Accelerated Optimization in the PDE Framework Formulations for the Active Contour Case.

Following the seminal work of Nesterov, accelerated optimization methods have been used to powerfully boost the performance of first-order, gradient based parameter estimation in scenarios where second-order optimization strategies are either inapplicable or impractical. Not only does accelerated gradient descent converge considerably faster than traditional gradient descent, but it also performs a more robust local search of the parameter space by initially overshooting and then oscillating back as it settles into a final configuration, thereby selecting only local minimizers with a basis of attraction large enough to contain the initial overshoot. This behavior has made accelerated and stochastic gradient search methods particularly popular within the machine learning community.

Accelerated Variational PDEs for Efficient Solution of Regularized Inversion Problems.

We further develop a new framework, called , by applying it to calculus of variation problems defined for general functions on , obtaining efficient numerical algorithms to solve the resulting class of optimization problems based on simple discretizations of their corresponding accelerated PDEs. While the resulting family of PDEs and numerical schemes are quite general, we give special attention to their application for regularized inversion problems, with particular illustrative examples on some popular image processing applications. The method is a generalization of momentum, or accelerated, gradient descent to the PDE setting.

SurfCut: Surfaces of Minimal Paths from Topological Structures.

We present SurfCut, an algorithm for extracting a smooth, simple surface with an unknown 3D curve boundary from a noisy 3D image and a seed point. Our method is built on the novel observation that ridge curves of the Euclidean length of minimal paths ending on a level set of the solution of the eikonal equation lie on the surface. Our method extracts these ridges and cuts them to form the surface boundary.

Shape Tracking with Occlusions via Coarse-to-Fine Region-Based Sobolev Descent.

We present a method to track the shape of an object from video. The method uses a joint shape and appearance model of the object, which is propagated to match shape and radiance in subsequent frames, determining object shape. Self-occlusions and dis-occlusions of the object from camera and object motion pose difficulties to joint shape and appearance models in tracking.

Tracking using motion estimation with physically motivated inter-region constraints.

We propose a method for tracking structures (e.g., ventricles and myocardium) in cardiac images (e.

Nonlinear registration of serial coronary CT angiography (CCTA) for assessment of changes in atherosclerotic plaque.

Purpose: Coronary CT angiography (CCTA) is a high-resolution three-dimensional imaging technique for the evaluation of coronary arteries in suspected or confirmed coronary artery disease (CAD). Coregistration of serial CCTA scans would allow precise superimposition of images obtained at two different points in time, which could aid in recognition of subtle changes and precise monitoring of coronary plaque progression or regression. To this end, the authors aimed at developing a fully automatic nonlinear volume coregistration for longitudinal CCTA scan pairs.

3D Topology Preserving Flows for Viewpoint-Based Cortical Unfolding.

We present a variational method for unfolding of the cortex based on a user-chosen point of view as an alternative to more traditional global flattening methods, which incur more distortion around the region of interest. Our approach involves three novel contributions. The first is an energy function and its corresponding gradient flow to measure the average visibility of a region of interest of a surface with respect to a given viewpoint.

Coarse-to-fine segmentation and tracking using Sobolev active contours.

Recently proposed Sobolev active contours introduced a new paradigm for minimizing energies defined on curves by changing the traditional cost of perturbing a curve and thereby redefining their gradients. Sobolev active contours evolve more globally and are less attracted to certain intermediate local minima than traditional active contours, and it is based on a well-structured Riemannian metric. In this paper, we analyze Sobolev active contours using scale-space analysis in order to understand their evolution across different scales.

Global regularizing flows with topology preservation for active contours and polygons.

Active contour and active polygon models have been used widely for image segmentation. In some applications, the topology of the object(s) to be detected from an image is known a priori, despite a complex unknown geometry, and it is important that the active contour or polygon maintain the desired topology. In this work, we construct a novel geometric flow that can be added to image-based evolutions of active contours and polygons in order to preserve the topology of the initial contour or polygon.