Curvelet-Based Snake for Multiscale Detection ... - Semantic Scholar

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 12, DECEMBER 2006

Curvelet-Based Snake for Multiscale Detection and Tracking of Geophysical Fluids Jianwei Ma, Anestis Antoniadis, and Francois-Xavier Le Dimet

Abstract—Detection and target tracking have an application to many scientific problems. The approach developed in this paper is motivated by the applications of detection and tracking characteristic deformable structures in geophysical fluids. We develop an integrated detection and tracking method of geophysical fluids based on a discrete curvelet representation of the information characterizing the targets. Curvelets are in some sense geometric wavelets, allowing an optimal sparse representation of twodimensional piecewise continuous objects with C 2 -singularities. The proposed approach first identifies a consistent vortex by a curvelet-based gradient-vector-flow snake and then establishes the motion correspondence of the snaxels between successive time frames by a constructed so-called semi-T or comp-T multiscale motion-estimation method based on the geometric wavelets. Furthermore, a combination of total-variation regularization and cycle-spinning techniques effectively removes false matches and improves significantly the estimation. Numerical experiments at each stage demonstrate the performance of the proposed tracking methodology for temporal oceanographic satellite image sequences corrupted by noise, with weak edges and submitted to large deformations, in comparison to conventional methods. Index Terms—Curvelets, motion estimation (ME), snake, total variation (TV), tracking of geophysical fluids.

I. I NTRODUCTION

T

RACKING motion of nonrigid targets can be an important step to handle some problems in data assimilation. Data assimilation is an ensemble of techniques combining in an optimal way the mathematical information provided by equations and the physical information given by observations, in order to retrieve as best as possible the state variables of a model. It has been used in meteorological, hydrological, and oceanographic applications, to name only a few. Data used for assimilation of geophysical fluids are usually direct or indirect measurements of model’s state variables. For example, in the meteorological domain, besides the observed data from ground stations, radiosondes and pilot balloons, there is a huge number of available satellite data that can be assimilated as a collection of individual measurements. But most often, the structured spatial information observed on these data/images is used in

Manuscript received December 5, 2005; revised August 15, 2006. This work was supported by the IDOPT Project (CNRS-INRIA-UJF-INPG). J. Ma was with the Laboratoire LMC-IMAG, University Joseph Fourier, 38041 Grenoble Cedex 9, France. He is now with the Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China (e-mail: [email protected]). A. Antoniadis and F.-X. Le Dimet are with the Laboratoire LMC-IMAG, University Joseph Fourier, 38041 Grenoble Cedex 9, France (e-mail: antonia@ imag.fr; [email protected]). Digital Object Identifier 10.1109/TGRS.2006.885017

a rather qualitative way. An interesting question is how to make use of the quantitative structured information contained in satellite observed data for data assimilation within geophysical prediction models in order to allow better forecasts. The potential of satellite images for assimilation in the oceanographic domain has been pointed out recently by coupling dynamic data such as the motion vector field of characteristic structures [1]. In this paper, we focus on the extraction of the quantitative structured information, i.e., Lagrangian trajectories by detection and tracking of boundaries of characteristic deformable structures, especially the vortexes, using a new designed curvelet-based multiscale gradient-vector-flow (GVF) snake/ active contour model and multiscale motion-estimation (ME) model combining total-variation (TV) minimization. The hope is that the extracted information could be blended into a corresponding data assimilation system for weather forecasting in subsequent stages. When no assumptions are made about the morphology of the structures to be estimated and the problem is confined to that of finding some individual image region, it is commonly referred to as contour estimation, and snakes (or active contours) and their descendants constitute the most often used class of approaches to smooth boundary estimation. As originally proposed by Kass et al. [2], a snake is a virtual object (living on the image plane) which can deform elastically (thus possessing an internal energy) and which is immersed in a potential field. Basically, there are two types of snake models: the implicit ones and the parametric ones. The emphasis in this paper is on parametric snake models in which the active contours are represented explicitly as parameterized curves, leading to a mathematical formulation that makes it easier to integrate image data and desired contour properties in a fast real-time implementation. There are two important and challenging issues in applying the snake model to a temporal sequences of images: the generation of an initial contour in each frame and the design of external forces. These two issues correspond to the problems of object tracking and object segmentation, respectively. However, there are several key difficulties with classical parametric snake algorithms. A first one concerns the generation of an initial contour in each frame, which must be close to the desired boundary, or else, it cannot be attracted to the true boundary by external forces. The second problem occurs for boundary concavities, i.e., the active contours have difficulties to move into boundary concavities. Some methods, including pressure force, distance potential force, and constrained clustering algorithm, have been proposed to separately overcome the above difficulties. To address both problems, Xu and Prince [3] successfully

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MA et al.: CURVELET-BASED SNAKE FOR MULTISCALE DETECTION AND TRACKING

constructed a force-field-based snake, so-called GVF snake, by employing a vector diffusion equation that diffuses the gradient of an edge map in regions distant from the boundary. Instead of directly using the gradient of image as external forces, their model uses a spatial diffusion of the edge gradient. This paper will further investigate the possibility of extending the above GVF-snake methodology in a multiscale setup based on the curvelet transform. The introduction of multiscale strategies has the advantage to reduce the problems caused by convergence to local minima, and allows tracking the best solution from a coarse scale to finer scales in a scale-space representation of an object; thus, it has the advantage of robustness with respect to image noise during the snake computation process. From an optimization view, at a coarse scale, the objective function involved in snake estimation usually displays fewer local and well-separated minima, allowing the identification of the global minimum efficiently before switching to a finer scale representation. Using the obtained results at a coarse scale as an initial value for the optimization at the finer scale results in updating the global minima just by a few iterations. Several multiscale-space-based approaches have been proposed previously in the literature. Mignotte and Meunier [4] proposed a multiscale approach for active contour optimization based on a multigrid minimization and a coarseto-fine relaxation algorithm; Tang and Acton [5] present a multiscale GVF snake to detect vessel boundaries based on a scale space produced by average filters; Bresson et al. [6] consider a parametric snake model in linear scale space; Liu and Hwang [7] propose a wavelet-based multiscale snake by adopting the modulus of wavelet-transform coefficients as the external forces; Wu et al. [8] propose a wavelet-frame-based directional image force model, which is the inner product of the zero crossing strength of wavelet frame coefficients, and the normal of a snake; and the approach by Keserci and Yoshida [9] involves a so-called edge-guided wavelet snake model using a wavelet snake to fit multiscale edges in medical images. Inspired by such approaches, we propose a curvelet-based geometric multiscale method for snake modeling and tracking, which appears to be superior to conventional scale-space techniques for detecting the edges and suppressing noise at the same time. One reason for this is that curvelets can represent edges and singularities along curves much more efficiently than the traditional wavelet transforms, i.e., using much fewer coefficients for a given accuracy of reconstruction (essential optimal representation of edges). Fig. 1 shows the framework of our multiscale tracking. We propose first to identify a consistent vortex by a curveletbased GVF snake and then to establish the motion correspondence of the snaxels between the successive frames by a constructed so-called semi-T or comp-T multiscale ME method based on the geometric wavelets. Furthermore, we blend the TV regularization and cycle-spinning techniques into the above framework to effectively recover the mismatches and make a significant improvement for the estimation. This paper is organized as follows. An overview of curvelets and the curvelet transform we have used for representing edges in a multiscale fashion is presented in Section II. In Section III, we formally state our curvelet-based snake model. The ME in

