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using a prior statistical model based on the alpha-stable and Bessel K Forms densities

Experimental Results

We assess the performance of our Bayesian denoiser with the scale-mixture approximation to the  alpha-stable prior, called alpha-stable mixture [1], and we compare it to other previously published denoising methods. For the comparaison to be fair, we only chose denoising methods using the same transforms, namely, the DWT. Extension to overcomplete representations which are translation and rotation invariant are the subject of our ongoing research. Six other denoising algorithms are considered: the universal threshold Hard and Soft thresholding [2], the Stein Unbiased Risk Estimator (SURE) [3], the Oracle threshold estimator (Oracle), the Bessel K forms (BKF) Bayesian denoiser [4] and the original version of the  alpha-stable Bayesian denoiser [5]. In the latter, no closed-form is available for the PCM Bayesian denoiser. We here used an equivalent form involving Fourier integrals as proven in [6]. The numerically Fourier integrals were implemented using FFT-based methods.

The following example shows the estimated images for each denoising methods for the Women image with an input SNRin = 15dB. One can clearly see that the visual quality of the alpha-stable mixture  Bayesian denoiser is superior to the other methods but remains comparable to the BKF Bayesian denoiser. Owing to its hyperparameter estimation method, our denoiser overcomes the limitations of the original exact alpha-stable Bayesian denoiser as used in [5,6]. Furthermore, our denoiser is faster and very stable numerically.

Maximum A Posterior (MAP) estimator

[1] L. Boubchir, J. Fadili, "Bayesian Denoising in the Wavelet-Domain Using an Analytical Approximate alpha-stable Prior", ICPR(4) 2004: 889-892
[2] D. L. Donoho, I. M. Johnstone, "Ideal spatial adaptation by wavelet shrinkage", Biometrika 81 (3) (1994) 425 455.
[3] D. L. Donoho, I. M. Johnstone, "Adapting to unknown smoothness via wavelet shrinkage", Journal of the American Statistical Association 90 (432) (1995) 1200  1224.
[4] J. Fadili, L. Boubchir, "Analytical form for a bayesian wavelet estimator of images using the bessel k forms densities", IEEE Transactions on Image Processing 14 (2) (February 2005) 231 240.
[5] A. Achim, A. Bezerianos, P. Tsakalides, "Novel bayesian multiscale method for speckle removal in medical ultrasound images", IEEE Trans. Med. Imag. 20 (2001) 772 783.
[6] J. Mathieu, "Transformée en ondelettes et régression non-paramétrique dans un contexte bayesien", Master's thesis, Ecole Nationale Supérieure d Ingénieur, Caen (2002).
[7] L. Boubchir, M.J. Fadili , "Bayesian Denoising Based on The MAP Estimation in Wavelet-domain Using Bessel K Form Prior" , IEEE International Conference on Image Processing ICIP'05, A paraître.
[8] P. Moulin and J. Liu,  "Analysis of multiresolution image denoising schemes using generalized gaussian and complexity priors", IEEE Transactions on Information Theory, vol. 45, no. 3, pp. 909-919, 1999.