Multiscale transforms : wavelets, ridgelets, curvelets, etc.

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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.

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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)

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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!

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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.

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Seeking sparse and generic representations

• Sparsity
• Why do we need sparsity?
• data compression
• Feature extraction, detection
• Image restoration

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Candidate analyzing functions for piecewise
smooth signals

• Windowed fourier transform or Gaborlets

• Wavelets

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Heisenberg uncertainty principle

• Localization in time and frequency requires a
  compromise

• Different tilings in time frequency space

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Windowed/Short term Fourier transform

• Decomposition

( with a gaussian window w, this is the Gabor
transform)

• Invertibility condition
• Reconstruction

with
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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.
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Continuous Wavelet Transform

• Example

The mexican hat wavelet
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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.
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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.
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Continuous Ridgelet Transform
Ridgelet Transform (Candes, 1998)
Ridgelet function
Transverse to these ridges, it is a wavelet.
The function is constant along lines.
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The ridgelet coefficients of an object f are
given by analysis of the Radon transform via
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Example application of Ridgelets
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SNR 0.1
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Undecimated Wavelet Filtering (3 sigma)
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Ridgelet Filtering (5sigma)
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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
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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.
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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 Length2
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Numerical Curvelet Transform
The Curvelet Transform for Image Denoising,
IEEE Transaction on Image Processing, 11, 6,
2002.
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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

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The Curvelet Transform
J.L. Starck, E. Candès and D. Donoho, "Astronomica
l Image Representation by the Curvelet
Transform, Astronomy and Astrophysics, 398,
785--800, 2003.
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NGC2997
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A trous algorithm
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PARTITIONING
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CONTRAST ENHANCEMENT
if
if
if
if
Modified curvelet coefficient
Curvelet coefficient
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Contrast Enhancement
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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