Multiscale transforms : wavelets, ridgelets, curvelets, etc. Outline : The Fourier transform Time-frequency analysis and the Heisenberg principle Cauchy Schwartz inequality The continuous wavelet transform 2D wavelet transform Anisotropic frames : Ridgelets, curvelets, etc.

The Fourier transform (1) Diagonal representation of shift invariant linear transforms. Truncated Fourier series give very good approximations to smooth functions. Limitations : Provides poor representation of non stationary signals or image. Provides poor representations of discontinuous objects (Gibbs effect)

The Fourier transform (2) A Fourier transform is a change of basis. Each dot product assesses the coherence between the signal and the basis element. Cauchy-Schwartz : The Fourier basis is best for representing harmonic components of a signal!

What is good representation for data? Computational harmonic analysis seeks representations of s signal as linear combinations of basis, frame, dictionary, element : Analyze the signal through the statistical properties of the coefficients The analyzing functions (frame elements) should extract features of interest. Approximation theory wants to exploit the sparsity of the coefficients. basis, frame coefficients

Seeking sparse and generic representations Sparsity Why do we need sparsity? data compression Feature extraction, detection Image restoration sorted index few big many small

Candidate analyzing functions for piecewise smooth signals Windowed fourier transform or Gaborlets : Wavelets :

Heisenberg uncertainty principle Localization in time and frequency requires a compromise Different tilings in time frequency space :

Windowed/Short term Fourier transform Decomposition : ( with a gaussian window w, this is the Gabor transform) Invertibility condition : Reconstruction : with

The Continuous Wavelet Transform decomposition reconstruction admissible wavelet : simpler condition : zero mean wavelet The CWT is a linear transform. It is covariant under translation and scaling. Verifies a Plancherel-Parceval type equation.

Continuous Wavelet Transform Example : The mexican hat wavelet

2D Continuous Wavelet transform either a genuine 2D wavelet function (e.g. mexican hat) or a separable wavelet i.e. tensor product of two 1D wavelets. example : Images obtained using the nearly isotropic undecimated wavelet transform obtained with the a trous algorithm.

Wavelets and edges many wavelet coefficients are needed to account for edges ie singularities along lines or curves : need dictionaries of strongly anisotropic atoms : ridgelets, curvelets, contourlets, bandelettes, etc.

Continuous Ridgelet Transform Ridgelet Transform (Candes, 1998): Ridgelet function: The function is constant along lines. Transverse to these ridges, it is a wavelet.

The ridgelet coefficients of an object f are given by analysis of the Radon transform via:

Example application of Ridgelets

SNR = 0.1

Undecimated Wavelet Filtering (3 sigma)

Ridgelet Filtering (5sigma)

Local Ridgelet Transform The ridgelet transform is optimal to find only lines of the size of the image. To detect line segments, a partitioning must be introduced. The image is decomposed into blocks, and the ridgelet transform is applied on each block. Partitioning Ridgelet transform Image

In practice, we use overlap to avoid blocking artifacts. Smooth partitioning Image Ridgelet transform The partitioning introduces a redundancy, as a pixel belongs to 4 neighboring blocks.

Edge Representation Suppose we have a function f which has a discontinuity across a curve, and which is otherwise smooth, and consider approximating f from the best m-terms in the Fourier expansion. The squarred error of such an m-term expansion obeys: In a wavelet expansion, we have In a curvelet expansion (Donoho and Candes, 2000), we have Width = Length^2

Numerical Curvelet Transform The Curvelet Transform for Image Denoising, IEEE Transaction on Image Processing, 11, 6, 2002.

The Curvelet Transform The curvelet transform opens us the possibility to analyse an image with different block sizes, but with a single transform. The idea is to first decompose the image into a set of wavelet bands, and to analyze each band by a ridgelet transform. The block size can be changed at each scale level. à trous wavelet transform Partitionning ridgelet transform . Radon Transform . 1D Wavelet transform

The Curvelet Transform J.L. Starck, E. Candès and D. Donoho, "Astronomical Image Representation by the Curvelet Transform, Astronomy and Astrophysics, 398, 785--800, 2003.

NGC2997

A trous algorithm:

PARTITIONING

CONTRAST ENHANCEMENT if if if if Modified curvelet coefficient

Contrast Enhancement

F

Multiscale Transforms Critical Sampling Redundant Transforms Pyramidal decomposition (Burt and Adelson) (bi-) Orthogonal WT Undecimated Wavelet Transform Lifting scheme construction Isotropic Undecimated Wavelet Transform Wavelet Packets Complex Wavelet Transform Mirror Basis Steerable Wavelet Transform Dyadic Wavelet Transform Nonlinear Pyramidal decomposition (Median) New Multiscale Construction Contourlet Ridgelet Bandelet Curvelet (Several implementations) Finite Ridgelet Transform Platelet (W-)Edgelet Adaptive Wavelet