Road vectors update using SAR imagery: a snake ... - IEEE Xplore

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 8, AUGUST 2003

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Road Vectors Update Using SAR Imagery: A Snake-Based Method Layachi Bentabet, Sylvie Jodouin, Djemel Ziou, and Jean Vaillancourt

Abstract—The paper presents an approach for roads detection based on synthetic aperture radar (SAR) images and road databases. The vectors provided by the database are refined using active contours (snakes). In this framework, we firstly develop a restoration filter based on the frost filter achieving an acceptable compromise between speckle elimination and lines preserving. This is followed by a line plausibility calculation step which is used to deform the snake from its initial location toward the final solution. The snake is reformulated using finite elements method. The setting of the snake parameters is not an obvious problem especially when they are tuned by trial-and-error process. We propose a new automatic computational rule for the snake parameters. Our approach is validated by a series of tests on synthetic and SAR images. Index Terms—Deformable models, finite elements, road database, roads detection, synthetic apeture radar (SAR) imagery, snakes parameterization, speckle reduction.

I. INTRODUCTION

S

PATIAL databases are gaining popularity as a reference tool in many fields of application. A current problem faced by spatial information providers is how to increase the accuracy of this data with limited resources. Indeed, the roads location on topographic databases is characterized by an inaccuracy ranging from 4 m to more than 200 m. Furthermore, the updating of road databases is ranging from today to more than 50 years ago. Fortunately, image processing offers increasing accuracy in terms of geometrical and temporal features extraction. In the last years, many approaches have been developed to deal with the detection of linear features on both optical images [1]–[4] and radar images [5]–[10]. Table I gives a nonexhaustive comparison between samples of proposed approaches in the literature treating the roads detection problem. The criteria distinguishing these methods could be summarized as follows: 1) the difficulty level: it depends on the nature of the used data. It is clear that the use of optical images or road databases increases the reliability of the detected roads. In Table I, the difficulty is ranging from “average”to “very difficult; 2) the objectives: they could be slightly different. Some approaches are essentially concerned with the detection of road features in the image; whereas, Manuscript received December 16, 2001; revised December 10, 2002. This work was supported by NSERC and Lockheed Martin under the CRD Grant 223322-98. L. Bentabet and D. Ziou are with the Département de Mathematiques et d’Informatique, Université de Sherbrooke, Sherbooke, QC J1K 2R1, Canada (e-mail: [email protected]; [email protected]). S. Jodouin is with Geomatics Canada, Natural Resources Canada, Sherbrooke, QC Canada (e-mail: [email protected]). J. Vaillancourt is with the Université du Québec en Outaouais, Hull, QC J8X 3X7 Canada. Digital Object Identifier 10.1109/TGRS.2003.813850

others extend the detection to allow a transaction process with a database; 3) the input data: the input may be images (SAR images, Optical images,…, etc.), road maps (numerical databases, …, etc.), or others; 4) the computational framework: the different approaches could be compared in terms of the proposed solutions; 5) preprocessing and postprocessing: another criterion is the need for preprocessing and/or postprocessing steps; in fact, some approaches involve both pre- and postprocessing and others need neither. This aspect is also related to the method proposed or/and to the data used; 5) the results: the results constitute an interesting criterion for evaluating the strength of each approach with respect to the difficulty level. In this paper, we are interested, like in [1]–[4], in the detection of roads using SAR images. The proposed method is similar to the method proposed by Auclair et al. [5]. The line detection is divided into two steps: a line plausibility calculation followed by an optimization process. The global knowledge is given by the road database which is handled as being an initial estimation of the roads location. By comparison with methods involving snake relocalization as [1], [2], [5], [6], the innovative aspects of this work consist in the use of an automated solution for snake parameterization, a finite elements-based implementation and a speckle reduction step with enhanced line preserving performances. The first innovation is made by considering that the roads’ databases provide direct knowledge on the topological properties of the targeted roads. As it will be shown later, this knowledge is directly integrated into the snake parameterization. In [2], it is proposed to reduce the speckle before the localization step, but no constraints are introduced to limit the roads smoothing. In this paper, we propose a restoration filter based on the statistical characteristics of SAR images in which are integrated the local characteristics of roads to be restored. An overview of our method is presented in the Section II. Section III focuses on the speckle suppression in SAR images. In Section IV, we present the snake-based process based on finite elements discretization. Section VI is devoted to the experimental results achieved on both synthetic and SAR images. II. OVERVIEW OF THE METHOD The context of our work offers two sources of knowledge: SAR imagery and road databases. The database provides high-level knowledge which could be considered as an initial estimation of the roads location. Indeed, the road coordinates in the database are near the real roads location in the image; consequently, the natural solution to be used is the snake approach. This approach is based on a minimization of a global energy function composed by an internal part and an external part [11]. The internal energy depends on the geometric characteristics

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TABLE I SURVEY OF WORKS ON ROADS EXTRACTION IN OPTICAL AND SAR IMAGES

of the targeted roads such as length and curvature. The external energy is based on the plausibility that a line is passing through a pixel or not. To achieve the line plausibility calculation step, the Ziou’s line detector is chosen because it provides rapid and accurate detection [12]. The snake is discretized using the finite elements scheme with a basis of Hermite functions. These functions are characterized by a low number of degrees of freedom which implies interesting smoothness properties. Hence, the snakes become more effective in tracking tasks involving noisy images where roads may exhibit significant gaps. As mentioned before, the information provided by the database is used in two different ways: firstly, the roads coordinates from the database are considered as being the initialization of the snake; secondly, the curvature of the initial roads is calculated and integrated in the snake parameterization. The general scheme could be presented as follows: 1) speckle reduction step; 2) extraction of line plausibility map using Ziou’s line-detector algorithm [12]; 3) relocalization of each road from the database using the snake approach.

