Download Beyond Wavelets by G. V. Welland PDF

By G. V. Welland

"Beyond Wavelets" provides cutting-edge theories, tools, algorithms, and purposes of mathematical extensions for classical wavelet research. Wavelets, brought twenty years in the past byMorlet and Grossmann and constructed very speedily through the 1980's and 1990's, has created a typical hyperlink among computational arithmetic and different disciplines of technological know-how and engineering.Classical wavelets have supplied potent and effective mathematical instruments for time-frequency research which boosts and replaces the Fourier approach.However, with the present advances in technology and know-how, there's an instantaneous have to expand wavelet mathematical instruments besides. "Beyond Wavelets" provides an inventory of principles and mathematicalfoundations for such extensions, together with: non-stop and electronic ridgelets, brushlets, steerable wavelet packets, contourlets, eno-wavelets, spline-wavelet frames, and quasi-affine wavelets. Wavelet subband algorithms are prolonged to pyramidal directional and nonuniform clear out banks. additionally, this quantity contains amethod for tomographic reconstruction utilizing a mechanical photo version and a statistical learn for self reliant adaptive sign representation.Investigators already conversant in wavelet equipment from parts comparable to engineering, data, and arithmetic will profit via possessing this quantity. *Curvelets, Contourlets, Ridgelets, *Digital Implementation of Ridgelet Packets*Steerable Wavelet Packets*Essentially Non-Oscillatory Wavelets*Medical Imaging*Non-Uniform filter out Banks*Spline-wavelet frames and *Vanishing second restoration services

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Edu A b s t r a ct The Ridgelet Packets library provides a large family of orthonormal bases for functions f{x,y) in L^{dxdy) which includes orthonormal ridgelets as well as bases deriving from tilings reminiscent from the theory of wavelets and the study of oscillatory Fourier integrals. An intuitively appealing fea› ture: many of these bases have elements whose envelope is strongly aligned along specified ’ridges’ while displaying oscillatory components across the main ’ridge’. There are two approaches to constructing ridgelet packets; the most direct is a frequency-domain viewpoint.

7). Then define the pseudopolar Fourier transform as the evaluation of the Fourier transform n-l H0= Yl ^(^i’^2)exp{-(xi6+X2C2)} a:i ,X2=0 at the An? points of this pointset. The pointset can be viewed as a set of "concentric squares" stacked inside each other (like Chinese boxes), with equispaced points along the boundary of the box. The half-width of a side functions as a pseudo radius, and the arclength along the perimeter of the box functions as a pseudo angular variable. As shown in [3] , the evaluation of the Fourier sum on this set of gridpoints can be performed in order A^ log(A^) flops, where N = n?

In the overcomplete case, what we can hope for is that the nonzero elements in each row, once rearranged in order of decreasing amplitude, decay rapidly. Owing to the obviously high inner products of true ridge functions oriented in neighboring directions, we should not expect rapid decay of the coefficients if we work in the ridgelet system, but should look instead to the ortho-ridgelet system. To check the sparsity condition, we consider the following computational exper› iment: we synthesize a single ortho-ridgelet and then analyze that ridgelet, inspect› ing the resulting transform coefficients for rapid decay.

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