Example-based interpolation for correspondence-based computer vision problems
Authors: Ronald Chi-Kit Chung
Venue:  (2006)

Example-Based Interpolation (EBI) is a powerful method to interpolate function from a set of input-output examples. The first part of the dissertation exams the EBI in detail and proposes a new enhanced EBI, indexed function Example-Based Interpolation (iEBI). The second part demonstrates the application of both EBI and iEBI to solve three well-defined problems of computer vision. First, the dissertation has analyzed EBI solution in detail. It argues and demonstrates that there are three desired properties for any EBI solution. To satisfy all three desirable properties, the EBI solution must have adequate degrees of freedom. This dissertation shows in details that, for the EBI solution to have enough degrees of freedom, it needs only be in a simple format: the sum of a basis function plus a linear function. This dissertation also presents that a particular EBI solution, in a certain least-squares-error sense, could satisfy exactly all the three desirable properties. Moreover, this dissertation also points out EBI's restriction and describes a new interpolation mechanism that could overcome EBI's restriction by constructing general indexed function from examples. The new mechanism, referred to as the general indexed function Example-Based Interpolation (iEBI) mechanism, first applies EBI to establish the initial correspondences over all input examples, and then interpolates the general indexed function from those initial correspondences. EBI and iEBI mechanism have all the desirable properties of a good interpolation: all given input-output examples are satisfied exactly, and the interpolation is smooth with minimum oscillations between the examples. The second part of the dissertation focuses on applying the EBI and iEBI methods to solve three correspondence-based problems in computer vision: (1) stereo matching, (2) novel view synthesis, and (3) viewpoint determination. Stereo matching, or the determination of corresponding image points projected by the same 3-D feature, is one of the fundamental and long-studied problems in computer vision. Yet, few have tried to solve it using interpolation. This dissertation presents an interpolation approach, Interpolation-based Iterative Stereo Matching (IISM), that could construct dense correspondences in stereo image from sparse initial correspondences. IISM improves the existing EBI to ensure that the established correspondences satisfy exactly the epipolar constraint of the image pair, and to a certain extent, preserve discontinuities in the stereo disparity space of the imaged scene. IISM utilizes the refinement technique of coarse-to-fine to iteratively apply the improved EBI algorithm, and eventually, produces the dense disparity map for stereo image pair. Novel View Synthesis (NVS) is an important problem in image rendering. It tries to synthesize an image of a scene at any specified (novel) viewpoint using only a few images of that scene at some sample viewpoints. To avoid explicit 3-D reconstruction of the scene, this dissertation formulates the problem of NVS as an indexed function interpolation problem by treating viewpoint and image as the input and output of a function. The interpolation formulation has at least two advantages. First, it allows certain imaging details like camera intrinsic parameters to be unknown. Second, the viewpoint specification need not be physical. For example, the specification could consist of any set of values that adequately describe the viewpoint space and need not be measured in metric units. This dissertation solves the NVS problem using the iEBI formulation and presents how the iEBI mechanism could be used to synthesize images at novel viewpoints and acquire quality novel views even from only a few example views. Viewpoint determination of image is the problem of, given an image, determining the viewpoint from which the image was taken. This dissertation demonstrates to solve this problem without referencing to or estimating any explicit 3-D structure of the imaged scene. Used for reference are a small number of sample snapshots of the scene, each of which has the associated viewpoint. By treating image and its associated viewpoint as the input and output of a function, and the given snapshot-viewpoint pairs as examples of that function, the problem has a natural formulation of interpolation. Same as that in NVS, the interpolation formulation allows the given images to be uncalibrated and the viewpoint specification to be not necessarily measured. This dissertation presents an interpolation-based solution using iEBI mechanism that guarantees all given sample data are satisfied exactly with the least complexity in the interpolated function. This dissertation also illustrates, for all the three problems, experimental results on a number of real and benchmarking image datasets, and shows that interpolation-based methods could be effective in arriving at good solution even with sparse input examples.