Multiresolution representations are effective for analyzing the information content of images. The properties operator which approximates a signal at given resolution were studied. It is shown that difference between approximation resolutions 2/sup j+1/ and j/ (where j an integer) can be extracted by decomposing this on wavelet orthonormal basis L/sup 2/(R/sup n/), vector space measurable, square-integrable n-dimensional functions. In 2/(R), family functions built dilating translating unique...

The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms are selected from redundant dictionary functions. These chosen in order to best match the structures. Matching pursuits general procedures compute adaptive representations. With Gabor functions pursuit defines time-frequency transform. They derive energy distribution plane, which does not include interference terms, unlike Wigner and Cohen class distributions. A...

The mathematical characterization of singularities with Lipschitz exponents is reviewed. Theorems that estimate local functions from the evolution across scales their wavelet transform are It then proven maxima modulus detect locations irregular structures and provide numerical procedures to compute exponents. fast oscillations has a particular behavior studied separately. frequency such measured maxima. been shown numerically one- two-dimensional signals can be reconstructed, good...

A multiscale Canny edge detection is equivalent to finding the local maxima of a wavelet transform. The authors study properties edges through theory. For pattern recognition, one often needs discriminate different types edges. They show that evolution across scales characterize shape irregular structures. Numerical descriptors are derived. completeness representation also studied. describe an algorithm reconstructs close approximation 1-D and 2-D signals from their images, reconstruction...

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The author reviews recent multichannel models developed in psychophysiology, computer vision, and image processing. In have been particularly successful explaining some low-level processing the visual cortex. expansion of a function into several frequency channels provides representation which is intermediate between spatial Fourier representation. describes mathematical properties such decompositions introduces wavelet transform. He classical multiresolution pyramidal transforms vision...

A wavelet scattering network computes a translation invariant image representation which is stable to deformations and preserves high-frequency information for classification. It cascades transform convolutions with nonlinear modulus averaging operators. The first layer outputs SIFT-type descriptors, whereas the next layers provide complementary that improves mathematical analysis of networks explains important properties deep convolution stationary processes incorporates higher order...

Abstract This paper constructs translation‐invariant operators on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$ , which are Lipschitz‐continuous to the action of diffeomorphisms. A scattering propagator is a path‐ordered product nonlinear and noncommuting operators, each computes modulus wavelet transform. local integration defines windowed transform, proved be C 2 As window size increases, it converges transform that translation invariant. Scattering coefficients...

This paper introduces a new class of bases, called bandelet which decompose the image along multiscale vectors that are elongated in direction geometric flow. flow indicates directions gray levels have regular variations. The decomposition basis is implemented with fast subband-filtering algorithm. Bandelet bases lead to optimal approximation rates for geometrically images. For compression and noise removal applications, optimized algorithms so resulting produces minimum distortion....

The completeness, stability, and application to pattern recognition of a multiscale representation based on zero-crossings is discussed. An alternative projection algorithm described that reconstructs signal from zero-crossing representation, which stabilized by keeping the value wavelet transform integral between each pair consecutive zero-crossings. reconstruction has fast convergence iteration requires O(N log/sup 2/ (N)) computation for N samples. define particularly well adapted solving...

A scattering transform defines a locally translation invariant representation which is stable to time-warping deformations. It extends MFCC representations by computing modulation spectrum coefficients of multiple orders, through cascades wavelet convolutions and modulus operators. Second-order characterize transient phenomena such as attacks amplitude modulation. frequency transposition obtained applying along log-frequency. State-the-of-art classification results are for musical genre...

A general framework for solving image inverse problems with piecewise linear estimations is introduced in this paper. The approach based on Gaussian mixture models, which are estimated via a maximum posteriori expectation-maximization algorithm. dual mathematical interpretation of the proposed structured sparse estimation described, shows that resulting estimate stabilizes when compared traditional problem techniques. We demonstrate that, number problems, including interpolation, zooming,...

An affine invariant representation is constructed with a cascade of invariants, which preserves information for classification. A joint translation and rotation image patches calculated scattering transform. It implemented deep convolution network, computes successive wavelet transforms modulus non-linearities. Invariants to scaling, shearing small deformations are linear operators in the domain. State-of-the-art classification results obtained over texture databases uncontrolled viewing conditions.

We introduce a class of inverse problem estimators computed by mixing adaptively family linear corresponding to different priors. Sparse weights are calculated over blocks coefficients in frame providing sparse signal representation. They minimize an l1 norm taking into account the regularity each block. Adaptive directional image interpolations wavelet with O(N logN) algorithm, state-of-the-art numerical results.

A rigid-motion scattering computes adaptive invariants along translations and rotations, with a deep convolutional network. Convolutions are calculated on the group, wavelets defined translation rotation variables. It preserves joint information, while providing global at any desired scale. Texture classification is studied, through characterization of stationary processes from single realization. State-of-the-art results obtained multiple texture data bases, important scaling variabilities.

Computing the optimal expansion of a signal in redundant dictionary waveforms is an NP-hard problem. We introduce greedy algorithm, called matching pursuit, which computes suboptimal expansion. The that best match signal's structures are chosen iteratively. An orthogonalized version pursuit also developed. Matching pursuits general procedures for computing adaptive representations. With Gabor functions, defines time-frequency transform. chaotic maps whose attractors define generic noise with...