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Fig. 1. Flow chart of the curvelet-based tracking; k denotes the kth frames.

the curvelet domain, which is an essential part of our algorithm, is described in Section IV. Translation-invariant regularization with TV penalty for ME is addressed in Section V. Numerical experiments at each stage of the resulting algorithm on a temporal sequence of meteorological images, to investigate the performance of the proposed methodology in comparison to conventional methods, are reported in Section VI, where we also clarify how the pieces of the processing all fit together. We then conclude in Section VII. II. M OTIVATION AND AN O VERVIEW ON C URVELETS AND THE C URVELET T RANSFORM A. Motivation The complexity of tracking structures in meteorological remotely sensed images generally arises from the following: 1) the edges of the object are very weak and blurry, i.e., present small image gradient values; 2) large motion displacements and deformations between successive frames; and 3) the existence of noise and environment interference around the target. Considerable work involving the tracking problem in the field of meteorology has been conducted using different approaches. Mukherjee and Acton [10] addressed the problem of cloud tracking within a sequence of geostationary satellite images using the technique of scale-space classification. In their work, the scale-space representation is generated by an areabased morphological operator. Cohen and Herlin [11] proposed a curve matching using geodesic paths to track vortices of clouds. However, most of the existing methods suffer from either heavy computational cost or weak results due to the aforementioned problems. Wavelet-based methods can improve the above challenging problems at least partially (see, e.g., [7]). It is known that wavelets have good performance at representing point singularities. Unfortunately, this is not the case in higher dimensions because wavelets ignore the geometric properties of objects with edges and do not exploit the regularity of the edge curves. For two-dimensional (2-D) functions, wavelets isolate well the discontinuities across the edges, but along the edges, they prevent edges to be smooth, which leads to unsatisfying results in the existing wavelet-based snakes. Recently, Candès et al. [12]–[15] developed a new geometric multiscale transform, the so-called curvelet transform, which allows an optimal sparse representation of objects with C 2 singularities. The elements of this geometric multiresolution

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Fig. 2. (a) Curvelets on different scales and directions in the spatial domain and (b) their log magnitude spectra. The waveform is oscillatory across the needle- and bell-shaped nonoscillatory along the needle.

transform own very high directional sensitivity and anisotropy. Fig. 2 shows a needle-shaped element of curvelets at fine scale in spatial and frequency domains, respectively. The transform can represent edges and singularities along curves much more efficiently than the traditional wavelet transforms, i.e., using much fewer coefficients for a given accuracy of reconstruction (essential optimal representation of edges). For a smooth object f with discontinuities along a generic C 2 -smooth curve, the best m-term approximation f˜m by wavelets thresholding obeys
f − f˜m
22 ≈ m−1 ,

m→∞

(1)

c achieves while a curvelet best m-term approximation f˜m c 2
2 ≤ Cm−2 (log m)3 .
f − f˜m

(2)

It has been shown that the new transform is superior to 2-D standard wavelet transforms in terms of edge detection [13], discontinuity-preserving denoising [16], and satellite image fusion [17]. It is therefore natural to apply the curvelet transform for edge-based snake tracking. For the sake of selfcontainedness and clarity, we next present a brief review of curvelets and the curvelet transform (for a more detailed account, see [12] and [14]). B. Curvelets In this paper, we use a so-called second-generation discrete curvelet transform (DCT) [14], [15], which is considerably simple and totally transparent to apply. Let µ be the collection of triple index (j, l, k), where j, l, and k = (k1 , k2 ) are scale, orientation, and translation parameters, respectively. The curvelets are defined as functions of x ∈ R2 by ϕµ (x) = ϕj (RθJ (x − kδ )) .

(3)

In the above, ϕ is a waveform which is oscillatory in the horizontal direction and bell shaped (nonoscillatory) along the vertical direction. RθJ is a rotation matrix with the rotation angle θJ = 2π · 2− j/2 · l, J = (j, l) indexing the scale/angle, with · denoting the integer part, while the translation para(k1 · 2−j , k2 · 2−j/2 ). The curvelet meter is given by kδ = Rθ−1 J

Fig. 3. Discrete curvelet tiling with parabolic pseudopolar support in the frequency plane.

frame elements are obtained by anisotropic dilations, rotations, and translations of a collection of unit scale oscillatory blobs. The curvelet coefficients are given by
(4) cµ = f, ϕµ = f (x)ϕ¯µ (x)dx R2

which can be evaluated directly in the frequency domain. Introduce the 2-D frequency window  j/2    2 θ Wj (ω) = 2−3j/4 w 2−j |ω| v (5) 2π where the radial window (e.g., Meyer wavelet window) partitions the frequency domain into annuli |x| ∈ [2j , 2j+1 ) and the angular window partitions the annuli into wedges θl = 2πl · 2− j/2 . By defining the curvelets in the frequency domain ϕˆµ (ω) = Wj (RθJ ω)e−ikδ ,ω , i.e., taking oriented local Fourier bases on each wedge, and using Plancherel’s theorem for (4), we get
1 cµ = fˆ(ω)ϕ¯ˆµ (ω)dω (2π)2
1 (6) fˆ(ω)Wj (RθJ ω)eikδ ,ω dω. = (2π)2 The window Wj locates the frequencies near a polar wedge, but this is not adapted to Cartesian arrays in practical implementations. In digital coronization, Candès et al. [15] applied ˜ j , which a pseudopolar grid by revising the window as W isolates the frequencies near a pseudopolar wedge (concentric squares, see Fig. 3). In the Cartesian case, the digital analog of coefficients can be given by
−T   ˜ j S −1 ω eiSθl b,ω dω cµ = fˆ(ω)W θl   1 0 Sθ = . (7) − tan θ 1

