Wavelets and digital image processing

The Wavelet transform (WT) is a mathematical tool for analyzing signals. Originally introduced in a group-theoretic setting, the WT was soon realized to have powerful applications in various elds, which include analysing acustic signals 36] or the construction of time-frequency localizing operators in quantum physics 20]. The purpose of this paper is to outline the present state of the art and possible future developements of applications of the WT in image processing. Basically the WT decomposes a given function f into its components on diierent scales or frequency bands. This is done by convolving f with the translated and dilated wavelet : L f(a; b) = 1 p a Z f(t) t ? b a dt : Depending on the choice of the tranformed function allows e.g. to extract the discontinuities or edges of f, to perform a pattern recognition task or to store eeciently a compressed version of f. More over discretizing the WT leads to very fast algorithms, which are needed for real-time applications with large data sets as in image processing.

[1]  Joachim M. Buhmann,et al.  Distortion Invariant Object Recognition in the Dynamic Link Architecture , 1993, IEEE Trans. Computers.

[2]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[3]  Edward H. Adelson,et al.  The Laplacian Pyramid as a Compact Image Code , 1983, IEEE Trans. Commun..

[4]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[5]  Arun N. Netravali,et al.  Digital Pictures: Representation and Compression , 1988 .

[6]  A. Grossmann,et al.  DECOMPOSITION OF HARDY FUNCTIONS INTO SQUARE INTEGRABLE WAVELETS OF CONSTANT SHAPE , 1984 .

[7]  Mary Beth Ruskai,et al.  Wavelets and their Applications , 1992 .

[8]  Dennis Gabor,et al.  Theory of communication , 1946 .

[9]  I. Daubechies,et al.  Time-frequency localisation operators-a geometric phase space approach: II. The use of dilations , 1988 .

[10]  Jean-Pierre Antoine,et al.  Image analysis with 2D wavelet transform: Detection of position, orientation and visual contrast of simple objects , 1991 .

[11]  Alfred K. Louis,et al.  Medical imaging: state of the art and future development , 1992 .

[12]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  Ronald R. Coifman,et al.  Wavelet analysis and signal processing , 1990 .

[14]  T. Eirola.  Sobolev characterization of solutions of dilation equations , 1992 .

[15]  A. Grossmann,et al.  Transforms associated to square integrable group representations. I. General results , 1985 .

[16]  J. Klauder,et al.  Unitary Representations of the Affine Group , 1968 .

[17]  Rieder Andreas.  The high frequency behaviour of continuous wavelet transforms , 1994 .

[18]  Y. Meyer.  Ondelettes sur l'intervalle. , 1991 .

[19]  J. Morel,et al.  Segmentation of images by variational methods: a constructive approach. , 1988 .

[20]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[21]  A. Calderón.  Intermediate spaces and interpolation, the complex method , 1964 .

[22]  W. Lawton.  Tight frames of compactly supported affine wavelets , 1990 .

[23]  Karlheinz Gröchenig,et al.  Multiresolution analysis, Haar bases, and self-similar tilings of Rn , 1992, IEEE Trans. Inf. Theory.

[24]  Hans G. Feichtinger,et al.  Coherent Frames and Irregular Sampling , 1990 .

[25]  C. Chui.  Wavelets: A Tutorial in Theory and Applications , 1992 .

[26]  L. Villemoes.  Energy moments in time and frequency for two-scale difference equation solutions and wavelets , 1992 .

[27]  J. Benedetto.  Irregular sampling and frames , 1993 .

[28]  Gregory K. Wallace,et al.  The JPEG still picture compression standard , 1992 .

[29]  Georges Koepfler.  Formalisation et analyse numérique de la segmentation d'images , 1991 .

[30]  P. Tchamitchian,et al.  Bases d'ondelettes sur les courbes corde-arc, Noyau de Cauchy et Espaces de Hardy Associés. , 1989 .

