d y , (6)where C ( x , y , a , b , u , v , ψ ) = ( ( x − u ) cos ψ + ( y − v ) sin ψ ) 2 a 2 + ( ( x − u ) sin ψ − ( y − v ) cos ψ ) 2 b 2 − 1 is the ellipse equation, with ( u , v ) the ellipse center coordinates, ( a , b ) the (semi-major, semi-minor) axes and ψ, the ellipse orientation. For the sake of simplification and without a loss of generality, the study of the ellipse SSRT behaviour would be done for an ellipse with (0,0) as center, and a null orientation, that is, ψ = 0 . In this case, C ( x , y , a , b ) = x 2 / a 2 + y 2 / b 2 − 1 . Here, because the use of SSRT requires to find ( a , b ) maximizing E ( ) , our goal is to study precisely the behaviour of the maximum (if any) in the scale space. For that purpose, we consider that the image I as the area between C ( x , y , a 0 , b 0 ) and C ( x , y , a 1 , b 1 ) where a 0 < a 1 and b 0 < b 1 ; that is, I ( x , y ) = H C ( x , y , a 0 , b 0 ) − H C ( x , y , a 1 , b 1 ) , where H is the Heaviside function. An example of the image I and the ellipse anchored Gaussian are shown in Figure 5b,c. The integral in Equation (6) is not tractable and therefore we will produce a rough idea about E ( ) by using numerical calculus. When σ → 0 , the ellipse anchored Gaussian is equal to δ ( C ( x , y , a , b ) ) , which yields the elliptical Radon transform (ERT) [18] as the optimal ellipse maximizing E is the largest ellipse; that is, ( a , b ) = ( a 1 , b 1 ) . As σ increases, the maximum of E ( ) involves the displacement of the resulting ellipse from C ( x , y , a 1 , b 1 ) towards C ( x , y , a 0 , b 0 ) without reaching it. Figure 6a shows the evolution of b in the scale space as a function of σ. The curve evolution is rapid for small values of σ, but it becomes slow as σ increases. The maximum location converges to a stable location ( a , b ) when σ is greater than some specific value. The last situation occurs when the volume of the ellipse anchored Gaussian within the ellipse fulfils ∫ 0 + ∞ ∫ 0 + ∞ ( H C ( x , y , a 0 , b 0 ) − H C ( x , y , a , b ) ) e − C ( x , y , a , b ) 2 / 2 σ 2 d x d y = ∫ 0 + ∞ ∫ 0 + ∞ ( H C ( x , y , a , b ) − H C ( x , y , a 1 , b 1 ) ) e − C ( x , y , a , b ) 2 / 2 σ 2 d x d y . (7) FIGURE 5Open in figure viewerPowerPoint (a) Concentric ellipses represented for ϕ ∈ [ 0 , π / 2 ] . (b) The image I, where ( a 1 , b 1 ) = ( 15 , 7.5 ) and ( a 0 , b 0 ) = ( 10 , 5 ) . (c) Gaussian on an ellipse a = 12 , b = 6 , and σ = 1 FIGURE 6Open in figure viewerPowerPoint (a) Evolution of E ( ) (the grid) and b (the red line) wrt σ for the structure of Figure 5b. (b) Plot of the solution of ∂ A ( b , σ ) ∂ b = 0 In this case, the ellipse C ( x , y , a , b ) is the one maximizing E ( ) under the constraints: a 0 ≤ a ≤ a 1 and b 0 ≤ b ≤ b 1 . The smallest value of σ leading to this configuration is the one allowing that the maximal volume of the ellipse anchored Gaussian is contained within the ellipse structure. To establish a relationship between σ and b analytically, let us represent the ellipse by polar coordinates ( r , ϕ ) with r and ϕ are, respectively, the radial distance and the polar angle;where, a geometric configuration of the two concentric ellipses E l l i , i = 0 , 1 , is plotted in Figure 5a. The SSRT is then given by E ( a , b , σ ) = 1 2 π σ ∫ 0 2 π ∫ r 0 ( ϕ ) r 1 ( ϕ ) e − ( r − d ( ϕ ) ) 2 / 2 σ 2 r d r d ϕ , (8)where d ( ϕ ) = b / 1 − c 2 cos 2 ( ϕ ) and r i ( ϕ ) = b i / 1 − c 2 cos 2 ( ϕ ) , i = 0 , 1 , with c = 1 − b 2 / a 2 the ellipse eccentricity. Straightforward manipulations of Equation (8) leads to Equation (9) (see the proof in Appendix D). It is noted that the solutions of the latter are given by a numerical integration method. E ( a , b , σ ) = ∫ 0 π / 2 4 σ 2 π e − ( r 0 ( ϕ ) − d ( ϕ ) ) 2 / 2 σ 2 − e − ( r 1 ( ϕ ) − d ( ϕ ) ) 2 / 2 σ 2 ∫ 2 + 2 d ( ϕ ) erf r 1 ( ϕ ) − d ( ϕ ) 2 σ − erf r 0 ( ϕ ) − d ( ϕ ) 2 σ d ϕ . (9) It is obvious that a and b values of the optimal ellipse that maximizes E are equal to the ones maximizing the volumes under the Gaussians within the structure for all ϕ, including ϕ = 0 and ϕ = π / 2 . Taking ϕ = π / 2 in Equation (9) and discarding the integral, the mentioned area A ( b , σ ) will be expressed as A ( b , σ ) = σ 2 π [ e − ( b 0 − b ) 2 / 2 σ 2 − e − ( b 1 − b ) 2 / 2 σ 2 ] + b 2 [ erf ( b 1 − b σ 2 ) − erf ( b 0 − b σ 2 ) ] . The b value maximizing A is given by ∂ A ( b , σ ) ∂ b = 1 σ 2 π b 0 e − ( b 0 − b ) 2 2 σ 2 + b 1 e − ( b 1 − b ) 2 2 σ 2 + 1 2 erf b 1 − b σ 2 − erf b 0 − b σ 2 = 0 . (10)The plot of the solution of Equation (10) is illustrated in Figure 6b, where it can be noted that the optimal value of b (where the b values are almost constant) is provided for σ exceeding a certain value σ o p t . In our case, where b 0 = 5 , b 1 = 7.5 ; so, we obtain σ o p t = 1.6 and b o p t = 6.34 . 