MA et al.: CURVELET-BASED SNAKE FOR MULTISCALE DETECTION AND TRACKING

Here, b = (k1 · 2−j , k2 · 2−j/2 ). A crucial difference between (7) and (6) is the use of the shear matrix Sθ instead of the rotation matrix Rθ . To numerically evaluate (7), one can: 1) take the 2-D FFT and obtain Fourier samples fˆ; 2) multiply fˆ with the parabolic ˜ j ; and then 3) apply the inverse Fourier pseudopolar window W b to collect the transform on the special sheared grid Sθ−T l coefficients. A difficulty here is to evaluate the inverse discrete Fourier transform on a nonstandard Cartesian grid in the third step, where the classical FFT algorithm does not work well. Unequispaced fast Fourier transform (USFFT) and a wrapping technique are two effective methods to solve this problem [14], [15]. The USFFT interprets the parallelepipedal grids as unequispaced Cartesian grids in the frequency domain where the fast algorithm can work directly. The wrapping technique uses a periodic warp around, instead of interpolation, to localize the Fourier samples in a rectangular grid where the inverse FFT can be applied. ˜ j [n1 , n2 ] is supported within a Assume that window W shifted rectangle of length Lj and width lj , say Pj = {(n1 , n2 ) : 0 ≤ n1 − n1,0 < Lj , 0 ≤ n2 − n2,0 < lj } (8) where n1,0 , n2,0 denote the initial location parameters. Then, the discrete curvelet coefficients can be obtained by the USFFTbased algorithm cµ =



fˆ[n1 , n2 − n1 tan θl ]

n1 ,n2 ∈Pj

˜ j [n1 , n2 ]ei2π(n1 k1 /Lj +n2 k2 /lj ) ×W

(9)

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III. C URVELET -B ASED S NAKE A traditional parametric snake defines the evolution of a curve r(s) = [x(s), y(s)], s ∈ Ω to minimize the energy functional [2]
E=

1 2 2 α |r (s)| + β |r (s)| + Eext (r(s)) ds 2

Ω

where α and β are weighting parameters of an internal energy function that controls the tension and the rigidity of the snake. The external energy functional Eext is derived from the image in order to guide the snake toward features of interest such as boundaries. A typically used function is Eext (x, y) = −|∇[G(x, y) ∗ I(x, y)]|2 for a given gray-level image I(x, y), where G(x, y) is a 2-D Gaussian kernel and ∇ denotes the gradient operator. By solving its Euler equation, we consider a dynamic evolution of the snake r(s, t) rt (s, t) = αr (s, t) − βr (s, t) − ∇Eext



 1   ˜ j fˆ [n1 , n2 ]ei2π(n1 k1 /Lj +n2 k2 /lj ) cµ = 2 ξ W n n =0 n =0 1 2 (10) where the wrapping transform ξ(·) is actually a simple reindexing of the array. The above algorithms can be performed at the cost of a computational complexity of O(n2 log(n)) flops for an n × n image. Another DCT algorithm can be derived by a so-called contourlet transform proposed by Do and Vetterli [18]. To solve the curve-singularity problem in a different way, the contourlet transform is obtained by first applying a Laplacian pyramid to capture the point discontinuities, and then using a directional filter bank (DFB) in each detail subband to link the point discontinuities into linear structures. In particular, if the number of directions in the DFB is doubled at every other finer scale to ensure a parabolic scaling relation, the contourlet can be regarded as a DCT. Since the contourlet transform is implemented via iterated filter banks, its computational complexity is O(n2 ) for an n × n image. In numerical experiments, we flexibly apply the DCT algorithms according to their respective characteristic to get an efficient tracking algorithm.

(12)

where rt denotes the partial derivative of r with respect to t. When the solution r(s, t) stabilizes, the term rt (s, t) vanishes and one achieves a solution to (12). In GVF snake [3], the traditional external force is replaced by a new force field, called GVF, produced by a spatial diffusion of an edge map f (x, y), which guides the propagation to boundaries from both sides. The GVF field refers to the vector field υ(x, y) = [u(x, y), v(x, y)] that minimizes the energy function

  Φ= µ u2x + u2y + vx2 + vy2 + |∇f |2 |υ − ∇f |2 dxdy. (13)

or by the wrapping-based algorithm Lj −1 lj −1

(11)

The parameter µ is a regularization parameter that balances the two terms in (13). The GVF field is obtained by solving υt = µ∇2 υ − (υ − ∇f )|∇f |2

(14)

where υt denotes the partial derivative of υ(x, y, t) with respect to t. Hence, the evolution of the GVF snake can be written as rt (s, t) = αr (s, t) − βr (s, t) + υ.

(15)

In our model, the edge map is produced by a curvelet thresholding  τ (cµ )ϕµ (16) f= µ

where τ is a thresholding function for local maxima of coefficients and cµ = I, ϕµ are the curvelet coefficients of the image. The procedure described in (16) is actually a denoising. Due to the fact that curvelet thresholding optimally represents edges and suppresses noise simultaneously, our method is well suited to extract the required edge map, even for an image I(x, y) corrupted with a strong noise.

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coarsest scale). In the curvelet domain, the edge structures are preserved analogously in the same directional subband at different scales, while noise decreases rapidly along the scales. The adjacent-scale multiplication in the first step uses the intrinsic relation of coefficients in the curvelet domain to sharpen the edges while removing the noise to some extent. IV. ME IN C URVELET D OMAIN

Fig. 4. Flow chart of curvelet-based GVF snake. The dotted line denotes the scale iteration.

In our curvelet-based model, the multiscale snake evolves from a coarse scale to fine scale. The detailed algorithm may be described as follows (also refer to Fig. 4). 1) Perform a multiscale decomposition of the observed image using the DCT. 2) Obtain the multiscale edge map by thresholding local maxima of the curvelet coefficients. 3) Get the gradient of the edge map in a sequence of scales, which will be used to compute the GVF field in (14). 4) Initialize a snake at the coarsest scale and evolve the contour using (15) to get the converging contour at this scale. 5) Take the resulting contour as an initial snake for the next finer scale, and evolve, until the finest scale (or expecting scale) is reached to get the final snake. The crucial step is how to obtain a desirable edge map in the presence of weak boundaries and noise. The wavelet-based snake described in [7] follows the contour in the coefficient modulus domain. In this case, the initial snaxels of the snake at the coarsest scale are down scaled and down sampled to match the down-sample modulus image at step 4). The merit of this strategy is its low computational burden because of the shorter length of the snake at the coarser scale. However, the algorithm looses accuracy and cannot preserve the topology due to the low resolution of the snaxels and the modulus images at coarse scales. On the contrary, in our approach, the subband reconstructed edge map is used to take advantage of the needle-shaped elements of curvelets by: 1) adjacent scale multiplication of subband coefficients [19], i.e., replacing the curvelet coefficients θj,l,k by the product of θj,l,k and θj+1,l,k after upsampling; 2) extracting the local maxima of the multiplied coefficients; 3) thresholding the local maxima; and 4) reconstructing the thresholded coefficients at scales j, j + 1, . . . , j0 as the edge map at scale j (j0 denotes the

