EM-BASED ESTIMATION OF SPATIALLY VARIANT CORRELATED IMAGE NOISE Bart Goossens, Aleksandra Piˇ zurica * , Wilfried Philips Ghent University - TELIN - IPI - IBBT Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium bart.goossens@telin.ugent.be ABSTRACT In image denoising applications, noise is often correlated and the noise energy and correlation structure may even vary with the position in the image. Existing noise reduction and estimation methods are usually designed for stationary white Gaussian noise and generally work less efficient in this case because of the noise model mismatch. In this paper, we propose an EM algorithm for the estimation of spatially variant (nonstationary) correlated image noise in the wavelet domain. In particular, we study additive white Gaussian noise filtered by a space-variant linear filter. This general noise model is applicable to a wide variety of practical situations, including noise in Computed Tomography (CT). Results demonstrate the effectiveness of the proposed solution and its robustness to signal structures. Index Terms— Noise estimation, Image restoration, Correlated noise I. INTRODUCTION Gaussian noise processes are characterized entirely by their second order statistical moments [1]. On the other hand recent studies (e.g. [2], [3]) have shown that signal features in the bandpass and highpass subbands of a given multiresolution representation are not Gaussian and require the specification of the fourth order moment, the kurtosis. This property can be exploited to distinguish signal information from noise and this has succesfully been applied to the estimation of stationary correlated noise [4]. However, in practice, we encounter many situations where the noise energy and correlation structure depends on the position in the image (nonstationary noise). Even for local stationary Gaussian noise processes, that have properties that change slowly in space, the estimation is still difficult because only local information can be used. Therefore it is useful to estimate the noise properties in well structured bases that approximately diagonalize noise covariance matrices, such that fewer observations are needed. An example are the local cosine bases in [1]. In this work, we assume an additive noise process, that is generated by sending white Gaussian noise through a linear spa- tially variant filter. We employ a wavelet basis that has similar ”sparsifying” properties as the local cosine bases, but that are better suited in representing nonstationary signal features like edges and textures. Wavelet bases provide a non-uniform partitioning of the time-frequency plane which allows retrieving information both in specific frequency bands and at spatial positions. We propose an Expectation-Maximization (EM) algorithm for the wavelet domain estimation of the noise covariance function. The estimated noise properties can be directly plugged in into recent wavelet domain denoising methods (e.g. [2], [5], [6], [7]). On the other hand, this allows us to study noise properties in regions where we have * A. Piˇ zurica is a postdoctoral researcher of the Fund for the Scientific Research in Flanders (FWO) Belgium no signal-free patches, e.g. in medical images. First we treat the case where the noise Power Spectral Density (PSD) is the same throughout the image but where the local noise energy is allowed to vary (we will call this separable space-varying spectrum, see further). Next, we study the more general case where the local PSD is position-dependent (denoted as nonseparable space-varying spectrum). The remainder of this paper is as follows: in Section II we introduce some basic concepts that are used throughout this paper. In Section III we explain the EM algorithm that is used in the wavelet domain, for both separable and nonseperable space-varying noise spectra. Implementation aspects are discussed in Section IV. Results and a discussion are given in Section V. Finally, Section VI concludes this paper. II. BASIC CONCEPTS II-A. Local stationarity and space-varying spectra Let Y (t), t ∈ Z 2 be a real-valued zero-mean random process with covariance function (r ∈ Z 2 ): R(t, r)=E {Y (t)Y (t + r)} (1) If the process is stationary then the covariance only depends on the distance between two points and not on their absolute positions: R(t, r)= R(0, r). Furthermore, we say that a process is locally stationary, if in the neighbourhood of any v ∈ Z 2 , there exists a square window δ(v) of size l(v), centered at position v, where the process can be approximated by a stationary one : for t ∈ δ(v) and for |r|≤ l(v)/2, the covariance is well approximated by [1] E {Y (t)Y (t + r)}≈ E {Y (v)Y (v + r)} = R(v, r) (2) We define the space-varying spectrum (SVS) of Y (t) as the Discrete Time Fourier transform (DTFT) of R(v, r) with respect to r: S(v, ω)= r∈Z 2 R(v, r) exp(−j 〈r, ω〉) (3) where 〈·, ·〉 denotes the inner product. For stationary processes, the SVS reduces to the Power Spectral Density (PSD). We say that the SVS is separable if it can be factored as S(v, ω)= S0(v)S1(ω) with 1 2π π -π S1(ω)dω =1. The first component S0(v) represents the variance at position v while the second component S1(ω) denotes the normalized Power Spectral Density (PSD). II-B. Spatially variant filtering of White Noise A specific class of locally stationary processes is obtained by the spatially variant filtering of white noise. Let ǫ(t) denote a white Gaussian noise process, then Y (t) is obtained as: Y (t)= v∈Z 2 ǫ(v)K(t, t − v) (4) with K(t, r) the impulse response of a linear spatially variant filter with DTFT A(t, ω). The covariance function of Y (t) is then given