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 :
coefficients basis, frame
• 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.
Seeking sparse and generic representations
• Sparsity
few big
many small
sorted index
• Why do we need sparsity?
– data compression
– Feature extraction, detection
– Image restoration
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
R f a,b,   a,b, x f xdx
Ridgelet Transform (Candes, 1998):
x1 cos( )  x 2 sin(  )  b 

Ridgelet function:  a,b, x   a 


a
 lines.Transverse to these ridges, it is a wavelet.
The function is constant along
1
2

The ridgelet coefficients of an object f are given by analysis
of the Radon transform via:

t b
R f (a,b, )   Rf (,t) (
)dt
a
Example application of Ridgelets
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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
Image
Ridgelet
transform
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:
f
–
F
fm
2
m
1
2
, m Œƒ
In a wavelet expansion, we have
–
f
f
W
m
2
1
m
, m Œƒ
In a curvelet expansion
(Donoho and Candes, 2000), we have
f
–
C
f m
2
log m
3
m
2
, m Œƒ
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:

I(k,l)  c J ,k,l  
J
j1
w j,k,l
PARTITIONING
CONTRAST ENHANCEMENT
I˜  CR y c CT I



Modified
curvelet
coefficient

yc (x, ) 1
if
x  c
x  c  m  2c  x
y c (x, ) 
  
c c 
c
p
m p
y c (x, )   
 x 
m s
y c (x, )   
 x 


Curvelet coefficient
i
f
i
f
if
x  2c
2c  x  m

xm
Contrast Enhancement
F
Multiscale Transforms
Critical Sampling
Redundant Transforms
(bi-) Orthogonal WT
Lifting scheme construction
Wavelet Packets
Mirror Basis
Pyramidal decomposition (Burt and Adelson)
Undecimated Wavelet Transform
Isotropic Undecimated Wavelet Transform
Complex Wavelet Transform
Steerable Wavelet Transform
Dyadic Wavelet Transform
Nonlinear Pyramidal decomposition (Median)
New Multiscale Construction
Contourlet
Bandelet
Finite Ridgelet Transform
Platelet
(W-)Edgelet
Adaptive Wavelet
Ridgelet
Curvelet (Several implementations)