On the Construction and Mathematical Analysis of the Wavelet Transform and its Matricial Properties

We study the properties of computational methods for the Wavelet Transform and its Inverse from the point of view of Linear Algebra. We present a characterization of such methods as matrix products, proving in particular that each iteration corresponds to the multiplication of an adequate unitary matrix. From that point we prove that some important properties of the Continuous Wavelet Transform, such as linearity, distributivity over matrix multiplication, isometry, etc., are inherited by these discrete methods. This work is divided into four sections. The first section corresponds to the classical theoretical foundation of harmonic analysis with wavelets; it is used for clarity only. The second section presents the construction of the Discrete Wavelet Transform for vectors and its Inverse, emphasizing on storage efficiency. The third section presents the generalization of the Transform to matrices. It is equivalent to section 2, but some methods and tools used are slightly different, showing an alternative approach to the subject. The fourth section presents the main results of this work