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

2. 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

3. Implementation

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

5. Conditional Distribution (1)
On parent (same position in next level)
P(X|PX)
Bow-tie shape uncorrelated but dependent

6. Conditional Distribution (2)
On parent
P(X|PX=px)
Kurtosis=3.51
~Gaussian

7. 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

8. Tree Structure of UDCT

9. 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)

10. 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)

11. 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

12. Denoising (2) Obtain Θ from EM algorithm
The variance of denoised coefficients is

13. Denoising (3)
The estimate of x

14. Denoising Results (1)
PSNR = Peak Signal to Noise Ratio

15. Denoising Results (2)
SSIM = Structure Similarity Index (Wang et. al 2004)

16. 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)

17. 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)

18. 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)