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EE565 Advanced Image Processing Copyright Xin Li 2008 1 Application of Wavelets (I): Denoising Problem formulation Frequency-domain solution: linear Wiener.

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Presentation on theme: "EE565 Advanced Image Processing Copyright Xin Li 2008 1 Application of Wavelets (I): Denoising Problem formulation Frequency-domain solution: linear Wiener."— Presentation transcript:

1 EE565 Advanced Image Processing Copyright Xin Li 2008 1 Application of Wavelets (I): Denoising Problem formulation Frequency-domain solution: linear Wiener filtering The need for a better basis Wavelet-domain solution: nonlinear thresholding Advanced denoising in wavelet space Go beyond wavelets Focus will be on ideas instead of technical details

2 EE565 Advanced Image Processing Copyright Xin Li 2008 2 Problem Formulation Noisy measurements signalN(0,σ w 2 ) Toy Example: how to measure Yao Ming’s height? Assume independent measurement errors: {y 1, y 2,…,y N } y k =x+w k

3 EE565 Advanced Image Processing Copyright Xin Li 2008 3 Real Challenges X is not a scalar but a vector or matrix (curse of dimensionality) X is not a single function but a class of functions (even multiple classes of functions) Uncertainty in W: Y=X+W=(X+z)+(W-z) We are taking a deterministic view in this course

4 EE565 Advanced Image Processing Copyright Xin Li 2008 4 Sample Function Classes Original data set Noisy data set

5 EE565 Advanced Image Processing Copyright Xin Li 2008 5 Easier Problem to Solve What if the signal x(t) is assumed to be piecewise constant (blocks)? Nearly optimal solution does exist x’(t) contains impulses in time domain Nonlinear diffusion (e.g., minimize the total variation) Graduated Non-Convexity (GNC) Algorithm (e.g., detect transition locations) Averaging N noisy data of the same X reduces the noise variance by a factor of N

6 EE565 Advanced Image Processing Copyright Xin Li 2008 6 Model-Data Mismatch What happened if we apply the techniques designed for blocks to bumps? Fails miserably – no transition point exists Class of piecewise constant functions is too small “All models are wrong, only some are useful”. – George Box

7 EE565 Advanced Image Processing Copyright Xin Li 2008 7 Pursuit of Better Models Sobolev space functions Google it if you want to learn about the mathematical definition Characterize the class of smooth functions Fourier transform offers nearly optimal approximation (the first K FT coefficients are the principal components) Wiener filtering has a linear frequency weighting interpretation (the larger spectral magnitude, the higher weight)

8 EE565 Advanced Image Processing Copyright Xin Li 2008 8 Besov-Space Function Again, we skip its mathematical definition Intuitively, it is a function that is smooth almost everywhere (allow finite exceptions) More suitable for modeling edges in images

9 EE565 Advanced Image Processing Copyright Xin Li 2008 9 Failure of Linear Frequency Weighting Why Fourier transform is suboptimal for Besov-space function? Every exception causes infinitely many large coefficients (spread across the whole spectrum) Difficult to tell signal (e.g. edges) apart from noise due to the overlapping in positions At the mercy of Gibbs phenomenon

10 EE565 Advanced Image Processing Copyright Xin Li 2008 10 Wavelets Save the Day Now think of the class of compactly-supported wavelets They would produce finite (localized) large coefficients around exceptions Easier to tell signal (e.g. edges) apart from noise due to better separation in positions Reduced Gibbs Phenomenon (depends on the length of wavelet filters)

11 EE565 Advanced Image Processing Copyright Xin Li 2008 11 Wavelet Shrinkage WTIWT Nonlinear Thresholding Noisy signal denoised signal Hard thresholding Soft thresholding

12 EE565 Advanced Image Processing Copyright Xin Li 2008 12 Why Thresholding Works? We need to distinguish spatially- localized events (edges) from noise components Think about noise components Wavelet is such a basis because exceptional event generates identifiable exceptional coefficients due to its good localization property in both spatial and frequency domain As long as it does not generate exceptions Additive White Gaussian Noise after WT is still AWGN

13 EE565 Advanced Image Processing Copyright Xin Li 2008 13 Tails of Distribution noise signal

14 EE565 Advanced Image Processing Copyright Xin Li 2008 14 Choice of Threshold Donoho and Johnstone’1994 Gives denoising performance close to the “ideal weighting” Reference: S. Mallat, “A Wavelet Tour of Signal Processing”, Section 10.2 (pp. 435-453)

15 EE565 Advanced Image Processing Copyright Xin Li 2008 15 Soft vs. Hard thresholding ● It can be also viewed as a computationally efficient approximation of ideal weighting soft ideal ● Soft-thresholding has the same upper bound as hard-thresholding asymptotically and larger error than hard-thresholding at the same threshold value, but perceptually it works better. ● Shrinking the amplitude by T guarantees with a high probability that.

