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.

Steerable Pyramids and Tight Wavelet Frames in $L_{2}({\BBR}^{d})$

Abstract: We present a functional framework for the design of tight steerable wavelet frames in any number of dimensions. The 2-D version of the method can be viewed as a generalization of Simoncelli's steerable pyramid that gives access to a larger palette of steerable wavelets via a suitable parametrization. The backbone of our construction is a primal isotropic wavelet frame that provides the multiresolution decomposition of the signal. The steerable wavelets are obtained by applying a one-to-many mapping (N th-order generalized Riesz transform) to the primal ones. The shaping of the steerable wavelets is controlled by an M × M unitary matrix (where M is the number of wavelet channels) that can be selected arbitrarily; this allows for a much wider range of solutions than the traditional equiangular configuration (steerable pyramid). We give a complete functional description of these generalized wavelet transforms and derive their steering equations. We describe some concrete examples of transforms, including some built around a Mallat-type multiresolution analysis of L2(Rd), and provide a fast Fourier transform-based decomposition algorithm. We also propose a principal-component-based method for signal adapted wavelet design. Finally, we present some illustrative examples together with a comparison of the denoising performance of various brands of steerable transforms. The results are in favor of an optimized wavelet design (equalized principal component analysis), which consistently performs best.

Combined line-based architecture for the 5-3 and 9-7 wavelet transform of JPEG2000

Abstract: The wavelet transform is a very promising tool for image compression. In JPEG2000, two filter banks are used, one an integer lossless 5-3 filter, and one a lossy 9-7. A combined architecture for the 5-3 and 9-7 transforms with minimum area is presented. The lifting scheme is used to realize a fast wavelet transform. Two lines are processed at a time. This line-based architecture allows minimum memory requirement and fast calculation. The pipeline and the optimization of the operations provide speed, while the combination of the two transforms in one structure contributes to saving the area. On a VIRTEXE1000-8 FPGA implementation, decoding of 2 pixels per clock cycle can be performed at 110 MHz. Only 19% of the total area of the VIRTEXE1000 is needed. Compared to existing architectures, memory resource and area can be reduced thanks to the proposed solution.

Spatial Methods for Multispectral Pansharpening: Multiresolution Analysis Demystified

Abstract: The majority of multispectral (MS) pansharpening methods may be labeled as spectral or spatial, depending on whether the geometric details that shall be injected into the interpolated MS bands are extracted from the panchromatic (P) image by means of a spectral transformation of MS pixels or a spatial transformation of the P image, achieved by means of linear shift-invariant digital filters. Spectral methods are known as component substitution; spatial methods are based on multiresolution analysis (MRA). In this paper, the authors show that, under the most general conditions, MRA-based pansharpening is characterized by a unique separable low-pass filter, which can be parametrically optimized based on the modulation transfer function (MTF) of the MS instrument, possibly followed by decimation and interpolation stages. This happens for the discrete wavelet transform (DWT) and its undecimated version (UDWT), for the “a-trous” wavelet (ATW) transform and its decimated version, i.e., the generalized Laplacian pyramid (GLP), and for nonseparable wavelet transforms, such as the nonsubsampled contourlet transform (NSCT). Hybrid methods, in which MRA fusion is performed on the intensity component derived from a spectral transformation, are equivalent to MRA fusion with a specific detail injection model. ATW and GLP are preferable to DWT, UDWT, and NSCT, because of computational benefits and of a looser choice of the low-pass filter, unconstrained from the requirement of generating a perfect reconstruction filter bank. Ultimately, GLP outperforms ATW, because its decimation and interpolation stages allow the aliasing impairments intrinsically present in the original MS bands to be removed from the pansharpened product.

Wavelet Leaders: A new method to estimate the multifractal singularity spectra

Abstract: Wavelet Leaders is a novel alternative based on wavelet analysis for estimating the Multifractal Spectrum It was proposed by Jaffard and co-workers improving the usual wavelet methods In this work, we analyze and compare it with the well known Multifractal Detrended Fluctuation Analysis The latter is a comprehensible and well adapted method for natural and weakly stationary signals Alternatively, Wavelet Leaders exploits the wavelet self-similarity structures combined with the Multiresolution Analysis scheme We give a brief introduction on the multifractal formalism and the particular implementation of the above methods and we compare their effectiveness We expose several cases: Cantor measures, Binomial Multiplicative Cascades and also natural series from a tonic–clonic epileptic seizure We analyze the results and extract the conclusions