Abstract
Denoising an image, while retaining the important features of the image, has been a fundamental problem in image processing. Dual-tree complex wavelet transform is a recently created transform that offers both near shift invariance and improved directional selectivity properties. This transform has been used in many techniques, including denoising. However, these techniques have used the real and imaginary components of the complex-valued sub-band coefficients separately. This paper proposes the use of coefficient magnitudes to provide an improvement in image denoising. Our proposed algorithm is based on the maximum a posteriori estimator, wherein the heavy-tailed scale mixtures of bivariate Rayleigh distribution are considered as the noise-free wavelet coefficient magnitudes’ prior distribution. Also, in our work, the necessary parameters of the bivariate distributions are estimated in a locally adaptive way to improve the denoising results via using the correlation between the amplitudes of neighbor coefficients. Simulation results delineate the performance of the proposed algorithm in both MSSIM and PSNR metrics.
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References
A. Achim, P. Tsakalides, A. Bezerianos, SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling. IEEE Trans. Geosci. Remote Sens. 41(8), 1773–1784 (2003). https://doi.org/10.1109/TGRS.2003.813488
A. Achim, D. Herranz, E.E. Kuruoglu, Astrophysical image denoising using bivariate isotropic cauchy distributions in the undecimated wavelet domain, in International Conference on Image Processing, 2004, ICIP ‘04, vol. 1222, 24–27 Oct 2004, pp. 1225–1228
A. Achim, E.E. Kuruoglu, Image denoising using bivariate α-stable distributions in the complex wavelet domain. IEEE Signal Process. Lett. 12(1), 17–20 (2005). https://doi.org/10.1109/LSP.2004.839692
A. Achim, A. Bezerianos, P. Tsakalides, Novel Bayesian multiscale method for speckle removal in medical ultrasound images. IEEE Trans. Med. Imaging 20(8), 772–783 (2001). https://doi.org/10.1109/42.938245
S.G. Chang, Y. Bin, M. Vetterli, Spatially adaptive wavelet thresholding with context modeling for image denoising. IEEE Trans. Image Process. 9(9), 1522–1531 (2000). https://doi.org/10.1109/83.862630
G. Chen, W. Zhu, W. Xie, Wavelet-based image denoising using three scales of dependency. IET Image Process. 6(6), 756–760 (2012). https://doi.org/10.1049/iet-ipr.2010.0408
D. Cho, T.D. Bui, Multivariate statistical modeling for image denoising using wavelet transforms. Signal Process. Image Commun. 20, 77–85 (2005)
M.S. Crouse, R.D. Nowak, R.G. Baraniuk, Wavelet-based statistical signal processing using hidden Markov models. IEEE Trans. Signal Process. 46(4), 886–902 (1998). https://doi.org/10.1109/78.668544
K. Dabov, A. Foi, V. Katkovnik, K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007). https://doi.org/10.1109/TIP.2007.901238
D.L. Donoho, I.M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90(432), 1200–1224 (1995). https://doi.org/10.2307/2291512
R.A. Fisher, Moments and product moments of sampling distributions. Proc. Lond. Math. Soc. s2-30(1), 199–238 (1930). https://doi.org/10.1112/plms/s2-30.1.199
D. Guang, Generalized Wiener estimation algorithms based on a family of heavy-tail distributions, in IEEE International Conference on Image Processing 2005, 14–14 Sept 2005, pp. I-457
P.R. Hill, C.N. Canagarajah, D.R. Bull, Image segmentation using a texture gradient based watershed transform. IEEE Trans. Image Process. 12(12), 1618–1633 (2003)
P. Hill, A. Achim, D. Bull, The undecimated dual tree complex wavelet transform and its application to bivariate image denoising using a Cauchy model, in 2012 19th IEEE International Conference on Image Processing, 30 Sept.–3 Oct. 2012, pp. 1205–1208
P.R. Hill, N. Anantrasirichai, A. Achim, M.E. Al-Mualla, D.R. Bull, Undecimated dual-tree complex wavelet transforms. Signal Process. Image Commun. 35, 61–70 (2015). https://doi.org/10.1016/j.image.2015.04.010