III. SPECKLE REDUCTION The speckle is usually modeled as a correlated signal-dependent random phenomenon resulting in a low signal-to-noiseratio [13], [14]. As shown in Fig. 1, the speckle acts as a strong multiplicative noise. This gives rise to noisy anisotropic structures in regions with high gray level which make the automated processing of speckle-corrupted images difficult. For example, the unexpected lines introduced by the speckle (see Fig. 1) could be detected by a usual differential edge detector as being true lines. To avoid this problem, we propose as in [15] to achieve this task throughout a two-step method. Indeed, the first stage is a prefiltering step which aims to ensure an acceptable compromise between speckle elimination and edge preserving. The second step achieves the line detection task that is performed on the filtered image. Many filters have been proposed in the literature for processing images corrupted by multiplicative noise [13], [15]–[18]. The filters that have shown superior performances for processing of speckle images are the local statistic Lee and Frost filters [16], [18], [19]. The latter one possesses even better edge/detail preserving properties that the local statistic Lee

BENTABET et al.: ROAD VECTORS UPDATE USING SAR IMAGERY: A SNAKE-BASED METHOD

Fig. 2.

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GEF line profiles for various values

 > 1, and leptokurtic if  < 1.

. The profile is platykurtic if

B. Directional Modified Frost Filter (DMFF)

Fig. 1.

SAR image containing roads.

filter with equal scanning window size [19]. However, the edge preserving properties of the Frost Filter decreases when a high RSB is requested. Furthermore, the filter is implemented as being a two-dimensional (2-D) separable filter without taking into account the orientation of the linear structures in the image. These issues gave us motivation to modify the frost filter in such a way to enhance its edge/detail preserving properties by achieving a better integration of the lines information such as orientation and profile. The first part of this section is devoted to the Standard Frost Filter (SFF). In the second part, the SFF is modified by integrating the line profile information to build what we call the Directional Modified Frost Filter (DMFF). The lines preserving properties of both filters are compared in the last part. A. The Standard Frost Filter (SFF)

In this section, the SFF is rebuilt by taking into account the local characteristics of lines such as profile and orientation. The first assumption made concerns the shape of a line profile in the image. Indeed, as presented in the literature [20]–[23], many one-dimensional (1-D) parametric forms could be used such as Gaussian, exponential, rectangular and triangular functions. To achieve this task, we choose a flexible form which is able to fit better the data. Indeed, we propose to model a cross-line profile using a 1-D generalized exponential function (GEF) as follows: with

(4)

The GEF introduced in the equation above is controlled by two parameters, and . The first parameter (i.e., ) is inversely proportional to the second moment of the GEF. It controls the width of the GEF. The smaller is , the larger is the profile. The second parameter (i.e., ) controls the flatness of the GEF, (also and hence could be connected to the fourth moment, called kurtosis) as follows: (5)

Woods and Biemond [17], and Frost et al. [16] have addressed the problem of restoration of images corrupted by multiplicative noise. They present an optimum minimum mean-square error (MMSE) estimation starting from the following image model:

(2)

From (5), we can distinguish three significant cases: 1) If (i.e., ), the GEF has a flat shape and is called platykurtic. In this case, the GEF tends to a rectangular func(i.e., ), the tion when tends to infinity. 2) if GEF has an acute shape and is called leptokurtic. In this case, the GEF tends to a kronecker pulse when tends to zero. 3) if (i.e., ), the GEF is called mesokurtic; in fact, the GEF is a Gaussian. In Fig. 2 are shown some GEF shapes including rectangular, Gaussian, exponential and other lepto- and platykurtic profiles. In (1), the speckle is considered as a stationary random process with the following 1-D power spectral density [24]:

The minimization leads to the MMSE restoration filter with the following impulse response:

(6)

(1) is the ground-truth where is the convolution operation, is a signal-independent fading noise compoimages, is the point-spread function of the sensor (annent, and tenna, receiver, and correlator). The MMSE restoration filter is obtained by minimizing the quadratic error

(3) where is a scale constant. and variance of the observed image

are the local mean and .

if , and zero otherwise, is the where the number of looks, and is the local average intensity, noise bandwidth. Woods and Biemond expressed the frost filter in the frequency domain in terms of the power spectral densities

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a sharp signal. For this latter, the pass-band of the filter should be larger to minimize its smoothing effect. This fact could be interpreted as being a compromise that the filter ensures between the speckle reduction and the lines smoothing. The amplitude of the secondary modes is lower than the one of the principal mode; hence, it results in a slight enhancing of higher frequency components. The assumption made on the shape of the 1-D line profile is valid only if the built filter is applied in the direction perpendicular to the line orientation. This requires the extraction of line orientation and plausibility maps which are not known a priori. To achieve this task, we use the Rotating-Kernel Transformation as proposed by Lee and Rhodes [20], [21]. Indeed, the input image is convoluted with a rotating GEF kernel , is narrow in the directhrough 360 . The used kernel, tion given by angle and broad in the direction perpendicular is described by to it. The convolution output

(8) As the kernel rotates, the angle which minimizes the convois stored to build the orientation map. The lution output line plausibility map is calculated from the convolution output as follows: (9) Fig. 3. Shape of the restoration filter based on platykurtic GEF function with (a) a = 1 and  = 5 and (b) a = 2 and  = 5.

and are the maximum and the where minimum values of the convolution output measured at each ). Henceforth, it is important to mention that the best of the ground-truth image, , and the noise model, , point ( fitting between the GEF model given by (4) and lines in the as follows: image is obtained only if the parameters and are locally estimated. This could be achieved using a pattern matching approach as in [23]. However, the drawback of such a task is to (7) be time consuming. As it will be seen in the next section, the where is the average of the speckle. In [25], it is showed GEF parameters are set at global values that ensure an acceptthat the speckle do not change the average gray level of the able lines preserving in an image containing lines with various ) which is equivalent to say that . The widths and profiles. Moreover, in regions known not to contain scene ( main question concerns the properties of the point-spread filter lines, the scale parameter, , should be set at zero to increase the . If we assume this filter to be of finite bandwidth, this pro- smoothing effect. In this case the DMFF is equivalent to a 1-D duces a blurring effect in the spatial domain and artifacts due mean filter; besides, the SFF for the same region tends to be a . Thus, we assume that has an infi- 2-D mean filter. This leads us to choose this latter if the conto the zeros of ). This allows us to focus on sidered pixel do not belongs to a line in order to achieve much nite bandwidth (e.g., the problem of restoration of images with multiplicative noise more smoothing. Nevertheless, such a filtering strategy requires only. The final expression of the 1-D DMFF is obtained by cal- a line location map which is not known a priori. As pointed out culating the power spectral density of the line profile in (4), and by Lee and Rhodes in [20], such a map could be computed by . and in (7). The filter is controlled by thresholding the line plausibility image replacing two parameters; the flatness parameter and the scale parameter . The first parameter defines the shape of the filter; whereas, C. Quantification of Line Preserving Performances of SFF the second parameter controls the bandwidth of the filter. Fig. 3 and DMFF shows the shape of the DMFF in the case of a platykurtic profile The SFF is applied on the image in Fig. 1 at different values and . for of which corresponds to different levels of SNR. In the case of As shown in Fig. 3, The DMFF is a low-pass filter with three the DMFF, the rotational kernel transformation step is achieved modes. The bandwidth of the filter corresponds to the width using a kernel of three pixels wide and seven pixels long. The of the principal mode. By comparing the principal modes for convolution is monitored throughout eight orientations and the and , we notice clearly that the larger is the line, output image is binarized by fixing the threshold at one standard the smaller is the bandwidth. This is due to the duality that exists deviation above the mean. Both platy- and leptokurtic GEF were between the spatial and frequency domains. So that, a flat signal used to model the line profile. However, in the following only will be presented, since in the spatial domain contains less frequency components than the platykurtic-based DMFF with

