Hidden Markov Tree Model of the Uniform Discrete Curvelet Transform Image for Denoising Yothin Rakvongthai.

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Presentation transcript:

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 = 23.71 Kurtosis = 24.42 Kurtosis = E[(x-μ)4]/σ4 . Kurtosis of Gaussian = 3

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 μm(j,k,n) and variance σ2m(j,k,n) (m=Ssmall variance, m=Llarge variance)

HMT (2) The total pdf P(S(j,k,n)), μm(j,k,n), σ2m(j,k,n) can be trained from the EM algorithm (Crouse et al 1998). Define Θ = set of P(S(j,k,n)), μm(j,k,n), σ2m(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 Θ 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) Wavelet (25.73dB) (SSIM 0.112) (SSIM 0.561) Contourlet (25.85dB) DT-CWT (26.54dB) UDCT (27.32dB) (SSIM 0.590) (SSIM 0.579) (SSIM 0.676)

Denoising Results (4) Original Noisy (14.14dB) Wavelet (23.38dB) (SSIM 0.184) (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) Wavelet (25.25dB) (SSIM 0.110) (SSIM 0.539) Contourlet (25.51dB) DT-CWT (25.99dB) UDCT (26.51dB) (SSIM 0.555) (SSIM 0.553) (SSIM 0.627)