P.R. Hill, A.M. Achim, D.R. Bull, M.E. Al-Mualla, Dual-tree complex wavelet coefficient magnitude modelling using the bivariate Cauchy–Rayleigh distribution for image denoising. Signal Process. 105, 464–472 (2014). https://doi.org/10.1016/j.sigpro.2014.03.028
W. Hu, Calibration of multivariate generalized hyperbolic distributions using the EM algorithm, with applications, in Risk Management, Portfolio Optimization And Portfolio Credit Risk (2005)
Q. Huynh-Thu, M. Ghanbari, Scope of validity of PSNR in image/video quality assessment. Electron. Lett. 44(13), 800–801 (2008). https://doi.org/10.1049/el:20080522
J. Ilow, D. Hatzinakos, Applications of the empirical characteristic function to estimation and detection problems. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). Signal Process. 65(2), 199–219 (1998). https://doi.org/10.1016/S0165-1684(97)00219-3
S. Kaur, N. Singh, Image denoising techniques: a review. Int. J. Innov. Res. Comput. Commun. Eng. 2(6), 699–705 (2014)
N. Kingsbury, The dual-tree complex wavelet transform: a new technique for shift invariance and directional filters, in IEEE Digital Signal Processing Workshop (1998)
S. Kotz, S. Nadarajah, Multivariate T-distributions and their applications (Cambridge University Press, Cambridge, 2004)
S. Kotz, T. Kozubowski, K. Podgorski, in The Laplace Distribution and Generalizations (2001)
R. Kwitt, A. Uhl, Lightweight probabilistic texture retrieval. IEEE Trans. Image Process. 19(1), 241–253 (2010). https://doi.org/10.1109/TIP.2009.2032313
M. Li, Z. Jia, J. Yang, Y. Hu, D. Li, An algorithm for remote sensing image denoising based on the combination of the improved BiShrink and DTCWT. Procedia Eng. 24, 470–474 (2011). https://doi.org/10.1016/j.proeng.2011.11.2678
S.G. Mallat, A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11(7), 674–693 (1989). https://doi.org/10.1109/34.192463
M.K. Mihcak, I. Kozintsev, K. Ramchandran, P. Moulin, Low-complexity image denoising based on statistical modeling of wavelet coefficients. IEEE Signal Process. Lett. 6(12), 300–303 (1999). https://doi.org/10.1109/97.803428
M. Miller, N. Kingsbury, Image modeling using interscale phase properties of complex wavelet coefficients. IEEE Trans. Image Process. 17(9), 1491–1499 (2008). https://doi.org/10.1109/TIP.2008.926147
M. Miller, N. Kingsbury, Image denoising using derotated complex wavelet coefficients. IEEE Trans. Image Process. 17(9), 1500–1511 (2008). https://doi.org/10.1109/TIP.2008.926146
H. Naimi, A.B.H. Adamou-Mitiche, L. Mitiche, Medical image denoising using dual tree complex thresholding wavelet transform and Wiener filter. J. King Saud Univ. Comput. Inf. Sci. 27(1), 40–45 (2015). https://doi.org/10.1016/j.jksuci.2014.03.015
B.N. Narayanan, R.C. Hardie, E.J. Balster, Multiframe adaptive Wiener filter super-resolution with JPEG2000-compressed images. EURASIP J. Adv. Signal Process. 2014(1), 55 (2014)
M. Nasri, H. Nezamabadi-pour, Image denoising in the wavelet domain using a new adaptive thresholding function. Neurocomputing 72(4), 1012–1025 (2009). https://doi.org/10.1016/j.neucom.2008.04.016
H. Om, M. Biswas, A generalized image denoising method using neighbouring wavelet coefficients. Signal Image Video Process 9(1), 191–200 (2015)
J. Portilla, V. Strela, M.J. Wainwright, E.P. Simoncelli, Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Trans. Image Process. 12(11), 1338–1351 (2003). https://doi.org/10.1109/TIP.2003.818640
H. Rabbani, M. Vafadust, S. Gazor, I. Selesnick, Image denoising employing a bivariate Cauchy distribution with local variance in complex wavelet domain, in 2006 IEEE 12th Digital Signal Processing Workshop and 4th IEEE Signal Processing Education Workshop, 24–27 Sept 2006, pp. 203–208
H. Rabbani, M. Vafadoost, Wavelet based image denoising based on a mixture of Laplace distributions. Iran. J. Sci. Technol. 30(6), 711 (2006)
H. Rabbani, M. Vafadust, Image/video denoising based on a mixture of Laplace distributions with local parameters in multidimensional complex wavelet domain. Signal Process. 88(1), 158–173 (2008)
Y. Rakvongthai, A.P.N. Vo, S. Oraintara, Complex Gaussian scale mixtures of complex wavelet coefficients. IEEE Trans. Signal Process. 58(7), 3545–3556 (2010). https://doi.org/10.1109/TSP.2010.2046698
