In this paper, MMSE estimator is employed for noise-free 3D OCT data recovery in 3D complex wavelet domain. Since the proposed distribution for noise-free data plays a key role in the performance of MMSE estimator, a priori distribution for the pdf of noise-free 3D complex wavelet coefficients is proposed which is able to model the main statistical properties of wavelets. We model the coefficients with a mixture of two bivariate Gaussian pdfs with local parameters which are able to capture the heavy-tailed property and inter- and intrascale dependencies of coefficients. In addition, based on the special structure of OCT images, we use an anisotropic windowing procedure for local parameters estimation that results in visual quality improvement. On this base, several OCT despeckling algorithms are obtained based on using Gaussian/two-sided Rayleigh noise distribution and homomorphic/nonhomomorphic model. In order to evaluate the performance of the proposed algorithm, we use 156 selected ROIs from 650 × 512 × 128 OCT dataset in the presence of wet AMD pathology. Our simulations show that the best MMSE estimator using local bivariate mixture prior is for the nonhomomorphic model in the presence of Gaussian noise which results in an improvement of 7.8 ± 1.7 in CNR. 1. Introduction Optical coherence tomography (OCT) is an optical signal acquisition and processing method that captures 3D images from within optical scattering media such as biological tissues [1–4]. For example, in ophthalmology, OCT is used to obtain detailed images from within the retina [4]. Similar to other optical tomographic techniques, OCT suffers from speckle noise that reduces the ability of image interpretation [5]. So, noise reduction is an essential part of OCT image processing systems. Until now, several techniques for OCT noise reduction have been reported [6–14]. Initial methods perform in complex domain [15], that is, before producing magnitude of OCT interference signal, while most introduced despeckling methods are applied after an OCT image is formed [6–14]. These methods, which usually suppose multiplicative noise for speckled data, also can be categorized into image domain and transform domain methods. As an example for image domain techniques, in [16] the rotating kernel transform (RKT) filters are applied on an image with a set of oriented kernels and keep the largest filter output for each pixel. Other image domain methods based on enhanced Lee filter [17], median filter [17], symmetric nearest neighbor filter [17], and adaptive Wiener filter [17], and I-divergence
References
[1]
D. Huang, E. A. Swanson, C. P. Lin et al., “Optical coherence tomography,” Science, vol. 254, no. 5035, pp. 1178–1181, 1991.
[2]
J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE Journal on Selected Topics in Quantum Electronics, vol. 5, no. 4, pp. 1205–1215, 1999.
[3]
M. A. Choma, M. V. Sarunic, C. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Optics Express, vol. 11, no. 18, pp. 2183–2189, 2003.
[4]
M. E. J. van Velthoven, D. J. Faber, F. D. Verbraak, T. G. van Leeuwen, and M. D. de Smet, “Recent developments in optical coherence tomography for imaging the retina,” Progress in Retinal and Eye Research, vol. 26, no. 1, pp. 57–77, 2007.
[5]
J. M. Schmitt, S. H. Xiang, and K. M. Yung, “Speckle in optical coherence tomography,” Journal of Biomedical Optics, vol. 4, no. 1, pp. 95–105, 1999.
[6]
D. L. Marks, T. S. Ralston, and S. A. Boppart, “Speckle reduction by I-divergence regularization in optical coherence tomography,” Journal of the Optical Society of America A, vol. 22, no. 11, pp. 2366–2371, 2005.
[7]
Z. Jian, Z. Yu, L. Yu, B. Rao, Z. Chen, and B. J. Tromberg, “Speckle attenuation in optical coherence tomography by curvelet shrinkage,” Optics Letters, vol. 34, no. 10, pp. 1516–1518, 2009.
[8]
Z. Jian, L. Yu, B. Rao, B. J. Tromberg, and Z. Chen, “Three-dimensional speckle suppression in optical coherence tomography based on the curvelet transform,” Optics Express, vol. 18, no. 2, pp. 1024–1032, 2010.
[9]
S. Chitchian, M. A. Fiddy, and N. M. Fried, “Denoising during optical coherence tomography of the prostate nerves via wavelet shrinkage using dual-tree complex wavelet transform,” Journal of Biomedical Optics, vol. 14, no. 1, Article ID 014031, 2009.
[10]
V. Zlokolica, Lj. Jovanov, A. Pizurica, and W. Philips, “Wavelet-based denoising for OCT images, Interaction between image processing, optics and photonics,” in Symposium on Optical Science and Technology, Proceedings of SPIE, August 2007.
