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. 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.

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 ?

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 0.2 4. 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 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.

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

Comparison of BayesShrink vs. SureShrink

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 ?

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 .

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!

Results for Elaine Compare with σ2 = 900 ! 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.

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

Results

Comparisons

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.