Mark Moyou

LBO-Shape Densities: Efficient 3D Shape Retrieval Using Wavelet Density Estimation

Driven by desirable attributes such as topological characterization and invariance to isometric transformations, the use of the Laplace-Beltrami operator (LBO) and its associated spectrum have been widely adopted among the shape analysis community. Here we demonstrate a novel use of the LBO for shape matching and retrieval by estimating probability densities on its Eigen space, and subsequently using the intrinsic geometry of the density manifold to categorize similar shapes. In our framework, each 3D shapes rich geometric structure, as captured by the low order eigenvectors of its LBO, is robustly characterized via a nonparametric density estimated directly on these eigenvectors. By utilizing a probabilistic model where the square root of the density is expanded in a wavelet basis, the space of LBO-shape densities is identifiable with the unit hyper sphere. We leverage this simple geometry for retrieval by computing an intrinsic Karcher mean (on the hyper sphere of LBO-shape densities) for each shape category, and use the closed-form distance between a query shape and the means to classify shapes. Our method alleviates the need for superfluous feature extraction schemes-required for popular bag-of-features approaches-and experiments demonstrate it to be robust and competitive with the state-of-the-art in 3D shape retrieval algorithms.

The Geometry of Orthogonal-Series, Square-Root Density Estimators: Applications in Computer Vision and Model Selection

We consider the geometry and model order specification of a class of density models where the square-root of the distribution is expanded in an orthogonal series. The simplicity of the resulting spherical geometry makes this framework ideal for many applications that rely on information geometric concepts like distances and manifold statistics. Specifically, we demonstrate applications of these models in the computer vision field of object recognition and retrieval. We illustrate how invariant shape representations can be used in conjunction with these probabilistic models to yield state-of-the-art classifiers. Moreover, the viability of formulating classification models that take into account shape deformation in an optimal transport context are investigated, yielding insight into the practicalities of working with the parameter space of the densities versus the Wasserstein measure space approach. The free parameters associated with these square-root estimators can be rigorously selected using the Minimum Description Length (MDL) criterion for model selection. Under these models, it is shown that the MDL has a closed-form representation, atypical for most applications of MDL in density estimation. Experimental evaluation of our techniques are conducted on one, two, and three dimensional density estimation problems in shape analysis, with comparative analysis demonstrating our approach to be state-of-the-art in object recognition and model selection.

Bayesian Fusion of Back Projected Probabilities BFBP: Co-occurrence Descriptors for Tracking in Complex Environments

Among the multitude of probabilistic tracking techniques, the Continuously Adaptive Mean Shift CAMSHIFT algorithm has been one of the most popular. Though several modifications have been proposed to the original formulation of CAMSHIFT, limitations still exist. In particular the algorithm underperforms when tracking textured and patterned objects. In this paper we generalize CAMSHIFT for the purposes of tracking such objects in non-stationary backgrounds. Our extension introduces a novel object modeling technique, while retaining a probabilistic back projection stage similar to the original CAMSHIFT algorithm, but with considerably more discriminative power. The object modeling now evolves beyond a single probability distribution to a more generalized joint density function on localized color patterns. In our framework, multiple co-occurrence density functions are estimated using information from several color channel combinations and these distributions are combined using an intuitive Bayesian approach. We validate our approach on several aerial tracking scenarios and demonstrate its improved performance over the original CAMSHIFT algorithm and one of its most successful variants.

A new energy minimization framework and sparse linear system for path planning and shape from shading

For over 30 years, the static Hamilton-Jacobi (HJ) equation, specifically its incarnation as the eikonal equation, has been a bedrock for a plethora of computer vision models, including popular applications such as shape-from-shading, medial axis representations, level-set segmentation, and geodesic processing (i.e. path planning). Numerical solutions to this nonlinear partial differential equation have long relied on staples like fast marching and fast sweeping algorithms—approaches which rely on intricate convergence analysis, approximations, and specialized implementations. Here, we present a new variational functional on a scalar field comprising a spatially varying quadratic term and a standard regularization term. The Euler-Lagrange equation corresponding to the new functional is a linear differential equation which when discretized results in a linear system of equations. This approach leads to many algorithm choices since there are myriad efficient sparse linear solvers. The limiting behavior, for a particular case, of this linear differential equation can be shown to converge to the nonlinear eikonal. In addition, our approach eliminates the need to explicitly construct viscosity solutions as customary with direct solutions to the eikonal. Though our solution framework is applicable to the general class of eikonal problems, we detail specifics for the popular vision applications of shape-from-shading, vessel segmentation, and path planning. We showcase experimental results on a variety of images and complex mazes, in which we hold our own against state-of-the art fast marching and fast sweeping techniques, while retaining the considerable advantages of a linear systems approach.