Thomas Hurtut

Farida Cheriet

Automatic closed edge detection using level lines selection

This paper presents a closed edge detection method based on a level lines selection approach. The proposed method is based on an unsupervised probabilistic scheme using an a contrario method. A level line is considered meaningful if its contrast and length is unlikely to be due to chance. Besides being unsupervised, this method exploits a tree structure. The first step of the proposed approach is to reduce the meaningful level lines set using this hierarchical structure. Compared with a previous method using the same principle, our method achieve a 67% reduction rate of irrelevant levels lines. The second step of the proposed approach illustrates the high flexibility of using closed edge boundaries such as levels lines. Using a rather simple curvature analysis, the proposed method detects anatomical structures boundaries from CT scan images.

Introduction

Edge detection challenges the vision community since its beginnings. Many solutions have been proposed. So called classic methods are based on gradient detection by filters [START_REF] Torre | On edge detection[END_REF], masks, Laplacian zero-crossing [START_REF] Marr | Thoery of edge detection[END_REF][START_REF] Haralick | Digital step edges from zero crossing of second directionnal derivatives[END_REF] and adaptive masks [START_REF] Hueckel | An operator which locates edges in digitized pictures[END_REF]. Analytic approaches follows, among them are Canny [START_REF] Canny | A computationnal approach to edge detection[END_REF] and Deriche [START_REF] Deriche | Using canny's criteria to derive recursively implemented optimal edge detector[END_REF] detectors. Unfortunately, they require fine and supervised parameters tuning and are also sensitive to image noise. They also consider edges as a local concept which is opposed to Gestalt laws [START_REF] Kanizsa | La grammaire du Voir[END_REF]. One of the biggest drawback of these methods is their open edge processing. This require a difficult and hazardous edge closing step [START_REF] Martelli | Edge detection using heuristic search methods[END_REF][START_REF] Montanari | On the optimal detection of curves in noisy images[END_REF][START_REF] Giraudon | Chainage efficace des contours[END_REF]. Besides these methods, Kass et al. [START_REF] Kass | Active contour models[END_REF][START_REF] Blake | Active Contours[END_REF] introduced the active contour model or snake. These semi-automatic methods use a deformable model which minimizes an internal deformation energy subject to external forces derived from images. Closely, level sets approaches allows topology flexibility [START_REF] Osher | Front propagating with curvature dependant speed: algorithm based on the hamilton-jacobi formulations[END_REF][START_REF] Sethian | Level-set methods[END_REF]. Here, so-called level sets are defined as the zeros levels of surfaces in a three dimensions space. The active contour model is also close to the geodesic model [START_REF] Caselles | Geodesic active contours[END_REF]. These three kind of deformable methods also require many parameters to be tuned, and are strongly dependent on the initialization.

Mathematic morphology proposed to use another kind of image level sets to extract object contours. These level sets are different from the ones previously defined but unfortunately hold the same name. Given an image u, a level set at the value λ is χ λ (u) = {x ∈ R 2 , u(x) ≥ λ}. One important property of this representation is that no information is lost, since we can reconstruct an image from the whole family of its level sets [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF]. It thus provides a complete representation of images. It is also worth noting how large shapes are already present with as few as 5 or 6 coarsely quantized levels. Levels sets boundaries are called level lines, and the whole family of the level lines of an image is called the topographic map [START_REF] Caselles | Topographic maps and local contrast changes in natural images[END_REF]. This representation enjoys several important advantages. It is contrast invariant. Levels lines are closed curves (except the ones meeting the image borders). It is a hierarchical representation. An important property is that object contours locally coincide with level lines [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF]. Unfortunately, the topographic map contains also the texture details and noise effects. Therefore Desolneux et al. presented a level lines selection method [START_REF] Desolneux | Edge detection by helmohltz principle[END_REF], which has been compared with active contour model [START_REF] Cao | Extracting meaningful curves from images[END_REF]. Desolneux et al. do not claim to propose a ready to use optimal edge detector. Instead they consider their method only provide a set of feasible, i.e. acceptable edges.