Wavelet-based ME or compensation procedures for video analysis (compression, tracking, and coding) have received a considerable attention in the last years (see [20] and [21]). Such methods have shown a superior performance to conventional ME methods in terms of the mean-squared error (mse), subjective quality, as well as low computational cost. In this section, a curvelet-based ME of snake is constructed by making use of the curvelets’ optimal properties at representation of edges and features. In our proposed multiscale-block-matching processing, we generate two types of multiscale information: boundary locations (snaxels) and its image values (gray) on the locations, k−1 = (xk−1 , yik−1 ) i.e., Sj,i = (xj,i , yj,i ) and I(Sj,i ). Let S0,i i be the ith snaxel of the converging snake at frame k − 1, and k−1 k−1 = (xk−1 Sj,i j,i , yj,i ) be the ith snaxel of the snake contour at scale j, (j = 1, . . . j0 ). The following equation shows how to get the multiscale locations by decimation:  k−1  Dj S0,i k−1 (17) Sj,i = 2j where Dj denotes a down-sample operator by a factor of 2j . k−1 , the motion vector can be found Centered at each snaxel Sj,i between the modulus of coefficients of the curvelet transform at the scale j of frames k − 1 and k, respectively denoted by Ijk−1 and Ijk , using the block-based ME (BME) B

B

2 2    k−1  k−1  1 k−1 I min xj,i + m, yj,i +n j 2 vx ,vy ∈Ω (B + 1) B B

m=−

2

n=−

2

 k

2 k−1  . (18) − Ij xk−1 j,i + vx + m, yj,i + vy + n Here, B + 1 is the size of matching block and Ω is the search region. Both parameters can be varied at different scales to get a high-quality estimation at a low computational cost [20]. Generally, the size of a block at scale j is increased by 2j0 −j times the initial size at the smoothest scale, while Ω is decreased by a factor of two at every refinement step of the procedure. The optimal value in (18) is assembled as the snaxel’s motion vector Vj,i = (vx∗ , vy∗ ). Applying the DCT for successive frames, we get the multiscale image information Ijk−1 and Ijk at smooth scale sj and detailed scale dj for the multiscale block matching. More precisely, we start from the smooth modulus image at the coarsest k−1 = Dj0 (S0,i )/2j0 , and the scale j0 . As to this matching, Sjk−1 0 ,i s obtained motion vector is denoted by Vj0 ,i . At the smoothest scale, there is more shape similarity between successive frames and smaller image sizes, so we can rapidly obtain the

MA et al.: CURVELET-BASED SNAKE FOR MULTISCALE DETECTION AND TRACKING

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Fig. 6. Flow chart of curvelet-based ME (comp-T). The dotted rectangle denotes scale iteration. Fig. 5. Flow chart of curvelet-based ME (semi-T). The dotted rectangle denotes scale iteration. sj and dj denote the smooth and detailed components, respectively.

initial-scale motion vectors {Vjs0 ,i }i∈Θj0 (Θj0 is the index of all snaxels at scale j0 ). Then, we refine the motion vectors using the detailed modulus image. The refinement is repeated until the finest scale is reached. Here, the detailed modulus image at each scale is a combination of all the curvelet coefficients at each angular subband. Due to the fact that the search regions Ω are very narrow during the refinement process, the motion vectors need only small adjustment. This multiscale strategy ensures the robustness and speediness of the algorithm. The detailed algorithm is described as follows (also refer to Fig. 5). 1) Perform the DCT for the images in successive frames k − 1 and k (needed to generate the multiscale information: the smooth component sj and detailed scale dj ). k−1 2) Get the snaxels Sjk−1 = Dj0 (S0,i )/2j0 , and estimate the 0 ,i s motion vectors {Vj0 ,i }i∈Θj0 using the smooth component of coefficients at the coarsest scale j0 3) Refine the motion vectors using the detailed component of the coefficients generated by the DCT at scale j0 Vj0 ,i = Vjs0 ,i + ∆j0 ,i ,

i ∈ Θj0 .

(19)

4) Iteratively refine the motion vectors {Vj,i }i∈Θj using the detailed component at the next finer scales j = j0 − 1, . . . , 1 Vj,i = 2U (Vj+1,i ) + ∆j,i ,

j = j0 − 1, . . . , 1,

i ∈ Θj (20)

where U denotes the upsampling operator by a factor of two. 5) The estimated snaxels at frame k are then given by k k−1 S0,i = S0,i + 2U (V1,i ),

i ∈ Θ0 .

(21)

It should be noted that the curvelet transforms are not shift invariant, although they own a good directional selectivity. In this paper, a cycle-spinning technique [22], in analogy to what is widely used in wavelet processing to achieve a shift invariance, is applied in the procedure of ME in order to reduce the false matching due to artifacts resulting from the lack of translation invariance. How to design an efficient shift-invariant curvelet transform is still underway. The strategy of the algorithm described above is similar to that of in [7], except that the curvelet is used here. The algorithm is simple and fast. Note however that the algorithm is processed in interlaced domains, i.e., the images are transformed into the curvelet domain, while the snaxels are left in the spatial domain. For example, in (18), the curvelet coefficient modulus values Ijk−1 and Ijk are used, but the index of pixels k−1 (xk−1 j,i , yj,i ) results directly from the downscaling and downk−1 sampling of snaxels S0,i , which live in the spatial domain. The approximate equivalence of snaxel location between the spatial and curvelet domain simplifies the algorithm, at a price of loss in the accuracy. We denote this algorithm as a fast semitransform algorithm (semi-T), which means that only the images are transformed into the curvelet domain. In the following, we design a socalled complete-transform algorithm (comp-T). This time, the matching is processed fully in the curvelet domain by mapping the snaxel from the spatial domain to snaxel in the coefficient domain by means of the intrinsic structure of the curvelet transform. Fig. 6 outlines the framework of the comp-T. The main differences between the comp-T and semi-T procedures can be summarized as follows. 1) The snaxels used in comp-T are mapped from spa(c)k−1 tial snaxels following the downsampling, i.e., Sj,w,i = k−1 Mj,w (Dj (S0,i )), instead of using a downscaling by a j factor of 2 in semi-T. Here, the superscript denotes the snaxels in the coefficient domain, the subscript j, w, and i respectively denoting the index of scale, directional

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subband, and snaxel. The Mj,w denotes the mapping operator from the spatial domain to the wth directional subband at scale j. 2) In comp-T, the refinement processing through all the directional detail subbands at each scale is applied, i.e., if there is the same number of directional subbands at suc(w) cessive scale (let Nj denote the number of directional subbands at scale j ), we have Vj,w,i = χ(Vj+1,w,i ) + ∆j,w,i

(w)

if Nj

(w)

= Nj+1

(22)

else Vj,w,i = χ(Vj+1,w/2,i ) + ∆j,w,i

(23)

where χ(·) denotes the operator with possible upscaling and upsampling at each scale; the operator · rounds its argument to the nearest integer greater than or equal to. (c)k 3) After we get the finest snaxel S1,w,i , we map back to the spatial domain, average, and upsample, to obtain the estimated initial snake at the current frame k   (w) N1      1 (c)k k −1 S1,w,i  S0,i = U  (w) M1,w (24) N1 w=1 −1 where M1,w denotes the inverse mapping operator from the wth directional subband at the finest scale to spatial domain. The semi-T can be seen as a simplified version of comp-T because: 1) it uses the decimating (down/upscaling) instead of a mapping operator by using the intrinsic structure of DCT and 2) it just applies the BME (matching) one time in each detailed scale (it sums the all directional curvelet subbands into one modulus image), while the comp-T applies the matching in every directional subband at each scale (i.e., subband by subband). Consequently, the comp-T is highly accurate, but is slower than the semi-T. A detailed experimental comparison between these two procedures will be given in Section VI-C.

