Hasan Abdulrahman
email: hasan.abdulrahman@mines-ales.fr

Baptiste Magnier
email: baptiste.magnier@mines-ales.fr

Philippe Montesinos
email: philippe.montesinos@mines-ales.fr

Oriented Asymmetric Kernels for Corner Detection

Keywords:

Corners and junctions play an important role in many image analysis applications. Nevertheless, these features extracted by the majority of the proposed algorithms in the literature do not correspond to the exact position of the corners. In this paper, an approach for corner detection based on the combination of different asymmetric kernels is proposed. Informations captured by the directional kernels enable to describe precisely all the grayscale variations and the directions of the crossing edges around the considered pixel. Compared to other corner detection algorithms on synthetic and real images, the proposed approach remains more stable and robust to noise than the comparative methods.

I. INTRODUCTION AND RELATED WORK

In digital images, corners and junctions constitute important landmarks because they get affluent information like delimitation of objects. A pixel is considered a corner or a junction when two or more edges meet each other and refer to the point where several different contour orientations exist. These orientations could be estimated using structure tensors, voting tensors or oriented kernels. Thereafter, a non exhaustive review of corner detection methods is presented. Then, oriented kernels are described. Finally, a new corner detection method based only on oriented kernels is described.

A. Corner detection techniques

In the literature, several approaches have been developed to detect corners and junctions: (i) involving contour chains, (ii) using templates or, (iii) by image filtering techniques.

Traditional contour based corners methods focus on the processing of binary edges, by searching points having curvature in contour chains or in line segment intersections [21] [2]. This type of approach might be insufficient for applications in natural images because the accuracy of corner localization relies on edge detection performance (filtering, threshold(s)).

Another way to extract these feature points involves gradient orientation histogram [24] or templates in which all pixel values are compared with that of center pixel(s) of the shape template. In the wellknow SUSAN (Smallest Univalue Segment Assimilating Nucleus [START_REF] Smith | SUSAN a new approach to low level image processing[END_REF]), a corner is extracted in comparing every pixel inside a circular mask with the central pixel. On the contrary, for the FAST (Features from Accelerated Segment Test [START_REF] Rosten | Faster and better: A machine learning approach to corner detection[END_REF]) method, a point is considered a corner only if there are several pixels in the path of a circular template which are higher or lower than the value determined by the central pixel. These corner detection methods do not require any spatial derivatives; however, they do not lead directly to the position of the corner (i.e. the pixel having several local orientations as in Fig. 1 (d) and (e)). (d) SUSAN [START_REF] Smith | SUSAN a new approach to low level image processing[END_REF] (e) FAST [START_REF] Rosten | Faster and better: A machine learning approach to corner detection[END_REF] (f) Kitchen-Rosenfeld [START_REF] Kitchen | Gray-level corner detection[END_REF] (g) λ2, Shi-Tomasi [START_REF] Shi | Good features to track[END_REF] (h) c1, Förstner [START_REF] Förstner | A feature based correspondence algorithm for image matching[END_REF] (i) c2, Harris [START_REF] Harris | A combined corner and edge detector[END_REF] (j) Köthe [START_REF] Köthe | Edge and junction detection with an improved structure tensor[END_REF] (k) Aach et al. [START_REF] Aach | Analysis of superimposed oriented patterns[END_REF] (l) Proposed method As far as, image filtering is oncerned, the feature detectors operate directly on image intensities. Corners are defined by the combination of the gradient magnitude and points having maximum curvature of the image surface. A pioneer work in this domain remains Kitchen-Rosenfeld algorithm which involves first and second order image derivatives in the cornerness computation [START_REF] Kitchen | Gray-level corner detection[END_REF]. Using of second derivatives introduces false alarms in the results of this method because second image derivatives are better adapted for ridge detection. Indeed, the computation of only the first image derivatives informs on the local structure at a pixel by examining also the data in the neighborhood. Moreover, integrating the gradient information in the neighborhood of the pixel (i.e. the correlation) brings indications about whether the pixel must be considered an edge or a corner. This selection is given by involving a 2×2 symmetrical structure tensor. The derivation of a scalar image I is called the image gradient and is noted by ∇I = (I x , I y ) t in which I x and I y represent the image derivatives in x and y directions respectively (usually calculated by means of Gaussian derivative filters with a standard deviation of σ ∈ R + , see Fig. 2(a)). Involving a smoothing Gaussian kernel G ρ of standard deviation ρ, the first-order structure tensor J ρ is given by J ρ (∇I) = G ρ * ∇I • ∇I t . The scale of the neighborhood information is given by G ρ . When J ρ possesses two positive eigenvalues, then the pixel is considered having at least two distinguished orientations, therefore a corner or a junction. This isotropic cornerness measurement is given by the second eigenvalue denoted λ 2 [20][3] or could be estimated by c 1 = det(Jρ) tr(Jρ)+k [START_REF] Förstner | A feature based correspondence algorithm for image matching[END_REF] or c 2 = det(J ρ ) -k • tr(J ρ ) [START_REF] Harris | A combined corner and edge detector[END_REF], with k>0 (see also the diffusion scheme in [START_REF] Riedel | Corner-preserving anisotropic diffusion and junction detection using the structure tensor[END_REF] which preserves corners using λ 2 ). Feature detection using the linear structure tensor J ρ is insufficient in the presence of more than one dominant direction. Depending on its smoothing parameter ρ, this tensor representation is robust under noise, but generally the localization of the detected corner lacks precision. Indeed, according to the scale ρ and the image derivatives estimated by convolution with Gaussian kernels, the detected location of a corner tends to shift as ρ increases [START_REF] Deriche | A computational approach for corner and vertex detection[END_REF], as illustrated Figs. 1(g), (h) and (i). To bypass this weakness, several solutions have been proposed in the literature as nonlinear structure tensors [START_REF] Brox | Nonlinear structure tensors[END_REF] or tensor voting [START_REF] Moreno | Adaptation of tensor voting to image structure estimation[END_REF]. In [START_REF] Köthe | Edge and junction detection with an improved structure tensor[END_REF], the cornerness measurement λ 2 is propagated using hourglass-shaped filter instead of the Gaussian mask for data-adaptive smoothing at the crossing edge position. Finally, in [START_REF] Aach | Analysis of superimposed oriented patterns[END_REF], double local orientations for corners and junctions are extracted by: (1) involving J ρ to detect regions containing double orientations and (2) the computation and combinations of the eigenvalues of a 3×3 tensor. This method is more precise than previous mentioned approaches, as shown in Fig. 1(k).

In this work, corners and junctions are directly extracted involving only a combination of asymmetric oriented kernels.

