# Introduction to the Curvelet Transform by moti

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Introduction to the
Curvelet Transform
By
Zvi Devir and Yanai Shpinner
Introduction
Curvelet Transform is a new multi-scale
representation most suitable for objects with
curves.
Developed by Candès and Donoho (1999).
Still not fully matured.
Seems promising, however.
Approximation Rates
Having an object in the domain [0,1][0,1],
how „fast‟ can we approximate it using certain
system of functions?

Using the Fourier Transform:
~
f  fm
2
2
 
O m    12

Using the Wavelet Transform:
~
f  fm
2
2
 
 O m1
Using the Curvelet Transform:
~
f  fm
2
2
            
 O m2 log 3 m  O m2
Point and Curve Discontinuities
A discontinuity point affects all the Fourier
coefficients in the domain.
   Hence the FT doesn‟t handle points discontinuities well.
Using wavelets, it affects only a limited number of
coefficients.
   Hence the WT handles point discontinuities well.
Discontinuities across a simple curve affect all the
wavelets coefficients on the curve.
   Hence the WT doesn‟t handle curves discontinuities well.
Curvelets are designed to handle curves using only a
small number of coefficients.
   Hence the CvT handles curve discontinuities well.
Curvelet Transform
The Curvelet Transform includes four stages:
Sub-band decomposition
Smooth partitioning
Renormalization
Ridgelet analysis
Sub-band Decomposition
f  P0 f , 1 f ,  2 f ,
Dividing the image into resolution layers.
Each layer contains details of different
frequencies:
   P0 – Low-pass filter.
   1, 2, … – Band-pass (high-pass) filters.
The original image can be reconstructed from
the sub-bands:
f  P0 P0 f     s  s f 
s
Energy preservation
f 2  P0 f 2    s f 2
2          2              2

s
Sub-band
Decomposition
f  P0 f , 1 f ,  2 f ,

f

P0 f             1 f        2 f
Sub-band Decomposition
Low-pass filter 0 deals with low frequencies
near ||1.
Band-pass filters 2s deals with frequencies
near domain ||[22s, 22s+2].
   Recursive construction – 2s(x) = 24s (22sx).
The sub-band decomposition is simply
applying a convolution operator:
P0 f   0  f    s f  2 s  f
Sub-band Decomposition
The sub-band decomposition can be
approximated using the well known wavelet
transform:
   Using wavelet transform, f is decomposed into S0,
D1, D2, D3, etc.
   P0 f is partially constructed from S0 and D1,
and may include also D2 and D3.
   s f is constructed from D2s and D2s+1.
Sub-band Decomposition
P0 f is “smooth” (low-pass), and can be
efficiently represented using wavelet base.
The discontinuity curves effect the high-pass
layers s f. Can they be represented
efficiently?
   Looking at a small fragment of the curve, it appears
as a relatively straight ridge.
   We will dissect the layer into small partitions.
Smooth Partitioning
Smooth Partitioning
A grid of dyadic squares is defined:
                                                  Q
k1       k1 1              k2         k 2 1
Q s ,k1 ,k2      2    s   ,       2   s
2   s   ,    2   s          s
Qs – all the dyadic squares of the grid.
Let w be a smooth windowing function with
„main‟ support of size 2-s2-s.
For each square, wQ is a displacement of w
localized near Q.
Multiplying s f with wQ (QQs) produces a
smooth dissection of the function into
„squares‟.          h  w  f      Q             Q               s
Smooth Partitioning
The windowing function w is a nonnegative
smooth function.
Partition of the energy:
   The energy of certain pixel (x1,x2) is divided
between all sampling windows of the grid.
   w2 x1  k1 , x2  k2   1
k1 , k 2
Example:
   An indicator of the dyadic square
(but not smooth!!).
   Smooth window function with
an extended compact support:
Expands the number of coefficients.
Smooth Partitioning
Partition of the energy:
 w2 x1  k1 , x2  k2   1
k1 , k 2

