Xavier Bresson

Fast Global Minimization of the Active Contour/Snake Model

AbstractnThe active contour/snake model is one of the most successful variational models in image segmentation. It consists of evolving a contour in images toward the boundaries of objects. Its success is based on strong mathematical properties and efficient numerical schemes based on the level set method. The only drawback of this model is the existence of local minima in the active contour energy, which makes the initial guess critical to get satisfactory results. In this paper, we propose to solve this problem by determining a global minimum of the active contour model. Our approach is based on the unification of image segmentation and image denoising tasks into a global minimization framework. More precisely, we propose to unify three well-known image variational models, namely the snake model, the Rudin–Osher–Fatemi denoising model and the Mumford–Shah segmentation model. We will establish theorems with proofs to determine the existence of a global minimum of the active contour model. From a numerical point of view, we propose a new practical way to solve the active contour propagation problem toward object boundaries through a dual formulation of the minimization problem. The dual formulation, easy to implement, allows us a fast global minimization of the snake energy. It avoids the usual drawback in the level set approach that consists of initializing the active contour in a distance function and re-initializing it periodically during the evolution, which is time-consuming. We apply our segmentation algorithms on synthetic and real-world images, such as texture images and medical images, to emphasize the performances of our model compared with other segmentation models.n

Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction

Bregman methods introduced in [S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, Multiscale Model. Simul., 4 (2005), pp. 460-489] to image processing are demonstrated to be an efficient optimization method for solving sparse reconstruction with convex functionals, such as the

Geometric Applications of the Split Bregman Method: Segmentation and Surface Reconstruction

ell^1

Local Histogram Based Segmentation Using the Wasserstein Distance

norm and total variation [W. Yin, S. Osher, D. Goldfarb, and J. Darbon, SIAM J. Imaging Sci., 1 (2008), pp. 143-168; T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2 (2009), pp. 323-343]. In particular, the efficiency of this method relies on the performance of inner solvers for the resulting subproblems. In this paper, we propose a general algorithm framework for inverse problem regularization with a single forward-backward operator splitting step [P. L. Combettes and V. R. Wajs, Multiscale Model. Simul., 4 (2005), pp. 1168-1200], which is used to solve the subproblems of the Bregman iteration. We prove that the proposed algorithm, namely, Bregmanized operator splitting (BOS), converges without fully solving the subproblems. Furthermore, we apply the BOS algorithm and a preconditioned one for solving inverse problems with nonlocal functionals. Our numerical results on deconvolution and compressive sensing illustrate the performance of nonlocal total variation regularization under the proposed algorithm framework, compared to other regularization techniques such as the standard total variation method and the wavelet-based regularization method.

A Variational Model for Object Segmentation Using Boundary Information and Shape Prior Driven by the Mumford-Shah Functional

Variational models for image segmentation have many applications, but can be slow to compute. Recently, globally convex segmentation models have been introduced which are very reliable, but contain TV-regularizers, making them difficult to compute. The previously introduced Split Bregman method is a technique for fast minimization of L1 regularized functionals, and has been applied to denoising and compressed sensing problems. By applying the Split Bregman concept to image segmentation problems, we build fast solvers which can out-perform more conventional schemes, such as duality based methods and graph-cuts. The convex segmentation schemes also substantially outperform conventional level set methods, such as the Chan-Vese level set-based segmentation algorithm. We also consider the related problem of surface reconstruction from unorganized data points, which is used for constructing level set representations in 3 dimensions. The primary purpose of this paper is to examine the effectiveness of “Split Bregman” techniques for solving these problems, and to compare this scheme with more conventional methods.

Completely Convex Formulation of the Chan-Vese Image Segmentation Model

We propose and analyze a nonparametric region-based active contour model for segmenting cluttered scenes. The proposed model is unsupervised and assumes pixel intensity is independently identically distributed. Our proposed energy functional consists of a geometric regularization term that penalizes the length of the partition boundaries and a region-based image term that uses histograms of pixel intensity to distinguish different regions. More specifically, the region data encourages segmentation so that local histograms within each region are approximately homogeneous. An advantage of using local histograms in the data term is that histogram differentiation is not required to solve the energy minimization problem. We use Wasserstein distance with exponent 1 to determine the dissimilarity between two histograms. The Wasserstein distance is a metric and is able to faithfully measure the distance between two histograms, compared to many pointwise distances. Moreover, it is insensitive to oscillations, and therefore our model is robust to noise. Axa0fast global minimization method based on (Chan et al. in SIAM J. Appl. Math. 66(5):1632–1648, 2006; Bresson et al. in J. Math. Imaging Vis. 28(2):151–167, 2007) is employed to solve the proposed model. The advantages of using this method are two-fold. First, the computational time is less than that of the method by gradient descent of the associated Euler-Lagrange equation (Chan et al. in Proc. of SSVM, pp.xa0697–708, 2007). Second, it is able to find a global minimizer. Finally, we propose a variant of our model that is able to properly segment a cluttered scene with local illumination changes.

