Topic
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.
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TL;DR: A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied, and a general proof of convergence in the Lp norm of all these approximations is given.
Abstract: A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between the approximated function and its piecewise polynomial approximation after T is bisected.
The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the Lp norm of all these approximations.
Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of of the approximated function in case of C2 functions).
23 citations
TL;DR: The proposed range segmentation method does not require initial depth estimates, it allows the analysis of scenes containing multiple objects, and it provides a rich description of the 3-D structure of a scene.
Abstract: This paper describes a novel system for computing a three-dimensional (3-D) range segmentation of an arbitrary visible scene using focus information. The process of range segmentation is divided into three steps: an initial range classification, a surface merging process, and a 3-D multiresolution range segmentation. First, range classification is performed to obtain quantized range estimates. The range classification is performed by analyzing focus cues within a Bayesian estimation framework. A combined energy functional measures the degree of focus and the Gibbs distribution of the class field. The range classification provides an initial range segmentation. Second, a statistical merging process is performed to merge the initial surface segments. This gives a range segmentation at a coarse resolution. Third, 3-D multiresolution range segmentation (3-D MRS) is performed to refine the range segmentation into finer resolutions. The proposed range segmentation method does not require initial depth estimates, it allows the analysis of scenes containing multiple objects, and it provides a rich description of the 3-D structure of a scene.
23 citations
26 Sep 2006
TL;DR: The results reported in this paper show that wavelet-networks have better prediction properties than its similar back-propagation networks.
Abstract: This paper presents a wavelet neural-network for chaotic time series prediction. Wavelet-networks are inspired by both the feed-forward neural network and the theory underlying wavelet decompositions. Wavelet-networks are a class of neural network that take advantage of good localization properties of multiresolution analysis and combine them with the approximation abilities of neural networks. This kind of networks uses wavelets as activation functions in the hidden layer and a type of backpropagation algorithm is used for its learning. Comparisons are made between a wavelet-network and the typical feedforward network trained with the back-propagation algorithm. The results reported in this paper show that wavelet-networks have better prediction properties than its similar back-propagation networks
23 citations
TL;DR: In this article, the authors introduced and studied multiresolution analysis based on the up function, which is an infinitely differentiable function supported on [0, 2] and showed that the approximation orders associated with the corresponding spaces are spectral, thus making the spaces attractive for the approximation of very smooth functions.
23 citations
17 May 2004
TL;DR: This paper uses a new type of non-separable 3D wavelet transform for video denoising and overcome the motion-mixture problem by using oriented complex wavelets.
Abstract: Video processing techniques using true 3D transforms are largely unexploited, partly because of the drawbacks of traditional separable 3D transforms. In this paper, we use a new type of non-separable 3D wavelet transform for video denoising and overcome the motion-mixture problem by using oriented complex wavelets. This wavelet transform is a 3D version of Kingsbury's 1D and 2D dual-tree wavelet transforms. We also investigate video denoising techniques using a combination of both 2D and 3D oriented wavelet transforms. The results are compared with those obtained by separable wavelet transforms.
23 citations