The purpose is to study qualitative and quantitative rates of image compression by using different Haar wavelet banks. The experimental results of adaptive compression are provided. The paper deals with specific examples of orthogonal Haar bases generated by multiresolution analysis. Bases consist of three piecewise constant wavelet functions with a support $[0,1] \times [0,1] $.

We compute the Riemannian volume of the moduli space of flat connections on a nonorientable 2-manifold, for a natural class of metrics. We also show that Witten’s volume formula for these moduli spaces may be derived using Haar measure, and we give a new proof of Witten’s volume formula for the moduli space of flat connections on a Riemann surface using Haar measure. ———————–

Recently, we have been developing general multireso-lution signal decomposition schemes that unify traditional (linear) approaches and allow use of nonlinear ltering techniques in the decomposition. This paper summarizes our approach and provides several simple examples. Some of these examples are known in the literature (e.g., Haar pyramid and wavelet) and some of them are new (e.g., morpholog...

Noiselets are a family of functions completely uncompressible using Haar wavelet analysis. The resultant perfect incoherence to the Haar transform, coupled with the existence of a fast transform has resulted in their interest and use as a sampling basis in compressive sampling. We derive a recursive construction of noiselet matrices and give a short matrix-based proof of the incoherence.

In order to get an efficient image representation we introduce a new adaptive Haar wavelet transform, calledTetrolet Transform. Tetrolets are Haar-type wavelets whose supports are tetrominoes which are shapes made by connecting four equal-sized squares. The corresponding filter bank algorithm is simple but enormously effective. Numerical results show the strong efficiency of the tetrolet transf...

A modification of the Haar wavelet method, for which the stepsize of the argument is variable, is proposed. To establish the efficiency of the method three test problems, for which exact solution is known, are considered. Computer simulations show clear preference of the suggested method compared with the Haar wavelet method of a constant stepsize. 2007 Elsevier Inc. All rights reserved.

Modified sign Haar transform with sign Walsh-like structure is introduced in this article. This nonlinear transform converts binary/ternary vectors into digital spectral domain and is invertible. Recursive definitions for the calculation of this transform have been developed. The properties of logic functions and variables in the spectral domain of the modified sign Haar transform are presented.

Based on hierarchical partitions, we provide the construction of Haar-type tight framelets any compact set $$K\subseteq \mathbb {R}^d$$ . In particular, unit block $$[0,1]^d$$ , such can be built to with adaptivity and directionality. We show that adaptive directional Haar framelet systems used for digraph signal representations. Some examples are provided illustrate results in this paper.