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

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

• Yothin Rakvongthai

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Introduction

• Curvelet Transform (CandesDonoho 1999)
• Implementation
• Fast Discrete Curvelet Transform (FDCT) (Candes
  et. al 2005) in frequency domain
• Contourlet (DoVetterli 2005) in time domain
  with wavelet-like tree structure
• Uniform Discrete Curvelet Transform (UDCT)
  (NguyenChauris 2008) in frequency domain with
  wavelet-like tree structure

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Implementation
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UDCT Marginal Statistics
Kurtosis 23.71
Kurtosis 24.42
Kurtosis E(x-µ)4/s4 . Kurtosis of Gaussian
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Conditional Distribution (1)

• On parent (same position in next level)

P(XPX)
Bow-tie shape? uncorrelated but dependent
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Conditional Distribution (2)

• On parent
• P(XPXpx)
• Kurtosis3.51
• Gaussian

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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

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Tree Structure of UDCT
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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
• s2m(j,k,n) (mS?small variance, mL?large
  variance)

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HMT (2)

• The total pdf
• P(S(j,k,n)), µm(j,k,n), s2m(j,k,n) can be trained
  from the EM algorithm (Crouse et al 1998).
• Define T set of P(S(j,k,n)), µm(j,k,n),
  s2m(j,k,n)

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Denoising (1)

• Problem formulation y xw
• 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

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Denoising (2)

• Obtain T from EM algorithm
• The variance of denoised coefficients is

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Denoising (3)

• The estimate of x

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Denoising Results (1)
PSNR Peak Signal to Noise Ratio
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Denoising Results (2)
SSIM Structure Similarity Index (Wang et. al
2004)
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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)
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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)
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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)