The scale concept and the notion of multiscale representation are of crucial importance in signal processing, pattern recognition and computer vision. Since the initial work by Rosenfeld and Thurston (1971) and others, this concept has been greatly developed and a series of methods have been proposed, such as pyramids and scale-space theory. One main problem in the multiscale theory is how to integrate the information at different scales. Most of the early work used the "from coarse to fine" (or similar) strategy to deal with this problem. Various methods have been proposed to implement the coarse-to-fine strategy. However, the existing methods have many limitations. In this thesis, we introduce some mathematical techniques in harmonic analysis and the regularization theory to solve this problem. My work includes two relatively independent parts: 1. Optical flow in the scale space. There exists an optical flow in the multiscale representation of an image if this representation is viewed as an image sequence in the "'time" domain. The ill-posed tracking problem in the scale-space can be robustly solved by this method. The perceptual effect of any multiscale representation can be greatly improved by the so-called "'pull-back" technique. 2. Matching Using Schwarz Integrals. We show that a 1-D signal and its derivative at different scales can be represented by one analytic function defined on the unit disc in the complex plane. This representation (called Schwarz representation) is applied to the matching problem. By using the theory of analytic functions, we are able to define the inverse of a signal. The matching function between two signals can be defined as the composition of one signal's Schwarz representation and the other's inverse. The matching function determined by this method has an algebraic structure and is in closed form.
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