Indeed the resulting set of boundaries is not optimal. As we shall see it contains too much irrelevant boundaries to be used directly as an object boundaries detector. Two contributions are proposed in this paper. The first one uses the hierarchical representation of the boundaries set to reduce the number of irrelevant boundaries and aims all kind of images. The second one is designed for CT scan images which handle some characteristics that require special methods as it often occurs in medical image analysis [START_REF] Duncan | Medical image analysis: progress over two decades and the challenges ahead[END_REF]. Indeed in the context of deformable organ modeling from volumetric data such as CT scan images, in order to simulate treatment, it is useful to detect only the boundaries of the main anatomical structures such as fat, muscles and bones. Therefore, the objective of the second contribution is to detect only relevant boundaries that satisfy some specific criteria defined with respect to some descriptors of the targeted structures. Furthermore the second contribution allows to eliminate some specific noise in CT scan images.

The plan of the paper is as follows. We shall give more details on Desolneux et al. probabilistic scheme at the section 2.1 followed by our contributions presented at the section 2.2 and 2.3. It consists on a recursive course of the level lines tree structure on homogeneous subtrees, using of a topological criterion between included shapes, and a curvature analysis. These contributions reduce and optimize the final boundaries set. We present one medical application of this method using CT scan images at the section 3.

Proposed method

Desolneux et al. recently introduced a parameter-free method to detect feasible edges in a digital image without any a priori information [START_REF] Desolneux | Edge detection by helmohltz principle[END_REF], only using a basic perceptual principle called Helmholtz principle. It affirms that an observed geometric structure is perceptually meaningful if its number of occurrences is very small in a random situation. This principle has been successfully applied to numerous vision problems such as alignments detection [START_REF] Desolneux | Meaningful alignments[END_REF], vanishing point detection [START_REF] Almansa | Vanishing points detection without any a priori information[END_REF], histogram modes analysis [START_REF] Desolneux | A grouping principle and four applications[END_REF], shape indexation [START_REF] Musé | Unsupervised thresholds for shape matching[END_REF], and spatialcolor image retrieval [START_REF] Hurtut | Spatial color image retrieval without segmentation using thumbnails and the earth mover's distance[END_REF].

Probabilistic scheme

The proposed method is based on the probabilistic scheme used in [START_REF] Desolneux | Edge detection by helmohltz principle[END_REF][START_REF] Cao | Extracting meaningful curves from images[END_REF]. A geometric event is said to be ε-meaningful if this event occurs less than ε times in a random situation. This is an application of the general principle of a contrario methods: we do perceive events that are unlikely in a white noise situation. Here, the geometric event is the presence of long enough and contrasted enough level line. Let u be a gray level image. We search the level lines of u along which contrast assumes unexpectedly high values. These values are compared to the distribution of contrast following the law of the random variable X:

∀µ > 0, P (X > µ) = #{x, |∇u| > µ} #{x, |∇u| > 0} (1)
We denote by H(µ) this empirical probability, and let n u be the number of level lines of u.

According to Desolneux et al. definition, a level line C is ε-meaningful if N F A(C) = n u H(min x∈C |∇u(x)|) l/2 < ε ( 2
)
where l is the length of C. This number is called number of false alarms (NFA). It measures the meaningfulness of an event, and it depends only on the length of C and its minimal contrast. Low NFAs indicate perceptually meaningful contrasted boundaries. This event definition is discussed in [START_REF] Cao | Extracting meaningful curves from images[END_REF]. It is worth noting that this selection method only depends on one parameter: ε. Practically, this parameter is very robust. Desolneux et al. show that it is logarithmically dependent and thus varies very slowly. Therefore it is always fixed to ε = 1 whatever the image and source are. Hence the method is said to be parameterless. At this point, the method selection can be summarized as keeping all the level lines that have a NFA inferior to ε = 1.