16 EE565 Advanced Image Processing Copyright Xin Li 2008 16 Edges are kept, but the noise wasn’t fully suppressed Edges aren’t kept. However, the noise was almost fully suppressed

17 EE565 Advanced Image Processing Copyright Xin Li 2008 17 Denoising Example noisy image (σ 2 =100) Wiener-filtering (ISNR=2.48dB) Wavelet-thresholding (ISNR=2.98dB) X: original, Y: noisy, X: denoised ~ Improved SNR

18 EE565 Advanced Image Processing Copyright Xin Li 2008 18 Advanced Wavelet Denoising Translation Invariant Denoising Redundancy helps Spatially adaptive Wiener filtering of wavelet high-frequency band coefficients Extending thresholding to weighting Statistical modeling of wavelet high-frequency band coefficients Hidden Markov Model (Crouse et al.’1998) Gaussian Scalar Mixture (Portilla et al.’2003)

19 EE565 Advanced Image Processing Copyright Xin Li 2008 19 What do We Buy from Redundancy? 0 1 N-1 … x(n) H1H1 T -T

20 EE565 Advanced Image Processing Copyright Xin Li 2008 20 Translation Invariance (TI) Denoising T oe T oe -1 Thresholding T ce T ce -1 Thresholding T ce T ce -1 Thresholding z + x(n) Implementation based on overcomplete expansion Implementation based on complete expansion z -1

21 EE565 Advanced Image Processing Copyright Xin Li 2008 21 Gain Brought by Redundancy

22 EE565 Advanced Image Processing Copyright Xin Li 2008 22 From Thresholding to Weighting Challenges with wavelet thresholding Determination of a global optimal threshold Spatially adjusting threshold based on local statistics How to go beyond thresholding? We need an accurate modeling of wavelet coefficients – nonlinear thresholding is a computationally efficient yet suboptimal solution

23 EE565 Advanced Image Processing Copyright Xin Li 2008 23 Spatially Adaptive Wiener Filtering in Wavelet Domain Wavelet high-band coefficients are modeled by a Gaussian random variable with zero mean and spatially varying variance Apply Wiener filtering to wavelet coefficients, i.e., estimated in the same way as spatial-domain (Slide 15)

24 EE565 Advanced Image Processing Copyright Xin Li 2008 24 Practical Implementation T T N=T 2 Recall Conceptually very similar to its counterpart in the spatial domain (ML estimation of signal variance) Better solution than ML is MAP (refer to GSM paper by Portilla et al.)

25 EE565 Advanced Image Processing Copyright Xin Li 2008 25 Example Translation-Invariant thresholding (ISNR=3.51dB) Spatially-adaptive wiener filtering (ISNR=4.53dB)

26 EE565 Advanced Image Processing Copyright Xin Li 2008 26 Beyond Wavelets Transient events are not everything in the natural world What are non-transient events? Pitch-related periodicity in speech Texture patterns in image Long-term dependency in stock market Need a better understanding of CHANGE

27 EE565 Advanced Image Processing Copyright Xin Li 2008 27 Patch-based Denoising (BM3D) WD T T -1 ThresholdingWD = Noisy patches Denoised patches

28 EE565 Advanced Image Processing Copyright Xin Li 2008 28 State-of-the-Art Image Denoising PSNR(DB) PERFORMANCE COMPARISON AMONG DIFFERENT SCHEMES FOR 12 TEST IMAGES ATσw = 100

29 EE565 Advanced Image Processing Copyright Xin Li 2008 29 Duality with Image Coding* DWT IWTThresholding DWT IWT Q Q -1 Channel Image denoising system Image coding system

30 EE565 Advanced Image Processing Copyright Xin Li 2008 30 Difference from Image Coding G0G0 G1G1 x(n) H0H0 H1H1 y 0 (n) y 1 (n) x(n) H0H0 H1H1 2 2 G0G0 2 2 G1G1 s(n) d(n) complete expansion (non-redundant) - suitable for image coding overcomplete expansion (redundant) - suitable for image denoising T ce T ce -1 T oe T oe -1


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