This document summarizes a seminar presentation on an image denoising method based on the curvelet transform. The presentation covered: 1) How image noise occurs and traditional denoising methods like linear filters and edge-preserving smoothing. 2) The curvelet transform process including sub-band decomposition, smooth partitioning, renormalization, and ridgelet analysis. 3) An image denoising algorithm that applies wavelet and curvelet transforms, then combines results using quad tree decomposition.

1. Seminar
on
“Image Denoising Method based on Curvelet
Transform”
Master of Engineering
(Electronics and Communication )
Year 2011-12.
Rajput Sandeep Kumar Jawaharlal (100370704036)
Prepared By: Guided By:
Rajput Sandeep J Prof. A.R. Yadav
ME (EC-213) Professor , EC Dept.
PIET, Limda. PIET, Limda.

2. Introduction
 Image acquired through sensors charge coupled device (CCD)
cameras may be influenced by noise sources.
 Image processing technique also corrupts image with noise,
leading to significant reduction in quality.
Traditionally,
 Linear filters
 Edge preserving smoothing algorithm
New Methods,
 Non-linear techniques : Wavelet Transform
: Curvelet Transform

3. Original Image
Sub-band
decomposition
Smooth partitioning
and
Renormalization
Each subzone of
each block to carry
out analysis of the
Ridgelet
Block image
n x n
Ridgelet TransformRadon Transform
WT 1D
Angle
Inverse FFT 1 D
FFT 2D
Frequency
WT 2D
Process of Curvelet Transform
Figure: 1 Curvelet transform flow block diagram

4. Sub-band Decomposition
fP0 f1∆ f2∆
f ( ) ,,, 210 fffPf ∆∆

5. Smooth Partitioning

6. Smooth Partitioning
 The windowing function w is a nonnegative smooth
function.
 Partition of the intensity:
The intensity of certain pixel (x1,x2) is divided between
all sampling windows of the grid.
( ) 1,
21,
2211
2
≡−−∑kk
kxkxw

7.  Ridgelet are an orthonormal set {ρλ} for L2
(R2
).
Ridgelet Analysis
2-s
2-2s
1
2-s
2s
radius 2s
2s
divisions
Ridge in Square It’s Fourier TransformRidge in Square
Ridgelet Tiling
Fourier Transform
within Tiling

8. Ridgelet Analysis
 The ridgelet element in the frequency domain:
where,
ωi,l are periodic wavelets for [-π, π ).
i is the angular scale.
ψj,k are wavelets for R.
j is the ridgelet scale and k is the ridgelet location.
( ) ( ) ( ) ( ) ( )( )πθωψθωψρ likjlikjλ +⋅−+⋅=
−
,,,,2
1 ξˆξˆξξˆ 2
1

9. Curvelet Transform
The four stages of the Curvelet Transform were:
 Sub-band decomposition
 Smooth partitioning
 Renormalization
 Ridgelet analysis
( ) ,,, 210 fffPf ∆∆
fwh sQQ ∆⋅=
QQQ hTg
1−
=
( ) λQQ,λ ρgα ,=

10. Image Reconstruction
The Inverse of the Curvelet Transform:
 Ridgelet Synthesis
 Renormalization
 Smooth Integration
 Sub-band Recomposition
( ) λ
λ
Q,λQ ραg ⋅= ∑
QQQ gTh =
∑∈
⋅=∆
sQ
QQs hwf
Q
( ) ( )∑ ∆∆+=
s
ss ffPPf 00

11. Thresholding methods
Window Shrink Method
 Set di, j is the parameter which is from curvelet transformed
noise image; choose a di, j centered window of n×n as the
processing subject.
3X 3 Window Shrink
The curvelet coefficients
to be thresholded

12. Set Symbolic function:
 σ is the variance of Gaussian white noise in the image , then
shrinking processing parameter is
Then the thresholded parameter can be calculated as:
Thresholding methods
The sum of all the parameter’s square in the n×n window is
calculated.

13. Bayes Shrink method
Thresholding methods
 In this method σ2
D
is the variance of an image containing
noise, σ2
is the variance of noise, and σ2
X
is the original image’s
variance.
Now, noise variance is:
The variance of original image is calculated by,
 Setting Threshold is σ2
/ σ2
X
then begin the processing of
removing noise.

14. Combination of Window shrink and Bayes shrink
 The variance σ2
X
is estimated of the original picture using Bayes
shrink theory, then η is calculated using σ2
X
instead of the noise
variance σ 2
,such as
 At last shrink factors αi, j
are known and the noise coefficient is
filtered out by taking advantage of αi, j
.
Thresholding methods
x

15. Thresholding methods

16. Image denoising Algorithm
Original image σ = 20 noise image
2-D Wavelet transform Traditional Curvelet transform

17. Image denoising Algorithm
Quad tree Decomposition algorithm
Now, The Q(x,y) that define the matrix
of mxm image and S(vi) denote the
element of the Q(x,y) where vi denote
the number of decomposition required
for that element.

18. Image denoising Algorithm
Algorithm :
Denote result image of improved algorithm as R, this pixel
fusion based algorithm is described as follows.
 Applying wavelet transform to obtain result image W.
 Applying curvelet transform to obtain result image C.
 Get quad tree matrix Q with applying quad tree decomposition
to C.
 R(x, y) is calculated as
R(x, y) = cW(x, y) + dC(x, y)
Where,

19. Image denoising Algorithm
Result of algorithm
Original image σ = 20 noise image Improved Curvelet transform

20. Image denoising Algorithm

21. Image denoising Algorithm

22. Conclusion
 To overcome the disadvantages of the wavelet
transform along the curves in the images the curvelet
transform is used and it gives high PSNR.
 A new method of combination of the Window Shrink
and Bayes Shrink based on Curvelet transform is used to
remove noise from image. It has better PSNR. So the
image we get by this method is better and that of the
traditional wavelet methods.

23. References
i. Introduction to Wavelet: Bhushan D Patil PhD Research Scholar Department of Electrical Engineering Indian Institute
of Technology, Bombay.
ii. Pixel Fusion Based Curvelets and Wavelets Denoise Algorithm, Liyong Ma, Member, IAENG, Jiachen Ma and Yi
Shen Advance online publication: 16 May 2007
iii. The Curvelet Transform - Jean-Luc Starck, Emmanuel J. Candès, and David L. Donoho IEEE transactions on
image processing, vol. 11, no. 6, june 2002.
iv. Image denoising using wavelet transform: an approach for edge Preservation Received 03 March 2009; revised 24
November 2009; accepted 25 November 2009
v. Image Denoising Method Based on Curvelet Transform -University of Science and Technology, IEEE transactions on
image processing, vol. 11, no. 6, june 2008.
vi. New Method Based on Curvelet Transform for Image Denoising Donglei Li, Zhemin Duan, Meng Jia
vii. Department of Electronics and Information Northwestern Polytechnical University, China, 2010 International
Conference on Measuring Technology and Mechatronics Automation
viii. Improved Image Denoising Method based on Curvelet Transform Proceedings of the 2010 IEEE International
Conference on Information and Automation June 20 - 23, Harbin, China
ix. Image Denoising Based on Curvelet Transform and Continuous Threshold YUAN Ruihong TANG Liwei WANG Ping
YAO Jiajun Department of Artillery Engineering Ordnance Engineering College Shijiazhuang ,China, 2010 First
International Conference on Pervasive Computing, Signal Processing and Applications.