New Possibilities with Sobolev Active Contours

Recently, the Sobolev metric was introduced to define gradient flows of various geometric active contour energies. It was shown that the Sobolev metric outperforms the traditional metric for the same energy in many cases such as for tracking where the coarse scale changes of the contour are important. Some interesting properties of Sobolev gradient flows include that they stabilize certain unstable traditional flows, and the order of the evolution PDEs are reduced when compared with traditional gradient flows of the same energies. In this paper, we explore new possibilities for active contours made possible by Sobolev metrics. The Sobolev method allows one to implement new energy-based active contour models that were not otherwise considered because the traditional minimizing method render them ill-posed or numerically infeasible. In particular, we exploit the stabilizing and the order reducing properties of Sobolev gradients to implement the gradient descent of these new energies. We give examples of this class of energies, which include some simple geometric priors and new edge-based energies. We also show that these energies can be quite useful for segmentation and tracking. We also show that the gradient flows using the traditional metric are either ill-posed or numerically difficult to implement, and then show that the flows can be implemented in a stable and numerically feasible manner using the Sobolev gradient. Sundaramoorthi and Yezzi were supported by NSF CCR-0133736, NIH/NINDS R01-NS-037747, and Airforce MURI; Sapiro was partially supported by NSF, ONR, NGA, ARO, DARPA, and the McKnight Foundation.     

Stereoscopic Segmentation

We cast the problem of multiframe stereo reconstruction of a smooth shape as the global region segmentation of a collection of images of the scene. Dually, the problem of segmenting multiple calibrated images of an object becomes that of estimating the solid shape that gives rise to such images. We assume that the radiance of the scene results in piecewise homogeneous image statistics. This simplifying assumption covers Lambertian scenes with constant albedo as well as fine homogeneous textures, which are known challenges to stereo algorithms based on local correspondence. We pose the segmentation problem within a variational framework, and use fast level set methods to find the optimal solution numerically. Our algorithm does not work in the presence of strong photometric features, where traditional reconstruction algorithms do. It enjoys significant robustness to noise under the assumptions it is designed for.     

Curve evolution implementation of the Mumford-Shah functional forimage segmentation, denoising, interpolation, and magnification

We first address the problem of simultaneous image segmentation and smoothing by approaching the Mumford-Shah (1989) 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.     

Modified curvature motion for image smoothing and enhancement

We formulate a general modified mean curvature based equation for image smoothing and enhancement. The key idea is to consider the image as a graph in some R(72), and apply a mean curvature type motion to the graph. We consider some special cases relevant to grey-scale and color images.     

Vessels as 4-D curves: global minimal 4-D paths to extract 3-D tubular surfaces and centerlines

In this paper, we propose an innovative approach to the segmentation of tubular structures. This approach combines all of the benefits of minimal path techniques such as global minimizers, fast computation, and powerful incorporation of user input, while also having the capability to represent and detect vessel surfaces directly which so far has been a feature restricted to active contour and surface techniques. The key is to represent the trajectory of a tubular structure not as a 3-D curve but to go up a dimension and represent the entire structure as a 4-D curve. Then we are able to fully exploit minimal path techniques to obtain global minimizing trajectories between two user supplied endpoints in order to reconstruct tubular structures from noisy or low contrast 3-D data without the sensitivity to local minima inherent in most active surface techniques. In contrast to standard purely spatial 3-D minimal path techniques, however, we are able to represent a full tubular surface rather than just a curve which runs through its interior. Our representation also yields a natural notion of a tube's "central curve." We demonstrate and validate the utility of this approach on magnetic resonance (MR) angiography and computed tomography (CT) images of coronary arteries.     

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. We emphasize that, unlike other methods for topology preservation, the proposed geometric flow continually adjusts the geometry of the original evolution in a gradual and graceful manner so as to prevent a topology change long before the curve or polygon becomes close to topology change. The flow also serves as a global regularity term for the evolving contour, and has smoothness properties similar to curvature flow. These properties of gradually adjusting the original flow and global regularization prevent geometrical inaccuracies common with simple discrete topology preservation schemes. The proposed topology preserving geometric flow is the gradient flow arising from an energy that is based on electrostatic principles. The evolution of a single point on the contour depends on all other points of the contour, which is different from traditional curve evolutions in the computer vision literature.     

3-D Reconstruction of Shaded Objects from Multiple Images Under Unknown Illumination

We propose a variational algorithm to jointly estimate the shape, albedo, and light configuration of a Lambertian scene from a collection of images taken from different vantage points. Our work can be thought of as extending classical multi-view stereo to cases where point correspondence cannot be established, or extending classical shape from shading to the case of multiple views with unknown light sources. We show that a first naive formalization of this problem yields algorithms that are numerically unstable, no matter how close the initialization is to the true geometry. We then propose a computational scheme to overcome this problem, resulting in provably stable algorithms that converge to (local) minima of the cost functional. We develop a new model that explicitly enforces positivity in the light sources with the assumption that the object is Lambertian and its albedo is piecewise constant and show that the new model significantly improves the accuracy and robustness relative to existing approaches.     

Euler characteristics of oceanic sea states

We present an application of a novel Variational Wave Acquisition Stereo System (VWASS) for the estimation of the wave surface height of oceanic sea states. Specifically, we show that VWASS video technology combined with statistical techniques based on Euler Characteristics of random fields provides a new paradigm for the prediction of wave extremes expected over a given area of the ocean.     

Deformotion: Deforming Motion,Shape Average and the Joint Registration and Approximation of Structures in Images

International Journal of Computer Vision - What does it mean for a deforming object to be “moving”? How can we separate the overall motion (a finite-dimensional group action) from the...