1. Image Denoising Using Wavelets
Ramji Venkataramanan Raghuram Rangarajan
Siddharth Shah

2. What is Denoising ?
“ Method of estimating the unknown signal from available noisy data”
Aims to remove whatever noise is present regardless of the signal’s frequency
content.
Denoising is not smoothing ! Smoothing removes high frequencies and keeps
lower ones.
Denoising by Wavelet Thresholding
1. Calculate linear Forward Wavelet Transform.
Y=W(X)
2. Threshold wavelets using one of available techniques
Non linear, non-parametric step
Z=D(Y,λ)
Y=W-1(X)
3. Calculate Inverse Wavelet Transform.

3. Why should I Threshold ?
Sparsity: Small Coefficients dominated by noise. Large ones by signal.
Why don’t we replace small coefficients by zero ? or
How does one decide this threshold below which we set everything to zero ?
Is this way of thresholding coefficients i.e. “keep or kill” the only way ?

4. How Do I discard wavelet coefficients?
Hard V/s Soft Thresholding
Hard “keep or kill”: Wavelet Coefficient with an absolute value below the threshold λ is replaced by 0.
Yj,k = Xj,k if |Xj,k|≥ λ
0 if |Xj,k|<λ
Soft: Set coefficients below λ to
zero and shrink those above λ
in absolute value.
Yj,k = sgn(Xj,k)(|Xj,k – λ) if |Xj,k| ≥ λ
= if |Xj,k| < λ

5. 1D Signal Analysis
1.Add white noise to each of these functions with σ=1.
2. Took wavelet transforms using Haar, Daubechies2, Daubechies4 and Daubechies 8 filters.
3. Performed hard and soft thresholding using a variety of thresholds from 0 to 5 in steps of Compared MSEs for each filter for all 4 types of signals.

6. 1D Signal Analysis: Results
Comparision with Universal Threshold
λUNIV is the optimal threshold to minimize MSE in the asymptotic sense(N→∞)
λUNIV=√2log(2048)=3.905 >> optimal thresholds obtained empirically

7. Image Denoising OUTLINE Same underlying principle as in 1D signals.
Subbands of the wavelet transform
Low resolution residual LL
HL3
HL2
LH3
HH3
HL1
details
LH2
HH2
LH1
HH1

8. Denoising of Images Goal : Determine thresholds to minimize MSE
Types of thresholding
VisuShrink
Universal Threshold.
Works asymptotically.
Denoised image is overly smooth.
SureShrink
Subband adaptive threshold
Based on Stein’s unbiased estimator for risk (SURE!)

9. Threshold Selection by SURE
Let wavelet coefficients in the jth subband be { Xi : i =1,…,d }
SURE proposes method for estimating loss.
For the soft threshold estimator , we have
Select threshold tS by
Does not perform well in Sparse Cases. The Solution ??
Hybrid Scheme
SURE threshold tS for dense cases.
Universal threshold tdF for sparse cases.

10. BayesShrink
Idea : Wavelet coefficients in each subband ~ Generalized Gaussian Distribution (GGD).
GGD ~ Gaussian for β =2 ; ~ Laplacian for β =1
Find T*(σX , β) that minimizes Bayesian Risk assuming this GGD.
No closed form solution to this threshold !
Set threshold as ; very close to actual minimum!
Intuitive appeal !!

11. VisuShrink
Why is VisuShrink not good? Overly smoothed images

12. Comparison of BayesShrink vs. SureShrink

13. Denoising and Compression
Denoising has been done…Can we compress the denoised coefficients?
Signal – contains redundancies.
Noise-Highly uncorrelated Good compression method can also distinguish between signal and noise.
Question: Can we have a model that facilitates denoising as well as efficient compression of the coefficients ?
GGD - A good model for distribution of coefficients in a subband.
Problem: Difficult to design an optimal quantizer for a GGD.
Is there a simpler way out ?

14. A Gaussian model
For most images, Gaussian distribution is found to be a satisfactory approximation.
Model :
We can denoise as well as compress using this model !
DENOISING
We use an MMSE estimator to get an estimate of X from the noisy observations Y .

15. Denoising Therefore, subband adaptive estimation.
are estimated as before for each detail subband.
Therefore, subband adaptive estimation.
Note the similarity with shrinkage – all coefficients are pulled towards zero!

16. Results for Elaine Compare with σ2 = 900 !
MSE of the denoised image =
Compare with σ2 = 900 !

17. Compression Denoised coefficients in each subband are iid as
Quantization scheme:
Fix a maximum allowable distortion D.
Calculate variance of each detail coefficient in the subband. (How?)
Choose the smallest quantizer to encode each coefficient from a set of available optimal quantizers for a Gaussian distribution, so that the distortion is less than D.
Repeat for all detail subbands.

18. Compression This cannot be done for the LL subband ! Why?
Coefficients in the LL band represent local averages of the signal- Not zero mean.
So we model the LL band as a uniform pdf .
So what have we done ? X
Quantize with
Local variance Ŷ YQ Y W
White Noise

19. Results

20. Comparisons

21. Conclusions Wavelet shrinkage is an effective method for denoising.
2. Subband adaptive thresholding performs better than universal thresholding since it adapts to the characteristics of each subband
3. BayesShrink is found to give the best threshold among those compared for denoising images.
4. Assuming a Gaussian distribution for wavelets enables one to perform
simultaneous denoising and compression using highly tractable
equations.