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
Renormalization is centering each dyadic
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).
                                     radius 2s

             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
                         (about 0.5% of the coefficients)
                                   Original      Adding Gaussian Noise
Example:
Noise Reduction
using Curvelet
transform.


     WT + Thresholding   WT + k- Thresholding       Curvelet Transform
                                   Original      Adding Gaussian Noise
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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