# Image denoising via the redundant wavelet transform  (Page 3/3)

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$\mathbf{y}=\mathbf{x}+\mathbf{w}=\sqrt{z}\mathbf{u}+\mathbf{w}.$

The BLS-GSM algorithm is as follows:

• Decompose the image into subbands
• For the HH, HL, and LH subbands:
• Compute the noise covariance, ${\mathbf{C}}_{\mathbf{w}}$ , from the image-domain noise covariance
• Estimate ${\mathbf{C}}_{\mathbf{y}}$ , the noisy neighborhood covariance
• Estimate ${\mathbf{C}}_{\mathbf{u}}$ using ${\mathbf{C}}_{\mathbf{u}}={\mathbf{C}}_{\mathbf{y}}+{\mathbf{C}}_{\mathbf{w}}$
• Compute $\Lambda$ and $\mathbf{M}$ , where $\mathbf{Q}$ , $\Lambda$ is the eigenvector/eigenvalue expansion of the matrix ${\mathbf{S}}^{-\mathbf{1}}{\mathbf{C}}_{\mathbf{u}}{\mathbf{S}}^{-\mathbf{T}}$ $\mathbf{S}$ is the symmetric square root of the positive definite matrix ${\mathbf{C}}_{\mathbf{w}}$ , and $\mathbf{M}=\mathbf{SQ}$
• For each neighborhood
• For each value $z$ in the integration range
• Compute $\mathbb{E}\left[{x}_{c}|\mathbf{y},z\right]={\sum }_{n=1}^{N}\frac{z{m}_{cn}{\lambda }_{n}{v}_{n}}{z{\lambda }_{n}+1}$ , where ${m}_{ij}\in \mathbf{M}$ , $v\in \mathbf{v}={\mathbf{M}}^{-1}\mathbf{y}$ , $\lambda =diag\left(\lambda \right)$ , and $c$ is the index of the reference coefficient.
• Compute the conditional density $p\left(\mathbf{y}|z\right)$
• Compute the posterior $p\left(z|\mathbf{y}\right)$
• Compute $\mathbb{E}\left[{x}_{c}|\mathbf{y}\right]$
Reconstruct the denoised image from the processed subbands and the lowpass residual

## Simulation description

In order to compare and evaluate the efficacies of the Bishrink and BLS-GSM algorithms for the purpose of denoising image data, a simulation was developed to quantitatively examine their performance after addition of random noise to otherwise approximately noiseless images with a variety of features representative of those found in astronomical images. Specifically, the images encoded in the widely available files Moon.tif, which primarily demonstrates smoothly curving attributes, and Cameraman.tif, which exhibits a range of both smooth and coarse features, distributed in the MATLAB image processing toolbox were considered.

As a preliminary preparation for the simulation, the images were preprocessed such that they were represented in the form of a grayscale pixel matrix taking values on the interval $\left[0,1\right]$ of square dimensions equal to a convenient power of two. Noisy versions of each image were generated by superposition of a random matrix with Gaussian distributed pixel elements on the image matrix, using noise variance values $\left\{.01,.1,1\right\}$ . For each noise variance level and original image, 100 contaminated images were created in this way using a set of 100 different random generator seeds, which was the same for each noise level and original image. A redundant discrete wavelet transform of each of these contaminated images was computed using the length 8 Daubechies filters, and the denoised wavelet coefficients were estimated using both the Bishrink and the BLS-GSM algorithms as previously described. Computation of the inverse redundant discrete wavelet transform using the denoised wavelet coefficients then yielded 100 images denoised with the Bishrink algorithm and 100 images denoised with the BLS-GSM algorithm for each original image and noise variance level.

Using this simulated data, the performance of the two denoising methods on each image at each noise contamination level were evaluated using the six statistical measures described here. The first of these was the mean square error $\left(M,S,E\right)$ , which is calculated by the average of

$\frac{1}{n}\sum _{i=1}^{n}{\left(f,\left({x}_{i}\right),-,\stackrel{^}{f},\left({x}_{i}\right)\right)}^{2}$

over all 100 denoisings. Related to the above was the root mean square error $\left(R,M,S,E\right)$ , which is calculated by computing the square root of the mean square error. A third was the root mean square bias $\left(R,M,S,B\right)$ , which is calculated by

$\sqrt{\frac{1}{n}{\sum }_{i=1}^{n}{\left(f,\left({x}_{i}\right),-,\overline{f},\left({x}_{i}\right)\right)}^{2}}$

where $\overline{f}\left({x}_{i}\right)$ is the average of $\stackrel{^}{f}\left({x}_{i}\right)$ over all 100 denoisings. Two more, the maximum deviation $\left(M,X,D,V\right)$ , calculated by the average of

$\underset{1

over all 100 denoisings, and L1, calculated by the average of

$\sum _{i=1}^{n}\left|\left(f,\left({x}_{i}\right),-,\stackrel{^}{f},\left({x}_{i}\right)\right)\right|$

over all 100 denoisings, were also examined. The results of this simulation now follow.

## Bishrink results

 Measure Cameraman Moon MSE 0.0019 0.0004 RMSE 0.0442 0.0188 L1 2019.9 3160.4 RMSB 0.0274 0.0117 MXDV 0.3309 0.2634
 Measure Cameraman Moon MSE 0.0063 0.0012 RMSE 0.0296 0.0345 L1 3612.4 5880.7 RMSB 0.0568 0.0213 MXDV 0.6147 0.4116
 Measure Cameraman Moon MSE 0.0173 0.0052 RMSE 0.1315 0.0722 L1 6183.7 11839 RMSB 0.0934 0.0389 MXDV 0.8991 0.9774

## Bls-gsm results

 Measure Cameraman Moon MSE 0.0015 0.0003 RMSE 0.0390 0.0165 L1 1711.0 2718.6 RMSB 0.0283 0.0141 MXDV 0.3192 0.2635
 Measure Cameraman Moon MSE 0.0052 0.0008 RMSE 0.0718 0.0288 L1 3111.5 4786.5 RMSB 0.0583 0.0224 MXDV 0.5862 0.3337
 Measure Cameraman Moon MSE 0.0136 0.0017 RMSE 0.1167 0.0410 L1 5283.5 1500.2 RMSB 0.0970 0.0346 MXDV 0.7750 0.4614

## Conclusions

The results obtained from this simulation now allow us to evaluate and comment upon the suitability of each of the two methods examined for the analysis of astronomical image data. As is clearly manifested in the quantitative simulation results, the BLS-GSM algorithm demonstrated more accurate performance than did the Bishrink algorithm in every measure consistently over all pictures and noise levels. That does not, however, indicate that it would be the method of choice in all circumstances. While BLS-GSM outperformed the Bishrink algorithm in the denoising simulation, the measures calculated for the Bishrink algorithm indicate that it also produced a reasonably accurate image estimate. Also, the denoised images produced by the Bishrink simulation exhibit a lesser degree of qualitative smoothing of fine features like the craters of the moon and grass of the field. The smoothing observed with the BLS-GSM algorithm could make classification of fine, dim objects difficult as they are blended into the background. Thus, the success of the Bishrink algorithm in preserving fine signal details while computing an accurate image estimate is likely to outweigh overall accuracy in applications searching for small, faint objects such as extrasolar planets, while the overall accuracy of the BLS-GSM algorithm recommend it for coarse and bright featured images.

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