The hierarchical structure of the level lines representation is preserved beyond this step. Two meaningful level lines are either disjoint or linked by inclusion. The meaningful level lines set is represented by a tree structure where each node contain one level line. A level line is closed or open (if it crosses the image border). Depending on the gradient sign along the level line, it also said to be positive or negative. An example of a synthetic image and its resulting level lines tree structure is presented on figure 1. A positive level line is represented by a white node. Inclusion is bottom-up. A child level line is included in its parent.

The ε-meaningful level lines set is much smaller than the whole topographic map, but still contain a lot of redundant lines. Digital sampled images present thick edges where many parallel level lines take place. Therefore Desolneux et al. define maximal ε-meaningful level lines. A maximal monotone section of a level line tree is a branch of the tree where any node has only one son, the gray level is monotone in this part (nodes are either positive or negative), and is not

Recursive maximality principle on monotone subtrees

We propose to change the maximality principle by first using a topological criterion. We observed that redundant level lines are close to one parent line or one child line. With sampled lines, this closeness is represented by the fact that no local point of child line C c is very far from a parent line C p . The asymmetrical Hausdorff distance d H represents well this kind of proximity:

d H (C c , C p ) = max Pe∈Cc ( min Pp∈Cp d(P c , P p )) (3)
where d(P e , P p ) is the euclidean distance between two points. The distance d H is used in our maximality principle.

We also improve the maximality principle by applying it to maximal monotone subtrees. We observed that missed redundant lines are often placed on different tree levels of a same monotone subtree. This happens when at least two close objects are included in a same level line. An example of this situation is shown on figure 4a). Redundancies is also due to texture effects, when at least two small contrasted level lines break the branch into at least two branches. An example of this effect is visible on figure 4b).

Therefore we propose a recursive course applied on monotone maximal subtrees and not only single-parent monotone branch. For every maximal monotone subtree S of the level lines meaningful tree, we recursively search the line C m having the minimal NFA among the subtree. Then, we look upward and downward Fig. 2. Branch maximality principle of Desolneux et al [START_REF] Desolneux | Edge detection by helmohltz principle[END_REF]. Maximal monotone section of a level line tree is a branch where any node has only one son, the gray level is monotone in this part (nodes are either positive or negative), and is not strictly contained in another monotone section. A meaningful level line is maximal if it has a minimal NFA in a maximal monotone section. in the subtree to check if no close line has been selected previously. Closeness is here defined by a threshold on d H . If so, the line is selected and removed from the subtree. If one close line is find upward or downward, all the lines on the tree path linking these two lines are removed from the subtree. We then recursively look forward the next minimal NFA in the subtree. The algorithm stops exploring the subtree when it is empty. On figure 1, this algorithm is clarified. The distance threshold h is linked to edge thickness on image. It can be empirically fixed for every image source. Typical values are h < 5 pixels.

Up to now, the algorithm is very general and can be applied to detect all meaningful objects contours in an image. An important advantage of this method is that it manipulates closed and sampled contours -the level lines. Hence can we easily improve the detection of specific objects or features by adding criteria on curvature, length, contrast sign, etc.

Curvature analysis on segmented CT scan images

CT scan images have a specific important noise. It is visible on Figure 5. This noise is called quantum mottle and is due to the fact that an X-ray beam is an uneven source of X-rays photons [START_REF] Murphey | Computed radiography in musculoskeletal imaging: state of the art[END_REF]. This uneven source has an effect on the film image via the fluorescent screen. The quantum mottle noise is scattered and missed by the detector. Bodies texture also sometimes leads the proposed method in selecting many boundaries which are not relevant regarding our application. Many denoising solutions have been proposed in the literature e.g. using wavelets