V. TV R EGULARIZATION FOR ME Efficient ME is an important problem in snake-based tracking. A false estimated motion vector can cause aberrant initial snake contour, leading to intertwists of snake evolution, thus increasing the length of snake and the computational time. More seriously, it can lead to a wrong snake, i.e., converge to undesirable features. Curvelet-based ME procedures not only reduce the computational time, but also improve the prediction performance in comparison to traditional BME methods. However, as any other multiresolution ME, a serious problem we have to face is that if a false motion vector is generated due to the low resolution of snaxels and coefficient subband image at the coarsest scale, it will propagate along the prediction path and pollute the refinement at the next finer scales. Since most motion fields associated with a temporal sequence of images of natural scenes are smooth and continuous, compared to the neighboring motion vectors, a false motion

vector always appears as a discontinuity in the motion field. Exploiting this fact, a study in [23] uses techniques of linear prediction, median filtering, and multicandidate prediction to recover the false ME. For relatively early work involving estimation criteria with regularization, we refer to a review in [24]. Note also that the study in [25] proposes a regular and filtering technique for dense and sparse vector fields, focusing on the application to nonrigid registration. In this section, the TV regularization for ME is proposed to eliminate the false motion vector resulting from mismatches. The above-described curvelet-based methods including the semi-T and comp-T exploit the interscale motion correlation only, but ignore the intrascale motion correlation. From this point of view, TV regularization just considers the intrascale correlation. In our multiscale BME, we need only to apply the TV to Vjs0 ,i , i.e., the motion vectors obtained in the smoothest subband. Due to the fact that search regions are limited to narrow bands, the refinement in detail subbands rarely produces a false motion increment. In case of a false motion increment, it will always be very small and can be corrected in next scales. Alternatively, the application of TV regularization to the final estimated snake can improve the results, because the final estimated snake is not refined using the original image in semi-T and comp-T in order to reduce the computational cost. The TV of a function is defined as the integral of the Euclidean norm of the gradient [26]
TV(V ) : = |∇V (x)| dx. (25) To overcome the technical difficulties in digital algorithms, it is common to consider the TV function with a slight perturbation
 |∇V |2 + β 2 dx (26) where β is a small positive parameter controlling the tradeoff between the suppression of staircase effect and the diffusion of high frequent components. In this paper, the motion vector V = (Vx , Vy ) is a vectorvalued one-dimensional function along the snake contour. For the vector-valued motion function, (25) can be further written as [27], [28]  TV(V ) : = [TV(Vx )]2 + [TV(Vy )]2  2 L−1 2 L−1     = |Vxi+1 − Vxi | + |Vyi+1 − Vyi | i=1

i=1

(27) where L denotes the total number of the snaxels. Every motion vector Vi is subject to minimization of block mse, as described in (18), which can be rewritten using a succinct notation as 2  arg min ws · I k−1 (Si ) − ws · I k (Si + Vi ) (28) Vi ∈Ω

where the factor ws denotes the pixels contained in the window ws , centered at snaxel Si . Considering the global error, i.e.,

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the summation of block mse along the snake, as a constraint, the TV regularization amounts in solving the following nonlinear optimization problem:
 2 min α |∇V |dl + λ ws · I k−1 (S) − ws · I k (S + V ) 2

V ∈BV

L

(29)



where the norm satisfies
Φ
22 = i
Φi
22 . The first term in (29) aims to reduce the oscillations of the estimated motion vectors by diminishing its TV norm. The second term is a classical L2 fitting term with a fidelity weight λ, which controls the total block mse. The regularization parameter α is used to balance the tradeoff between the suppression of oscillations and fitting. When α tends to infinity, the motion vector tends to a constant. Equation (29) can be solved by a projected subgradient method V l+1 = P (V l − tl g l )

Fig. 7. Three successive frames used in the experiments.

shrink toward a point. In our experiments, the simplified version has much improved the performance of the curvelet-based ME, while the additional computation cost is almost negligible. Due to its low complexity, the proposed technique seems suitable for real-time implementations. Also, it is quite straightforward to embed the technique into the existing ME methods. VI. E XPERIMENTS

(30)

A. Experimental Data

where the initial V0 is the motion vector obtained using the existing ME methods. The operator P is the projection related l to the constraints, while ∞ tl > 0 is the step size satisfying l liml→+∞ t = 0 and l=0 t = ∞ in order √ to obtain the convergence. A typical step size is tl = t0 / l. The subgradient is of the form g l = (gxl , gyl ). Taking for example the x direction, we have the discrete subgradient     TV Vxl ∂TV Vxl l · gxi = . (31) TV(V l ) ∂Vxli l

The second term of the right side in (31), denoted as g xi , can be evaluated as follows:   l  l l − sgn Vxi+1 , − Vxi−1 − Vxi i = 2, . . . , L − 1   l l l g x1 = − sgn Vx2 − Vx1   l l l . g xL = sgn VxL − VxL−1  l l g xi = sgn Vxi

(32)

An analogous formulation holds for the y direction. For a convenient implementation, the following simplification can be adopted. First, we consider the TV of the estimated k k instead of the motion vectors Vj,i . Second, we snaxels Sj,i apply the TV regularization channel by channel, i.e., direct application of TV separately to the x-dimension and then to the y dimension, instead of the coupling version described in l

(27). In this case, the discrete subgradient is g xi instead of l . Comparing the difference, we can see that the coupling gxi version takes a global channelwise scaling by a factor of TV(Vxl )/TV(V l ). Therefore, a channel with larger TV will be smoothed more than a channel with smaller TV. In some cases, the channel-by-channel version shows better performance [28]. Third, considering that only a few iterations are needed to reach the results required in this paper, the fidelity term in (29) can be ignored. Hence, the problem is solved simply by a classical subgradient method. Note, however, that too many iterations without the fidelity term make the estimated snake contour