[11]
V. Gupta, C. C. Chan, C.-L. Poh, T. H. Chow, T. C. Meng, and N. B. Koon, “Computerized automation of wavelet based denoising method to reduce speckle noise in OCT images,” in Proceedings of the 5th International Conference on Information Technology and Applications in Biomedicine, pp. 120–123, May 2008.
[12]
P. Puvanathasan and K. Bizheva, “Speckle noise reduction algorithm for optical coherence tomography based on interval type II fuzzy set,” Optics Express, vol. 15, no. 24, pp. 15747–15758, 2007.
[13]
A. Pi？urica, L. Jovanov, B. Huysmans et al., “Multiresolution denoising for optical coherence tomography: a review and evaluation,” Current Medical Imaging Reviews, vol. 4, no. 4, pp. 270–284, 2008.
[14]
H. M. Salinas and D. C. Fernández, “Comparison of PDE-based nonlinear diffusion approaches for image enhancement and denoising in optical coherence tomography,” IEEE Transactions on Medical Imaging, vol. 26, no. 6, pp. 761–771, 2007.
[15]
K. M. Yung, S. L. Lee, and J. M. Schmilt, “Phase-domain processing of optical coherence tomography images,” Journal of Biomedical Optics, vol. 4, no. 1, pp. 125–136, 1999.
[16]
J. Rogowska and M. E. Brezinski, “Evaluation of the adaptive speckle suppression filter for coronary optical coherence tomography imaging,” IEEE Transactions on Medical Imaging, vol. 19, no. 12, pp. 1261–1266, 2000.
[17]
A. Ozcan, A. Bilenca, A. E. Desjardins, B. E. Bouma, and G. J. Tearney, “Speckle reduction in optical coherence tomography images using digital filtering,” Journal of the Optical Society of America A, vol. 24, no. 7, pp. 1901–1910, 2007.
[18]
D. C. Adler, T. H. Ko, and J. G. Fujimoto, “Speckle reduction in optical coherence tomography images by use of a spatially adaptive wavelet filter,” Optics Letters, vol. 29, no. 24, pp. 2878–2880, 2004.
[19]
S. G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, Calif, USA, 1998.
[20]
B. Vidakovic, Statistical Modeling by Wavelets, John Wiley & Sons, New York, NY, USA, 1999.
[21]
E. P. Simoncelli and E. H. Adelson, “Noise removal via Bayesian wavelet coring,” in Proceedings of the IEEE International Conference on Image Processing, pp. 379–382, Lausanne, Switzerland, September 1996.
[22]
E. P. Simoncelli, “Bayesian denoising of visual images in the wavelet domain,” in Bayesian Inference in Wavelet Based Models, P. Muller and B. Vidakovic, Eds., pp. 291–308, Springer, New York, NY, USA, 1999.
[23]
A. Achim, A. Bezerianos, and P. Tsakalides, “Novel Bayesian multiscale method for speckle removal in medical ultrasound images,” IEEE Transactions on Medical Imaging, vol. 20, no. 8, pp. 772–783, 2001.
[24]
J. M. Fadili and L. Boubchir, “Analytical form for a Bayesian wavelet estimator of images using the Bessel K form densities,” IEEE Transactions on Image Processing, vol. 14, no. 2, pp. 231–240, 2005.
[25]
P. A. Khazron and I. W. Selesnick, “Bayesian estimation of bessel K form random vectors in AWGN,” IEEE Signal Processing Letters, vol. 15, pp. 261–264, 2008.
[26]
M. S. Crouse, R. D. Nowak, and R. G. Baraniuk, “Wavelet-based statistical signal processing using hidden Markov models,” IEEE Transactions on Signal Processing, vol. 46, no. 4, pp. 886–902, 1998.
[27]
J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,” IEEE Transactions on Image Processing, vol. 12, no. 11, pp. 1338–1351, 2003.
[28]
H. Rabbani and S. Gazor, “Image denoising employing local mixture models in sparse domains,” IET Image Processing, vol. 4, no. 5, pp. 413–428, 2010.
[29]
A. Elmzoughi, A. Benazza-Benyahia, and J.-C. Pesquet, “An interscale multivariate statistical model for map multicomponent image denoising in the wavelet transform domain,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '05), pp. II45–II48, March 2005.
[30]
H. Rabbani and M. Vafadust, “Image/video denoising based on a mixture of Laplace distributions with local parameters in multidimensional complex wavelet domain,” Signal Processing, vol. 88, no. 1, pp. 158–173, 2008.
[31]
H. A. Chipman, E. D. Kolaczyk, and R. E. McCulloch, “Adaptive Bayesian wavelet shrinkage,” Journal of the American Statistical Association, vol. 92, no. 440, pp. 1413–1421, 1997.