B. Oriented Kernels and Edges Directions

Oriented filters were designed to capture multidirectional gray intensity variations [START_REF] Freeman | The design and use of steerable filters[END_REF] [START_REF] Jacob | Design of steerable filters for feature detection using Canny-like criteria[END_REF]. Indeed, they consist in finding the orientation where the derivative corresponds to the maximum response. The concept was generalized in [START_REF] Perona | Steerable-scalable kernels for edge detection and junction analysis[END_REF] by decomposing a given filter kernel optimally in a set of basis filters which approximates an Anisotropic Gaussian Kernel (AGK):

AGKσ ξ ,ση (x, y) = 1 2π • σ ξ • ση • e -1 2 x 2 σ 2 ξ + y 2 σ 2 η , (1)
where (x, y)∈R 2 represent the pixel coordinates, σ η and σ ξ are referred to the Gaussian scale and to the anisotropic factor respectively (Fig. 2(b)). The AGK can be oriented [START_REF] Shui | Corner detection and classification using anisotropic directional derivative representations[END_REF] but, possesses a common shortcoming, as a matter of fact, only one π-periodic orientation is extracted efficiently [START_REF] Perona | Steerable-scalable kernels for edge detection and junction analysis[END_REF], so the impossibility of these filters to estimate in a relevant way several coexisting orientations at the same pixel (see Fig. 1 (c)). Contrary to the templates remembered above, wedge [START_REF] Simoncelli | Steerable wedge filters for local orientation analysis[END_REF] [START_REF] Mühlich | Doubly-rotated matched filtering[END_REF] or asymmetric oriented filters [START_REF] Magnier | Fast anisotropic edge detection using gamma correction in color images[END_REF][14] sound better suited for a purpose like multiple edge directions detection or modeling a template. Thus, corner analysis requires finding maxima in filter responses when these 2π-periodic filters are successively steered in different directions. The equation of the anisotropic edge detector based on Half Gaussian Kernels (HGK) derivative is given by:

HGK σ ξ ,ση (x, y) = -H(y) • x σ η • G σ ξ ,ση (x, y), (2)
where H corresponds to the Heaviside function, illustrated in Fig. 2(d). By rotating the image [START_REF] Magnier | Fast anisotropic edge detection using gamma correction in color images[END_REF], maxima of the filter responses indicate the directions of the edges from 0 to 2π [START_REF] Magnier | Fast anisotropic edge detection using gamma correction in color images[END_REF] [START_REF] Magnier | Oriented half gaussian kernels and anisotropic diffusion[END_REF]. In addition, HGK and wedge steerable filters responses easily caracterize corners and junctions. Nevertheless, inside homogeneous/noisy regions, due to the isophotes, i.e., curves of the image surface of constant intensity, these directions become unpredictable. Perona shows in [START_REF] Perona | Steerable-scalable kernels for edge detection and junction analysis[END_REF] that the combination of the endstop kernel (see Fig. 2 (c)) with the HGK enables a junction and corner characterization. Indeed, the response of the combination is null along a straight contour whereas the response has maxima along the directions of the contours forming a junction or a corner (illustrated in Fig. 3). In order to avoid false corners or junctions, another solution proposed in [START_REF] Magnier | Oriented half gaussian kernels and anisotropic diffusion[END_REF] is to align the directions of the HGK when the gradient value is weak. However, near edges, the gradient generated by the HGK remains not so weak and disturbs the corner detection by creating a halo of acute angles around the contours (note that this remains the same problem using the endstop kernel). Fig. 3 (bottom) and Fig. 4 (d) illustrate this phenomenon, corners are correctly localized, however, close to the edge, HGK creates an angle which can be considered a corner.

II. A NEW METHOD OF CORNER EXTRACTION

The main idea of this new approach is to combine the HGK with an asymmetric filter computing the homogeneity along edges. On the one hand, the maxima responses of the HGK indicate the directions (2π periodic) of the edges. On the other hand, the oriented variance determines if the directions of the maxima of the HGK corresponds to edges or other types of pixels (texture, homogeneous region etc.).

A. Oriented Filter of Grayscale Homogeneity

The asymmetric IRON (Isotropic and Recursive Oriented Network) filter estimates the homogeneity in multiple local directions [START_REF] Michelet | Estimating local multiple orientations[END_REF]. This filter consists in a network of several parallel lines in which a homogeneity is computed and enables an estimation of edge directions modulo 2π. The variance for a pixel located at (x, y) on the network is computed by:

IRON(x, y) = 1 L L j=0   1 P P i=0 I(i, j) 2 - 1 P P i=0 I(i, j) 2   . (3)
Here, L represents the number of lines where the variance is computed and P the number of points per line. Fig. 2(e) represents an example of an asymmetric IRON filter. Computationally, the rotation of the image is applied at some discretized orientations from 0 to 2π before applying the IRON filters. Some examples of IRON filter signals are available in Fig. 3 fifth column, values of the IRON are close to 0 in the edges directions and it is shown in [START_REF] Michelet | Estimating local multiple orientations[END_REF] that the detections of edges directions stay precise in the presence of noise.

B. A Combination of Homogeneity and Edge Strength

The HGK and the asymmetric IRON are 1-side kernels, so they are steered in 2π directions. Moreover, the response of the HGK corresponds to maxima along the directions of the edges while the response of the IRON has minima along these same directions. Hence, the combination at the orientation θ ∈ [0; 2π[ between the HGK and IRON is straightforward:

S(x, y, θ) = HGK(x, y, θ) + IRON(x, y, θ) , with ∈ R + * , (4)
where HGK(x, y, θ) and IRON(x, y, θ) represent, respectively the HGK and IRON responses in a rotated image of angle θ.

Finally, corresponds to a constant avoiding a division by 0. The corner detection is equivalent to analyze the resulting signal S for each pixel. The polar curves in Fig. 3 indicate the modulus of the different kernels responses. Moreover, the extrema of S indicate the precise directions of the edges for contour and corner points. As illustrated in Fig. 4 (e), at a distance between 1 and 2σ η pixels, the extrema of S correspond to directions which are parallel to the edge directions. Finally, in order to compute the variance and the oriented derivative on the same neighborhood, the spacial influence of the IRON filter is inserted in the support of the HGK, i.e. 3P σ ξ and 3L σ η .

C. Angle Selection and Corner Extraction

The IRON energy is always positive while the HGK filter corresponds to an oriented derivative, so its responses are either positive, or negative. Consequently, the signal S possesses positive/negative values when HGK is positive/negative (see Fig. 5). To obtain the cornerness measure C as in Fig. 4 (b), the global extrema of S are combined:

                 C(x, y) = max θ∈[0,2π[ S(x, y, θ) -min θ∈[0,2π[ S(x, y, θ) θ 1 (x, y) = arg max θ∈[0,2π[ (S(x, y, θ)) θ 2 (x, y) = arg min θ∈[0,2π[ (S(x, y, θ)) β(x, y) = |θ 1 (x, y)-θ 2 (x, y)|, if |θ 1 (x, y)-θ 2 (x, y)| π 2π -|θ 1 (x, y) -θ 2 (x, y)| elsewhere.
(5) Once C, θ 1 and θ 2 have been obtained, the corners can be easily extracted in two steps: (i) by selecting pixels where β, the angle formed by θ 1 and θ 2 , corresponds to a desired angular sector followed by (ii) thresholding the local maxima of C. Finally, for a pixel belonging to a straight contour, the value of C is high (see Fig. 4 (b)) while β corresponds to an open angle ≈ π ((illustrated in Fig. 5 and4(c)); so it is not considered a corner point. Curves in Figs. 6(d-e) show the RMSE in function of the SNR when the standard deviation for the Gaussian derivative is the same for all methods excepted FAST (σ η =1 in (d) and σ η =2 in (e) in our case). Indeed, the standard deviation is the same for all detectors in order to compare them together, even for gradient orientation histogram technique [24] and contourbased method [START_REF] Shui | Corner detection and classification using anisotropic directional derivative representations[END_REF]. The proposed method achieves the best results in term of RMSE for all the noise levels, red curves in Figs. 6(d-e). The shape of the considered filters enables to locate the corners at the correct position even though the noise is strong (see Fig. 6(d) with SNR= 4dB). Note that results for [START_REF] Köthe | Edge and junction detection with an improved structure tensor[END_REF] are in sub-pixels resolution and merged to the original size for the evaluation process without lose detected points.