In this section, sea surface temperature (SST) satellite data obtained from the NOAA Advanced Very High Resolution Radiometer are taken for example. SST data are acquired from a thermal imagery in the infrared domain and are very often provided by meteorological satellites to observe oceans’ biologic activities. The SST estimation relies on the analysis of brightness temperatures obtained by inverting the Planck blackbody function from radiance measurements and by combining different infrared channels so as to discard the contribution of the atmosphere to the signal. Three successive frames are used to demonstrate the performance of the proposed curvelet-based tracking methods in comparison to the conventional methods. Fig. 7 shows a typical sequence of images: strong temporal variation between frames and large spatial variation of the local mean. The size of image in each frame is 256 × 256. However, in the following figures, we only show the interesting part of the images. B. Detection Fig. 8 shows the results of the snake evolution at two scales using the various discrete curvelet algorithms described in Section III. Our purpose here is to show the slight differences between the various DCT algorithms. The background in each subimage is an edge map. The left-hand-side pictures display the results at coarse scale, while the right-handside ones are those at the finest scale. From top to bottom, the pictures in every row are obtained in turn by warppingbased DCT, USFFT-based DCT, the contourlet-used DCT, and the contourlet-used DCT with scale multiplication. The basic process to obtain the converging snake is first to decompose the input image using the various DCTs mentioned above, then to get the edge map using a scale-dependent thresholding, and finally to evolve the GVF snake from coarse-to-fine scales. The same initial circle snake contour [see the dotted ring in Fig. 8(a)] was used for the four algorithms. In the first row, we can see from the results by warppingbased DCT that a reconstruction based only on a few significant

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Fig. 9. Estimated snake contour between the first and second frames using the conventional single-scale BME (a) without the use of TV and (b) with the use of TV.

relation, the contourlet transform may be regarded as a DCT (see the end of Section II-B, as well as [15] and [18]). Of course, the elements and data structure of contourlets and curvelets are not exactly same. In the third and forth row, we can see that the anisotropic elements produced by the contourlet transform also lead to smooth snake. Contourlets are not sharply localized in frequency, which means that the elements lack the smoothness along the ridge in the spatial domain and exhibit spurious oscillations, i.e., the digital contourlet transform lacks the faithful needle-shaped geometry in comparison to the DCT. However, the slight artifacts of the digital contourlet elements do not affect our method to obtain smooth snakes. The contourlet transform combined with scale multiplication, used in the forth row, results in a strong edge map, thus a reliable snake contour. As already mentioned in Section II-B, the contourlets-based snake algorithm is faster than the warpping/USFFT DCT algorithm with a CPU time of 17.7 s opposed to 39.9 s/80.0 s, using a notebook PC with PIII CPU and 128 MB memory. C. ME

Fig. 8. Snake evolution in two scales (left column: coarse scale, right column: fine scale) using different DCT algorithms.

coefficients has already represented the edges that are required to get a satisfactory converging snake. This is mainly due to the fact that the anisotropy of curvelets is well preserved in this discrete version. The second row is a comparison between curvelets and wavelets. In the USFFT-based DCT, the wavelet transform is used at the finest scale. It can be seen that in Fig. 8(c), only a few needle-shaped elements are needed to get the smooth snake in this scale. However, due to the fact that wavelets at the finest scale are used, numerous point-shaped elements are observed, leading to discontinuous edges and then to disordered GVF and an anomalous snake contour as shown in Fig. 8(d). This clearly reveals that the performance of curvelets is superior to wavelets for edge-based snake. Our final comparison is between curvelets and contourlets; since, when the number of directions in contourlets is doubled at every other finer scale to ensure a parabolic scaling

In this section, we present the experimental results regarding the ME of snaxels between the frame 1 and 2 using conventional and the techniques proposed in this paper. Note that the lefthand-side results in the following figures are obtained by methods without TV regularization, while the right-hand-side ones use the TV regularization. The background image is the current frame. The dotted line denotes the snake of the previous frame used for estimation, and the solid line denotes the estimated snake in the current frame. Fig. 9 is a simulation carried out in the spatial domain using conventional BME. The block size used here is 11 × 11, and the search region is 15. Obviously, the estimated contour shown as a solid line in Fig. 9(a) fails to be used as the initial snake in subsequent stages. The Fig. 9(b) is the estimated contour using the conventional BME combined with TV regularization. The results are much improved by correcting the discontinue mismatching arising in Fig. 9(a). We have used a large iterative time (1200) in TV regularization in order to show the converging trend of the contour when using our simplified version of TV model, i.e., without fitting terms. Generally, the contour converges to a shrinking rectangle with the increase of the iterative time. Fig. 10 shows the results obtained using the BME in curvelet subband reconstructed multiresolution space. The images in

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Fig. 10. Estimated snake using the curvelet subband reconstructions BME. (Top) Coarse scale. (Bottom) Fine scale. (Left column) Without the use of TV. (Right column) With the use of TV.

successive frames are first decomposed by a four-level DCT and then reconstructed at each scale. Therefore, the BME is still processed in the spatial domain from a coarse scale (shown in the upper row) to a fine scale (shown in the lower row). In the smoothest matching, the block size is 11 × 11 and the search region is 15, while in the detail matching, the search region is one. An advantage of this approach is that the upsampling and upscaling of the motion vector can be avoided, but at the price of heavy computational burden. Another advantage is that artifacts due to the lack of shift invariance of the curvelet transform can be reduced to some extent in comparison to coefficient domain-based methods. However, the mismatches are still obvious using such an approach. There is a significant improvement both in the smoothest and finest estimations when combining the TV regularization with a few iterations, as shown in the right-hand-side pictures. It should be noted that the TV is just applied in the smoothest estimation phase in this case. Fig. 11 shows the results obtained using our proposed semi-T procedure with a two-level decomposition. The block size is 9 × 9 and the search region is six in the smoothest estimation phase. The search region is one in the refined estimation phase. The variable rule of block size satisfies the conditions described in Section IV. We can see that the estimated snake in the smoothest coefficient modulus image, as shown in Fig. 11(a), is almost free from serious mismatches. However, the final estimation shown in Fig. 11(c) was not expected. The most plausible reason is that the estimations using semi-T are not accurate, because the approach uses some approximations in order to get a real-time speedup. Another reason is due to the lack of shift invariance of the transform. Moreover, the imprecise estimations are magnified due to the upscaling at each fine scale. The TV embedded semi-T makes a significant improvement, as shown in Fig. 11(b) and (d). The lowest row is the result obtained using the approximate shift-invariant semi-T by a cycle-spinning technique, especially the formulation given

Fig. 11. Estimated results using the semi-T combined with TV and cyclespinning techniques. (Left column) Without the use of TV. (Right column) With the use of TV.