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it provides the better preserving performances. As mentioned ) belongs to a line, the previously, if the pixel at position ( DMFF is applied in the direction given by the orientation map. Elsewhere, the SFF is applied. In order to measure the line preserving performances of the SFF and the DMFF, we consider two regions from the image given by Fig. 1 (Regions 1 and 3). Region 1 contains a narrow road; whereas, region 3 is homogeneous. In this context, it is clear that the filter ensuring the best compromise between speckle reduction and line preserving is the one which preserves well the road’s information in region 1, and eliminates effectively the speckle in region 3. To quantify the speckle elimination in region 3, a SNR measure is provided. The line preserving performances are quantified using the quality image index introduced by Wang et al. [26]. Basically, this measure is used to model the loss of information between two images as a combination of three factors: loss of correlation, radiometric distortion, and contrast distortion. The loss of information is issued from various phenomena including smoothing effect. Let be a region from the original image and be the same region from the smoothed image. The proposed quality index is defined as (10) The first component is the correlation coefficient between and . The second component measures how close the mean gray level is between and , whereas the third component measures how similar the contrasts of the images are. The dynamic , the similarity between and range of is [0,1]. If is maximal; . When the smoothing effect increases, the loss of information increases and hence decreases. In order to examine the effects of both filters on roads, the similarity between region 1 in the original image and the same region in the restored image is computed as function of the SNR of region 3. This latter is inversely proportional to the value of the scale parameter of the SFF in (3). The experiments are carried out by varying within the range [0.05,2]; consequently, the SNR of region 3 varies between 15.96 and 13.75. The DMFF is tested and . In Fig. 4(a) with two scale values for the GEF; are reported the similarity curves for region 1. As shown clearly in this figure, the measured similarity for the SFF is lower than the measured one for the DMFF for all SNR values. For SNR , the difference between the similarity measures of SFF (respectively, 0.19 for ). and DMFF equals 0.15 for This difference increases rapidly as the SNR increases. Indeed, when a high SNR is requested in a homogenous region, the loss of variance within roads neighborhood is higher in the case of SFF due to the diminution of the scale parameter . For exthe difference equals 0.6 for (reample, if SNR ). The global attenuation of the simispectively, 0.66 for Similarity Similarity larity (i.e., ) equals 83.1% for the SFF and 11.25% for the DMFF (respectively, 8.4% if ). The higher similarity with at a given SNR is obtained when the scale value of the GEF equals 2. In this case, the GEF used to model the road profile is narrower which increases the similarity by a global factor of 7.5%. The same experiment is carried out for a road with a large width (Region 2 in Fig. 1). In Fig. 4(b) is shown the similarity

Fig. 4. Variation of the Similarity measure as function of SNR in region 3 in (a) region 1 and (b) region 2.

curves for the road in region 2. As for region 1, the similarity is higher when the DMFF is employed. In addition, the similarity decreases more rapidly in the case of the SFF in comparison with the DMFF. In this case, the global attenuation equals (re91.6% for the SFF and 18.39% for the DMFF with ). Besides, since the road in region 2 is spectively, 20% if larger, the best similarity is obtained when the scale value of the GEF equals 1. Precisely, the global increase of the similarity and equals 3.06%. Fig. 5(a) shows the between which corimage of Fig. 1 restored using the SFF with , and Fig. 5(b) shows the same image responds to SNR and . As one could restored using the DMFF with see, the speckle is significantly suppressed in both images, however, relevant road are better preserved in the image restored by the DMFF. IV. ROAD DETECTION A. Snake Modeling As mentioned earlier, we want to refine the road localization provided by the database using active contours. Active-contour models were introduced by Kass et al. [11] as a novel solution to the low-level imaging task of finding step edges. A snake is a parametric curve which is allowed to deform from some arbitrary initial location toward the desired final location by minimizing an energy function. The minimization of

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eter. We admit that the energy function is built such that its global minimum coincides with the expected solution (12) The regularizing term has a stabilizer role, since the external term is usually irregular and shows a large amount of local minima. Tikhonov stabilizers are often used to measure the curve smoothness. Thus, the internal energy of a curve is given by (13) are positive functions. Second order stabilizers are where ). The stabilized curve must thus be twice mostly used ( differentiable. A curve regularized by a second order Tikhonov stabilizer is identical to a thin membrane with mechanical en. Function measures the tension while measures ergy its rigidity. In the remainder part of the paper, we assume that the thikonov stabilizers remain constant throughout the snake. and Hence, the parameter in (11) can be absorbed by as follows: (14) The external energy depends on a potential computed from ) atthe data. This term represents the image forces ( tracting the curve. In our case, the force is the line plausibility ) given by the line-detector algorithm [12] ( (15) More details on the Ziou’s line detector are provided in Appendix C. Fig. 6 shows a synthetic image with a multiplicative noise and the corresponding line plausibility image. As it can be seen from the figure below, the highest line plausibility is obtained for pixels where a line is passing through. The problem is to determine the curve such as