The same evaluation is led by changing the standard deviation for the Gaussian derivative (σ = σ η = 2) for all tested methods. The curve in Fig. 6 (f) illustrates the error measures. As pointed out in Section I-A, the scale of isotropic detectors affects the localization of detected corners. Note that the contour based on anisotropic Gaussian kernels is not robust to the detection of acute corners because these kernels delocalize strongly the corner points in the edge detection stage (see [START_REF] Magnier | Fast anisotropic edge detection using gamma correction in color images[END_REF]). On the contrary, the proposed method remains stable (less than 1 pixel RMSE measure compared to σ η =1 in the previous case). As far as acute angles are concerned, the half kernels are able to select the two directions of the edges and then qualify these pixels as corners. Finally, the performance of the new method is due to the HGK combined with the IRON filter which corresponds to thin filters, allowing a precise direction of contours, and thus of corners.

The first real image in Fig. 1(a) is composed of thin structures with a blur. Despite that, Fig. 1(l) illustrates very clearly that the new corner detection method has better accuracy than the ten other corner detectors.

The last experiment presented in this study focuses on the 'lab' image in Fig. 7(a). This image contains several corners of different type of angles and blurred edges. To make the comparison easier, the 280 best interest points are extracted for each algorithm. On the one hand, the orientation histogram technique [24], the contour-based method [START_REF] Awrangjeb | Performance comparisons of contourbased corner detectors[END_REF] and the tensorial approaches [START_REF] Shi | Good features to track[END_REF][5][7] [START_REF] Köthe | Edge and junction detection with an improved structure tensor[END_REF][1] fail to detect the majority of corners of obtuse angles (e.g. carpets on the floor) while they detect a lot of features concerning small objects as in the top right-hand of the image. On the other hand, Fig. 7 (f) shows that the performance of the proposed method is more efficient to detect the features. In order to capture acute and obtuse corner, the angular sector belongs to π 6 , 5π 6 . Such an angular sector is enough to detect only desired features only if the spacial support of the IRON filter is include in the spacial (Fig. 5 top right) support of the half Gaussian kernel (Fig. 5 top middle). These results using the combination of oriented half kernels and IRON filter illustrate reliable and promising results, even when dealing with blurred images.

  (a) Original image 63×44 (b) Contour-based [24](c) Contour-based[START_REF] Shui | Corner detection and classification using anisotropic directional derivative representations[END_REF]

Fig. 1 .

 1 Fig. 1. Corner detected involving several methods. The standard deviation used for the image derivatives and for the structure tensor Jρ are: σ = ρ = 1. For the developed method: ση = 1 and σ ξ = 3, L = 3 and P = 5.

  (e) IRON, L = 3 σ = 1 kernel and P = 5

Fig. 2 .

 2 Fig. 2. Different discretized 2D derivative Gaussian kernels and representation of the IRON filter. (b), (c) and (d) with ση = 1 and σ ξ = 3. The Anisotropic endstop filter is equivalent to the derivative of the AGK along the Y direction.

Fig. 3 .

 3 Fig. 3. Modulus of the energy of the different oriented kernels and their combinations (in degrees and normalized signals).

3 β 2π 3 (Fig. 4 .

 334 Fig. 4. Direction field. (a) Cropped image of Fig. 1(a).

Fig. 5 .

 5 Fig. 5. The minimum and maximum of the signal S corresponds to the two directions of the edges and β to the angular sector between the θ 1 and θ 2 directions.

Fig. 6 . 2 2

 622 Fig. 6. RMSE in function of the noise level. (a)-(c) contains the corner detected by the new algorithm with L=3 and p=5.

For a pixel p∈D

  c , d Tc (p) represents the minimal Euclidean distance between p and T c , whereas if p∈T c , d Dc (p) corresponds to the minimal distance between p and D c . Note that the two distances d Tc and d Dc are recorded [13]. Indeed, only the calculation of D can favour an algorithm where the detected corners are agglutinated around a single true point. Thus, ten corner detection approaches are compared in terms of the noise level, which is indicated by the Signal-to-Noise Ratio (SNR), as illustrated in Figs. 6 (a), (b), (c) and (d).


Hasan Abdulrahman

Baptiste Magnier

Philippe Montesinos

From Contours to Ground Truth: How to Evaluate Edge Detectors by Filtering

Keywords: Edge detection, ground truth, supervised evaluation, distance measure, objective evaluation

Edge detection remains a crucial stage in numerous image processing applications. Thus, an edge detection technique needs to be assessed before use it in a computer vision task. As dissimilarity evaluations depend strongly of a ground truth edge map, an inaccurate datum in terms of localization could advantage inaccurate precise edge detectors or/and favor inappropriate a dissimilarity evaluation measure. Hence, in this work, we demonstrate how to label these ground truth data in a semi-automatic way. Moreover, several referenced-based boundary detection evaluations are detailed and applied toward an objective assessment. Thus, each measure is compared by varying the threshold of the thin edges. Indeed, theoretically, the minimum score of the measure corresponds to the best edge map, compared to the ground truth. Finally, experiments on many images using six edge detectors show that the new ground truth database allows an objective comparison of numerous dissimilarity measures.

INTRODUCTION

Over the last decades, edge detection remains a crucial role in the computer vision community [START_REF] Canny | A computational approach to edge detection[END_REF] [START_REF] Freeman | The design and use of steerable filters[END_REF][1] [START_REF] Geusebroek | Fast anisotropic gauss filtering[END_REF][4] [41][8].

This segmentation is considered as a fundamental step in many image processing applications or analysis, pattern recognition, as well as in human vision. Moreover, contours include the most important structures in the image. Typically, edges occur on the boundary between two different regions in an image. In other words, an edge is the boundary between an object and the background or between two different objects.

There exist many different edge detection methods. Nevertheless, an important problem in image processing remains an efficient edge detector comparison and which parameter(s) correspond(s) to the best setting to obtain an accurate edge detection results. Indeed, a robust boundary detection method should create a contour image containing edges at their correct locations with a minimum of misclassified pixels. In order to objectively quantify the performance of an edge detector, a supervised measure computes a similarity/dissimilarity Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. between a segmentation result and a ground truth obtained from synthetic data or a human judgment [START_REF] Martin | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics[END_REF].

