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MOBK039-01 MOBK039-Median.cls November 9, 2006 21:33
INTRODUCTION 7
We beginby illustrating thetensor voting framework in 2D and 3D.We show how tokens,
which are simple primitives that potentially belong to perceptual structures, can be represented
by second-order, symmetric, nonnegative definite tensors. Then, we present the mechanism
for information propagation we have designed. The results of this propagation are structural
saliency values for each potential structure type along with the preferred normal and tangent
orientations at each token. Experimental results demonstrate the effectiveness of our approach,
as well as its robustness to noise.
A large part of our research efforts is devoted to the development of a stereo reconstruction
approach, which is presented in Chapter 3. Stereovision can be cast as a perceptual organization
problem under the premise that solutions must comprise coherent structures. These structures
become salient due to the alignment of potential pixel correspondences reconstructed in a 3D
space. Tensor voting is performed to infer the correct matches that are generated by the true
scene surfaces as inliers of smooth perceptual structures. The retained matches are grouped
into smooth surfaces and inconsistent matches are rejected. Disparity hypotheses for pixels
that remain unmatched are generated based on the color information of nearby surfaces and
validated by ensuring the good continuation of the surfaces via tensor voting. Thus, information
is propagated frommore to less reliablepixels considering both geometric and color information.
Arecent,major enhancement of the frameworkisanefficient N D implementation, which
is described in Chapter 4. We present a new implementation of tensor voting that significantly
reduces computational time and storage requirements, especially in high-dimensional spaces,
and thus can be applied to machine learning problems, as well as a variety of new domains.
This work is based on the observation that the Gestalt principles still apply in spaces of higher
dimensionality. The computational and storage requirements of the original implementation
prevented its wide application to problems in high dimensions. This is no longer the case with
the new implementation which opens up an array of potential applications mostly in the field
of instance-based learning.
Chapter 5 presents our approach to machine learning problems. We address unsuper-
vised manifold learning from observations in high-dimensional spaces using the new efficient
implementation of tensor voting. We are able to estimate local dimensionality and structure,
measure geodesic distances, and perform nonlinear interpolation. We first show that we can
obtain reliable dimensionality estimates at each point. Then, we present a quantitative eval-
uation of our results in the estimation of a local manifold structure using synthetic datasets
with known ground truth. We also present results on datasets with varying dimensionality and
intersections under severe noise corruption, which would have been impossible to process with
current state-of-the-art methods. We also address function approximation from samples, which
is an important problem with many applications in machine learning. We propose a noniter-
ative, local, nonparametric approach that can successfully approximate nonlinear functions in