Multi-scale approaches to signal deblurring and reconstruction from zero-crossings

Abstract

This thesis addresses two inverse problems in computer vision:

Solving a class of ill-posed convolution equations We analyze the problem of deblurring a Gaussian-blurred signal, which belongs to the class of ill-posed inverse problems. Prior attempts in the literature [1, 2], based on single-scale methods, have either suffered from instability or have established stable inversion procedures for highly restricted domains. We first solve this problem for a separable Gaussian kernel, in a multi-scale setting. To this end, we employ a basis of Hermite functions [3] parameterized by a scale factor. The algorithm we propose is shown to be stable and computationally efficient [4, 5]. Then we extend the method to separable and to arbitrary, non-separable convolution kernels in higher dimensions. Finally, we propose a method based on effective spectral concentration of the signal for blur parameter estimation, and illustrate it by applying it to one- and two-dimensional signal analysis.

Reconstruction of one-dimensional signals from their zero-crossings Traditionally, representing one-dimensional signals by their zeros has been intimately connected with the theory of entire functions [6]. We invoke results from this theory to show that a class of these signals is completely determined by their zeros. Using a basis of harmonic wavelets [7, 8], we formulate the reconstruction (from zeros) problem for a class of one-dimensional signals in an optimization framework. We solve this optimization problem using a team of stochastic learning automata [9], and present some experimental results.

Although each of the two problems deserves to be treated independently, the unifying factor is the common motivation drawn from investigations of Marr [10] who proposed that human visual perception is based on the LoG zero-crossings which define the boundaries of the objects in a scene. The problem of modelling human perception (à la Marr) can be decomposed into two subproblems, viz, (i) Deconvolving the Laplacian of the Gaussian; and (ii) Reconstructing a signal from its zero-crossings. However, in the thesis, we have treated these subproblems independently. The results of our deconvolution procedure are applicable to both one- and two-dimensional signals, while those of the reconstruction procedure apply only to one-dimensional signals. We hope that the framework employed and the method suggested by us would be useful in solving Marr’s problem of modelling the human vision system. The thesis is organized as follows: Abstract The first chapter formulates the problems addressed in the thesis and gives a brief introduction to wavelet transforms. In the second chapter, we propose a new algorithm to deconvolve Gaussian-blurred signals. In Chapter 3, we extend this method to separable and non-separable kernels in higher dimensions. We also propose a method for the related problem of blur parameter estimation. Chapter 4 deals with the reconstruction of signals from their zero-crossings. In Chapter 5, we conclude by highlighting the main contributions of the thesis, and by pointing out some unsolved problems for future research.