In this paper, we detail several edge dissimilarity measures and present how to evaluate filtering edge detection technique involving these considerate measures. In a second time, we demonstrate how to build a new ground truth database which can be used in supervised contour detection evaluation. Indeed, results presented show the importance of the choice of the ground truth. Finally, considering these new ground truth images, results obtained by the measures are exposed.

SUPERVISED ERROR MEASURES

To assess an edge detector, the confusion matrix remains a cornerstone in boundary detection evaluation methods. Let G t be the reference contour map corresponding to ground truth and D c the detected contour map of an original image I. Comparing pixel per pixel G t and D c , the first criterion to be assessed is the common presence of edge/non-edge points, as illustrated in Fig. 1

P * m (G t , D c ) = 1 - T P T P + FP + FN Complemented Φ measure [5] Φ * (G t , D c ) = 1 - T PR • T N T N + FP Complemented χ 2 measure [6]
T N = |¬D c ∩ ¬G t |.
Computing only FPs and FNs enables a segmentation assessment to be performed [START_REF] Magnier | Fast anisotropic edge detection using gamma correction in color images[END_REF]. The complemented Performance measure P * m presented in Table 1 considers directly and simultaneously the three entities T P, FP and FN to assess a binary image [3] [4]. The measure is normalized and decreases with improved quality of detection, with P * m = 0 qualifying perfect segmentation.

By combining FP, FN, T P and T N, another way to display evaluations is to create Receiver Operating Characteristic (ROC) [START_REF] Bowyer | Edge detector evaluation using empirical roc curves[END_REF] curves or Precision-Recall (PR) [START_REF] Martin | Learning to detect natural image boundaries using local brightness, color, and texture cues[END_REF], involving True Positive Rates (T PR) and False Positive Rates (FPR): T PR = T P T P+FN and FPR = FP FP+T N . Derived from T PR and FPR, the three measures Φ, χ 2 and F α (detailed in Table 1) are frequently used in edge detection assessment. Using the complement of these measures, a score close to 1 indicates a poor segmentation, whereas a value close to 0 a good segmentation. Among these three measures, F α remains the most stable because it does not consider the TNs, which are dominant in edge maps. Indeed, taking into consideration T N in Φ and χ 2 influences solely the measurement (as is the case in huge images).

These measures evaluate the comparison of two edge images, pixel per pixel, tending to severely penalize a (even slightly) misplaced contour, as illustrated in Fig. 2 (g) and (h). Thus, to perform an edge evaluation, the assessment should penalize a misplaced edge point proportionally to the distance from its true location.

FoM (G t , D c ) = 1 - 1 max (|G t | , |D c |) • ∑ p∈Dc 1 1 + κ • d 2 Gt (p)
FoM of over-segmentation [START_REF] Strasters | Three-dimensional image segmentation using a split, merge and group approach[END_REF] FoM e (G t , D c ) = 1 -1 max (e -FP , FP)

• ∑ p∈Dc∩¬Gt 1 1 + κ • d 2 Gt (p)
FoM revisited [START_REF] Pinho | Edge detection filters based on artificial neural networks[END_REF]

F (G t , D c ) = 1 -1 |Gt ∪Dc| • ∑ p∈Gt 1 1 + κ • d 2 Dc (p)
Combination of FoM and statistics [START_REF] Boaventura | Method to evaluate the performance of edge detector[END_REF]

d 4 (G t , D c ) = 1 2 • S + FoM (G t , D c ) with S = (T P -max (|G t | , |D c |)) 2 + FN 2 + FP 2 (max (|G t | , |D c |)) 2 Symmetric Figure of Merit [14] SFoM (G t , D c ) = 1 2 • FoM (G t , D c ) + 1 2 • FoM (D c , G t ) Maximum Figure of Merit [14] MFoM (G t , D c ) = max (FoM (G t , D c ) , FoM (D c , G t ))
Edge map quality measure [START_REF] Panetta | A new reference-based edge map quality measure[END_REF]

D p (G t , D c ) = 1/2 |I|-|Gt | • L + 1/2 |Gt | • R L = ∑ p∈Dc 1 - 1 1 + κ •d 2 Gt (p) and R = ∑ p∈Gt 1 - 1 1 + κ •d 2 Gt ∩Dc (p)
A reference-based edge map quality measure requires that a displaced edge should be penalized in function not only of FPs and/or FNs but also of the distance from the position where it should be located. Tables 2 and3 review the most relevant measures in the literature. The common feature between these evaluators corresponds to the error distance d G t (p) or/and d D c (p). Indeed, for a pixel belonging to the desired contour p ∈ D c , d G t (p) represents the minimal euclidian distance between p and G t . On the contrary, if a pixel p belongs to the ground truth G t , d D c (p) is the minimal euclidian distance between p and D c . On the one hand, some distance measures are specified in the evaluation of oversegmentation (i.e. presence of FPs), like: FoM e , ϒ, D k , Θ and Γ (see also [START_REF] Goumeidane | New discrepancy measures for segmentation evaluation[END_REF]). On the other hand, Ω measure assesses an edge detection by computing only an under segmentation (i.e. missing ground truth points, see also [START_REF] Goumeidane | New discrepancy measures for segmentation evaluation[END_REF]). Other edge detection evaluation measures consider both FPs and FNs.

First, to achieve a quantitative index of edge detector performance, one of the most popular descriptors is the Figure of Merit (FoM). This distance measure ranges from 0 to 1, where 0 corresponds to a perfect segmentation [START_REF] Abdou | Quantitative design and evaluation of enhancement/thresholding edge detectors[END_REF]. Widely utilized for comparing several different segmentation methods, in particular thanks to its normalization criterion, this assessment approach nonetheless suffers from a main drawback. Whenever FNs are created, the distance of FNs (d D c (p)) are not recorded. Indeed, FoM can be rewritten as:

FoM (G t , D c ) = 1 - ∑ p∈Dc ∩Gt 1 1 + κ • d 2 Gt (p) + ∑ p∈Dc ∩¬Gt 1 1 + κ • d 2 Gt (p) max (|G t | , |D c |) = 1 - T P + ∑ p∈Dc ∩¬Gt 1 1 + κ • d 2 Gt (p) max(|Gt |,|Dc |) , (1)
because, for

p ∈ D c ∩ G t , d 2 G t (p) = 0 and 1 1+κ•d 2 Gt (p) = 1.
Knowing that T P = |G t | -FN, for the extreme cases, the FoM measures takes the following values:

         if FP = 0 : FoM (G t , D c ) = 1 - T P |G t | , if FN = 0 : FoM (G t , D c ) = 1 - 1 max(|Gt |,|Dc |) • ∑ p∈Dc ∩¬Gt 1 1 + κ • d 2 Gt (p) . (2)
When FP = 0, FoM behaves like matrix-based error assessments. Moreover, for FP > 0, as