Fig. 5. SAR image after restoration with (a) SFF (SNR = 15:21) and (b) DMFF (a = 2,  = 5, and SNR = 15:21).

the energy function gives rise to spatial forces that act upon the contour to deform it by translation, stretching and bending. Constraints upon the snake are introduced to provide smooth localization. The Kass model incorporates two internal energy terms which act to constrain the solution by imposing conditions related to contour smoothness and regularity. In the continuous domain, the snake is defined as a parametric curve, , where is a parameter advancing along the snake and is related to arc length. The snake minimizes an energy function based on internal and external constraints at time [11] (11) is the external energy defined through the data, where the internal regularizing energy and a regularization param-

(16) The equation above does not admit any analytical solution. An iterative scheme is required to solve it such as greedy algorithms or simulated annealing. The most common method is to rewrite this equation as an Euler equation corresponding to force equilibrium. The equilibrium equation is then discretized using finite elements or finite differences. Besides, the energy is usually a nonconvex function that possesses several local minima. The goal is to make the model evolve toward one of ), it those minima. Using Euler-Lagrange equation ( can be proved that the local minimum of energy must satisfy (17) In (17), it is assumed that and remain constant along the in the snake. It should be noticed that the derivative term right-hand side of the previous equation is a vector derivative

BENTABET et al.: ROAD VECTORS UPDATE USING SAR IMAGERY: A SNAKE-BASED METHOD

Fig. 7.

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Discretizing a curve with finite elements.

Fig. 8. Calculation of the average curvature of a road.

derivatives, leading to greater accuracy. Discretizing the para, for , into finite elements leads to an metric curve expression for each element, given by (19) denotes row vectors and matrices or where is a set of shape or basis functions column vectors. defining the interpolating curves is a two-column matrix of control nodes. and can be approximated Thus, as shown in Fig. 7, the curve using finite elements Fig. 6.

(a) Synthetic image with simulated roads. (b) Line plausibility map.

(20)

which can be decomposed in terms of and . This equation might be seen as an equilibrium state between internal regularizing forces and external image forces. It requires the computation of the curve fourth-order derivative which might cause numerical stability problems. A Lagrangian evolution is obtained by introducing a first-order derivative term which is considered as an inertial term

By the application of the Rayleigh-Ritz element method [31], [32] and referring to the energy function, we derive for each element the following:

(18)

(21)

where represents the viscosity of the background. The higher the viscosity is, the greater are the inertia and the slower is curve evolution. The initial curve evolves until the inertial term vanishes; consequently, the curve satisfies the stationary equilibrium between internal and external forces.

(22)

B. Snake Discretization In computer vision, partial derivative equations, such as (18), are usually implemented using finite differences [27]–[29]. Few works have been devoted to finite elements [30], [31]. The finite elements method is a simple approach widely used in engineering. This method offers an accurate discretization of the

and , respectively, are the first and second where . columns of the elemental control nodes, and and , represent the first and second derivatives of the elemental shape functions. Details on the differentiation of the snake energy are given in Appendix A. The minimization of leads to ; thus (23)

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Fig. 9.

Fig. 10.

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 8, AUGUST 2003

Synthetic image with simulated roads. (a) The roads template. (b) The template after applying multiplicative noise. (c) The restored image.

Line plausibility map of (a) the noisy image and (b) the restored image.

where (24) is the external forces vector applied on the control nodes of element of the snake. The choice of Hermite polynomials is -continuity properties. The discretization motivated by their scheme of the curve is explained in Appendix B. Performing the necessary calculus with the cubic Hermite polynomials yields

where is the vector containing and , respectively, are the stiffthe nodes of the snake, ness matrix and the vector of external forces. The matrices and are obtained by assembling , , and for all snake elements. For a better understanding, it is important to and the external forces mention that the stiffness matrix are time independent. Indeed, the stiffness matrix is calculated before iterating using the Hermite basis and the snake parameis given, ters (i.e., and ). The vector of external forces for all iterations, by the values of line plausibility at the current snake location. C. Time Discretization The dynamic system given in (18) must be discretized in time by the finite differences scheme. Given a time step, partial derivatives might be approximated by

If we consider all the snake elements, we obtain the following global system: (25)

(26)

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TABLE II THE AVERAGE DISTANCE BETWEEN THE OBTAINED SOLUTION AND THE GROUND-TRUTH

To discretize (18), both explicit and implicit schemes could be used. From (18) and (25), we deduce the implicit numerical scheme as follows: (27) This scheme converges only if the selected time step is small [33]. However, (27) gives an explicit way of computing from (28) identity matrix and is an matrix where is the which depends on , , and the shape of the interpolating funcgives the position of the element nodes at iteration . tions. An implicit numerical scheme (29) could not be solved, since it would require the knowledge of at iteration . In practice, a semi-implicit scheme is used (30) The semi-implicit model is more stable than the explicit scheme and insures the numerical convergence in the case of large time step. The semi-implicit model yields an iterative solution

Fig. 11. Initial road vectors.

1) Parameter : As mentioned before, is a positive parameter which is used to model the viscosity. In fact, the bigger this parameter is, the higher is the inertia of the snake and as a result a better regularity is obtained. On the other hand, with a small , the snake has a very weak inertia, and thus, it becomes too sensitive to the forces of the image which results in oscillating behaviors. Quite often, at the beginning of a process, a small value of is used in order to allow the snake to advance rapidly and thereby reach a strong value allowing a smooth evolution when the snake is quite close to the targeted boundary. An adaptive proposal of is put forward in [34] which allows to control the mean displacement, , of the snake by reducing it at each iteration. The parameter is fixed in order to balance the level of change in the energy of the snake and its displacement. This allows the minimization of the snake’s energy to be more pronounced if the snake evolves more rapidly. Fua [34] has modeled this behavior by the following equation: (32) However

(31) The use of semi-implicit model allows us to select large values of time step, and as a consequence the number of iteration is reduced. Nevertheless, the semi-implicit scheme requires the costly inversion of matrix . The matrix inversion is computed each time the snake coefficients change. D. Automatic Snake Parameterization According to (18), the snake model is controlled by three parameters. First, controls the speed of evolution of the snake. Secondly, controls the stretching of the snake during the localization process. Finally, impacts on the bending capability of the snake (rigidity parameter). These parameters are difficult to establish correctly. Indeed, they are generally set in a lengthy trial-and-error testing process. In addition, the final snake location is highly sensitive to small variations in these values. In the remainder part of this section, the impact of each parameter we will be discussed. Computation rules associated with each parameter will be proposed and discussed thereafter.