Algorithm 1 Maximal subtree filtering

Require: Meaningful level lines tree Ensure: Filtered level lines tree Main function: h ← fixed threshold for all monotone subtree S do filter(S) end for Function filter(S): while S = ∅ do Cm ← minimal NFA node of S if an upward or downward already selected node C ud is closer to Cm than h then remove all nodes between C ud and Cm remove Cm else Cm is kept for output, and removed from S for the rest of the algorithm end if filter(Sm) with Sm: subtree beginning with Cm root end while [START_REF] Pizurica | A Review of Wavelet Denoising in MRI and Ultrasound Brain Imaging[END_REF], adaptive scale space [START_REF] Scharcanski | Adaptive image denoising using scale and space consistency[END_REF], Wiener filtering [START_REF] Lee | Digital image enhancement and noise filtering by use of local statistics[END_REF] or air gap placement on the imaging plate [START_REF] Kottamasu | Musculoskeletal computed radiography in children: scatter reduction and improvement in bony trabecular sharpness using air gap placement of the imaging plate[END_REF]. This geometric noise occurs at a specific granularity scale (related the X-ray beam). This explains why proposed solutions in the literature uses wavelets. We propose to use our level lines set. To eliminate noisy boundaries, we propose a method relying on a two curvature thresholds. Curvature is easily calculated along the selected boundaries: c i = (θ (P i-1 , P i ) -θ (P i , P i+1 )) / (P i-1 P i -P i P i+1 ) on every point P i . Curvature is seen as the discrete differential of tangent angle divided by the sampling rate which is fixed and constant for all images and all boundaries. For every boundary B we measure the standard deviation σ B on its whole length, and also the standard deviation σ L B on the partial length L where the standard deviation is maximized. Typically we take L = 10% of the image diagonal. Our goal here is to detect the boundaries which have at least L part that have a noisy geometrical structure hence high standard deviations.

We manually analyzed relevant boundaries of 5 CT scans to fix these thresholds. The relevant and non relevant boundaries from this set composed of 896 boundaries are shown on Figure 6. Besides a few outlayers, the thresholds σ L < 0.45 and σ < 0.35 achieve best results (see Section 3). The diagonal alignment on this figure is due to boundaries whose length is smaller than L.

Results

Experiments have been done on a set of 20 CT scan images from the Human Visible Project [START_REF] Banvard | The Visible Human Project R Image Data Set From Inception to Completion and Beyond[END_REF]. In the following experiments we always used h = 5. This value is linked to edge thickness on image and can be empirically fixed for every image source.

Comparison results with Canny-Deriche open edge detector [START_REF] Deriche | Using canny's criteria to derive recursively implemented optimal edge detector[END_REF] and the closededge selection method using the maximal monotone selection principle from Desolneux et al. [START_REF] Desolneux | Edge detection by helmohltz principle[END_REF] are shown on Figure 7 from first row to fourth row. The Canny-Deriche detector produces many open edges or misses some contours. An edge closing step remains hazardous in this situation. The average size reduction rate from the boundaries set obtained with the branch maximality principle and the set obtained with the recursive subtrees maximality principle is 67%. This set reduction is not visually obvious because most of the boundaries which have been eliminated are multiple very close (inside the h=5 pixels distance threshold) boundaries. On all our experiments this reduction of the set size is lossless, meaning absolutely no relevant boundaries are lost.

With the curvature thresholds the reduction raises to 93%. It is visible on the second row of Figure 8 that this 24% improvement focuses on noisy and texture boundaries. Resulting boundaries meet the objective of the second contribution, that is only detect the main anatomical structures boundaries. Thanks to the flexibility of the closed edge representation, these results are achieved with a quite simple curvature analysis. Future work will include more sophisticated pattern recognition criteria with respect to some descriptors characterizing the targeted structures. Further work is undertaken to demonstrate the clinical relevance of the proposed method by validating the detected boundaries by an expert on actual data of patients.

