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Image Denoising Using Wavelets. Ramji Venkataramanan Raghuram Rangarajan Siddharth Shah. 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.
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Image DenoisingUsing Wavelets Ramji VenkataramananRaghuram Rangarajan Siddharth Shah
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 Y=W(X) 1. Calculate linear Forward Wavelet Transform. Z=D(Y,λ) 2. Threshold wavelets using one of available techniques Non linear, non-parametric step Y=W-1(X) 3. Calculate Inverse Wavelet Transform.
How does one decide this threshold below which we set everything to zero ? or Is this way of thresholding coefficients i.e. “keep or kill” the only way ? Why should I Threshold ? Sparsity: Small Coefficients dominated by noise. Large ones by signal. Why don’t we replace small coefficients by zero ?
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 = 0 if |Xj,k| < λ
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 0.24. Compared MSEs for each filter for all 4 types of signals.
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
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
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!)
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 tSby • Does not perform well in Sparse Cases. The Solution ?? Hybrid Scheme • SURE threshold tSfor dense cases. • Universal threshold tdFfor sparse cases.
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 !!
VisuShrink Why is VisuShrink not good? Overly smoothed images
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 ?
DENOISING We use an MMSE estimator to get an estimate of X from the noisy observations Y . 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 are estimated as before for each detail subband. Therefore, subband adaptive estimation. Note the similarity with shrinkage – all coefficients are pulled towards zero!
Results for Elaine MSE of the denoised image =123.76 . Compare with σ2 = 900 !
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.
Quantize with Local variance YQ Ŷ 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 Y W White Noise
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.
