Anthony J. Yezzi

A shape-based approach to the segmentation of medical imagery using level sets

We propose a shape-based approach to curve evolution for the segmentation of medical images containing known object types. In particular, motivated by the work of Leventon, Grimson, and Faugeras (2000), we derive a parametric model for an implicit representation of the segmenting curve by applying principal component analysis to a collection of signed distance representations of the training data. The parameters of this representation are then manipulated to minimize an objective function for segmentation. The resulting algorithm is able to handle multidimensional data, can deal with topological changes of the curve, is robust to noise and initial contour placements, and is computationally efficient. At the same time, it avoids the need for point correspondences during the training phase of the algorithm. We demonstrate this technique by applying it to two medical applications; two-dimensional segmentation of cardiac magnetic resonance imaging (MRI) and three-dimensional segmentation of prostate MRI.

Gradient flows and geometric active contour models

In this paper, we analyze the geometric active contour models discussed previously from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new feature-based Riemannian metrics. This leads to a novel snake paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus the snake is attracted very naturally and efficiently to the desired feature. Moreover, we consider some 3-D active surface models based on these ideas.<<ETX>>

A geometric snake model for segmentation of medical imagery

We employ the new geometric active contour models, previously formulated, for edge detection and segmentation of magnetic resonance imaging (MRI), computed tomography (CT), and ultrasound medical imagery. Our method is based on defining feature-based metrics on a given image which in turn leads to a novel snake paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus, the snake is attracted very quickly and efficiently to the desired feature.

Conformal curvature flows: From phase transitions to active vision

In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new feature-based Riemannian metrics. This leads to a novel edge-detection paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus an edge-seeking curve is attracted very naturally and efficiently to the desired feature. Comparison with the Allen-Cahn model clarifies some of the choices made in these models, and suggests inhomogeneous models which may in return be useful in phase transitions. We also consider some 3-dimensional active surface models based on these ideas. The justification of this model rests on the careful study of the viscosity solutions of evolution equations derived from a level-set approach.

A statistical approach to snakes for bimodal and trimodal imagery

We describe a new region based approach to active contours for segmenting images composed of two or three types of regions characterizable by a given statistic. The essential idea is to derive curve evolutions which separate two or more valves of a pre-determined set of statistics computed over geometrically determined subsets of the image. Both global and local image information is used to evolve the active contour. Image derivatives, however, are avoided, thereby giving rise to a further degree of noise robustness compared to most edge based snake algorithms.

A nonparametric statistical method for image segmentation using information theory and curve evolution

In this paper, we present a new information-theoretic approach to image segmentation. We cast the segmentation problem as the maximization of the mutual information between the region labels and the image pixel intensities, subject to a constraint on the total length of the region boundaries. We assume that the probability densities associated with the image pixel intensities within each region are completely unknown a priori, and we formulate the problem based on nonparametric density estimates. Due to the nonparametric structure, our method does not require the image regions to have a particular type of probability distribution and does not require the extraction and use of a particular statistic. We solve the information-theoretic optimization problem by deriving the associated gradient flows and applying curve evolution techniques. We use level-set methods to implement the resulting evolution. The experimental results based on both synthetic and real images demonstrate that the proposed technique can solve a variety of challenging image segmentation problems. Furthermore, our method, which does not require any training, performs as good as methods based on training.

On the relationship between parametric and geometric active contours

Geometric active contours have many advantages over parametric active contours, such as computational simplicity and the ability to change the curve topology during deformation. While many of the capabilities of the older parametric active contours have been reproduced in geometric active contours, the relationship between the two has not always been clear. We develop a precise relationship between the two which includes spatially-varying coefficients, both tension and rigidity, and non-conservative external forces. The result is a very general geometric active contour formulation for which the intuitive design principles of parametric active contours can be applied. We demonstrate several novel applications in a series of simulations.

Integral Invariants for Shape Matching

For shapes represented as closed planar contours, we introduce a class of functionals which are invariant with respect to the Euclidean group and which are obtained by performing integral operations. While such integral invariants enjoy some of the desirable properties of their differential counterparts, such as locality of computation (which allows matching under occlusions) and uniqueness of representation (asymptotically), they do not exhibit the noise sensitivity associated with differential quantities and, therefore, do not require presmoothing of the input shape. Our formulation allows the analysis of shapes at multiple scales. Based on integral invariants, we define a notion of distance between shapes. The proposed distance measure can be computed efficiently and allows warping the shape boundaries onto each other; its computation results in optimal point correspondence as an intermediate step. Numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts, missing parts and noise. As a quantitative analysis, we report matching scores for shape retrieval from a database

Model-based curve evolution technique for image segmentation

We propose a model-based curve evolution technique for segmentation of images containing known object types. In particular, motivated by the work of Leventon et al. (2000), we derive a parametric model for an implicit representation of the segmenting curve by applying principal component analysis to a collection of signed distance representations of the training data, The parameters of this representation are then calculated to minimize an objective function for segmentation. We found the resulting algorithm to be computationally efficient, able to handle multidimensional data, robust to noise and initial contour placements, while at the same time, avoiding the need for point correspondences during the training phase of the algorithm. We demonstrate this technique by applying it to two medical applications.

Tracking Deforming Objects Using Particle Filtering for Geometric Active Contours

Tracking deforming objects involves estimating the global motion of the object and its local deformations as a function of time. Tracking algorithms using Kalman filters or particle filters have been proposed for finite dimensional representations of shape, but these are dependent on the chosen parametrization and cannot handle changes in curve topology. Geometric active contours provide a framework which is parametrization independent and allow for changes in topology, in the present work, we formulate a particle filtering algorithm in the geometric active contour framework that can be used for tracking moving and deforming objects. To the best of our knowledge, this is the first attempt to implement an approximate particle filtering algorithm for tracking on a (theoretically) infinite dimensional state space.