1. Imola K. Fodor, Chandrika Kamath Center for Applied Scientific Computing Lawrence Livermore National Laboratory IPAM Workshop January, 2002 Exploring the use of wavelet thresholding for denoising images UCRL-JC-145671. This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract W-7405-Eng-48.

2. Sapphire/IKF 2 CASC Many data mining applications require removing noise from the data l Noise contaminates many scientific data sets –instrumental noise –data acquisition process –interfering natural phenomena l Denoising is a first pre-processing step in such cases l Many different denoising techniques –spatial filters, wavelets, simple thresholding, level sets, total variational methods, essentially non- oscillatory schemes, hidden Markov models, ridgelets, curvelets, … l No comprehensive evaluation of all methods exists

3. Sapphire/IKF 3 CASC We explore wavelet-based and spatial filter-based 2D denoising methods l Denoising –estimate signal from noisy observation –we assume i.i.d. Gaussian noise, independent from the signal: –find estimate with desired properties, e.g. minimum mean square error (MSE)

4. Sapphire/IKF 4 CASC Wavelets decompose the data into different multiresolution levels l Decimated wavelet transform with J=2 levels Horizontal Decomposed: level 1, level2 VerticalDiagonal Original

5. Sapphire/IKF 5 CASC Denoising by thresholding of wavelet coefficients entails several steps Image Wavelet transform Threshold detail wavelet coefficients Inverse wavelet transform De-noised image Wavelet: Haar Daublets Symmlets Coiflets... Boundary treatment: Zero-pad Periodic Symmetric Reflective Constant Shrinkage rule: calculate threshold Shrinkage function: apply threshold Noise scale: Number of levels: J

6. Sapphire/IKF 6 CASC The shrinkage function determines how the threshold is applied Semisoft Garrote Soft Hard

7. Sapphire/IKF 7 CASC The calculation of some thresholds requires an estimate of the noise scale l Choice of coefficients D: global or level-dependent? or for l Choice of estimator –median absolute deviation (MAD) –sample standard deviation – norm

8. Sapphire/IKF 8 CASC The shrinkage rule determines how the threshold is calculated l Universal: –global l minFDR: minimize false discovery rate –global l Top: based on quantiles of the coefficients –global l HypTest: test hypothesis of zero coefficients –level dependent l SURE: minimize Stein’s Unbiased Risk Estimate –adapt: combines SURE and Universal –level and shrinkage function dependent l Bayes: use a Bayesian framework –level dependent

9. Sapphire/IKF 9 CASC Our implementation is more general than conventional methods in literature l Conventional: either –1 (global) or –J (level-dependent) or –3J (subband-dependent) threshold(s) l Sapphire: choice of –1 (global) or –J (level-dependent) or –3J (subband-dependent) threshold(s) l Preliminary observation –subband-dependent de-noising outperforms level- dependent de-noising on some test images

10. Sapphire/IKF 10 CASC Spatial filtering and simple thresholding are alternative ways to denoise images l Spatial filters are commonly used in the signal processing community. Our software includes –mean filters and alpha-trimmed mean filters –Gaussian filters –(scaled) unsharp masking filters –median filters –mid-point filters –minimum mean squared error filters –combinations of the above filters l Simple thresholding –drop the pixels below 5*RMS in the FIRST images

11. Sapphire/IKF 11 CASC Results with Lena using soft shrinkage, symm12 wavelet, periodic boundary, J=3 S_ : single global threshold P_ : pyramid of thresholds (J or 3J)

12. Sapphire/IKF 12 CASC Results with Lena using hard shrinkage, symm12 wavelet, periodic boundary, J=3 S_ : single global threshold P_ : pyramid of thresholds (J or 3J)

13. Sapphire/IKF 13 CASC Results with Lena using garrote and semisoft shrinkage, symm12, periodic boundary, J=3 S_ : single global thresholdP_ : pyramid of thresholds (J or 3J)

14. Sapphire/IKF 14 CASC The Lena test image with simulated Gaussian noise added OriginalNoise added, sigma=20 MSE=399.50

15. Sapphire/IKF 15 CASC Wavelet denoising on the Lena test image SureShrink MSE=61.59 Global universal hard thresholding MSE=103.95 Notice the artifacts and ringing near the edges

16. Sapphire/IKF 16 CASC Results with the Lena image using various spatial filters

17. Sapphire/IKF 17 CASC Spatial filtering on the Lena test image Min-MSE (5x5) followed by Gaussian (3x3) MSE=56.80 Min-MSE (5x5) followed by mean (3x3) MSE=64.80

18. Sapphire/IKF 18 CASC Summary l SureShrink and BayesShrink are best wavelet-based denoisers across a range of images and noise levels l Soft is the best shrinkage function l MAD is the best noise estimator l Choice of wavelets, number of multiresolution levels, and boundary treatment rule have little influence l Combination of spatial filters (5x5 minimum MSE filter followed by 3x3 Gaussian filter) often yields smaller error rates than SureShrink and BayesShrink