(33) where is the average displacement of the snake. Since therefore

,

(34) where the number of pixels of the snake. Nevertheless, we have to choose the decreasing sequence of the step that should be used. This requires prior knowledge on the way in which the snake will evolve. Furthermore, at the equilibrium point [from (32)] we deduce that (35) This is true only if the local energy of the snake remains constant along the detected road. Since, the external energy depends on the plausibility of the line, which is not necessarily constant along the detected road; we deduce that (32) is not necessarily ), we can only verified. At the equilibrium status (

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Final snake location in (a) the restored image and (b) the noisy image.

state that the global energy of the snake does not change; and . Thus, we propose to rewrite (32) only in hence, terms of temporal derivatives as follows: (36) Rewriting (36) in terms of the partial derivatives of energy using the chain derivative rule yields (37) During the snake relocalization

, thus (38)

We obtain a new formulation of the parameter . Unlike [34] a decreasing sequence during the adaptive evaluation of the parameter is not needed. 2) Rigidity and Elasticity: The internal snake force serves as a smoothing constraint and acts on the snake shape. Indeed, the tension parameter, , defines the capability of the snake to stretch and the rigidity parameter, , constraints the snake curvature. In [35], a method is developed to estimate upper and lower bounds for the elasticity and the rigidity parameters. The parameters are defined through a ratio between the internal energy and the external energy in a way to keep the snake in the neighborhood of the targeted contour. In [36], upper bounds for the snake parameters are defined using the snake shape. Indeed, the snake curvature is used to determine the maximum values of the rigidity and the elasticity parameters which provide a smooth solution. When the snake expands, the parameters are updated to maintain a constant ratio between the external and internal forces. A similar approach is given in [37] where the snake parameters are dynamically adjusted to avoid instability in the deformation process. The previous approaches give a range for the rigidity and elasticity parameters without giving an effective computation. In this section, we give a set of rules for the determination of the snake parameters. We focus on the rigidity parameter where a direct estimation method is developed. Since

Fig. 13.

Localization with setting using Fua method.

Fig. 14. road.

Variation of the global energy of the snake attracted by the high curved

the snake initialization comes from the existing database, an a priori knowledge on the snake shape is then available. This latter is used to set the snake parameters with the assumption that the global shape and length of the road do not change considerably.

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Fig. 15. (a) and (b) STARLOS SAR images. (c) and (d) Line maps.

a) Parameter : The elasticity parameter controls the rate of change of the snake length. Indeed, this parameter is pro, where is the accepted change rate in the portional to global length of the snake. The idea is to give a direct relation . In our framework, we do the assumpbetween and tion that the snake do not stretch significantly from its initial length given in the database. In this case, a small value (near zero) is assigned to in order to constrain the snake to limit its stretching. The snakes are initialized from a simulated road database. The initialization is shown in white in the image (see Fig. 11). The rigidity parameter for each road is calculated according to (42). We obtain the following values: for the straight road , for the road with average curvature and for the high curved road. The parameter was set according to

(38). In order to measure the impact of the restoration step, the snake localization process is achieved before and after restoration. Since the ground-truth is known, a quantification of the localization error is possible for both cases. Indeed, the obtained results are sampled and compared to the ground-truth vectors. This is achieved by measuring the average distance between the ground-truth and the obtained results as follows:

where ( ) are the coordinates of the obtained solution using ) are the coordinates of the snake relocalization, and ( the ground-truth roads. The final snake locations are given in Fig. 12(a) and (b). In Fig. 12(a) are shown the localized roads

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Fig. 16. (a) and (b) Restored SAR images. (c) and (d) Line maps.

in the restored image; whereas, Fig. 12(b) shows the localized road in the noisy image. If we compare the localization results with the ground-truth we obtain the following. b) Parameter : The rigidity parameter is estimated from the curvature of the initial roads given by the database. The rigidity parameter can be estimated globally from either the average curvature of the road, or locally from the local curvature. The local curvature of the snake can be estimated using the finite elements method according to the nodes coordinates and the interpolation curve as follows: curv

(39)

The curvature is ranging from zero to infinity. Generally, the absolute value of the angle between two roads do not exceed

(the considered angles are shown in Fig. 8). With such an assumption, the maximum of the snake curvature will correspond . In this case, the maxto the case of a road angle equal to given by (39) equals 5 when Hermite imum curvature curv polynomials are used. The rigidity parameter is inversely proportional to the curvature measure. Thus, the parameter could be estimated as follows: curv curv

with

(40)

. is the maximum value assigned where to the rigidity parameter which corresponds to a straight road. ), quadratic ( ), or The parameter allows a linear ( higher order relationships between the rigidity parameter and the curvature. Equations (39) and (40) allow a dynamic update

BENTABET et al.: ROAD VECTORS UPDATE USING SAR IMAGERY: A SNAKE-BASED METHOD

Fig. 17.

Ground-truth.

Fig. 18.

Initial vectors.

of . This implies the recomputation of the stiffness matrix at the end of each iteration; however, this is a time consuming task. Let us now consider a global setting of . The rigidity parameter can be directly estimated from the road characteristics in the database. Indeed, the average curvature of a road composed by several segments could be calculated as follows:

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line segments equals or ; the minimum is obtained when the angle equals zero. As in (40), the rigidity is calculated as follows: (42) where parameter.

is the maximum value assigned to the rigidity

(41) V. ALGORITHM is the number of segments forming the road. The angle where , between two successive segments is shown in Fig. 8. We assume that the angle between two road’ segments does not ex(or ). For , the average curvaceed ture varies within the range [0, 1]. Indeed, the maximum curvature is reached when the angle between each two successive

The developed algorithm takes as input the road database coordinates and a SAR image of the same region after the restoration has been applied. The output could be the SAR image with highlighted roads and/or new road definition coordinates which serves to update the database. To summarize the correction step, for each road segment we do the following:

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Fig. 19.