1 1+κ•d 2
Gt (p) < 1, the FoM measure penalizes the over-detection very low compared to the under-detection. The curve in Fig. 2 shows that the penalization of missing points (FNs) becomes higher whereas it is weaker concerning FP. On the contrary, the F measure computes the distances of FNs:

F (G t , D c ) = 1 - T P + ∑ p∈¬Dc ∩Gt 1 1 + κ • d 2 Dc (p) |G t ∪ D c | . (3)
F behaves inversely to FoM:

           if FP = 0 : F (G t , D c ) = 1 - |D c | + ∑ p∈¬Dc ∩Gt 1 1 + κ • d 2 Dc (p) |G t | , if FN = 0 : F (G t , D c ) = 1 -|Gt | |Dc | . (4)
Also, d 4 measure depends particularly on T P, FP, FN and FoM. Nonetheless, this measure penalizes FNs like the FoM measure, as shown in Fig. 2 (j). SFoM and MFoM take into account both distances of FNS and FPs, so they can compute a global evaluation of a contour image, but as illustrated in Figs. 2 (i) and (j), MFoM does not considers FPs and FNs a the same time, contrary to SFoM. Another way to compute a global measure is presented in [START_REF] Panetta | A new reference-based edge map quality measure[END_REF] with the edge map

ϒ (G t , D c ) = 100 |I| • ∑ p∈Dc d 2 Gt (p)
Hausdorff distance [START_REF] Huttenlocher | A multiresolution technique for comparing images using the hausdorff distance[END_REF] H

(G t , D c ) = max max p∈Dc d Gt (p), max p∈Gt d Dc (p) Distance to G t [18] [17][19][20] D k (G t , D c ) = 1 |Dc| • k ∑ p∈Dc d k Gt (p), k ∈ R + , k = 1 for [18]
Maximum distance [START_REF] Dubuisson | A modified hausdorff distance for object matching[END_REF]

f 2 d 6 (G t , D c ) = max 1 |D c | • ∑ p∈Dc d Gt (p), 1 |G t | • ∑ p∈Gt d Dc (p) Oversegmentation [21][22] Θ (G t , D c ) = 1 FP • ∑ p∈Dc d Gt (p) δ T H k , k ∈ R + and δ T H ∈ R + * [22], k = δ T H = 1 for [21]
Undersegmentation [START_REF] Haralick | Digital step edges from zero crossing of second directional derivatives[END_REF][22]

Ω (G t , D c ) = 1 FN • ∑ p∈Gt d Dc (p) δ T H k , k ∈ R + and δ T H ∈ R + * [22], k = δ T H = 1 for [21]
Baddeley's Delta Metric [START_REF] Baddeley | An error metric for binary images[END_REF] ∆

k (G t , D c ) = k 1 |I| • ∑ p∈I |w(d Gt (p)) -w(d Dc (p))| k , k ∈ R + and a convex function w : R → R Symmetric distance [19][20] S k (G t , D c ) = k ∑ p∈Dc d k Gt (p)) + ∑ p∈Gt d k Dc (p) |D c ∪ G t | , k ∈ R + , k = 1 for [19]
Magnier et al. measure [START_REF] Magnier | A quantitative error measure for the evaluation of roof edge detectors[END_REF] Γ

(G t , D c ) = FP + FN |G t | 2 • ∑ p∈Dc d 2 Gt (p)
Symmetric distance measure [START_REF] Magnier | Edge detection: a review of dissimilarity evaluations and a proposed normalized measure[END_REF] [14]

Ψ(G t , D c ) = FP + FN |G t | 2 • ∑ p∈Gt d 2 Dc (p) + ∑ p∈Dc d
m = min ( min( f (G 1 , D 1 )), min( f (G 2 , D 2 ))), M = max(max( f (G 1 , D 1 )), max( f (G 2 , D 2 )));
then the normalization N of a measure is computed by:

N ( f ) =          0 if M = m = 0 1 if M = m = 0 f -m M -m if M > 1 and m = 0 f otherwise. (5)
Other details and behaviors of the different measures are available in [START_REF] Magnier | Edge detection: a review of dissimilarity evaluations and a proposed normalized measure[END_REF] and [START_REF] Abdulrahman | A new normalized supervised edge detection evaluation[END_REF]. In the rest of this communication and the supplementary material 1 , the values indicated in the Tables or curves correspond to the true scores of each measure.

HOW TO CREATE PRECISE GROUND TRUTH IMAGES? HOW TO EVALUATE A FILTERING TECHNIQUE?

An edge detector is considered as robust when the evaluation score of the dissimilarity with a given G t is close to 0. Table in Fig. 3 reports different assessments for four edge detection methods on a real image (color): Sobel, Canny [START_REF] Canny | A computational approach to edge detection[END_REF], Steerable Filters (S-F) [START_REF] Freeman | The design and use of steerable filters[END_REF] and Half Gaussian Kernels (H-K) [START_REF] Magnier | Fast anisotropic edge detection using gamma correction in color images[END_REF]. Only the comparison of D c with a G t is studied here. Segmentations are classified together by comparing the scores of the dissimi- 1 The supplementary material is available at http://media.wix.com/ugd/c95124_ b9338752ae3a4e47852e0fa7bcc58b28.pdf. (d) Canny edge detection [START_REF] Canny | A computational approach to edge detection[END_REF] (f) Half Gaussian Filters [START_REF] Magnier | Fast anisotropic edge detection using gamma correction in color images[END_REF] Meas. larity measures and the smallest score for a given measure indicates the best segmentation. Indeed, for example, Sobel corresponds to the best edge detector for ϒ, Canny for ∆ k , S-F for Ω and H-F for FoM. However, this assessment suffers from two main drawbacks. Firstly, segmentations are compared using the threshold (voluntary) chosen by the user, this evaluation is very subjective and not reproducible [START_REF] Abdulrahman | A new normalized supervised edge detection evaluation[END_REF]. Secondly, some deficiencies appear in real ground truth contour maps, which could disturb the evaluation of efficient segmentation methods, or, on the contrary, advantage weak/biased edge detectors. Thus, according to the used measure or threshold any detector is classified the first one or the last one.

Ground truth images

In edge detection assessment, the ground truth is considered as a perfect segmentation. The most common method for ground-truth definition in natural images remains manual labeling by humans [2] [32]. These data sets are not optimal in the context of the definition of low-level segmentation. Firstly, labelers have marked mainly edges of salient objects, whereas equally strong edges in the background or around less important objects are missing. Moreover, errors may be created by human labels (oversights or supplements); indeed, an inaccurate ground truth contour map in terms of localization penalizes precise edge detectors and/or advantages the rough algorithms. There is another problem with the perception. In human perception, images can be ambiguous [START_REF] Peterson | Mental images can be ambiguous: Reconstruals and reference-frame reversals[END_REF], image structures tend to retain their initial reference (desired shapes) frames, even when rotated or with scale variation(s). In manual segmentation, the perception is influenced by the effects of the particular expectations, the labeler tends to mark easier the contours of a desired object which should be labeled and it influences the result. Finally, in [START_REF] Hou | Boundary detection benchmarking: Beyond f-measures[END_REF], the question is raised concerning the reliability of the datasets regarded as ground truths for novel edge detection methods. Thus, an incomplete ground truth penalizes an algorithm detecting true boundaries and efficient edge detection algorithms obtain between 30% and 40% of errors. Furthermore, when G t maps are built from a consensus which consists in the combination of several human-labelled images [START_REF] Fernández-García | Automatic generation of consensus ground truth for the comparison of edge detection techniques[END_REF][27][29], the deficiencies recalled above remain present. These reasons accentuate the importance of the relevant development of the ground truth labeling.