 wQ  1
2

QQ s

Reconstruction:
 wQ  hQ 
QQ s

QQ s
wQ  h  h
2

Parserval relation:
h                 w       h      w          h  h  h 2
2
           2     2               2    2     2    2
Q 2               Q                     Q
QQ s             QQ s                 QQ s
Renormalization
square to the unit square [0,1][0,1].
For each Q, the operator TQ is defined as:
T f x , x   2 f 2 x  k , 2 x
Q     1   2
s    s
1   1
s
2    k2   
Each square is renormalized:
1
g Q  TQ hQ
Before the Ridgelet Transform
The s f layer contains objects with
frequencies near domain ||[22s, 22s+2].
   We expect to find ridges with width  2-2s.
Windowing creates ridges of width  2-2s and
length  2-s.
The renormalized ridges has an aspect ratio
of width  length2.
We would like to encode those ridges
efficiently
   Using the Ridgelet Transform.
The Ridgelet Transform
Ridgelet are an orthonormal set {} for L2(2).
Developed by Candès and Donoho (1998).

2-s
2-2s

2s
2s                                divisions
2-s
1
Ridge in Square     It‟s Fourier Transform   Ridgelet Tiling     Fourier Transform
within Tiling

   Divides the frequency domain to dyadic coronae
||[2s, 2s+1].
   In the angular direction, samples the s-th corona at
least 2s times.
   In the radial direction, samples using local wavelets.
The Ridgelet Transform
The ridgelet element has a formula in the
frequency domain:
ρλ ξ   ξ ψ j ,k  ξ  ωi,l θ   ψ j ,k  ξ  ωi,l θ  π 
ˆ         1
2
1
ˆ
2
                  ˆ                          
where,
 i,l are periodic wavelets for [-,  ).

 i is the angular scale and l[0, 2i-1–1] is the
angular location.
 j,k are Meyer wavelets for .

 j is the ridgelet scale and k is the ridgelet location.
Ridgelet Analysis
Each normalized square is analyzed in the
ridgelet system:
αQ,λ   g Q , ρλ
   The ridge fragment has an aspect ratio
of 2-2s2-s.
   After the renormalization, it has localized
frequency in band ||[2s, 2s+1].
   A ridge fragment needs only a very few ridgelet
coefficients to represent it.
Digital Ridgelet Transform (DRT)
Unfortunately, the (current) DRT is not truly
orthonormal.
An array of nn elements cannot be fully
reconstructed from nn coefficients.
The DRT uses n2n coefficients for almost
perfect reconstruction

Still a lot of research need to be done…
Curvelet Transform
The four stages of the Curvelet Transform were:
Sub-band decomposition
f  P0 f , 1 f ,  2 f , 
Smooth partitioning
hQ  wQ   s f
Renormalization
1
g Q  TQ hQ
Ridgelet analysis
αQ,λ   g Q , ρλ
Image Reconstruction
The Inverse of the Curvelet Transform:
Ridgelet Synthesis
g Q   αQ,λ   ρλ
λ
Renormalization
hQ  TQ g Q
Smooth Integration
s f        w
QQ s
Q    hQ
Sub-band Recomposition
f  P0 P0 f     s  s f 
s
Example:
Roy Lichtenstein: “In The Car” 1963

Original Image            Approximation with only
(256256)           64 wavelets and 256 curvelets
Example:
Noise Reduction
using Curvelet
transform.

WT + Thresholding   WT + k- Thresholding       Curvelet Transform
Example:
Noise Reduction
using Curvelet
transform.

WT + Thresholding   WT + k- Thresholding       Curvelet Transform
References
[1] D.L. Donoho and M.R. Duncan. Digital Curvelet Transform:
Strategy, Implementation and Experiments; Technical Report,
Stanford University 1999
[2] E.J. Candès and D.L. Donoho. Curvelets – A Surprisingly
Effective Non-adaptive Representation for Objects with Edges;
Curve and Surface Fitting: Saint Malo 1999
[3] Lenna examples from
http://www-stat.stanford.edu/~jstarck/comp.html

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