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 8, AUGUST 2003

Final snakes’ location.

1) Initialize a snake curve with roads from the database. 2) Compute the parameters for the snake. 3) Compute the line plausibility map 4) Compute V(n+1)T from VnT using (31). 5) If the total energy of the curve decreases, then 6) If (EnT 0 E(n+1)T ) < "EnT , stop the snake; else, increment t and go to step 4. 7) If the total energy of the curve increases, recalculate , increment t and go to step 4.

The snake is stopped if the relative difference between two successive iterations becomes smaller than a fixed threshold .

TABLE III AVERAGE DISTANCE BETWEEN THE OBTAINED SOLUTION AND THE GROUND-TRUTH

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Fig. 20.

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(a) RADARSAT SAR image. (b) Restored image. (c) The line plausibility map of the noisy image. (c) The line plausibility map of the restored image.

VI. EXPERIMENTAL RESULTS The algorithm is validated through a series of tests on both synthetic and SAR images. The images are restored with the DMFF using a platykurtic GEF. The GEF parameters, and , are set at 5 and 2, respectively. The rigidity parameter is automatically set using the average curvature of roads segments is set empirically as given by (42); where, equals 2, and at at 100. The initial snake is generated by inserting nodes regular distance intervals along the road segments. It is important to mention that the features database scale and the images resolution could be different, and in such a case, we should bring them into equivalent values. A. Synthetic Images The roads localization was firstly carried out on the synthetic image in Fig. 9(b). This image is obtained by applying a multiplicative noise on the template given by Fig. 9(a). The image contains three simulated roads with different curvatures. The first road is straight, the second contains average curvatures, and

the third is a high curved one. The restored template is given by Fig. 9(c). The line plausibility map is established for both noisy and restored images. The obtained maps are shown in Fig. 10(a) and (b). As we could see, the use of the DMFF improves the lines detection, and suppresses significantly the speckle. This will improves the accuracy of the snake localization; indeed, the number of local minima in the roads neighborhood is significantly reduced. From Table II we could see that the restoration filter improves the accuracy for all roads. Indeed, the localization error, when the DMFF is employed, is at the subpixel level. In order to demonstrate the effectiveness of our method for calculation, a comparison between the Fua’s method and our approach is carried out. Fig. 13 shows the localization of the high curved road if is calculated using Fua’s method. As we can see, the snake converge toward the road’s location; but, since the ), line plausibility is not constant along the road ( the snake moves along the road and collapses in some parts (see the right bottom part of the road in Fig. 13). The rate of global collapsing of the snake length equals 9%. he curve presented in

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Fig. 21.

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 8, AUGUST 2003

Initialization is in white and the final snakes’ location is in black.

Fig. 14 gives the variation of the total energy of the snake during the localization process of the high curved road. We can see that the total energy decreases until it reaches a constant level which corresponds to the final location. B. SAR Images In this section, we illustrate the proposed algorithm on real SAR images, showing the potential of the method. Fig. 15(a) and (b) present MSTAR SAR images collected using the STARLOS sensor. The images are collected near Huntsville (Alabama) with 1-ft resolution. The images contain straight roads with different widths. At some places, the roads are partially gapped due to trees and houses shadows. For those regions, we expect the snake to recover the lacking information. The line plausibility maps before applying the DMFF are presented in Fig. 15(c) and (d). The images after restoration and their line plausibility maps are presented in Fig. 16(a)–(d). As for the synthetic images, the speckle is significantly reduced and the lines are well detected after applying the DMFF. For both presented scenes, the ground-truth information is not available. Thus, it is not obvious to quantify the accuracy or to compare the snake localization results obtained before and after restoration. However, since the resolution of the images is high, this permits us to use a segmentation of a human expert. Another factor, which encouraged us to use such segmentation, is that roads are generally straight. The human-expert roads localization is given in Fig. 17. The initialization is shown in white in Fig. 18; whereas, the final results are given by Fig. 19. For both scenes, the best results are obtained in the restored image. The table below gives the obtained values for and for each road. By analyzing Table III, we could see that except roads 4 and 6, the localization accuracy is better on the restored images, especially for road 5 where the improvement is considerable due to the efficient reduction of the local minima in the road’s neighborhood. For the shadowed region on road 2, the behavior of the snake is more interesting in Fig. 20(a). Indeed, the snake keeps a smooth shape which minimizes the impact of the shadow. For

the shadowed regions on roads 6, 7, and 8, the snakes recover the lacking information and converge to the right location. This was guaranteed by the smoothness of the Hermite basis and the direct integration of geometric road characteristics into the snake parameterization. The last test is undergone on a RADARSAT SAR image centered on Stephenville (Newfoundland) with 25-m resolution [see Fig. 20(a)]. At this resolution, it is difficult to use a human expertise. Indeed, some roads (road 2, for example) are completely corrupted by speckle, and hence not visible. In this scene, roads are bright or dark lines. This is taken into account when calculating the external energy of the snake. We calculate the external energy of the bright roads which is merged with the energy of the dark roads. The parameter for the first road equals 90; for the second road, equals 50; for the third road, it equals 20. Fig. 21 shows the snake initialization in white and the final result in black. We could see that the roads and their intersections are accurately localized. For shadowed regions, as region 4, the snake converges, and as a result, gives an accurate localization of the road. An interesting behavior could be also noted for road 2 where no information on the road’s location is available, and for which the snake do not diverge and gives an interesting solution. VII. CONCLUSION In this paper, a road detection method has been developed for SAR images. Our method combines high and low level information to achieve accurate roads localization. In order to suppress the speckle, a restoration filter, called DMFF, based on the frost filter was designed and applied. The use of the DMFF improves the restoration of roads by ensuring an acceptable compromise between speckle suppression and roads smoothing. The high knowledge level is provided by a road database. This latter gives an initial estimation of the roads location which is refined through active contour process. Our snakes were discretized using the finite elements method providing more accuracy and effectiveness for roads exhibiting significant gaps. We have also presented an original integration of the geometrical features of roads calculated from the database in snakes’ parameterization. Experimental results on synthetic and SAR images showed that our method is able to localize reliable roads, especially, when the roads are immersed in the speckle. This work could be extended to the detection of the new roads by using the junctions of roads and a roads following step. APPENDIX A The differentiation of the snake, from (21) and (22), is performed with respect to the control nodes as follows:

