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. 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

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

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:
The function is constant along lines.
Transverse to these ridges, it is a wavelet.

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, , 2003.

26. NGC2997

27. A trous algorithm:

28. PARTITIONING

29. CONTRAST ENHANCEMENT if if if if Modified curvelet coefficient

30. Contrast Enhancement

31. F

34. 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