1. Introduction to the Curvelet TransformBy
Zvi Devir and Yanai Shpinner

2. 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.

3. 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:
Using the Wavelet Transform:
Using the Curvelet Transform:

4. 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.

5. Curvelet Transform The Curvelet Transform includes four stages:
Sub-band decomposition
Smooth partitioning
Renormalization
Ridgelet analysis

6. Sub-band Decomposition
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:
Energy preservation

7. Sub-band Decomposition

8. 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:

9. 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.

10. 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.

11. Smooth Partitioning

12. Smooth Partitioning A grid of dyadic squares is defined:
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’.

13. 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.
Example:
An indicator of the dyadic square (but not smooth!!).
Smooth window function with an extended compact support:
Expands the number of coefficients.

14. Smooth Partitioning Partition of the energy: Reconstruction:
Parserval relation:

15. Renormalization
Renormalization is centering each dyadic square to the unit square [0,1][0,1].
For each Q, the operator TQ is defined as:
Each square is renormalized:

16. 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.

17. The Ridgelet Transform
Ridgelet are an orthonormal set {} for L2(2). Developed by Candès and Donoho (1998).
Ridgelet Tiling
radius 2s 2s
divisions
Ridge in Square
2-s
2-2s
Ridge in Square 1
2-s
It’s Fourier Transform
It’s Fourier Transform 2s
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.

18. The Ridgelet Transform
The ridgelet element has a formula in the frequency domain: 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.

19. Ridgelet Analysis
Each normalized square is analyzed in the ridgelet system:
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.

20. 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…

21. Curvelet Transform The four stages of the Curvelet Transform were:
Sub-band decomposition
Smooth partitioning
Renormalization
Ridgelet analysis

22. Image Reconstruction The Inverse of the Curvelet Transform:
Ridgelet Synthesis
Renormalization
Smooth Integration
Sub-band Recomposition

23. Example: Roy Lichtenstein: “In The Car” 1963 Approximation with only
64 wavelets and 256 curvelets
(about 0.5% of the coefficients)
Original Image
(256256)

24. Example: Noise Reduction using Curvelet transform. Original
Adding Gaussian Noise
Example:
Noise Reduction using Curvelet transform.
WT + Thresholding
WT + k- Thresholding
Curvelet Transform

25. Example: Noise Reduction using Curvelet transform. Original
Adding Gaussian Noise
Example:
Noise Reduction using Curvelet transform.
WT + Thresholding
WT + k- Thresholding
Curvelet Transform

26. 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