Conclusion

In this paper we propose an automatic unsupervised edge detector inspired from the Desolneux et al. level lines selection approach. We propose a new maximality principle and apply it to maximal monotone subtrees. During our experiments, it performed a detected boundaries set reduction of 67%. We also illustrate the high flexibility of this kind of approach addressing an important CT scans image segmentation problem, by using simple and intuitive criteria on level lines such as a two thresholds curvature analysis. In the situation of our application this improvement raised the size reduction of the set to 93%. Our goal is to show the interest of this approach to medical imaging. It is very robust to noise effect, contrast invariant [START_REF] Caselles | Topographic maps and local contrast changes in natural images[END_REF], parameter-less (besides the topological criterion), very flexible and ready-to-use as a pattern recognition preprocessing. It provides closed contours easy to handle for various medical applications, and it doesn't require any shape initialization like active contour model.

Fig. 1 .

 1 Fig. 1. A synthetic image (left) and its resulting level lines tree (right). Black nodes (resp. white) represent negative line (resp. positive). The doted line represents an open line, crossing the image border

Fig. 3 .Fig. 4 .

 34 Fig. 3. Two examples using Desolneux et al. branch maximality principle. Left, one thorax MRI. Right, a CT Scan image of the trunk.

Fig. 5 .

 5 Fig. 5. Quantum mottle noise effect. The X-ray source is uneven and so is the visible image reflected on the fluorescent screen and film during CT scanning.

Fig. 6 .

 6 Fig. 6. Partial curvature standard deviation σ L graph of relevant and irrelevant boundaries according to the standard deviation σ. Circle are irrelevant boundaries, and stars are relevant boundaries.

Fig. 7 .

 7 Fig. 7. First row: original CT-Scan. Second row: Canny-Deriche open edge detection. Third row: closed edge detection with the maximal monotone selection principle. Fourth row: closed edge detection with the proposed maximality selection principle without the curvature analysis. The average size reduction rate from the boundaries set obtained with the branch maximality principle and the set obtained with the recursive subtrees maximality principle is 67%.

Fig. 8 .

 8 Fig. 8. First row: closed edge detection with the proposed selection principle prior to the curvature analysis. Second row: final boundaries set using curvature analysis.


Thomas Hurtut
email: thomas.hurtut@polymtl.ca

Farida Cheriet
email: farida.cheriet@polymtl.ca

Julie Joncas

Jean Dansereau
email: jean.dansereau@polymtl.ca

Enhancement and Segmentation of Scar Color Images after a Scoliosis Surgery

In this paper, a color-based segmentation scheme applied to dermatoscopic scar images is proposed. The image is first preprocessed via a partial differential equations (PDE)-based filter which combines in the RGB color space coherent anisotropic diffusion and shock filters. Then the scar image is segmented using a segmentation algorithm based on color distributions. A post-processing algorithm was furthermore developed to extract the scar contour, based on spline interpolations, smoothed with the generalized-cross validation method. Experiments on real patient images show the performance of the proposed method.

Introduction

Scoliosis is a complex 3D deformity of the spine, rib cage and pelvis. About one of a thousand adolescent diagnosed with scoliosis require surgery, which involves a long scar on the back surface of the patient trunk. To avoid some clinical and esthetic complications due to the length of the scar and its position on the back, a follow-up is necessary. This follow-up should give information such as the area of the scar, its depth and its contrast based on color information. These clinical indices are intended to support the dermatologist in the supervised quantification of diagnosis features, during the time evolution of the scar.

Several segmentation schemes applied to color images of pigmented skin lesions were proposed. They deal with a few problems that are scar's -noise and small structural components elimination. A method based on two dimensional color histogram and clustering was presented in [START_REF] Schmidt | Segmentation of digitized dermatoscopic images by twodimensional color clustering[END_REF] to extract skin lesions contours. A color geodesic active contours method was presented in [START_REF] Chung | Segmenting skin lesions with partial-differentialequations-based image processing algorithms[END_REF] to solve the same problem with hair. They both exploit the high contrast and the regular size of skin lesions. Unfortunately, scars are too thin and scattered to be detected either by active contour or by histogram procedures.

