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.

Author Bio

Diego Sejas is a Mathematical Engineer from Bolivia, graduated from Universidad Mayor de San Simón. He has worked on many undergraduate research projects, and has been recognized as the best Mathematical Engineering student of his generation by the Bolivian Mathematical Society. After his graduation, he started teaching at Universidad Simón I. Patiño and Universidad Privada Boliviana. He currently holds a position as Invited Researcher at Universidad Simón I. Patiño.