Time-varying Finite Dimensional Basis for Tracking Contour Deformations

Abstract

Abstract-We consider the problem of tracking the boundary contour of a moving and deforming object from a sequence of images. If the motion of the "object" or region of interest is constrained (e.g. rigid or approximately rigid), the contour motion can be efficiently represented by a small number of parameters, e.g. the affine group. But if the "object" is arbitrarily deforming, each contour point can move independently. Contour deformation then forms an infinite (in practice, very large), dimensional space. Direct application of particle filters for large dimensional problems is impractical, due to the reduction in effective particle size as dimension increases. But in most real problems, at any given time, "most of the contour deformation" occurs in a small number of dimensions ("effective basis") while the residual deformation in the rest of the state space ("residual space") is "small". The effective basis may be fixed or time varying. Based on this assumption, we modify the particle filtering method to perform sequential importance sampling only on the effective basis dimensions, while replacing it with deterministic mode tracking in residual space (PF-MT). We develop the PF-MT idea for contour tracking. Techniques for detecting effective basis dimension change and estimating the new effective basis are presented. Tracking results on simulated and real sequences are shown and compared with past work.

Learning the Dynamics and Time-Recursive Boundary Detection of Deformable Objects

“…This would, of course, require generating one particle pair for each particle in , a computationally intensive task. Instead, we have found it effective to generate a single particle pair by optimizing an average over all particles of the right-hand sides of (16), i.e., by solving (17) Substituting expressions from (1), (3), (5), and (15) into (17), and using the fact that (18) we obtain (19) where (20) where and . Practically, we initialize the minimization using the mean shape from the training data.…”

“…While an approach to doing this has recently been developed (for models in which curve dynamics are described by linear-quadratic variational formulations) [18]- [20], we have chosen to avoid this additional complexity through an approximation that involves solving level-set based curve evolutions during the sampling procedure.…”

“…In particular, the mean shapes at ED and ES are determined from training data, and, given the shape estimate in a particular frame, the prediction is assumed to be Gaussian with mean shape given by a linear combination of the current estimate and a fraction of the difference between the mean shapes at ED and ES (hence, driving the means toward these pinned values at the end points). Vaswani et al [18]- [20] also use a particle-based method for causal shape tracking, but they do this directly in a variational framework that leads to level-set based representations and particles that represent samples of the intrinsically infinite-dimensional curve. Finally, Cremers [21] uses level sets and finite-dimensional representations using principal component analysis to represent shapes and learns linear, Gaussian dynamic models for this finite-dimensional representation.…”

Geometric Observers for Dynamically Evolving Curves

“…Blake and Isard [10] proposed a particle-filter-based curve tracking framework for evolving the coefficients of a B-spline representation of a curve. More recent work has focused on combining particle filters with shape models, both for curve representations [11], [12], [13] and in combination with active shape models [14], [15]. Learned shape-based priors have been combined with jointly learned shape-dynamics using Gaussian probability distributions in [16].…”

“…Of note, the solution of (10) is sufficient to define the error vector field. However, to facilitate easy numerical computations of the error vector field, we extend the solution to the remainder of the image domain 3 by solving an additional Laplace equation on R lo and a Poisson equation 4 on R pi (11) (12) with boundary conditions (13) The combined solution (14) defines the error vector field X err on Ω, 3 Note that the combined solution will be continuous everywhere but not necessarily differentiable on C 0 and C 1 . However, by construction, the gradient directions will align.…”