(A.1)

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It is helpful at this stage to rewrite making use of the and given by (B.4). Thus, we obtain definition for the following modified set of cubic Hermite shape functions: (A.2) (B.6) The constant factor 2 in (A.1) and (A.2) could be absorbed by and (as for in (14). Thus, this constant will be omitted in the following. Equations (A.1) and (A.2) may be combined into a single matrix equation

where the vector of control nodes is now defined as (B.7)

(A.3)

Without regard to the specific interpolation functions to be used, (A.3) can be rewritten as follows:

APPENDIX C Let us consider an image line profile without noise,

as the sum of two components: , and white noise (C.1)

and are positive constant. where must maximize According to Canny, a good edge detector the signal-to-noise ratio, the localization, and the multiple-response criterion. In this way, the search of the optimal detector could be handled by the maximization of

where and

APPENDIX B In this appendix we describe the steps to follow for the curve discretization. The cubic Hermite basis is defined as follows [38]:

(C.2) A particular solution of this problem is given by (C.3)

(B.1) This basis defines a vector of the above shape functions (B.2) The vector of control nodes defining on element of a piecewise parametric curve is given by (B.3) where the end-derivatives, slopes

, are specified as the default

and (B.4) Thus any point in one element of the curve is defined by (B.5)

where is obtained by normalization and is a scale parameter. In order to detect lines in an image, we need to extend the filter to two dimensions. This is done by using two directional filters. The filter in the direction ( respectively) is the result of the product of in the direction ( respectively) and a function in the direction ( respectively). Usually, is defined by . from the normalized The computation of the orientation and plausibility of the line requires the use of another filter which operate in both and directions. This filter is given by . Thus . Symmetrithe filter in the direction is . The cally, the filter in the direction is . The extraction filter in both directions is of lines requires the estimation of plausibility and orientation at each pixel of the image. Since the local maxima in the input signal corresponds the edges sought, the edge orientation can be defined as the orientation of the eigenvector of the following matrix which corresponds to the greatest eigenvalue. The plausibility is the value of that eignevalue. The matrix is (C.4)

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The eigenvalue could be easily calculated by

If the eigenvalue with greatest maximum absolute value is negative, the line is bright on a dark background, and if it is positive, the line is dark on a light background. ACKNOWLEDGMENT The images were provided by the Canadian Space Agency under the ADRO-2 program. REFERENCES [1] F. Tupin, H. Maître, J. F. Mangin, J. M. Nicolas, and E. Pechersky, “Detection of linear features in SAR images: Application to road network extraction,” IEEE Trans. Geosci. Remote Sensing, vol. 36, pp. 434–452, Mar. 1998. [2] B. K. Jeon, J. J. Jang, and K. S. Hong, “Map-based road detection in spaceborne synthetic aperture radar images based on curvilinear structures extraction,” Opt. Eng, vol. 39, no. 9, Sept. 2000. [3] J. Chanussot, G. Mauris, and P. Lambert, “Fuzzy fusion techniques for linear features detection in multitemporal SAR images,” IEEE Trans. Geosci. Remote Sensing, vol. 37, pp. 1292–1305, May 1999. [4] Y. Wang and Q. Zheng, “Recognition of roads and bridges in SAR images,” Patt. Recognit., vol. 31, no. 7, pp. 965–977, 1998. [5] M. F. Auclair-Fortier, D. Ziou, C. Armenakis, and S. Wang, “Automated correction and updating of road databases from high-resolution imagery,” Can. J. Remote Sens., vol. 27, no. 1, pp. 76–89. [6] H. Mayer, I. Laptev, A. Baumgartner, and C. Steger, “Automatic road extraction based on multi-scale modeling, context and snakes,” Int. Arch. Photogramm. Remote Sens., pt. 3–2W3, vol. 32, pp. 106–113, Sept. 1997. [7] A. Baumgartner, C. T. Steger, H. Mayer, and W. Eckstein, “Multi-Resolution. Semantic objects, and context for road extraction,” in Semantic Modeling for the Acquisition of Topographic Information from Images and Maps, W. Fröstner and L. Plümer, Eds. Basel, Switzerland, 1997, pp. 140–156. [8] N. Merlet and J. Zerubia, “Integration of Global Information for Roads Detection in Satellite Images,” INRIA, 3239, 1997. [9] C. Hivernat, X. Descombes, S. Randriamasy, and J. Zerubia, “Mise en Corespondance et Recalage de Graphes: Application aux Réseaux Routiers Extraits d’un Couple Carte/Image,” INRIA, 3529, 1998. [10] D. Klang, “Automatic detection of changes in road database using sattelite imagery,” in Proc. Int. Archives Photogrammetry and Remote Sensing, vol. 32, 1998, pp. 293–298. [11] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” Int. J. Comput. Vis., vol. 1, no. 4, pp. 321–331, 1988. [12] D. Ziou, “Finding lines in grey-level images,” in Proc. Int. Conf. Pattern Recognition, 2000, pp. 534–537. [13] D. T. Kuan, A. A. Sawchuk, T. C. Strand, and P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Pattern Anal. Machine Intell., vol. PAMI-7, pp. 165–177, Mar. 1985. [14] R. N. Czerwinski, D. L. Jones, and W. D. O’Brien, “Line and boundary detection in speckle images,” IEEE Trans. Image Processing, vol. 7, pp. 1700–1712, Dec. 1998. [15] P. Bolon, J. Chanussot, I. Issa, and P. Lambert, “Comparison of prefiltering operators for road network extraction in SAR images,” Proc. IEEE Conf. on Image Processing, ICIP99, vol. 2, pp. 924–928, Oct. 1999. [16] V. S. Frost, J. A. Stiles, K. S. Shanmugan, and J. C. Holtzman, “A model for radar images and its application to adaptive digital filtering of multiplicative noise,” IEEE Patt. Anal. Machine Intell., vol. PAMI-4, pp. 157–166, Mar. 1982.

