SCIENTIA SINICA Informationis, Volume 49, Issue 6: 708(2019) https://doi.org/10.1360/N112017-00293

## Image NSST-HMT model with associated multi-state coefficients

• AcceptedSep 5, 2018
• PublishedJun 6, 2019
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### Abstract

In recent years, due to its anisotropy, multi-directional capture characteristics, and translation invariance, the non-subsampled shearlet transform (NSST) hasplayed an important stabilizing role inthe process of image restoration. In this study, weanalyze an image's NSST coefficients, including the relationship betweencoefficients in the same subband, the relationship between“father-son” coefficients, and the relationship between “brotherhood”coefficients in different subbands. The results reveal that thecoefficients in the NSST subbands are sparse, and both “father-sonrelationship” and “brotherhood relationship” coefficients exhibitaggregation and transitivity. On this basis, a hidden Markov tree (HMT)model with associated multi-state coefficients (M-NSST-HMT) isproposed. This model estimates the reconstructed coefficientsusing “father-son relationship” and “brotherhood relationship”of the NSST subband coefficients as joint states of guiding thecoefficients' transfer between subbands. In addition, the model integrates thereconstructed coefficients using the mutual information betweenthese two associated states. Finally, the proposed model is appliedto image denoising with favorable results. The results indicatethat the proposed model can reveal the relationship ofcoefficients in NSST subbands and improve the prediction accuracy ofcoefficients more effectively than the traditional HMT model.

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• Table 1   TheKolmogorv-Smirov of GMM fitting for test images in Figure
 2*Test image The 1st direction subband The 1st direction subband The 1st direction subband in the 1st layer in the 2nd layer in the 3rd layer Test-image1 0.0154 0.0211 0.0366 Test-image2 0.0126 0.0125 0.0399 Bridge 0.0173 0.0275 0.0497
• Table 2   The parameters of Test-image1's M-NSST-HMT model
 2* $K=1$ $K=2$ 2* $K=1$ $K=2$ State1 State2 State1 State2 State1 State2 State1 State2 $P_{1,k,\rho}$ 0.35 0.65 0.30 0.70 $P_{1,k,c}$ 0.70 0.30 0.28 0.72 2*$A_{2,2k-1,\rho}$ State1 0.38 0.94 0.71 0.01 2*$A_{2,2k-1,c}$ State1 0.97 0.02 0.20 0.85 State2 0.62 0.06 0.29 0.99 State2 0.03 0.98 0.80 0.15 2*$A_{2,2k,\rho}$ State1 0.63 0.04 0.24 0.99 2*$A_{2,2k,c}$ State1 0.89 0.13 0.01 0.93 State2 0.37 0.96 0.76 0.01 State2 0.11 0.87 0.99 0.07 2*$A_{3,2k-1,\rho}$ State1 1.00 0.01 0.22 0.90 2*$A_{3,2k-1,c}$ State1 0.96 0.01 0.00 1.00 State2 0.00 0.99 0.78 0.10 State2 0.04 0.99 1.00 0.00 2*$A_{3,2k,\rho}$ State1 1.00 0.01 0.25 0.91 2*$A_{3,2k,c}$ State1 0.08 0.66 0.00 0.99 State2 0.00 0.99 0.75 0.09 State2 0.92 0.34 1.00 0.01 2*$A_{3,2k+3,\rho}$ State1 0.99 0.00 0.01 0.39 2*$A_{3,2k+3,c}$ State1 0.01 0.95 0.07 0.80 State2 0.01 1.00 0.99 0.61 State2 0.99 0.05 0.93 0.20 2*$A_{3,2k+4,\rho}$ State1 0.99 0.00 0.01 0.34 2*$A_{3,2k+4,c}$ State1 0.00 0.99 0.02 1.00 State2 0.01 1.00 0.99 0.66 State2 1.00 0.01 0.98 0.00 $\sigma_{1,k,\rho}$ 12.69 6.14 12.06 5.95 $\sigma_{1,k,c}$ 4.23 1.81 4.28 1.67 $\sigma_{2,2k-1,\rho}$ 9.08 4.82 9.56 4.21 $\sigma_{2,2k-1,c}$ 4.63 2.55 3.70 2.92 $\sigma_{2,k,\rho}$ 9.10 4.90 8.90 4.09 $\sigma_{2,k,c}$ 4.49 2.57 3.87 2.90 $\sigma_{3,2k-1,\rho}$ 5.74 2.30 4.86 2.22 $\sigma_{3,2k-1,c}$ 4.53 2.03 5.44 1.92 $\sigma_{3,2k,\rho}$ 5.74 2.30 4.92 2.23 $\sigma_{3,2k,c}$ 4.75 2.24 5.33 2.06 $\sigma_{3,2k+3,\rho}$ 5.63 2.26 6.48 2.53 $\sigma_{3,2k+3,c}$ 5.29 2.23 4.56 2.10 $\sigma_{3,2k+4,\rho}$ 5.63 2.23 6.93 2.62 $\sigma_{3,2k+4,c}$ 5.32 2.06 4.57 1.88
• Table 3   The PSNR value ofdifferent denoising models
 Image Noise PSNR (dB) level BM3D MSDCT BLS-GSM NSCT-HMT NSST-HMT Proposed 6*Test-image1 25 26.49 25.94 26.38 25.64 25.59 27.07 30 25.77 25.20 25.63 25.30 25.24 26.63 35 25.19 24.61 25.03 24.94 24.89 26.32 40 24.65 24.13 24.53 24.56 24.58 25.71 45 24.22 23.72 24.11 24.17 24.21 25.16 50 23.89 23.38 23.74 23.78 23.82 24.54 6*Test-image2 25 25.59 24.93 25.48 24.72 24.70 26.15 30 24.95 24.29 24.81 24.49 24.50 25.75 35 24.47 23.83 24.31 24.24 24.26 25.30 40 24.06 23.48 23.92 23.94 24.00 24.72 45 23.68 23.20 23.61 23.61 23.65 24.41 50 23.44 22.97 23.34 23.39 23.46 24.07 6*Test-image3 25 24.77 24.12 24.37 23.21 23.25 24.92 30 23.84 23.13 23.45 22.89 22.86 24.39 35 23.08 22.35 22.74 22.54 22.43 23.83 40 22.37 21.74 22.17 22.18 21.97 22.78 45 21.95 21.25 21.70 21.81 21.72 22.47 50 21.57 20.86 21.31 21.44 21.35 22.04 6*Bridge 25 26.23 25.69 26.03 24.53 24.67 26.50 30 25.46 24.92 25.25 24.24 24.38 26.07 35 24.86 24.33 24.64 23.91 24.03 25.48 40 24.30 23.86 24.15 23.56 23.68 25.97 45 23.90 23.48 23.93 23.20 23.29 24.53 50 23.59 23.15 23.38 22.83 22.94 24.08 6*Boat 25 29.91 29.15 29.27 26.97 27.31 29.96 30 29.12 28.29 28.43 26.52 26.81 29.42 35 28.43 27.58 27.72 26.03 26.23 28.75 40 27.74 26.99 27.12 25.53 25.60 28.32 45 27.24 26.49 26.60 25.03 25.24 27.72 50 26.78 26.05 26.15 24.53 24.60 26.94 6*Hill 25 29.85 29.20 29.27 27.60 27.43 29.71 30 29.16 28.51 28.55 27.33 27.18 29.34 35 28.56 27.94 27.96 27.02 26.91 28.88 40 27.98 27.46 27.47 26.68 26.62 28.22 45 27.57 27.05 27.05 26.31 26.26 27.89 50 27.18 26.69 25.92 25.92 25.94 27.48

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