Multiresolution analysis

About: Multiresolution analysis is a research topic. Over the lifetime, 4032 publications have been published within this topic receiving 140743 citations. The topic is also known as: Multiresolution analysis, MRA.

A wavelet Galerkin method for the stokes equations

Abstract: The purpose of this paper is to investigate Galerkin schemes for the Stokes equations based on a suitably adapted multiresolution analysis. In particular, it will be shown that techniques developed in connection with shift-invariant refinable spaces give rise to trial spaces of any desired degree of accuracy satisfying the Ladysenskaja-Babuska-Brezzi condition for any spatial dimension. Moreover, in the time dependent case efficient preconditioners for the Schur complements of the discrete systems of equations can be based on corresponding stable multiscale decompositions. The results are illustrated by some concrete examples of adapted wavelets and corresponding numerical experiments.

The Shiftable Complex Directional Pyramid—Part I: Theoretical Aspects

Abstract: This paper presents an over-complete multiscale decomposition by combining the Laplacian pyramid and the complex directional filter bank (DFB). The filter bank is constructed in such a way that each complex directional filter is analytical using the dual-tree structure of real fan filters. Necessary and sufficient conditions in order for the resulting multirate filter bank to be shift-invariant in energy sense (shiftability) are derived in terms of the magnitude and phase responses of these filters. Their connection to 2D Hilbert transform relationship is established. The proposed transform possesses several desirable properties including multiresolution, arbitrarily high directional resolution, low redundant ratio, and efficient implementation.

Adaptive image denoising using scale and space consistency

Abstract: This paper proposes a new method for image denoising with edge preservation, based on image multiresolution decomposition by a redundant wavelet transform. In our approach, edges are implicitly located and preserved in the wavelet domain, whilst image noise is filtered out. At each resolution level, the image edges are estimated by gradient magnitudes (obtained from the wavelet coefficients), which are modeled probabilistically, and a shrinkage function is assembled based on the model obtained. Joint use of space and scale consistency is applied for better preservation of edges. The shrinkage functions are combined to preserve edges that appear simultaneously at several resolutions, and geometric constraints are applied to preserve edges that are not isolated. The proposed technique produces a filtered version of the original image, where homogeneous regions appear separated by well-defined edges. Possible applications include image presegmentation, and image denoising.

Building nonredundant adaptive wavelets by update lifting

Abstract: textAdaptive wavelet decompositions appear useful in various applications in image and video processing, such as image analysis, compression, feature extraction, denoising and deconvolution, or optic flow estimation. For such tasks it may be important that the multiresolution representations take into account the characteristics of the underlying signal and do leave intact important signal characteristics such as sharp transitions, edges, singularities or other regions of interest. In this paper, we propose a technique for building adaptive wavelets by means of an extension of the lifting scheme. The classical lifting scheme provides a simple yet flexible method for building new, possibly nonlinear, wavelets from existing ones. It comprises a given wavelet transform, followed by a prediction and an update step. The update step in such a scheme computes a modification of the approximation signal, using information in the detail band. It is obvious that such an operation can be inverted, and therefore the perfect reconstruction property is guaranteed. In this paper we propose a lifting scheme including an adaptive update lifting and a fixed prediction lifting step. The adaptivity consists hereof that the system can choose between two different update filters, and that this choice is triggered by the local gradient of the original signal. If the gradient is large (in some seminorm sense) it chooses one filter, if it is small the other. In this paper we derive necessary and sufficient conditions for the invertibility of such an adaptive system for various scenarios.