[31]  O. Rioul,et al.  Wavelets and signal processing , 1991, IEEE Signal Processing Magazine.

[32]  David A. Huffman,et al.  A method for the construction of minimum-redundancy codes , 1952, Proceedings of the IRE.

[33]  G. Battle.  A block spin construction of ondelettes. Part I: Lemarié functions , 1987 .

[34]  P Perona,et al.  Preattentive texture discrimination with early vision mechanisms. , 1990, Journal of the Optical Society of America. A, Optics and image science.

[35]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[36]  P. P. Vaidyanathan,et al.  Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M, having the perfect-reconstruction property , 1987, IEEE Trans. Acoust. Speech Signal Process..

[37]  Hans-Georg Stark.  Multiscale Analysis, Wavelets and Texture Quality , 1990 .

[38]  Robert Hummel,et al.  Reconstructions from zero crossings in scale space , 1989, IEEE Trans. Acoust. Speech Signal Process..

[39]  Mark J. T. Smith,et al.  Exact reconstruction techniques for tree-structured subband coders , 1986, IEEE Trans. Acoust. Speech Signal Process..

[40]  M.,et al.  Statistical and Structural Approaches to Texture , 2022 .

[41]  P. Lions,et al.  Image selective smoothing and edge detection by nonlinear diffusion. II , 1992 .

[42]  H. Stark.  Continuous wavelet transform and continuous multiscale analysis , 1992 .

[43]  Gilbert Strang,et al.  Wavelets and Dilation Equations: A Brief Introduction , 1989, SIAM Rev..

[44]  Azriel Rosenfeld,et al.  Multiresolution image processing and analysis , 1984 .

[45]  Andreas Rieder.  The wavelet transform on Sobolev spaces and its approximation properties , 1990 .

[46]  Charles K. Chui,et al.  An Introduction to Wavelets , 1992 .

[47]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[48]  Bruce A. Draper,et al.  ISR: a database for symbolic processing in computer vision , 1989, Computer.

[49]  R. Murenzi.  Wavelet Transforms Associated to the n-Dimensional Euclidean Group with Dilations: Signal in More Than One Dimension , 1990 .

[50]  S. Mallat,et al.  Image coding from the wavelet transform extrema , 1989, Sixth Multidimensional Signal Processing Workshop,.

[51]  A. Grossmann,et al.  TRANSFORMS ASSOCIATED TO SQUARE INTEGRABLE GROUP REPRESENTATION. 2. EXAMPLES , 1986 .

[52]  I. Weinreich,et al.  Wavelet-Galerkin methods: An adapted biorthogonal wavelet basis , 1993 .

[53]  A. Grossmann,et al.  Cycle-octave and related transforms in seismic signal analysis , 1984 .

[54]  Christopher Heil,et al.  Continuous and Discrete Wavelet Transforms , 1989, SIAM Rev..

[55]  Azriel Rosenfeld,et al.  Digital Picture Processing , 1976 .

[56]  C. R. Smith,et al.  Maximum-Entropy and Bayesian Methods in Inverse Problems , 1985 .

[57]  Romain Murenzi,et al.  Isotropic and anisotropic multidimensional wavelets: Applications to the analysis of two-dimensional fields , 1991 .

[58]  J.-C. Feauveau.  A new approach for subband processing , 1990, International Conference on Acoustics, Speech, and Signal Processing.

[59]  D. Esteban,et al.  Application of quadrature mirror filters to split band voice coding schemes , 1977 .

[60]  Richard Kronland-Martinet,et al.  Analysis of Sound Patterns through Wavelet transforms , 1987, Int. J. Pattern Recognit. Artif. Intell..

[61]  Jelena Kovacevic,et al.  Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for Rn , 1992, IEEE Trans. Inf. Theory.

[62]  Paul Wintz,et al.  Instructor's manual for digital image processing , 1987 .

[63]  William R. Zettler,et al.  Application of compactly supported wavelets to image compression , 1990, Other Conferences.