Multiscale transforms : wavelets, ridgelets, curvelets, etc.

1. 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.

2. 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)

3. 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!

4. coefficients basis, frame 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.

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

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

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

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

9. 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.

10. Continuous Wavelet Transform • Example : The mexican hat wavelet

11. 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.

12. 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.

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

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

15. Example application of Ridgelets

16. SNR = 0.1

18. Undecimated Wavelet Filtering (3 sigma)

19. Ridgelet Filtering (5sigma)

20. 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

21. 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.

22. 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

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

24. 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

25. 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.

26. NGC2997

27. A trous algorithm:

28. PARTITIONING

29. CONTRAST ENHANCEMENT if if if if Modified curvelet coefficient Curvelet coefficient

30. Contrast Enhancement

31. F

34. Multiscale Transforms Critical SamplingRedundant 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