Analysis of finite Haar wavelet transform and its implementation

Abstract

In this paper, we give frame‐based finite Haar wavelet transform of ‐scale and ‐level (FHWT( ). The formulation of FHWT is based on the finite nonuniform wavelet transform (FNUWT), and the reconstruction is done using ‐frame reconstruction property. We formulate the FNUWT based on the technique of nonuniform sampling and study the reconstruction property of FNUWT in terms of the ‐frame reconstruction property. FHWT is a particular model of FNUWT based on critical sampling. Using the principles of ‐frame, we prove that FHWT( is a unitary operator. Also, our formulation enables us to define the finite Haar wavelet basis (FHWB) explicitly. In our study, we establish another variant of the fast Haar wavelet transform with varying rate of sampling, and with fast computations. Finally, we formulate the algorithms of finite Haar wavelet transform and its inverse through block implementations of ‐frame. It is observed that the number of computations involved in FHWT( ) is far less than the number of computations required in the fast Fourier transform.