We propose a manifold learning approach to fiber tract clustering using a novel similarity measure between fiber tracts constructed from dual-rooted graphs. In particular, to generate this similarity measure, the chamfer or Hausdorff distance is initially employed as a local distance metric to construct minimum spanning trees between pairwise fiber tracts. These minimum spanning trees are effective in capturing the intrinsic geometry of the fiber tracts. Hence, they are used to capture the neighborhood structures of the fiber tract data set. We next assume the high-dimensional input fiber tracts to lie on low-dimensional non-linear manifolds. We apply Locally Linear Embedding, a popular manifold learning technique, to define a low-dimensional embedding of the fiber tracts that preserves the neighborhood structures of the high-dimensional data structure as captured by the method of dual-rooted graphs. Clustering is then performed on this low-dimensional data structure using the k-means algorithm. We illustrate our resulting clustering technique on both synthetic data and on real fiber tract data obtained from diffusion tensor imaging.

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, 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.

In this paper, we introduce a novel approach for simultaneous restoration and segmentation of blurred noisy images by approaching a variant of the Mumford-Shah functional from a curve evolution perspective. In particular, by viewing the active contour as the set of discontinuities in the image, we derive a gradient flow to minimize an extended Mumford-Shah functional where the known blurring function is incorporated as part of the data fidelity term. Each gradient step involves solving a discrete approximation of the corresponding partial differential equation to obtain a smooth and deblurred estimate of the observed image without blurring across the curve. The experimental results based on both synthetic and real images demonstrate that the proposed method segments and restores the blurred images effectively. We conclude that our work is an edge-preserving image restoration technique that couples segmentation, denoising, and deblurring within a single framework. In addition, this framework provides an intellectual connection between regularization theory (used to solve the deblurring inverse problem) and the theory of curve evolution.

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.

In this work, we first address the problem of simultaneous image segmentation and smoothing by approaching the Mumford-Shah paradigm from a curve evolution perspective. In particular, we let a set of deformable contours define the boundaries between regions in an image where we model the data via piecewise smooth functions and employ a gradient flow to evolve these contours. Each gradient step involves solving an optimal estimation problem for the data within each region, connecting curve evolution and the Mumford-Shah functional with the theory of boundary-value stochastic processes. The resulting active contour model offers a tractable implementation of the original Mumford-Shah model (i.e., without resorting to elliptic approximations which have traditionally been favored for greater ease in implementation) to simultaneously segment and smoothly reconstruct the data within a given image in a coupled manner. Various implementations of this algorithm are introduced to increase its speed of convergence. We also outline a hierarchical implementation of this algorithm to handle important image features such as triple points and other multiple junctions. Next, by generalizing the data fidelity term of the original Mumford-Shah functional to incorporate a spatially varying penalty, we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing. This more general model leads us to a novel PDE-based approach for simultaneous image magnification, segmentation, and smoothing, thereby extending the traditional applications of the Mumford-Shah functional which only considers simultaneous segmentation and smoothing.

In this work, we approach the classic Mumford-Shah problem from a curve evolution perspective. In particular we let a given family of curves define the boundaries between regions in an image within which the data are modeled by piecewise smooth functions plus noise as in the standard Mumford-Shah functional. The gradient descent equation of this functional is then used to evolve the curve. Each gradient descent step involves solving a corresponding optimal estimation problem which connects the Mumford-Shah functional and our curve evolution implementation with the theory of boundary-value stochastic processes. The resulting active contour model, therefore, inherits the attractive ability of the Mumford-Shah technique to generate, in a coupled Mumford-Shah a smooth reconstruction of the image and a segmentation as well. We demonstrate applications of our method to problems in which data quality is spatially varying and to problems in which sets of pixel measurements are missing. Finally, we demonstrate a hierarchical implementation of our model which leads to a fast and efficient algorithm capable of dealing with important image features such as triple points.

We present an estimation-theoretic approach to curve evolution for the Mumford-Shah problem. By viewing an active contour as the set of discontinuities in the Mumford-Shah problem, we may use the corresponding functional to determine gradient descent evolution equations to deform the active contour. In each gradient descent step, we solve a corresponding optimal estimation problem, connecting the Mumford-Shah functional and curve evolution with the theory of boundary-value stochastic processes. In employing the Mumford-Shah functional, our active contour model inherits its attractive ability to generate, in a coupled manner, both a smooth reconstruction and a segmentation of the image. Next, by generalizing the data fidelity term of the original Mumford-Shah functional to incorporate a spatially varying penalty, we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing. This more general model leads us to a novel PDE-based approach for simultaneous image magnification, segmentation, and smoothing, thereby extending the traditional applications of the Mumford-Shah functional which only considers simultaneous segmentation and smoothing.

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 novel region-based approach to snakes is introduced in this paper for the segmentation of images composed of two or three types of regions where each region may be distinguished by a given statistic. The basic idea behind this technique is to formulate curve evolutions which separate two or more values of a predetermined set of statistics computed over geometrically determined subsets of the image data. Our methodology provides a natural framework for incorporating both global and local image information in the active contour motion while avoiding the use of image derivatives. As such, this technique possesses a robustness to noise which is noncharacteristic of most edge-based snake algorithms.