H. Sadreazami, M.O. Ahmad, M.N.S. Swamy, A study on image denoising in contourlet domain using the alpha-stable family of distributions. Signal Process. 128, 459–473 (2016). https://doi.org/10.1016/j.sigpro.2016.05.018
I.W. Selesnick, R.G. Baraniuk, N.C. Kingsbury, The dual-tree complex wavelet transform. IEEE Signal Process. Mag. 22(6), 123–151 (2005). https://doi.org/10.1109/MSP.2005.1550194
L. Sendur, I.W. Selesnick, Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency. IEEE Trans. Signal Process. 50(11), 2744–2756 (2002). https://doi.org/10.1109/TSP.2002.804091
L. Sendur, I.W. Selesnick, Bivariate shrinkage with local variance estimation. IEEE Signal Process. Lett. 9(12), 438–441 (2002). https://doi.org/10.1109/LSP.2002.806054
E.P. Simoncelli, Bayesian denoising of visual images in the wavelet domain, in Bayesian Inference in Wavelet-Based Models, ed. by P. Müller, B. Vidakovic (Springer, New York, 1999), pp. 291–308
M. Smith, C. Rose, Mathematical Statistics with Mathematica (Springer, Berlin, 2002)
C. Su, L.K. Cormack, A.C. Bovik, Closed-form correlation model of oriented bandpass natural images. IEEE Signal Process. Lett. 22(1), 21–25 (2015). https://doi.org/10.1109/LSP.2014.2345765
J. Wang, M.R. Taaffe, Multivariate mixtures of normal distributions: properties, random vector generation, fitting, and as models of market daily changes. INFORMS J. Comput. 27(2), 193–203 (2015)
X.Y. Wang, N. Zhang, H.-L. Zheng, Y.-C. Liu, Extended shearlet HMTmodel-based image denoising using BKF distribution. J Math Imaging Vis 54(3), 301–319 (2015)
M. Yin, W. Liu, X. Zhao, Q.-W. Guo, R.-F. Bai, Image denoising using trivariate prior model in nonsubsampled dual-tree complex contourlet transform domain and non-local means filter in spatial domain. Optik 124(24), 6896–6904 (2013). https://doi.org/10.1016/j.ijleo.2013.05.132
W. Zhou, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli, Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004). https://doi.org/10.1109/TIP.2003.819861
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Appendices
Appendix 1
1.1 Univariate Rayleigh–Student’s t Distribution
Using Eq. (12) and by assuming gamma distribution for \( u \) as follows:
we obtain
Let
Also, we have
Therefore,
Using Eqs. (53), (54), and (56), we have
By substituting \( A \) into Eq. (57) and after some algebra, we have
1.2 Univariate Rayleigh–Laplace Distribution
Using Eq. (12) and by assuming \( u \sim {\text{IG}}\left( {1,\frac{1}{2}} \right) \) as follows:
we will have:
Also, we have [17]
Using Eqs. (60) and (61), we derive
Appendix 2
Using Eq. (19) and by the assuming gamma distribution for \( u, \) we have
Let
therefore
By substituting \( A \) into Eq. (65), we have
2.1 Bivariate Rayleigh–Laplace Distribution
Using Eq. (19) and by assuming \( u \sim {\text{IG}}\left( {1,\frac{1}{2}} \right), \) we derive
Using Eqs. (61) and (67), we derive
Appendix 3
In probability theory and statistics, kurtosis is a measure of the tailedness of the probability distribution of a random variable and variance is the expectation of the squared deviation of a random variable from its mean and is defined as:
where \( M_{4} \) and \( M_{2} \) are fourth and second central moments, respectively.
The cumulants \( k_{n} \) of a random variable \( x \) are defined via the cumulant generating function \( (K(t)) \), which is the natural logarithm of the moment generation function:
The fourth and second cumulants are related to the central moments by the following equations [11, 44]:
Kurtosis and variance in terms of cumulants are:
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Saeedzarandi, M., Nezamabadi-pour, H. & Jamalizadeh, A. Dual-Tree Complex Wavelet Coefficient Magnitude Modeling Using Scale Mixtures of Rayleigh Distribution for Image Denoising. Circuits Syst Signal Process 39, 2968–2993 (2020). https://doi.org/10.1007/s00034-019-01291-y
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DOI: https://doi.org/10.1007/s00034-019-01291-y