Hidden Markov Tree Model of the Uniform Discrete Curvelet Transform Image for Denoising Yothin Rakvongthai Introduction Curvelet Transform Candes Donoho 1999 Implementation Fast Discrete Curvelet Transform FDCT Candes et al 2005 in frequency domain Contourlet Do Vetterli 2005 in time domain with wavelet like tree structure Uniform Discrete Curvelet Transform UDCT Nguyen Chauris 2008 in frequency domain with wavelet like tree structure Implementation UDCT Marginal Statistics Kurtosis 24 42 Kurtosis E x 4 4 Kurtosis of Gaussian 3 Kurtosis 23 71 Conditional Distribution 1 On parent same position in next level P X PX Bow tie shape uncorrelated but dependent Conditional Distribution 2 On parent P X PX px Kurtosis 3 51 Gaussian Hidden Markov Tree HMT Model Conditional distribution is Gaussian X depends on PX Use HMT to model the coefficients HMT model links between the hidden state variables of parent and children HMT parameters parameters of the density function can be trained using the expectation minimization EM algorithm Tree Structure of UDCT HMT 1 c j k n coefficient in scale j direction k position n S j k n hidden state taking on values m S or L with density function P S j k n Conditioned on S j k n m c j k n is Gaussian with mean m j k n and variance 2m j k n m S small variance m L large variance HMT 2 The total pdf P S j k n m j k n 2m j k n can be trained from the EM algorithm Crouse et al 1998 Define set of P S j k n m j k n 2m j k n Denoising 1 Problem formulation y x w y noisy coefficients x denoised coefficients w noise coefficients with known variance Want to estimate x from the knowledge of y and variance of w Denoising 2 Obtain from EM algorithm The variance of denoised coefficients is Denoising 3 The estimate of x Denoising Results 1 PSNR Peak Signal to Noise Ratio Denoising Results 2 SSIM Structure Similarity Index Wang et al 2004 Denoising Results 3 Original Noisy 14 14dB SSIM 0 112 Contourlet 25 85dB DT CWT 26 54dB SSIM 0 590 SSIM 0 579 Wavelet 25 73dB SSIM 0 561 UDCT 27 32dB SSIM 0 676 Denoising Results 4 Original Noisy 14 14dB SSIM 0 184 Wavelet 23 38dB SSIM 0 508 Contourlet 22 94dB DT CWT 24 15dB UDCT 24 35dB SSIM 0 479 SSIM 0 557 SSIM 0 570 Denoising Results 5 Original Noisy 14 14dB SSIM 0 110 Contourlet 25 51dB DT CWT 25 99dB SSIM 0 555 SSIM 0 553 Wavelet 25 25dB SSIM 0 539 UDCT 26 51dB SSIM 0 627
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