PhD, Colorado State University, Aug. 2009
Major Professor: Anthony A. Maciejewski
Eigenspace decomposition represents one computationally efficient approach for dealing with object recognition and pose estimation, as well as other vision-based problems, and has been applied to sets of correlated images for this purpose. The general idea behind eigenspace decomposition is that a large set of highly correlated images can be approximately represented by a much smaller subspace. Unfortunately, determining the dimension of the subspace, as well as computing the subspace itself is computationally prohibitive. To make matters worse, this off-line expense increases drastically as the number of correlated images becomes large (which is the case when doing fully general three-dimensional pose estimation or illumination invariant pose estimation). However, previous work has shown that for data correlated in one-dimension, Fourier analysis can help reduce the computational burden of this off-line expense.
The first part of this dissertation extends some of the ideas developed for one-dimensionally correlated image data to data correlated in two- and three-dimensions making fully general three-dimensional pose estimation possible (assuming the object is illuminated from a single distant light source). The first step in this extension is to determine how to capture training images of the object by sampling the two-sphere (S2), and the rotation group (SO(3)) appropriately. Therefore, a thorough analysis of spherical tessellations is performed as applied to the problem of pose estimation. An algorithm is then developed for reducing the off-line computational burden associated with computing the eigenspace by exploiting the spectral information of this spherical data set. The algorithm is based on the fact that, similar to Fourier analysis on the line or circle, spherically correlated functions can be expanded into a finite series using spherical harmonics. It is then shown that the algorithm can be extended to higher dimensions by applying a proper rotation to each of the samples defined on the surface of the sphere. Using this sampling technique, a parameterization of SO(3) is obtained. It is shown that SO(3) correlated functions can be expanded into a finite series by applying a rotation to the set of spherical harmonics and expanding the function using Wigner-D matrices. Experimental results are presented to compare the proposed algorithm to the true eigenspace decomposition, as well as assess the computational savings.
The second part of this dissertation deals with the problem of pose estimation when variations in illumination conditions exist. It is shown that the dimensionality of a set of images of an object under a wide range of illumination conditions and fixed pose can be significantly reduced by expanding the image data in a series of spherical harmonics. This expansion results in a reduced dimensional set of "harmonic images". It is shown that the set of harmonic images are capable of recovering a significant amount of information from a set of images captured when both single and multiple illumination sources are present. An algorithm is then developed to estimate the eigenspace of a set of images that contain variation in both illumination and pose. The algorithm is based on projecting the set of harmonic images onto a set of Fourier harmonics by applying Chang's eigenspace decomposition algorithm. Finally, an analysis of eigenspace manifolds is presented when variations in both illumination and pose exist. A technique for illumination invariant pose estimation is developed based on eigenspace partitioning. Experimental results are presented to validate the proposed algorithm in terms of accuracy in estimating the eigenspace, computational savings, and the accuracy of determining the pose of three-dimensional objects under a range of illumination conditions.