[32]
M. J. Borran and R. D. Nowak, “Wavelet-based denoising using hidden Markov models,” in Proceedings of the IEEE Interntional Conference on Acoustics, Speech, and Signal Processing, pp. 3925–3928, May 2001.
[33]
J. K. Romberg, H. Choi, and R. G. Baraniuk, “Bayesian tree-structured image modeling using wavelet-domain hidden Markov models,” IEEE Transactions on Image Processing, vol. 10, no. 7, pp. 1056–1068, 2001.
[34]
A. Pi？urica, W. Philips, I. Lemahieu, and M. Acheroy, “A joint inter- and intrascale statistical model for Bayesian wavelet based image denoising,” IEEE Transactions on Image Processing, vol. 11, no. 5, pp. 545–557, 2002.
[35]
A. A. Bharath and J. Ng, “A steerable complex wavelet construction and its application to image denoising,” IEEE Transactions on Image Processing, vol. 14, no. 7, pp. 948–959, 2005.
[36]
L. ？endur and I. W. Selesnick, “Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency,” IEEE Transactions on Signal Processing, vol. 50, no. 11, pp. 2744–2756, 2002.
[37]
H. Rabbani, M. Vafadust, S. Gazor, and I. Selesnick, “Image denoising employing a bivariate cauchy distribution with local variance in complex wavelet domain,” in Proceedings of the 12th IEEE Digital Signal Processing Workshop and 4th IEEE Signal Processing Education Workshop, pp. 203–208, Grand Teton National Park, Wy, USA, September 2006.
[38]
J. Portilla, “Full blind denoising through noise covariance estimation using gaussian scale mixtures in the wavelet domain,” in Proceedings of the International Conference on Image Processing (ICIP '04), pp. 1217–1220, October 2004.
[39]
S. Lyu and E. P. Simoncelli, “Modeling multiscale subbands of photographic images with fields of Gaussian scale mixtures,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 31, no. 4, pp. 693–706, 2009.
[40]
H. Rabbani, R. Nezafat, and S. Gazor, “Wavelet-domain medical image denoising using bivariate laplacian mixture model,” IEEE Transactions on Biomedical Engineering, vol. 56, no. 12, pp. 2826–2837, 2009.
[41]
I. Selesnick and K. Li, “Video denoising using 2d and 3d dual-tree complex wavelet transforms,” in Wavelet Applications in Signal and Image Processing, vol. 5207 of Proceedings of SPIE, August 2003.
[42]
Z. Cai, T. H. Cheng, C. Lu, and K. R. Subramanian, “Efficient wavelet-based image denoising algorithm,” Electronics Letters, vol. 37, no. 11, pp. 683–685, 2001.
[43]
M. K. Mih？ak, I. Kozintsev, K. Ramchandran, and P. Moulin, “Low-complexity image denoising based on statistical modeling of wavelet coefficients,” IEEE Signal Processing Letters, vol. 6, no. 12, pp. 300–303, 1999.
[44]
H. Rabbani, M. Vafadust, P. Abolmaesumi, and S. Gazor, “Speckle noise reduction of medical ultrasound images in complex wavelet domain using mixture priors,” IEEE Transactions on Biomedical Engineering, vol. 55, no. 9, pp. 2152–2160, 2008.
[45]
L. ？endur and I. W. Selesnick, “Bivariate shrinkage with local variance estimation,” IEEE Signal Processing Letters, vol. 9, no. 12, pp. 438–441, 2002.
[46]
H. Rabbani, M. Vafadust, and S. Gazor, “Image denoising based on a mixture of bivariate gaussian distributions with local parameters in complex wavelet domain,” in Proceedings of the International Conference on Biomedical and Pharmaceutical Engineering, pp. 174–179, December 2006.
[47]
V. Katkovnik, K. Egiazarian, and J. Astola, “Adaptive window size image de-noising based on intersection of confidence intervals (ICI) rule,” Journal of Mathematical Imaging and Vision, vol. 16, no. 3, pp. 223–235, 2002.
[48]
I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Processing Magazine, vol. 22, no. 6, pp. 123–151, 2005.
[49]
B. Wang, Y. Wang, I. Selesnick, and A. Vetro, “Video coding using 3D dual-tree wavelet transform,” Eurasip Journal on Image and Video Processing, vol. 2007, Article ID 42761, 15 pages, 2007.
[50]
W. Jingwei, G. Xinbo, and Z. Juanjuan, “A video watermarking based on 3-D complex wavelet,” in Proceedings of the 14th IEEE International Conference on Image Processing, pp. V493–V496, September 2007.