White matter fiber tract segmentation in DT-MRI using geometric flows.

In this paper, we propose a new variational model to segment an object belonging to a given shape space using the active contour method, a geometric shape prior and the Mumford-Shah functional. The core of our model is an energy functional composed by three complementary terms. The first one is based on a shape model which constrains the active contour to get a shape of interest. The second term detects object boundaries from image gradients. And the third term drives globally the shape prior and the active contour towards a homogeneous intensity region. The segmentation of the object of interest is given by the minimum of our energy functional. This minimum is computed with the calculus of variations and the gradient descent method that provide a system of evolution equations solved with the well-known level set method. We also prove the existence of this minimum in the space of functions with bounded variation. Applications of the proposed model are presented on synthetic and medical images.

Nonlocal Mumford-Shah Regularizers for Color Image Restoration

The active contours without edges model of Chan and Vese (IEEE Transactions on Image Processing 10(2):266–277, 2001) is a popular method for computing the segmentation of an image into two phases, based on the piecewise constant Mumford-Shah model. The minimization problem is non-convex even when the optimal region constants are known a priori. In (SIAM Journal of Applied Mathematics 66(5):1632–1648, 2006), Chan, Esedoḡlu, and Nikolova provided a method to compute global minimizers by showing that solutions could be obtained from a convex relaxation. In this paper, we propose a convex relaxation approach to solve the case in which both the segmentation and the optimal constants are unknown for two phases and multiple phases. In other words, we propose a convex relaxation of the popular K-means algorithm. Our approach is based on the vector-valued relaxation technique developed by Goldstein et xa0al. (UCLA CAM Report 09-77, 2009) and Brown etxa0al. (UCLA CAM Report 10-43, 2010). The idea is to consider the optimal constants as functions subject to a constraint on their gradient. Although the proposed relaxation technique is not guaranteed to find exact global minimizers of the original problem, our experiments show that our method computes tight approximations of the optimal solutions. Particularly, we provide numerical examples in which our method finds better solutions than the method proposed by Chan etxa0al. (SIAM Journal of Applied Mathematics 66(5):1632–1648, 2006), whose quality of solutions depends on the choice of the initial condition.

A level set method for segmentation of the thalamus and its nuclei in DT-MRI

In this paper, we present a 3D geometric flow designed to segment the main core of fiber tracts in diffusion tensor magnetic resonance images. The fundamental assumption of our fiber segmentation technique is that adjacent voxels in a tract have similar properties of diffusion. The fiber segmentation is carried out with a front propagation algorithm constructed to fill the whole fiber tract. The front is a 3D surface that evolves with a propagation speed proportional to a measure indicating the similarity of diffusion between the tensors lying on the surface and their neighbors in the direction of propagation. We use a level set implementation to assure a stable and accurate evolution of the surface and to handle changes of topology of the surface during the evolution process. The fiber tract segmentation method does not need a regularized tensor field since the surface is automatically smoothed as it propagates. The smoothing is done by an intrinsic surface force, based on the minimal principal curvature. This segmentation can be used for obtaining quantitative measures of the diffusion in the fiber tracts and it can also be used for white matter registration and for surgical planning.

Scale Space Analysis and Active Contours for Omnidirectional Images

We propose here a class of restoration algorithms for color images, based upon the Mumford-Shah (MS) model and nonlocal image information. The Ambrosio-Tortorelli and Shah elliptic approximations are defined to work in a small local neighborhood, which are sufficient to denoise smooth regions with sharp boundaries. However, texture is nonlocal in nature and requires semilocal/non-local information for efficient image denoising and restoration. Inspired from recent works (nonlocal means of Buades, Coll, Morel, and nonlocal total variation of Gilboa, Osher), we extend the local Ambrosio-Tortorelli and Shah approximations to MS functional (MS) to novel nonlocal formulations, for better restoration of fine structures and texture. We present several applications of the proposed nonlocal MS regularizers in image processing such as color image denoising, color image deblurring in the presence of Gaussian or impulse noise, color image inpainting, color image super-resolution, and color filter array demosaicing. In all the applications, the proposed nonlocal regularizers produce superior results over the local ones, especially in image inpainting with large missing regions. We also prove several characterizations of minimizers based upon dual norm formulations.