4 EXPERIMENTS In order to show the effectiveness of the proposed transform, two experiments have been carried out. The first experiment consists in the detection of linear and elliptical structures generated synthetically and corrupted by blur and noise. Comparisons in terms of detection accuracy and computational time with well-known transforms and a recent work, dedicated to detection of linear structures corrupted by strong blur and noise, are conducted. The second experiment uses linear and elliptical structures issued from real-world application such as road traffic, road detection in satellite images and weld radiographic testing where the robustness to image noise is also analyzed. The implementations are performed in Matlab environment on a workstation HP Z8 with Intel Xeon Silver 4210 as processor, 2.2 GHz as processor clock speed and 64 GB as random access memory (RAM). Synthetic linear structure images To compare the SSRT behaviour to some other known transforms behaviours like the Radon transform (RT) and the Hough transform (HT), a set of synthetic images, consisting of a linear structure with known location and orientation, are firstly used. The input image represented on the right side of Figure 7 is of size 201 × 201 and contains a linear structure with a known centerline location and orientation ρ = − 155.968 and θ = 150 ∘ , respectively. This structure is of 21 pixels width ( 21 = 2 b ) and 125 pixels length ( 125 = 2 a ) where a and b, representing the half of width and length, respectively, are illustrated in Figure 1. Before beginning, it is necessary to present some SSRT implementation aspects we have adopted. The space ( ρ , θ ) is sampled with discrete steps of 1 pixel for ρ and 1 ∘ for θ. Since we handle digital images, the integral in the SSRT expressions are replaced, during implementation, by summation. To apply the SSRT on an image, we use an explicit method where F ( ) in Equation (4) is estimated for each cell ( ρ i , θ i ) using a large value of σ. By large, it means that σ ≥ 2 b / 9 . The SSRT F ( ) of the input image computed with σ = 10 is illustrated at left of Figure 7a, where F ( ) presents only one maximum. The area borders of the integrals in Equation (3) are induced by the image support limits. It follows that the maximum of F ( ) gives ρ o p t and θ o p t . Furthermore, to estimate σ o p t , F ( ρ o p t , θ o p t , σ ) is computed by decreasing the values of σ. F ( ) increases as σ decreases until reaching some value of σ, noted σ o p t where F ( ) becomes almost constant as it can be viewed in Figure 8. It follows that, as shown in Section 2, the approximated structure width and length are given by 2 b ∼ ≃ 8.85 σ o p t and 2 a ∼ ≃ F ( ρ o p t , θ o p t , σ o p t ) , respectively. Thus, the SSRT maximum permits the computation of the structure orientation and location which are equal to ρ ∼ = − 155.969 and θ ∼ = 150 ∘ , respectively. The corresponding directional line is depicted on this structure at right of Figure 7a. The estimated structure width 2 b ∼ and length 2 a ∼ for the computed σ o p t = 2.43 are equal to 21.6 ( b ∼ = 10.8 ) and 125.16 ( a ∼ = 62.58 ) , respectively. Applying SSRT on the same image, for σ = 0.01 , provides two directional lines, as shown in Figure 7b, namely the two structure diagonals which correspond to the two SSRT maxima. Moreover, the two lines issued form the Radon transform RT maxima depicted in Figure 7c, are exactly the same as the ones obtained for the SSRT with small σ, as the two computed directional lines parameters ρ ∼ 1 , 2 = { − 157.771 , − 151.481 } and θ ∼ 1 , 2 = { 142 ∘ , 158 ∘ } are the same, for both cases. These results were expected since the SSRT, when σ tends to 0, is nothing but the Radon transform, as demonstrated in Appendix B. Figure 7d concerns the outcome of Hough transform (HT) applied on the image, where the 3D representation of HT is depicted and the highlighted maximum of HT is used the recover the line segment on the image, of which location and orientation are equal to ρ ∼ = − 154.4031 and θ ∼ = 148 ∘ , respectively. FIGURE 7Open in figure viewerPowerPoint 3D transform view and linear structure detection results for a) SSRT, b) SSRT for σ = 0.01, c) RT and d) HT FIGURE 8Open in figure viewerPowerPoint F ( θ o p t , ρ o p t , σ ) evolution wrt σ Figures 9-11 show the transform spaces and detected lines when adding blur, noise, or both to the image in Figure 7. The image is blurred by using the convolution of the sharp image in Figure 7 and a Gaussian kernel Θ ( x , y ; κ ) = ( 2 π κ 2 ) − 1 / 2 exp ( − ( x 2 + y 2 ) / ( 2 κ 2 ) ) of size 15 × 15 with spread κ (spatial standard deviation). An additive white Gaussian noise (AWGN) with standard deviation ζ is applied to the blurred image, that is (AWGN) ε → N ( 0 , ζ ) . Here, we take κ = 3 and ζ = 100 . Figures 9a, 10a and 11a show the 3D representation of the SSRT for each case, as well as the corresponding estimated lines, of which the parameters computed by the SSRT maximum are equal to ρ ∼ = − 155 , 969 and θ ∼ = 150 ∘ for all cases. Figures 9b, 10b and 11b show the res