in [7] and [22]: Averageh∈H S−h (semi-T(Sh )), where Sh denotes the circulant shift by a vector h, H denotes the shift range, and semi-T denotes the proposed ME. Fig. 11(e) and (f) shows the results obtained by applying the cycle spinning to the methods used in Fig. 11(c) and (d), respectively. We can see from Fig. 11(f) that the estimated snake is markedly better by combining the cycle spinning and TV regularization with semi-T, compared to Fig. 11(c). Fig. 12 shows the performance of the proposed comp-T method (warpping DCT used). Fig. 12(a) shows the estimated snake at the smoothest subband. The TV embedded comp-T further improves the estimation, as shown in the right-handside pictures. The comp-T is more accurate and reliable to get the estimated snake compared to the semi-T. The estimated snake in Fig. 12(d) approximates well the contour of the target. On the other hand, it requires more computational time than semi-T with comp-T requiring for this example 34.4 s, while semi-T only requires 3.9 s in a notebook with PIII CPU and 128 MB memory. To summarize: 1) The proposed comp-T and semi-T both show a promising performance. The comp-T appears highly accurate and stable, while semi-T is attractive to its high computational speed that is useful for real-time processing. 2) The proposed TV regularization of motion vectors/snakes is a significant improvement for all methods. It is especially effective to remove spurious matches with just few more

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Fig. 14. Tracking for noisy image sequence using the proposed curvelet-based method.

Fig. 12. (Left column) Estimated results using the comp-T without TV and (right column) with TV. (Upper row) Coarse scale. (Lower row) Fine scale.

Fig. 13. Tracking using the proposed curvelet-based method. The solid line shows the final detected results by the curvelet-based snake, and the dotted line shows the intermediate results by the curvelet-based ME.

iterations with very limited additional computation cost. 3) Cycle spinning improves the estimation to some extent. D. Tracking In Fig. 13, we exhibit the proposed framework for tracking in the sequences. The dotted lines in the three pictures denote the initial snake contours at the respective frame. Here, the initial snake contours in second and third frames are obtained by a comp-T-based ME combined with the TV regularization. The solid lines denote the final estimated snakes obtained by a curvelet-based snake evolution. The results demonstrate the performance of the proposed curvelet-based tracking approach even for image sequences with weak edges and large deformations. Traditional single-scale methods fail to complete the tracking for such sequences. Furthermore, we consider a case of tracking for noisy image sequence using the proposed curvelet-based method, as shown in Fig. 14. The original images used in the last experiment are contaminated by an additive zero-mean Gaussian white noise. The results, including the final snakes shown as real line in two frames and the estimated snake shown as the dotted line in the second frame, are directly obtained using our curvelet-

based methodology by simply adjusting the parameters without any other denoise preprocessing. The results clearly show that the proposed methodology involving the snake evolution and ME has a good performance and is quite well immunized to the noise. To end this section, we now summarize and give a clear picture on how all the steps involved in the final result, namely snake, ME, and TV regularization of motion vectors, fit together. We start first with an initial contour, say contour1, in the first frame [the dotted circle in Fig. 13(a)]. Using this contour as an initial snake contour, we then apply the snake algorithm (specially, the curvelet-based GVF snake) to reach the converging snake contour on the same frame [the solid line in Fig. 13(a)], say contour2. It follows then the application of our ME curvelet-based and TV-regularization procedures in order to estimate the edges of contour2 in the next frame [shown as the dotted line in Fig. 13(b)], which we denote by contour3. Using contour3 as an initial snake contour, we apply again the snake algorithm to get the converging snake contour in this frame [the solid line shown in Fig. 13(b)]. The procedure is then repeated from frame to frame in order to complete the tracking in the sequence of images. The ME step is useful in obtaining a better (closer to the expected edges) initial snake contour in each frame, and it is worthwhile noting that it is applied on the pixels of snake (snaxels) instead than on all the pixels of the image, following the general idea in most snakebased tracking methods (see e.g., [7]). The TV is embedded within our ME estimation step in order to smooth the estimated motion of snaxels. VII. C ONCLUSION Aiming at collecting the quantitative information from meteorological satellite images for potential data assimilation, this paper presents an integrated approach to tracking based on the curvelet transform, an optimal representation of piecewise 2-D objects with C 2 -singularities. At a first stage, the curvelet-based GVF snake efficiently detects the interesting target weak boundaries. At a second stage, two so-called semi-T and comp-T methods combined with TV regularization and cycle-spinning techniques are proposed for ME of the snaxels based on the geometric wavelets. The comp-T is shown to be highly accurate, while the semi-T procedure is faster which is useful for real-time implementations. The simple and applicable TV regularization makes a significant improvement

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for the ME. It is straightforward to apply the proposed TV model of motion vector to other existing methods of ME. The numerical experiments demonstrate the performance of our curvelet-based tracking approach for image temporal sequences with possible noise, weak edges, and large deformations. It provides the potential application of satellite images for assimilation within geophysical models. The main limitation of the curvelet-based method is the edge effect and complicated data structure of current DCT. The latter makes the manipulation in the curvelet domain to be very complicated. This is why we consider other alternative DCT algorithms to use. Another limitation is that we consider only a parametric snake framework in our current implementation, which cannot deal with the topography change, i.e., splitting and merging of objects. An important future extension would involve applications of hidden Markov model for curvelet-based ME to further make use of the intrascale, interscale, and interframe dependencies of motion vectors.