The goal of this paper was to develop a system that automatically extract the borders of such scars, using images captured with an active vision system designed to acquire 3D textured surface views of the trunk. Such an automatic system allows precise follow-up, and make databases on scars evolution buildingup possible. Such databases allow objective comparison between different wound closing techniques, industrial tools, and wire involved in plastic surgery procedures.

2 The proposed method

Image acquisition procedure

When a surgery is prescribed, the patient is scanned before and after the surgery3 for different applications besides the one described here. The system is composed of three 3D monoscopic cameras 4 which work in visible light. A view of the system is given on Fig. 1. Only the color texture of the back of the patient is used in this paper since the scar is always in the back. Due to the calibration of the system, and the need to capture the whole scar in one shot, the camera is placed quite far from the patient. Hence the resolution of the scar is quite small. Its contrast and level noise compared to the width of the scar is rather high too. Consequently, an important preprocessing step is necessary to enhance the contrast and denoise the image.

Preprocessing Algorithm

In order to eliminate small structure components and noise, a PDE-based filter was designed. This filter is composed of an anisotropic diffusion filter and a shock filter, both using a measure of local coherence on color space.

Coherent anisotropic diffusion

In opposition to the classic anisotropic circular diffusion of Perona and Malik [START_REF] Perona | Scale space and edge detection using anisotropic diffusion[END_REF], coherent diffusion allows elliptic diffusion. The coherent diffusion model takes the form :

∂ t u = div[D∇u] (1)
where D ∈ R 2×2 and ∂ t is the partial derivative on t. D defines the diffusion along the direction and the normal to ∇u. There were many works on different ways to define the diffusity matrix D for since. Here the regularized structure tensor matrix J ρ (∇u σ ) is used ( [START_REF] Weickert | Coherence-enhancing diffusion filtering[END_REF]):

J ρ (∇u σ ) = N ρ * (∇u σ ⊗ ∇u σ ) (2)
where u σ is the regularized image of u by convolving with a Gaussian N σ (x, y) = (2πσ 2 ) exp( x 2 +y 2 2σ 2 ), and ρ is the integration scale. The convolution with N ρ is done component-wise mainly to average a feature over a known neighborhood and over which the orientation information is average. In (2) a measure of ∇u on scalar images is used, but it can be generalized to color images ( [START_REF] Weickert | Coherence-enhancing diffusion of colour images[END_REF]):

J ρ (∇ -→ u σ ) = 3 i=1 J ρ (∇u i,σ ) (3)
where u i,σ denotes the ith channel on RGB space. Since this matrix is symmetric positive semi-definite, using eigenvalue decomposition, the formulation in ( 2) can be put as :

J ρ (∇ -→ u σ ) = (w 1 |w 2 ) µ 1 0 0 µ 2 w 1 w 2 (4)
where the eigenvalues are µ 1 ≥ µ 2 ≥ 0, and the eigenvectors are then w 1 ∇ -→ u σ and w 2 ⊥ ∇ -→ u σ . Since µ 1 and µ 2 measure the variation of the color values within a neighborhood of scale O(ρ), they describe the average contrast in the eigendirections. Constant area are characterized by µ 1 ≈ µ 2 ≈ 0. Strong and straight edges give µ 1 µ 2 ≈ 0, and corners or less structured regions give µ 1 ≥ µ 2 0. Hence, it offers a good way to measure the local coherence. Different ways to combine µ 1 and µ 2 exist. Here ∆ = √ µ 1 + µ 2 is used, which is more respectful with the corner points, in comparison with √ µ 1µ 2 or √ µ 1 , see [START_REF] Tschumperle | Restauration d'images vectorielles par EDP[END_REF] for more details. Finally D was defined as:

D(J ρ (∇ -→ u σ )) = (w 1 |w 2 ) λ 1 0 0 λ 2 w 1 w 2 (5)
where:

