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!

• 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.

What is good representation for data?

basis, framecoefficients

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

• Different tilings in time frequency space :

• Localization in time and frequency requires a compromise

Windowed/Short term Fourier transform

• Invertibility condition :

• Reconstruction :

• Decomposition :

with

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

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 function:

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

Ridgelet Transform (Candes, 1998):

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R f a,b,θ( ) = ψ a,b,θ∫ x( ) f x( )dx

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ψa,b,θ x( ) = a1

2ψx1 cos(θ) + x2 sin(θ) − b

a

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The ridgelet coefficients of an object f are given by analysis

of the Radon transform via:

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R f (a,b,θ) = Rf (θ, t)ψ (t − b

a∫ )dt

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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 isdecomposed into blocks, and the ridgelet transform is applied on each block.

Image

Partitioning

Ridgelet transform

In practice, we use overlap to avoid blocking artifacts.

The partitioning introduces a redundancy, as a pixel belongs to 4 neighboringblocks.

Smooth partitioning

Image

Ridgelettransform

Edge Representation

Suppose we have a function f which has a discontinuity across a curve, andwhich is otherwise smooth, and consider approximating f from the best m-terms in the Fourier expansion. The squarred error of such an m-termexpansion obeys:

f–f m

F 2 m12 ,mŒƒ

f–f m

W 2 m 1 ,mŒƒ

In a wavelet expansion, we have

f–f m

C 2 log m 3m 2 ,mŒƒ

In a curvelet expansion (Donoho and Candes, 2000), we have

Width = Length^2

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

Numerical Curvelet Transform

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, andto analyze each band by a ridgelet transform. The block size can be changedat 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:

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I(k, l) = cJ ,k,l + w j,k,lj=1

J

∑

PARTITIONING

CONTRAST ENHANCEMENT

Curvelet coefficient

Modifiedcurvelet coefficient

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˜ I = CR yc CT I( )( )

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yc (x,σ ) =x − cσ

cσ

m

cσ

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+2cσ − x

cσ€

yc (x,σ ) =1

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yc (x,σ ) =m

x

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yc (x,σ ) =m

x

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if

if

if

if

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x < cσ

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x < 2cσ

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2cσ ≤ x < m

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x > m

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Contrast Enhancement

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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)

Multiscale Transforms

New Multiscale Construction

Contourlet RidgeletBandelet Curvelet (Several implementations)Finite Ridgelet TransformPlatelet(W-)Edgelet Adaptive Wavelet