ACKNOWLEDGMENT The authors would like to thank the Editor, J. A. Benediktsson, the Associate Editor, and the three referees for their careful reading, their suggestions, and criticisms, that resulted to this improved version. R EFERENCES [1] I. Herlin, F. X. Le Dimet, E. Huot, and J. P. Berroir, “Coupling models and data: Which possibilities for remotely-sensed images?” in E-Environement: Progress and Challenges. México City México: Instituto Politécnico Nacional, 2004, pp. 365–383. [2] M. Kass, A. Witkin, and D. Terzopulos, “Snakes: Active contour models,” Int. J. Comput. Vis., vol. 1, no. 4, pp. 321–331, 1987. [3] C. Xu and J. L. Prince, “Snakes, shapes, and gradient vector flow,” IEEE Trans. Image Process., vol. 7, no. 3, pp. 359–369, Mar. 1998. [4] M. Mignotte and J. Meunier, “A multiscale optimization approach for the dynamic contour-based boundary detection issue,” Comput. Med. Imaging Graph., vol. 25, no. 3, pp. 265–275, 2001. [5] J. Tang and S. T. Acton, “Vessel boundary tracking for intravital microscopy via multiscale gradient vector flow snakes,” IEEE Trans. Biomed. Eng., vol. 51, no. 2, pp. 316–324, Feb. 2004. [6] X. Bresson, P. Vandergheynst, and J. Thiran, “Multiscale active contours,” Int. J. Comput. Vis., vol. 70, no. 3, pp. 197–211, Dec. 2006. [7] J. C. Liu and W. L. Hwang, “Active contour model using wavelet modulus for object segmentation and tracking in video sequences,” Int. J. Wavelets, Multiresolution Inf. Process., vol. 1, no. 1, pp. 93–112, 2003. [8] H. Wu, J. Liu, and C. Chui, “A wavelet-frame based image force model for active contouring algorithms,” IEEE Trans. Image Process., vol. 9, no. 11, pp. 1983–1988, Nov. 2000. [9] B. Keserci and H. Yoshida, “Computerized detection of pulmonary nodules in chest radiographs based on morphological features and wavelet snake model,” Med. Image Anal., vol. 6, no. 4, pp. 431–447, 2002. [10] D. Mukherjee and S. T. Acton, “Cloud tracking by scale space classification,” IEEE Trans. Geosci. Remote Sens., vol. 40, no. 2, pp. 405–415, Feb. 2002. [11] I. Cohen and I. Herlin, “Tracking meteorological structures through curves matching using geodesic paths,” in Proc. Int. Conf. Comput. Vis., Bombay, India, 1998, pp. 396–401. [12] E. J. Candès and D. L. Donoho, “Curvelets—A surprisingly effective nonadaptive representation for objects with edges,” in Curves and Surface Fitting: Saint-Malo 1999, A. Cohen, C. Rabut, and L. Schumaker, Eds. Nashville, TN: Vanderbilt Univ. Press, 2000, pp. 105–120. [13] E. J. Candès and F. Guo, “New multiscale transforms, minimum total variation synthesis: Applications to edge-preserving image reconstruction,” Signal Process., vol. 82, no. 11, pp. 1519–1543, 2002.

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[14] E. J. Candès and D. L. Donoho, “New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities,” Commun. Pure Appl. Math., vol. 57, no. 2, pp. 219–266, 2004. [15] E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transforms,” Multiscale Model. Simul., vol. 5, no. 3, pp. 861–899, 2006. [16] J. L. Starck, E. J. Candès, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process., vol. 11, no. 6, pp. 670–684, Jun. 2002. [17] M. Choi, R. Y. Kim, M. R. Nam, and H. O. Kim, “Fusion of multispectral and panchromatic satellite images using the curvelet transform,” IEEE Geosci. Remote Sens. Lett., vol. 2, no. 2, pp. 136–140, Apr. 2005. [18] M. N. Do and M. Vetterli, “Contourlets,” in Beyond Wavelets, J. Stoeckler and G. Welland, Eds. New York: Academic, 2003, pp. 1–27. [19] L. Zhang and P. Bao, “Edge detection by scale multiplication in wavelet domain,” Pattern Recognit. Lett., vol. 23, no. 14, pp. 1771–1784, 2002. [20] Y. Q. Zhang and S. Zafar, “Motion-compensated wavelet transform coding for color video compression,” IEEE Trans. Circuits Syst. Video Technol., vol. 2, no. 3, pp. 285–296, Sep. 1992. [21] S. Kim, S. Rhee, J. G. Jeon, and K. T. Park, “Interframe coding using twostage variable block-size multiresolution motion estimation and wavelet decomposition,” IEEE Trans. Circuits Syst. Video Technol., vol. 8, no. 4, pp. 399–410, Aug. 1998. [22] R. R. Coifman and D. L. Donoho, “Translation-invariant de-noising,” in Wavelets and Statistics, A. Antoniadis and G. Oppenheim, Eds. New York: Springer-Verlag, 1995. [23] J. Zan, M. Ahmad, and M. Swamy, “New techniques for multi-resolution motion estimation,” IEEE Trans. Circuits Syst. Video Technol., vol. 12, no. 9, pp. 793–802, Sep. 2002. [24] C. Striller and J. Konrad, “Estimating motion in image sequences, a tutorial on modeling and computation of 2D motion,” IEEE Signal Process. Mag., vol. 16, no. 4, pp. 70–91, Jul. 1999. [25] P. J. Cachier and N. J. Ayache, “Isotropic energies, filters and splines for vector field regularization,” J. Math. Imaging Vis., vol. 20, no. 3, pp. 251–265, 2004. [26] L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, no. 1, pp. 259–268, 1992. [27] G. Sapiro and D. L. Ringach, “Anisotropic diffusion of multivalued images with applications to color filtering,” IEEE Trans. Image Process., vol. 5, no. 11, pp. 1582–1586, Nov. 1996. [28] P. Blomgren and T. Chan, “Color TV: Total variation methods for restoration of vector-valued images,” IEEE Trans. Image Process., vol. 7, no. 3, pp. 304–309, Mar. 1998.

Jianwei Ma received the Ph.D. degree in solid mechanics from Tsinghua University, Beijing, China, in 2002. He was a Visiting Scholar, Research Fellow, Postdoctoral Scientist, and Guest Professor with the University of Cambridge, University of Oxford, University of Huddersfield, University of Mannheim, University of Joseph Fourier (INRIA, Grenoble), University of Duisburg-Essen, and the Swiss Federal Institute of Technology Lausanne (EPFL). Since 2006, he has been an Assistant Professor with the Department of Engineering Mechanics, and a Research Director with the Institute of Seismic Exploration, Tsinghua University. His research interests include wavelets, nonlinear diffusion, image processing, computer vision, and applications in seismic exploration and aerospace engineering.

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Anestis Antoniadis received the Doctorate degree in applied mathematics from the University of Grenoble I, Grenoble, France, in 1983. He is a University Distinguished Professor with the Department of Applied Mathematics (Laboratoire LMC), University Joseph Fourier, Grenoble. His research interests include wavelet theory, nonparametric function estimation, abstract inference of stochastic processes, statistical pattern recognition, and statistical methodology in meteorology and crystallography. He coedited the book Wavelet in Statistics (Springer Verlag, 1995). He was a joint Editor-in-Chief of the journal ESAIM: Probability and Statistics from 2001 to 2005. Dr. Antoniadis is a Fellow of the American Statistical Association and of the Institute of Mathematical Statistics.

Francois-Xavier Le Dimet received the “Doctorat d’Etat” from University Blaise-Pascal, ClermontFerrand, France. He was a Visiting Professor with many universities in the U.S., Russia, China, and Vietnam. Since 1991, he has been a Professor with University JosephFourier, Grenoble, France, and since 2006, he has been a Directeur de Recherche with the National Institute for Research in Informatics and Automatics (INRIA). He is currently in charge of a national group “ASSIMAGE” funded by ANR and devoted to the assimilation of images in mathematical models for geophysical flows. His main field of interest is the theory of data assimilation for geophysical fluids. In 1982, he proposed to use the optimal control of PDEs for this purpose, and presently, this method is used by the main meteorological operational centers.