λ 1 = α 1 -∆ 2 s 2 if ∆ 2 ≤ s 2 and |∠w 2 | > θ min 0 else ( 6
)
λ 2 = α (7)
λ 1 defines the diffusion intensity along ∇ -→ u σ , and λ 2 along the structure. The parameter s is called the coherence threshold. Above it, no diffusion occurs in ∇ -→ u σ direction, therefore not corrupting the edge. On the other hand, it always diffuses along the edge. θ min represents the prior information on objets, that is scars are unidirectional, along a dimension of the image thanks to the fixed acquisition procedure.

Shock Filters: Shock filters are hyperbolic partial differential equations based filters which were first proposed by Rudin and Osher. Here a recent improved version of Remaki and Cheriet [START_REF] Remaki | Numerical schemes of shock filter models for image enhancement and restoration[END_REF] is used, since it allows a good control and parametrization of the shock velocity. Roughly speaking, this filter creates a shock in the position of zero-crossing of second derivative of the image, along the color gradient direction. A velocity parameter a let control the speed of the shock. This shock filter is used to balance the smoothing drawback of anisotropic diffusion, and to enhance the scar along its contour. Here it is made color coherence dependant by the use of ∆ with a Tukey biweight function. The model is:

∂ t u = -a(∆)F (∂ ηη u, ∂ η u)u η (8)
where

a = 1 -exp -(∆ 2 -s 2 ) if ∆ 2 ≥ s 2 0 else (9)
and F (a, b) = sign(a) • sign(b). ( 1) and ( 8) are eventually combined to get the general model :

∂ t u = α d • div[D∇u] -α c • a(∆)F (∂-→ ηη u, ∂-→ η u)u-→ η (10)
where α c and α d allows to control the importance of each filter.

Segmentation Algorithm

The segmentation algorithm was based on the core of the edge detector of Ruzon and Tomasi [START_REF] Ruzon | Edge, junction, and corner detection using color distributions[END_REF]. The main idea is the use of color distributions instead of simple intensity neighborhood. Combined with the use of Earth Mover's Distance (EMD) [START_REF] Rubner | A metric for distributions with applications to image databases[END_REF] to measure distance between color distributions, it gives an efficient way to exploit channels correlation. On each point, a local circular weighted window of pixels is concerned. The size of this window has to be related to the characteristic size of a scar edge which is constant. This disk is divided in two equal portions with a rotating diameter. The color distribution of each is computed, and then the EMD between them. This EMD distance is bounded by [0, 1], and represents the minimal amount of work to go from one distribution to the other in the color space, which is an instance of the transportation problem. The maximum EMD indicates the most asymmetric configuration, hence the precise direction of the local scar edge in the [0, π] interval. The value of this maximum gives the strength of this local edge. This is how it also called. The minimum EMD indicates from global point of view how asymmetric the local structure is. A corner or a junction for example will produce a higher minimum EMD than a strong binary edge. This minimum is called the asymmetry.

Postprocessing Algorithm

Since the scars are often scattered, and also because we are looking for an automatic scheme that only give a scar contour, we cannot use the simple maximum value postprocessing of Ruzon and Tomasi [START_REF] Ruzon | Edge, junction, and corner detection using color distributions[END_REF]. We have to process the segmentation algorithm outputs. Three measures are saved during the segmentation algorithm for every pixel of the image: the strength of the local edge, its orientation, and its asymmetry. These maps feed a K-means classification algorithm [START_REF] Duda | Pattern Classification[END_REF] with K = 3 classes: the scar, the background, the dermatoscopic stains.