[17] J. W. Woods and J. Biemond, “Comments on “A model for radar images and its application to adaptive digital filtering of multiplicative noise”,” IEEE Pattern Anal. Machine Intell., vol. PAMI-6, pp. 658–659, Sept. 1984. [18] J. Lee, “A simple speckle smoothing algorithm for synthetic aperture radar images,” IEEE Trans. Syst. Man, Cybern., vol. SMC-13, pp. 85–89, 1983. [19] V. V. Lukin, V. P. Melnik, A. B. Pogrebniak, A. A. Zelensky, J. T. Astola, and K. P. Saarinen, “Digital adaptive robust algorithms for radar image filtering,” Electron. Imag. J., vol. 5, no. 3, pp. 410–421, July 1996. [20] Y. K. Lee and W. T. Rhodes, “Rotating-kernel min-max algorithms for straight-line feature enhancement,” Appl. Opt., vol. 34, no. 2, pp. 290–298, Jan. 1995. , “Invariant pattern recognition using angular signature functions,” [21] Appl. Opt., vol. 32, no. 23, pp. 4372–4377, Aug. 1993. [22] R. N. Czerwinski, D. L. Jones, and W. D. O’Brien, “An approach to boundary detection in ultrasound imaging,” Proc. 1993 IEEE Ultrasonics Symp., Nov. 1993. [23] S. K. Nayar, S. Baker, and H. Murase, “Parametric feature detection,” in Proc. Computer Vision and Pattern Recognition, CVPR’96, San Francisco, CA, June 1996, pp. 471–477. [24] A. C. Bovic, “On detecting edges in speckle imagery,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp. 1618–1627, Oct. 1988. [25] F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive. Norwood, MA: Artech House, 1981. [26] Z. Wang and A. C. Bovic, “A universal image quality index,” IEEE Signal Processing Lett., vol. 9, pp. 81–84, Mar. 2002. [27] M. Kisworo, S. Venkatesh, and G. A. W. West, “Detection of curved edges at subpixel accuracy using deformable models,” IEE Proc. Inst. Elect. Eng., vol. 142, no. 5, pp. 304–312, Oct. 1995. [28] J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science. Cambridge, U.K.: Cambridge Univ. Press, 1999. [29] R. Deriche and O. Faugeras, “Tracking line segments,” Image Vis. Comput., vol. 8, no. 4, pp. 261–270, Nov. 1990. [30] L. D. Cohen and I. Cohen, “Finite element methods for active contour models and balloons for 2–D and 3–D images,” IEEE Trans. Pattern Anal. Machine Intell., vol. 15, pp. 1131–1147, Nov. 1993. [31] J. Liang, T. Mclnerny, and D. Terzopoulos, “United snakes,” in IEEE 7th Int. Conf. Computer Vision, Sept. 1999, pp. 933–940. [32] J. N. Reddy, An Introduction to the Finite Element Method, 2nd ed. New York: McGraw-Hill, 1993. [33] J. Montagnat, H. Delinguette, N. Scapel, and N. Ayache, “Representation, shape, topology and evolution of deformable surfaces. Application to 3D medical image segmentation,” INRIA, France, 3954, 2000. [34] P. Fua and Y. Leclerc, “Model driven edge detection,” Mach. Vis. Applicat., 1989. [35] O. V. Larsen and P. Radeva, “Bounds on the optimal parameters of elasticity for a snake,” in Proc. 8th Int. Conf. Image Analysis and Processing, Sanremo, Italy, 1995. [36] M. S. Horritt, “A statistical active contour model for SAR image segmentation,” Image Vis. Comput., vol. 17, pp. 213–224, 1999. [37] R. Samadani, “Adaptive snakes: Control of damping and material parameters,” Proc. SPIE, vol. 1570, 1991. [38] W. B. Bickford, A First Course in the Finite Element Method. Homewood, IL: Irwin, 1990.

Layachi Bentabet received the B.Eng. degree in electrical engineering from the National Polytechnic School, Algiers, Algeria, in 1998, and the Diplôme d’études approfondies degree in signal and image processing from the Institut National des Sciences Appliquées, Lyon, France, in 1999. He is currently pursuing the Ph.D. degree at the University of Sherbooke, Sherbooke, QC, Canada. His research interests include image processing, computer vision, and deformable shape models.

BENTABET et al.: ROAD VECTORS UPDATE USING SAR IMAGERY: A SNAKE-BASED METHOD

Sylvie Jodouin received the B.S. degree in computer sciences in 1987, and the M.S. degree in computer sciences in 2001, both from the University of Sherbrooke, Sherbooke, QC, Canada. She has been working for Geomatics Canada, Natural Resources Canada, Sherbrooke, since 1988. She is now working on framework data models for the Canadian government. The updating of spatial data remains her greatest subject of interest.

Djemel Ziou received the B.Eng. degree in computer sciences from Université d’Annaba, Annaba, Algeria, in 1984, and the Ph.D. degree in computer sciences from the Institut National Polytechnique de Lorraine, Lorraine, France, in 1991. From 1987 to 1993, he served as Lecturer in several universities in France. During the same period, he was a Researcher in the Centre de Recherche en Informatique de Nancy (CRIN), Nancy, France, and the Institut National de Recherche en Informatique (INRIA), Paris, France. He is currently a Professor at the University of Sherbrooke, Sherbrooke, QC, Canada. His research interests include image processing, computer vision, and pattern recognition.

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Jean Vaillancourt received the B.S. degree from Université de Laval, Canada, in 1981, and the Ph.D. degree from Carleton University, Ottawa, ON, Canada, in 1987. He is currently Dean of research at the Université du Québec en Outaouais, Gatineau, QC, Canada. After starting his career as a Junior Methodologist at Statistics Canada, he became Professor of mathematics in 1986 and was Associate Dean of sciences from 1997 to 2001, at the University of Sherbrooke, Sherbrooke, QC, Canada. During this period, he served as member of the Board of Directors of the Montreal Mathematics Research Center (CRM), the Management Committee of the Quebec Mathematical Sciences Institute (ISM), the Board of Directors of the Statistical Society of Canada, as well as several committees of the Natural Sciences and Engineering Research Council of Canada. He is a founding member of the multiuniversity consortium CoRIMedia for automated content-based image searches. He is also a member of the Board of Directors of the Quebec Institute for the sustainable management of deciduous forests (IQAFF) His research interests include statistical estimation in the context of the numerical treatment of images.