Candes, E. J., On the representation of mutilated Sobolev functions, Tech. Report, Department of Statistics, Stanford University, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is supported in a band of angular resolution O(2 s ) Hence it also takes order 2 s coe#cients to represent a single ridge fragment. Only the ridgelet basis has the required angular localization to mimic the ridge fragment signatures. For rigorous analysis, see the references, for example, [5]. 3 Implementation Strategy We now describe a strategy for realizing a digital implementation of the curvelet transform. 8 3.1 Specific Assumptions Our strategy is based on a series of assumptions. Image Size 256 256. We have preferred to deal with images of size n by n, where n = 2 8 ....

Candes, E. (1999) On the Representation of Mutilated Sobolev Functions. Technical Report, Statistics, Stanford.

....### is a complete orthonormal system for L 2 (IR 2 ) Hence, we have a new decomposition of the form f = X # #f, # # ## # . Ridgelets turn out to be optimal for representing functions with linear singularities. Indeed, let us consider the template (2) The following theorem is proved in [4]. 4 Emmanuel J. Candes Theorem 1. Let g # W s 2 (IR 2 ) and f(x 1 , x 2 ) H(x 1 cos # 0 x 2 sin # 0 t 0 ) g(x 1 , x 2 ) Then the sequence (# # = #f, # # #) of orthonormal ridgelet coe# cients of f satisfies # # # # 1 n # C n 1 (s 1) #g#W s 2 for some constant C ....

Candes, E. J., On the representation of mutilated Sobolev functions, Tech. Report, Department of Statistics, Stanford University, 1999.

....for the Representation of Images with Edges Emmanuel J. Cand es Department of Statistics Stanford University Stanford, California 94305 4065 In a previous paper [1] the author introduced a new system for representing multivariate functions, namely, the ridgelets. In a following article [3], ridgelets were shown to be optimal for representing functions that are smooth away from hyperplanes, e.g. in two dimensions ridgelets provide optimally sparse representations of smooth functions that are discontinuous along lines, i.e. straight edges. This is unlike Fourier or wavelet methods. ....

....compact notation ( 2 A) and, therefore, we will keep in mind that the index runs through an enumeration of the triples (j; k) It will then be handy to use the notation A j to refer to the subset of indices for which the scale is equal to j. 1. 3 Ridgelets and singularities Previous work [3] showed that 2 dimensional ridgelets were optimal for representing functions that are smooth away from lines (and in higher dimensions, for representing functions that are smooth away from hyperplanes) For instance, suppose that g is a C r function that may be discontinuous along an arbitrary ....

E. J. Candes, On the Representation of Mutilated Sobolev Functions. Stanford Technical Report, Department of Statistics, Stanford University, 1999. Submitted for publication.

.... lines at all locations and orientations and ranging though a variety of scales (localization widths) It has been shown that for these schemes, simple thresholding of the discrete ridgelet transform provides nearoptimal N term approximations to smooth functions with discontinuities along lines [8, 5, 18]. In short, discrete ridgelet representations solve the problem of sparse approximation to smooth objects with straight edges. 3 In image processing, edges are typically curved rather than straight and ridgelets alone cannot yield ecient representations. However at suciently ne scales, a curved ....

E. J. Candes. On the representation of mutilated Sobolev functions. Technical report, Department of Statistics, Stanford University, 1999. Submitted for publication.

....such Fourier transforms by using wavelets in the angular direction, so that the Fourier ridge is captured neatly by one or a few wavelets. In the radial direction, the Fourier ridge is actually oscillatory, and this is captured by local cosines. A precise quantitative treatment is given in [4]. Multiscale Ridgelets Think of ortho ridgelets as objects which have a length of about 1 and a width which can be arbitrarily fine. The multiscale ridgelet system renormalizes and transports such objects, so that one has a system of elements at all lengths and all finer widths. In a light ....

Candes, E. J., On the representation of mutilated Sobolev functions, Technical Report, Statistics, Stanford, 1999.

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