In real digital images, various profile edge types determine contours such as: step, ramp, roof of peaks. Pure step edges are seldom present in real image scenes, but they can be created in synthetic data. As illustrated in Fig. 4, edge positions correspond to the points of the higher gradient magnitude. For a 2D signal, pixels of contours are measured having the higher slope and are localized in the perpendicular direction of the slope of the image function. Considering synthetic data, true edges are positioned between two different colors/gray levels. Nevertheless, the edge position of an object could be interpreted in different ways: for a vertical step edge, an edge can be located either on the left, or on the right. In Fig. 5, several white shapes are immersed in a black background, creating step edges. To avoid the problem of edge pixel placements, a blur must be voluntary created by adding a 1 pixel width of gray around each shape. Thus, the ground truth corresponds to the points where the slope of the image surface is maximum, i.e. to this gray. These points could be extracted involving odd filters (derivative filters of order 1). In the one hand, a [ -1 0 1 ] mask allows to extract the edges at the correct position, i.e. the gray pixels in Fig. 5, contrary to edge detector involving smoothing parameters which delocalize edge positions (especially corners and small objects [START_REF] Ziou | Edge detection techniques: an overview[END_REF]). In the other hand, using an odd filter, two edges are extracted corresponding to the two boundaries of the roof/peak (see Fig. 4 and [START_REF] Magnier | A quantitative error measure for the evaluation of roof edge detectors[END_REF]); however, human labelers, in majority, indicate only one edge. The new database of contour images issued of real images takes into account all these properties. This paper presents ground truth edge maps which are labeled in a semi-automatic way in order to evaluate the performance of filtering step/ramp edge detectors. Therefore, the motivations to create new ground truth edge images are: 1. To obtain contours accurately localized, 2. To extract edges of the secondary objects or in the background, 3. To exclude boundaries inside noisy/textured regions.

In fact, this new label processes in return to hand made ground truth. Indeed, in a first time, the contours are detected involving the convolution of the image with [ -1 0 1 ] vertical and horizontal masks followed by a computation of a gradient magnitude and a suppres-sion of local non-maxima in the gradient direction [START_REF] Rosenfeld | Edge and curve detection for visual scene analysis[END_REF].

Concerning color images, [ -1 0 1 ] vertical and horizontal masks are applied to each channel of the image followed by a structure tensor [START_REF] Zenzo | A note on the gradient of a multiimage[END_REF]. In a second time, undesirable edges are deleted while missing points are added both by hand. Fig. 6 illustrates the steps to obtain new ground truth images. Using the [ -1 0 1 ] mask enables to capture the majority of edge points and corners without deforming small objects, contrary to edge detectors involving Gaussian filters (see for example Fig. 6 in [START_REF] Perona | Detecting and localizing edges composed of steps, peaks and roofs[END_REF]). Moreover, this process enables to detect the good positions of the contours while avoiding the addition of too much imprecise ground truth points, as shown in Fig. 4 and Fig. 5.

Minimum of the measure

Instead of thresholding manually or automatically [START_REF] Otsu | A threshold selection method from graylevel histograms[END_REF][39] and then comparing the segmentation of several edge detectors, as in Fig. 8 (c) and (d), the dissimilarity measures are used for an objective assessment. Indeed, the purpose is to compute the minimal value of a dissimilarity measure by varying the threshold T h of the thin edges computed by an edge detector (thin edges are created after the non-maximum suppression of the absolute gradient [START_REF] Rosenfeld | Edge and curve detection for visual scene analysis[END_REF]). Indeed, compared to a ground truth contour map, the ideal edge map for a measure corresponds to the desired contour at which the evaluation obtains the minimum score for the considered measure among the thresholded gradient images. Theoretically, this score corresponds to the threshold at which the edge detection represents the best edge map, compared to the ground truth contour map [START_REF] Fernández-Garcıa | Characterization of empirical discrepancy evaluation measures[END_REF][27] [START_REF] Magnier | Edge detection: a review of dissimilarity evaluations and a proposed normalized measure[END_REF]. Fig. 7 illustrates all this process. Since a small threshold leads

Original image

Gradient image

Gradient angle image

Normalized thin edges, after non-maximum suppression of the absolute gradient and before thresholding.

Scores of a measure

Best edge map for the considerated edge map quality measure, with Th = 0.4.

Ground truth edge map

Edges, Th = 0.1

Research of the ideal edge map for a measure corresponings to the segmentation at which the evaluation obtains the minimum score among the thresholded images, compared to the ground truth image.

}

Minimum score of the measure Edges, Th = 0.5 Edges, Th = 0.9

Edges, Th = 0.3 Edges, Th = 0.7 (i) FoM scores with G t in Fig. 3 (j) Contours, T h = 0.03 (k) FoM scores with G t in Fig. 6 (l) Contours, T h = 0.01

Figure 8: Scores of the measures depending on the threshold of the thin gradient image [START_REF] Canny | A computational approach to edge detection[END_REF].

to heavy over-segmentation and a strong threshold may create numerous false negative pixels, the minimum score of an edge detection evaluation should be a compromise between under-and over-segmentation.

As illustrated in Fig. 8 (e) the best score for the under-segmentation evaluation corresponds to T h = 0, because false negative points penalize the Ω measure. On the contrary, false positive points penalize oversegmentation dissimilarity measures, as FoM e , ϒ, D k , Θ and Γ measures, see Fig. 8 (g). Consequently, the best score concerning an over-segmentation measure corresponds to T h ≈ 1. As G t are not the same for the evaluation in Fig. 8 (g) and (h), the two curves are different.

EXPERIMENTAL RESULTS

The purpose of the experiments presented here is to obtain the best edge map in a supervised way. In order to study the performance of the contour detection evaluation measures, each measure is compared by varying the threshold of the thin edges computed by until six edge detectors: Sobel, Canny [START_REF] Canny | A computational approach to edge detection[END_REF], Steerable Filters of order 1 (SF 1 ) [START_REF] Freeman | The design and use of steerable filters[END_REF], Steerable Filters of order 5 (SF 5 ) [START_REF] Jacob | Design of steerable filters for feature detection using canny-like criteria[END_REF], Anisotropic Gaussian Kernels (AGK) [START_REF] Geusebroek | Fast anisotropic gauss filtering[END_REF] and Half Gaussian Kernels (HK) [START_REF] Magnier | Fast anisotropic edge detection using gamma correction in color images[END_REF]. In the one hand, experiments are led on two synthetic noisy images. In the other hand, contour detection evaluations are compared on seven real images where G t edge maps are labelled by a semi-automatic way (section 3.1). Finally, compared to a ground truth contour map, the ideal edge map for a measure corresponds to the desired contour at which the evaluation obtains the minimum score for the considered measure among the thresholded thin gradient images [START_REF] Canny | A computational approach to edge detection[END_REF].

