A Unified Variational Volume Registration Method Based on Automatically Learned Brain Structures

We introduce a new volumetric registration technique that effectively combines active surfaces with the finite element method. The method simultaneously aligns multi-label automatic structural segmentation results, which can be obtained by the application of existing segmentation software, to produce an anatomically accurate 3D registration. This registration is obtained by the minimization of a single energy functional. Just like registering raw images, obtaining a 3D registration this way still requires solving a fundamentally ill-posed problem. We explain through academic examples as well as an MRI dataset with manual anatomical labels, which are hidden from the registration method, how the quality of a registration method can be measured and the advantages our approach offers.     

Active contours without edges

We propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by the gradient. We minimize an energy which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. We give a numerical algorithm using finite differences. Finally, we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.     

Self-repelling snakes for topology-preserving segmentation models.

The implicit framework of the level-set method has several advantages when tracking propagating fronts. Indeed, the evolving contour is embedded in a higher dimensional level-set function and its evolution can be phrased in terms of a Eulerian formulation. The ability of this intrinsic method to handle topological changes (merging and breaking) makes it useful in a wide range of applications (fluid mechanics, computer vision) and particularly in image segmentation, the main subject of this paper. Nevertheless, in some applications, this topological flexibility turns out to be undesirable: for instance, when the shape to be detected has a known topology, or when the resulting shape must be homeomorphic to the initial one. The necessity of designing topology-preserving processes arises in medical imaging, for example, in the human cortex reconstruction. It is known that the human cortex has a spherical topology so throughout the reconstruction process this topological feature must be preserved. Therefore, we propose in this paper a segmentation model based on an implicit level-set formulation and on the geodesic active contours, in which a topological constraint is enforced.     

Image segmentation using a multilayer level-set approach

We propose an efficient multilayer segmentation method based on implicit curve evolution and on variational approach. The proposed formulation uses the minimal partition problem as formulated by D. Mumford and J. Shah, and can be seen as a more efficient extension of the segmentation models previously proposed in Chan and Vese (Scale-Space Theories in Computer Vision, Lecture Notes in Computer Science, Vol. 1682, pp. 141–151, 1999, IEEE Trans Image Process 10(2):266–277, 2001), and Vese and Chan (Int J Comput Vis 50(3):271–293, 2002). The set of unknown discontinuities is represented implicitly by several nested level lines of the same function, as inspired from prior work on island dynamics for epitaxial growth (Caflisch et al. in Appl Math Lett 12(4):13, 1999; Chen et al. in J Comput Phys 167:475, 2001). We present the Euler–Lagrange equations of the proposed minimizations together with theoretical results of energy decrease, existence of minimizers and approximations. We also discuss the choice of the curve regularization and conclude with several experimental results and comparisons for piecewise-constant segmentation of gray-level and color images.     

Mathematical Modeling of Textures: Application to Color Image Decomposition with a Projected Gradient Algorithm

In this paper, we are interested in texture modeling with functional analysis spaces. We focus on the case of color image processing, and in particular color image decomposition. The problem of image decomposition consists in splitting an original image f into two components u and v. u should contain the geometric information of the original image, while v should be made of the oscillating patterns of f, such as textures. We propose here a scheme based on a projected gradient algorithm to compute the solution of various decomposition models for color images or vector-valued images. We provide a direct convergence proof of the scheme, and we give some analysis on color texture modeling.     

(Φ,Φ<Superscript>*</Superscript>) Image Decomposition Models and Minimization Algorithms

We propose in this paper minimization algorithms for image restoration using dual functionals and dual norms. In order to extract a clean image u from a degraded version f=Ku+n (where f is the observation, K is a blurring operator and n represents additive noise), we impose a standard regularization penalty Φ(u)=∫ φ(|Du|)dx<∞ on u, where φ is positive, increasing and has at most linear growth at infinity. However, on the residual f−Ku we impose a dual penalty Φ^*(f−Ku)<∞, instead of the more standard fidelity term. In particular, when φ is convex, homogeneous of degree one, and with linear growth (for instance the total variation of u), we recover the (BV,BV ^*) decomposition of the data f, as suggested by Y. Meyer (Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, vol. 22, Am. Math. Soc., Providence, 2001). Practical minimization methods are presented, together with theoretical, experimental results and comparisons to illustrate the validity of the proposed models. Moreover, we also show that by a slight modification of the associated Euler-Lagrange equations, we obtain well-behaved approximations and improved results.

Image Denoising and Decomposition with Total Variation Minimization and Oscillatory Functions

In this paper, we propose a new variational model for image denoising and decomposition, witch combines the total variation minimization model of Rudin, Osher and Fatemi from image restoration, with spaces of oscillatory functions, following recent ideas introduced by Meyer. The spaces introduced here are appropriate to model oscillatory patterns of zero mean, such as noise or texture. Numerical results of image denoising, image decomposition and texture discrimination are presented, showing that the new models decompose better a given image, possible noisy, into cartoon and oscillatory pattern of zero mean, than the standard ones. The present paper develops further the models previously introduced by the authors in Vese and Osher (Modeling textures with total variation minimization and oscillating patterns in image processing, UCLA CAM Report 02-19, May 2002, to appear in Journal of Scientific Computing, 2003). Other recent and related image decomposition models are also discussed.     

A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model

We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is also a generalization of an active contour model without edges based 2-phase segmentation, developed by the authors earlier in T. Chan and L. Vese (1999. In Scale-Space'99, M. Nilsen et al. (Eds.), LNCS, vol. 1682, pp. 141–151) and T. Chan and L. Vese (2001. IEEE-IP, 10(2):266–277). The multiphase level set formulation is new and of interest on its own: by construction, it automatically avoids the problems of vacuum and overlap; it needs only log n level set functions for n phases in the piecewise constant case; it can represent boundaries with complex topologies, including triple junctions; in the piecewise smooth case, only two level set functions formally suffice to represent any partition, based on The Four-Color Theorem. Finally, we validate the proposed models by numerical results for signal and image denoising and segmentation, implemented using the Osher and Sethian level set method.