Ideally the first class has no or few left stains. In practice, oblique and perpendicular stains, and also regular stains are very well eliminated thanks to the orientation measure and the fact that the scar is mainly unidirectional, and also the asymmetry measure. Unfortunately small irregular stains are often still present in the scar class. Therefore a second algorithm had to be designed in order to eliminate the most difficult stain. This algorithm is strongly designed for scars. It is based on an iterative study of perpendicular window histograms. From i = 1 to n (n is fixed by the length of the image), i windows are studied, and the histogram of each is calculated. See figure 2 for an example of windows cutting. The model of a scar histogram and its possible stains is given on figure 3. Regions numbered on figure 2 referred to portions of the histogram of the figure 3. The region number 3 is quite interesting, and every time it can be detected, its position on the image is saved and will be reused in the spline approximation further. Two parameters are used during this step: τ x which is related to the nominal width of a scar, and τ m which is related to the nominal height of the scar histogram extrema versus the stains. Basically, with this two parametersfixed for all scars -stains are detected and rejected of the image. The iterative concept helps to first eliminate the biggest stains, and increasingly reject smaller and closer to the scar stains.

Next, in order to split up the two parts of the scar, a smoothed approximation spline is calculated, using the positions of the regions numbered 3 in the iterative histogram algorithm. Then, each side of the scar is modelled with a weighted regression spline, whose smoothing parameter and coefficients are calculated with generalized cross-validation [START_REF] Craven | Smoothing noisy dara with spline fonctions[END_REF]. This fully automatic technique allows smooth regression without prior information on the signal and noise, and with very flexible conditions. For instance, the signal has not to be equally spaced. This proprieties of generalized cross-validation is useful here, since scar are often scattered, and since we do not have information on actual noise.

Results and Discussion

The algorithm has been tested on real patient images from St-Justine Hospital. One example of the pre-processing algorithm impact is given on figure 4. We can see that it preserves efficiently the scar while making the background more homogenous, and also improving the contrast. Details of three different scars enhancement are visible on figure 5. On figure 6, the regions where a shock is created and its intensity. We see here that the filter is very selective and succeed in applying its wanted effect only on the object we want to enhance and further analyze. If shocks were uniformly created on the image, the background noise would be enhanced. Here diffusion and shock processes balance their own drawback very well. An example of the segmentation algorithm strength output is given on figure 7. Notice how the scar is quite scattered, and thus a simple maxima research would give open edges. and red). Notice that the vertical scale is bigger than the horizontal scale, to help the reader to see the performance between automatic and manual contours. The results demonstrated that average deviation between the manual contours and the automatic contour is 0.67 ± 0.24 mm, which a very encouraging result knowing the resolution of the cameras (1.8 mm), and also noticing that the deviation between the two operators is 0.75 ± 0.21 mm.

Conclusion

The segmentation scheme proposed in this paper is the first one that have been proposed for the segmentation of scar images, though this idea has been brought by the clinical department itself. The results obtained with the segmentation scheme proposed in this paper have been visually assessed by a trained nurse, and are promising for less than one year old scars. One major future work that is in progress is the use of the 3D information to evaluate the depth of the scar, and an objective validation using some clinical indices on a large group of scar images.

Fig. 1 .

 1 Fig.1. Active vision image acquisition system[START_REF] Pazos | 3D reconstruction and analysis of the whole trunk surface for non-invasive follow-up of scoliotic deformities[END_REF]

Fig. 2 .

 2 Fig. 2. The windows for the iterative histogram algorithm. i = 5

Fig. 3 .

 3 Fig. 3. The histogram model. Portions of the histogram on figure 3 refers to regions numbered on the figure 2

Fig. 4 .

 4 Fig. 4. An example of original and pre-processed images

Fig. 5 .

 5 Fig. 5. Details of three scars, before (top) and after (bottom) the preprocessing filtering

Fig. 6 .

 6 Fig. 6. Example of regions where a shock is created, i.e. where the contours are enhanced

Fig. 7 .

 7 Fig. 7. An example of the segmentation algorithm strength output

Fig. 8 .

 8 Fig. 8. Example of extracted contours (blue), compared with manually drawn contours by two different operators (green and red)

this system is used at St-Justine Hospital, Montreal, Canada