Firstly, to evaluate the performances of the dissimilarity measures, the original image in Fig. 5(a) is disturbed with random Gaussian noise and edges are extracted from the noisy images (4dB and 3.3dB, see supplementary material). Generally, the scores of Φ * , d 4 and D p measures allow to correctly extract the edges at the price of numerous FPs. Moreover, ∆ k is more sensitive to FPs than the other dissimilarity measures and the best score corresponds to a contour edge map with many discontinuous contours. As pointed out in section 2, concerning the image corrupted by a noise at a level of 4dB, FoM penalizes strongly FNs to the detriment of FPs apparitions, and it considers that anisotropic edge detectors are less performant than the Canny edge detector. Other measures classify the Sobel method as the less efficient one and the H-K as the best one.

The second experiment concerns a real image presented in Fig. 6(a); G t is available in Fig. 6(c). The best edge maps based on minimum of the scores of different measures are presented in Fig. Numerous experiments show that S k k = 1 or k = 2 and Ψ dissimilarity measures are best fitted in the problem of supervised edge evaluation. Indeed, the minimum evaluation scores are coherent and the edge detectors are qualified as best when the filtering technique is adapted to the image structure (blur, noise, small objects). Moreover, the edge map corresponding to the minimum score delimit correctly the object with a majority of continuous contours points without much undesirable points.

CONCLUSION

This study presents a review of supervised edge detection assessment methods in details. Moreover, based on the theory of these dissimilarity evaluations, a technique is proposed to evaluate filtering edge detection methods involving the minimum score of the considerate measures. Indeed, to evaluate an edge detection technique, the result which obtains the minimum score of a measure is considerate as the best one and represents an objective evaluation. Theoretically and with the backing of many experiments is demonstrated that the minimum score of the S k k = 1 or k = 2 and Ψ dissimilarity measures correspond to the best edge quality map evaluations. These two measures take into account both the distances of false positive and false negative points. Many experiments of edge detection on synthetic and real images involving several edge detectors illustrate this conclusion. Experiments show the significance of the ground truth map choice: an inaccurate ground truth contour map in terms of localization penalizes precise edge detectors and/or advantages the rough algorithms. That is the reason why is described in this conversation how to build a new ground truth edge map labelled in semi-automatic way in real images. Firstly, the contours are detected involving the convolution of the image with [ -1 0 1 ] masks. Secondly, undesirable edges are removed while missing points are added both by hand, thus a more accuracy ground truth edge map image is built and can be used for supervised contour detection evaluation. By comparison with a real image where contours points are not precisely labelled, experiments illustrate that the new ground truth database allows to evaluate the performance of edge detectors by filtering. Finally, the advantage to compute the minimum score of a measure involving this new ground truth database is that it does not require tuning parameters. For this purpose, we plan in a future study to compare the robustness several edge detection algorithms by adding noise and blur on real images presented in the supplementary material and then using the optimum threshold computed by the minimum of the evaluation.

Figure 1 :

 1 Figure 1: Ground truth vs. desired contour. In (b), D c is contaminated with 6 FPs ans 4 FNs.

χ 2 *

 2 (G t , D c ) = 1 -T PR -T P -FP 1 -T P -FP • T P + FP + FPR T P + FP Complemented F α measure [7] F * α (G t , D c ) = 1 -PREC • T PR α • T PR + (1α) • PREC ,with PREC = T P T P + FP and α ∈]0; 1]

  G 1 vs. D 1 : Evolution of the confusion matrix-based error assessments (h) G 2 vs. D 2 : Evolution of the confusion matrix-based error assessments in function of the distances of the FPs. in function of the distances of the FNs. Φ * and P * m overlap.

  G 1 vs. D 1 : Evolution of the normalized dissimilarity measures (j) G 2 vs. D 2 : Evolution of the normalized dissimilarity measures in function of the distances of the FPs. F and MFoM overlap. in function of the distances of the FNs. FoM e = 0. FoM and MFoM overlap.

  G 1 vs. D 1 : Evolution of the non-normalized dissimilarity measures (l) G 2 vs. D 2 : Evolution of the non-normalized dissimilarity measures in function of the distances of the FPs. Ω = 0. H and Θ δ T H = 5 overlap. in function of the distances of the FNs. ϒ = Θ = D k = Γ = 0. ϒ, H 5% , D k , Θ δ T H = 1, f 2 d 6 , S k and Ψ overlap. H and Ω δ T H = 5 overlap. H 5% , Ω δ T H = 1, f 2 d 6 and S k overlap.

Figure 2 :

 2 Figure 2: Evolution of dissimilarity measures in function of the the distance of the false positive/negative points. A vertical line of false positive points (b) or false negative points (d) is shifted by a maximum distance of 16 pixels and the measure scores are plotted in function of the displacement of the desired/undesired contour.

  Number of pixels in % (a) Color real image, 240×160[START_REF] Martin | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics[END_REF] (c) Sobel edge detection (e) Steerable edge detection[START_REF] Freeman | The design and use of steerable filters[END_REF] Number of pixels in % (b) Gt , weak consensus image[START_REF] Lopez-Molina | Twofold consensus for boundary detection ground truth[END_REF]

Figure 3 :

 3 Figure 3: Edge detection after the non-maximum suppression [31] and comparison with a ground truth image.

  (d) Peak edge (e) Roof edge (f) Legend

Figure 4 :

 4 Figure 4: The different types of edges and result of a convolution with a [ -1 0 1 ] mask (absolute value).

Figure 5 :

 5 Figure 5: Synthetic data with a 1 pixel width gray around each shape: value of white pixels = 1, values of black pixels = 0, values of gray pixels = 0.5.

