IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010 3545 Complex Gaussian Scale Mixtures of Complex Wavelet Coefficients Yothin Rakvongthai, Student Member, IEEE, An P. N.Vo, Student Member, IEEE, and Soontorn Oraintara, Senior Member, IEEE Abstract—In this paper, we propose the complex Gaussian scale mixture (CGSM) to model the complex wavelet coefficients as an extension of the Gaussian scale mixture (GSM), which is for real- valued random variables to the complex case. Along with some re- lated propositions and miscellaneous results, we present the prob- ability density functions of the magnitude and phase of the com- plex random variable. Specifically, we present the closed forms of the probability density function (pdf) of the magnitude for the case of complex generalized Gaussian distribution and the phase pdf for the general case. Subsequently, the pdf of the relative phase is derived. The CGSM is then applied to image denoising using the Bayes least-square estimator in several complex transform do- mains. The experimental results show that using the CGSM of com- plex wavelet coefficients visually improves the quality of denoised images from the real case. Index Terms—Complex Gaussian scale mixtures (CGSMs), com- plex wavelets, magnitude, phase. I. INTRODUCTION I T IS admitted that statistical modeling in the wavelet do- main is favorable for many image-processing applications, such as denoising, compression, and classification, because of the wavelet’s capability of analyzing and representing images. This ability can be further improved by using complex-valued wavelets rather than real-valued wavelets. It is mainly because the complex wavelets are based on complex-valued sinusoids constituting an analytic signal [1]. Therefore, the real and imag- inary parts of a complex-valued wavelet form a Hilbert trans- form pair. In addition, the advantages of complex wavelets over real wavelets are directly related to the complex magnitude and phase. For example, the magnitude of a complex coefficient pos- sesses the shift-invariance property while a real wavelet coef- ficient is shift varying. Furthermore, it is well known that the magnitude of a complex wavelet coefficient better represents a singularity than either the real/imaginary part of the complex coefficient or the value of a real wavelet coefficient. Besides magnitude information, phase information plays a key role in Manuscript received July 21, 2009; accepted February 11, 2010. Date of pub- lication March 25, 2010; date of current version June 16, 2010 The associate editor coordinating the review of this manuscript and approving it for publica- tion was Dr. Ivan W. Selesnick. Y. Rakvongthai and S. Oraintara are with the Department of Electrical Engi- neering, University of Texas at Arlington, Arlington, TX 76019 USA (e-mail: yothin.rakvongthai@mavs.uta.edu; oraintar@uta.edu). A. P. N. Vo is with the Feinstein Institute for Medical Research, North Shore LIJ Health System, Manhasset, NY 11030 USA (e-mail: an- phuocnhu.vo@mavs.uta.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2010.2046698 image processing as well. A famous example that shows the im- portance of phase is in [2], where the Fourier phase is shown to contain more information about image features than the mag- nitude. Moreover, there is also a connection between the phase of a complex wavelet coefficient and image features, such as edges. Specifically, the phase of a complex coefficient in each scale near an edge varies linearly with its distance to the edge [3]. In addition, the coefficient phases across scales at an edge are aligned [4], [5]. These intrascale and interscale relationships have been used in some image-processing applications (e.g., in [3]–[7]). All of these point out the significance of the magni- tude and phase information of complex coefficients. In the prob- abilistic framework, an appropriate model for complex random variables is required to fully utilize complex wavelet coefficients as well as their magnitude and phase information. Indeed, the probability density function (pdf) of a complex random vector (as a general form of a complex random vari- able) is the joint pdf of two real-valued random vectors repre- senting the real and imaginary parts. In order to express the pdf of a complex random vector as an analytic function of the com- plex vector itself, we need an additional assumption. For a class of distributions whose pdfs depend only on the covariance ma- trix, such as the Gaussian distribution, the assumption is that the covariance matrices of the real and imaginary parts are equal, and that the sum of the two cross-covariance matrices is zero. With such an assumption, the pdf of a complex random vector whose real and imaginary parts are jointly Gaussian can be ex- pressed as a function of a complex vector, and has been studied in [8]–[10] as the complex Gaussian pdf. There are a number of research studies on the statistical mod- eling based on complex wavelets. For instance, in [11], the com- plex hidden Markov tree (CHMT) model is proposed for com- plex coefficients obtained from the dual-tree complex wavelet transform (DT- WT) [12]. The bivariate model with the bi- variate shrinkage in [13] and the bivariate -stable distribution in [14] are used for image denoising. In [15], the Cartesian and polar forms of marginal densities of DT- WT coefficients due to the Gaussian signals are studied. In [16], the complex general- ized Gaussian distribution (CGGD) is used for image modeling in the complex wavelet domain. The Gaussian scale mixture (GSM) [17] model characterizes the set of real-valued random vectors that can be expressed as the product of a zero-mean Gaussian random vector and an independent positive random variable (i.e., a GSM random vector is a mixture of a possibly infinite number of zero-mean Gaussian random vectors). The GSM distribution encompasses many known distributions as special cases, such as the Stu- dent’s t-distribution, the -stable distribution, the generalized 1053-587X/$26.00 © 2010 IEEE