[51]
J. Yang, Y. Wang, W. Xu, and Q. Dai, “Image and video denoising using adaptive dual-tree discrete wavelet packets,” IEEE Transactions on Circuits and Systems for Video Technology, vol. 19, no. 5, pp. 642–655, 2009.
[52]
I. Bayram and I. W. Selesnick, “A dual-tree rational-dilation complex wavelet transform,” IEEE Transactions on Signal Processing, vol. 59, no. 12, pp. 6251–6256, 2011.
[53]
K. Lee, M. Niemeijer, M. K. Garvin, Y. H. Kwon, M. Sonka, and M. D. Abramoff, “Segmentation of the optic disc in 3-D OCT scans of the optic nerve head,” IEEE Transactions on Medical Imaging, vol. 29, no. 1, pp. 159–168, 2010.
[54]
W. Y. Lo and I. W. Selesnick, “Wavelet-domain soft-thresholding for non-stationary noise,” in Proceedings of the IEEE International Conference on Image Processing, pp. 1441–1444, October 2006.
[55]
R. M. Willett and R. D. Nowak, “Fast multiresolution photon-limited image reconstruction,” in Proceedings of the 2nd IEEE International Symposium on Biomedical Imaging: Macro to Nano (ISBI '04), vol. 2, pp. 1192–1195, April 2004.
[56]
A. Pi？urica, W. Philips, I. Lemahieu, and M. Acheroy, “A versatile wavelet domain noise filtration technique for medical imaging,” IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 323–331, 2003.
[57]
S. Gupta, R. C. Chauhan, and S. C. Saxena, “Robust non-homomorphic approach for speckle reduction in medical ultrasound images,” Medical and Biological Engineering and Computing, vol. 43, no. 2, pp. 189–195, 2005.
[58]
S. Gupta, L. Kaur, R. C. Chauhan, and S. C. Saxena, “A versatile technique for visual enhancement of medical ultrasound images,” Digital Signal Processing, vol. 17, no. 3, pp. 542–560, 2007.
[59]
S. Yan, J. Yuan, M. Liu, and C. Hou, “Speckle noise reduction of ultrasound images based on an undecimated wavelet packet transform domain nonhomomorphic filtering,” in Proceedings of the 2nd International Conference on Biomedical Engineering and Informatics, pp. 1–5, October 2009.
[60]
R. F. Wagner, M. F. Insana, and D. G. Brown, “Statistical properties of radio-frequency and envelope-detected signals with applications to medical ultrasound,” Journal of the Optical Society of America A, vol. 4, no. 5, pp. 910–922, 1987.
[61]
P. M. Shankar, “Ultrasonic tissue characterization using a generalized Nakagami model,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 48, no. 6, pp. 1716–1720, 2001.
[62]
P. M. Shankar, J. M. Reid, H. Ortega, C. W. Piccoli, and B. B. Goldberg, “Use of non-Rayleigh statistics for the identification of tumors in ultrasonic B-scans of the breast,” IEEE Transactions on Medical Imaging, vol. 12, no. 4, pp. 687–692, 1993.
[63]
P. M. Shankar, “A model for ultrasonic scattering from tissues based on the K distribution,” Physics in Medicine and Biology, vol. 40, no. 10, pp. 1633–1649, 1995.
[64]
I. K. Eom and Y. S. Kim, “Wavelet-based denoising with nearly arbitrarily shaped windows,” IEEE Signal Processing Letters, vol. 11, no. 12, pp. 937–940, 2004.
[65]
W. Drexler and J. G. Fujimoto, “State-of-the-art retinal optical coherence tomography,” Progress in Retinal and Eye Research, vol. 27, no. 1, pp. 45–88, 2008.
[66]
M. D. Abramoff, M. K. Garvin, and M. Sonka, “Retinal imaging and image analysis,” IEEE Reviews in Biomedical Engineering, vol. 3, pp. 169–208, 2010.
[67]
C. Ercole, A. Foi, V. Katkovnik, and K. Egiazarian, “Spatio-temporal pointwise adaptive denoising of video: 3D non-parametric approach,” in Proceedings of the 1st International Workshop on Video Process and Quality Metrics for Consumer Electronics (VPQM '05), 2005.
[68]
Y. M. Lu and M. N. Do, “Multidimensional directional filter banks and surfacelets,” IEEE Transactions on Image Processing, vol. 16, no. 4, pp. 918–931, 2007.
[69]
L. Ying, L. Demanet, and E. Candès, “3D discrete curvelet transform,” in Wavelets XI, vol. 5914 of Proceedings of SPIE, pp. 1–11, August 2005.