Figure 6 :

 6 Figure 6: Image of our database are built after an edge detection involving a [-1 0 1] mask and concluded by hand.

Figure 7 :

 7 Figure 7: The most relevant edge map for a dissimilarity measure is indicated by its minimum score.

1 0

 1 Ω k=1 δ T H=1 scores with G t in Fig.3(f) Contours, T h = 0 (g) Θ k=1 δ T H=1 scores with G t in Fig.3(h) Contours, T h = = 0.01 score = 0.14776

Figure 9 :

 9 Figure 9: Best edge maps for each dissimilarity measure concerning a real image and G t in Fig. 6.

Table 1 :

 1 List of error measures involving only statistics.

Complemented Per f ormance measure

[3] [4]

  . A basic evaluation is compounded from statistics; to that effect, G t and D c are combined. Afterwards, denoting | • | as the cardinality of a set, all points are divided into four sets:

-True Positive points (TPs), common points of G t and D c : T P = |D c ∩ G t |, -False Positive points (FPs), spurious detected edges: FP = |D c ∩ ¬G t |, -False Negative points (FNs), missing boundary points of D c : FN = |¬D c ∩ G t |, -True Negative points (TNs), common non-edge points of G t and D c :

Table 2 :

 2 List of normalized error measures compared in this work, with the parameter κ ∈ ]0; 1].

	Figure of Merit (FoM) [10]

Table 3 :

 3 List of non-normalized error measures. In the literature, the most common values are k = 1 or k = 2.Yasnoff measure[START_REF] Yasnoff | Error measures for objective assessment of scene segmentation algorithms[END_REF]

  , an average distance from the edge pixels in the candidate image to those in the ground truth is more appropriate for matching purposes than H and H n% . To achieve this task, D k , ϒ, Θ and Γ which represent errors of distance only in function of d G t , they correspond to a measure of over-segmentation (only FPs), as indicated by the curves in 2 (l) where the curves stagnate at 0. On the contrary, the sole use of a distance d D c instead of d G t enables an estimation of the FN divergences, representing an under-segmentation (as in Ω). Nevertheless, as concluded in[START_REF] Chabrier | Comparative study of contour detection evaluation criteria based on dissimilarity measures[END_REF], a complete and optimum edge detection evaluation measure should combine assessments of both over-and under-segmentation, as f 2 d 6 , S k and Ψ. Also, combining both d D c and d G t , Baddeley's Delta Metric (∆ k )[START_REF] Baddeley | An error metric for binary images[END_REF] is a measure derived from the Hausdorff distance which is intended to estimate the dissimilarity between each element of two binary images. Finally, curves in Figs.2 (k) and (l) illustrate that H, H 5% , ∆ k , f 2 d 6 and S k behave similarly in function of the FPs or FNs distances. Note that the Ψ measure is more sensitive to the distance of the FPs.

	A second measure widely computed in matching
	techniques is represented by the Hausdorff distance
	H, which measures the mismatch of two sets of points
	[17]. This max-min distance could be strongly deviated
	by only one pixel which can be positioned sufficiently
	far from the pattern. To improve the measure, one idea

2

Gt (p) quality measure D p . The over-segmentation measure (left term) evaluates d D c , the distances between the FPs and G t . The under-segmentation measure (right term) computes the distances of the FNs between the closest correctly detected edge pixel, i.e. G t ∩ D c . That means that FNs and their distances are not counted without the presence of TP(s), and D p is more sensitive to FNs than FPs, see Figs.

2 (i) and (j)

. is to compute H with a proportion of the maximum distances (for example 5% of the values

[START_REF] Huttenlocher | A multiresolution technique for comparing images using the hausdorff distance[END_REF]

); let us note H 5% this measure. Nevertheless, as pointed out in

[START_REF] Dubuisson | A modified hausdorff distance for object matching[END_REF]

The scores of the non-normalized measures in Figs.

2 (k

) and (l) are normalized using the following equation for easy visual comparison. Denoting by f ∈ [0; +∞[ the score vectors of a distance measure such that:

  [START_REF] Bowyer | Edge detector evaluation using empirical roc curves[END_REF]. Statistical measures and d 4 consider that Sobel is the best edge detector for this image because edges are well localized. Even though edge maps are different, the scores obtained by FoM and F are similar for the different filtering techniques. Oriented kernels, however, are qualified as reliable by distance measures and edge maps corresponding to the minimum scores are less noisy. In the supplementary material are compared the best edge maps obtains with our G t and G t of Berkeley segmentation image (Fig.3(b)). Excepted for Φ * , d 4 and D p measures, the best edge map for all the other measures contains many holes in the contour chains and it is clearly impossible to conclude which edge detector is the most efficient. Table4mentions scores involving the two different G t : by hand made, and semi-automatic. It is important to note that the scores for each measure is smaller concerning G t built in a semi-automatic way.

	Other results presented in the supplementary material
	show that the minimum scores concerning the distance
	measures. When objects appear clear, like in image 56
	and buildings, most of the measure scores indicate that
	the edge detectors are equivalent. By contrast, as soon
	as images contain blur or/and noise, as in image 109
	and parkingmeter, the evaluation measures involving
	error distances considerate that oriented and anisotropic
	filters produce better-defined contours. Finally, image
	109 is a noisy image, however ∆ k and D p evaluate
	that Sobel detects better edge, whereas it creates many
	undesirable contour points, contrary to filtering tech-
	niques involving smoothing effects.

Table 4 :

 4 Comparison of scores of dissimilarity measures using a ground truth from[START_REF] Martin | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics[END_REF] (Fig.3 (b)) image and a constructed ground truth by a semi-automatic way. Contour images and curves for all the measures are available in the supplementary material.Our Gt Berkeley Gt Our Gt Berkeley Gt Our Gt Berkeley Gt Our Gt Berkeley Gt

	Meas.	Sobel Berkeley Gt	Canny	SF1 [28]	AGK [42]	H-K [8] Our Gt
	Φ *	0.738	0.298	0.757	0.430	0.971	0.447	0.813	0.496	0.761	0.504
	χ 2 *	0.979	0.635	0.975	0.725	0.983	0.712	0.982	0.759	0.973	0.502
	P * m F * α	0.901 0.820	0.530 0.360	0.901 0.819	0.603 0.432	0.909 0.834	0.594 0.422	0.917 0.847	0.637 0.468	0.893 0.808	0.778 0.483
	FoM	0.303	0.168	0.310	0.147	0.309	0.164	0.299	0.154	0.277	0.146
	F	0.592	0.346	0.579	0.352	0.572	0.310	0.589	0.337	0.589	0.367
	d4	0.675	0.333	0.671	0.379	0.687	0.375	0.695	0.412	0.667	0.424
	SFoM 0.297	0.145	0.289	0.134	0.270	0.111	0.271	0.119	0.268	0.128
	DP	0.173	0.036	0.184	0.058	0.193	0.056	0.208	0.065	0.183	0.072
	H	40.02	29.52	19.41 15.175 18.97	18.02	35.35	14.76	36.87	15.03
	H5%	13.72	9.406	11.89	9.142	11.53	6.781	14.18	6.048	14.56	7.165
	∆ k	6.632	4.094	5.039	3.000	4.844	2.462	6.044	2.040	6.562	2.576
	f2d6	2.851	1.066	2.498	1.294	2.467	0.900	2.625	0.895	2.582	0.983
	S k k=1 S k k=2	2.584 4.270	1.005 2.323	2.315 3.725	0.990 2.361	2.316 3.690	0.877 1.819	2.471 4.172	0.866 1.667	2.432 4.281	0.966 2.029
	Ψ	0.213	0.041	0.181	0.044	0.173	0.032	0.224	0.032	0.222	0.038

ACKNOWLEDGEMENTS

The authors would like to thank the Iraqi Ministry of Higher Education and Scientific Research for funding and supporting this work, and, Philippe Borianne for its helpful comments and suggestions.