Denoising Magnetic Resonance Images using Fourth Order Complex Diffusion Jeny Rajan, Ben Jeurissen, Jan Sijbers IBBT Vision Lab University of Antwerp Wilrijk, 2610 Antwerp, Belgium jeny.rajan, ben.jeurissen,
[email protected] Abstract Complex diffusion is a comparatively new Partial Differential Equations (PDE) based method introduced for removing noise from images. The efficiency of 2nd order complex diffusion for image denoising is already proved by many researchers. 2nd order non linear complex diffusion can behave like 3rd and 4th order real PDEs enabling a variety of new options with standard 2nd order numerical schemes. Extending 2nd order non linear complex diffusion to 4th order can produce a much better result. In this paper we present a 4th order non linear complex diffusion. Our experimental results show that this 4th order complex PDE is a good choice for denoising Magnetic Resonance images. The efficacy of the algorithm is demonstrated on both simulated and real Magnetic Resonance images.
1. Introduction Denoising is a crucial step required for correct interpretation of Magnetic Resonance (MR) data. Depending on specific diagnostic tasks, high spatial resolution and high contrast may be required for MR images, whereas for image processing applications, a high SNR is usually necessary because most of the algorithms are very sensitive to noise [5]. It has been shown that noise in Diffusion Weighted (DW) MR Images introduces errors in the estimation and sorting of the diffusion tensor eigenvalues and the derived anisotropy measures [6]. The quality of the estimated tensor field in Diffusion Tensor Imaging (DTI) also depends on the noise level. Hence, it is clear that an efficient denoising algorithm can have a significant impact on MR image processing tasks. Various approaches for removing noise from MRI have been proposed. Most of these methods are based on Wavelets or Partial Differential Equations (PDEs) [2][1],[14]. The goal of wavelet domain filtering is to obtain a better estimate of the noise free image wavelet co-
K. Kannan Medical Imaging Research Group NeST (P) Ltd Trivandrum 56, India
[email protected] efficients by filtering the observed coefficients drj (k) at any level j, spatial position k, and wavelet orientation r [5]. In PDE based methods, the basic idea is to deform an image, a curve, or a surface with a PDE and obtain the expected results as a solution to this equation [2]. Both wavelet and PDE based methods have their own merits and demerits, even though wavelet based methods seem to be more flexible for MR denoising. A comparative study of both methods is not in the scope of this work. Interested readers can refer to [5],[4]-[7]. In the proposed method, we used a PDE based approach for denoising MR images. PDE based methods, especially anisotropic diffusion (Perona-Malik), have proved to be particularly effective in pre-filtering MR images [5]. A variant of standard anisotropic diffusion method was extended by Yang [14] using both the local intensity orientation and an anisotropic measure of level counters, instead of utilizing local gradients to control the anisotropism of the filters. Even though anisotropic diffusion seems attractive, it assumes image noise to be Gaussian distributed. When processing magnitude MR data, a Gaussian assumption for image noise is not acceptable as it can be shown to be Rice distributed [9]. Using anisotropic diffusion or its variants can generate a bias in the magnitude MR data (which increases with decreasing SNR). Sijbers et al [11] proposed an adaptive anisotropic diffusion method to tackle this problem. However, several papers [3],[15],[13] have noted that 2nd order anisotropic diffusions are ill posed in the sense that images close to each other are likely to diverge during the diffusion process. Solutions like 4th order PDEs and complex PDEs are suggested to solve this problem. In this paper, we propose a 4th order, non linear, complex diffusion algorithm for denoising MR images. Complex diffusion is studied as an efficient tool for image denoising, which certainly has an edge over the ordinary PDEs. Simulation and real experiments based on the proposed 4th order complex PDE show that its performance with respect to the Peak signal-to-noise ratio (PSNR) and the Structural Similarity Index Matrix (SSIM) is superior compared to 2nd
order complex diffusion. This paper is organized as follows. Section 2 gives an overview of bias removal in MR images and explains the proposed 4th order complex PDE. In Section 3, we will discuss the results of the proposed method. Finally, conclusions and remarks are drawn in Section 4.
2. Proposed Method The proposed 4th order complex PDE is an improvement over the one proposed by Gilboa et al [3]. Since the method assumes the noise to be Gaussian distributed, we first did a bias reduction from the squared magnitude MR image as proposed in [7].
the maximum-minimum principle, it preserves other desirable mathematical and perceptual properties. When an image is processed with complex diffusion, we will get low frequency components (plateaus) of the image in the real plane and high frequency components (edges) in the imaginary plane [8]. The components in the real and imaginary plane are almost equivalent to that of the image convolved with a Gaussian and Laplacian of Gaussian (LOG) at various scales. Here we propose a 4th order non linear complex diffusion, which is an improvement over its 2nd order counterpart in terms of preserving edges. The method is based on: I t+1 = −∇2 (c(=(I t ))∇2 I t (5) where =(.) takes the imaginary part and c(=(I t )) =
2.1. Bias Removal Nowak [7] proposed a method to remove the bias, making use of the properties of the squared magnitude image. If M is the Rician distributed magnitude image, S is the unknown, noiseless image of interest, then using the moments of the non central chi square distribution, the mean and variance of the squared magnitude image M 2 are E M 2 = S 2 + 2σ 2 (1) Var[M 2 ]
=
4σ 2 (S 2 + σ 2 )
(2)
It is important to realize that Eq. (1) has a fixed, signal independent bias [10]. This property can be exploited to reduce the bias by subtracting an estimate of the bias from each pixel in the squared magnitude image as follows: c2 = M 2 − 2b M σ2 where σ b2 =
1 2 M 2
,
.
(3)
(4)
Even though this method does not remove the bias in the magnitude image completely, there will be a clear contrast enhancement.
2.2. Denoising using 4th order complex diffusion The concept of complex diffusion in image processing was introduced by Gilboa et al. [3] as an alternative to 2nd order anisotropic diffusion, which introduces blocky effects in images while processing. This blocky effect is inherent in the nature of ordinary second order equations; it can be avoided by using complex diffusion. Complex diffusion is derived by combining the standard diffusion equation with the free Schr¨odinger equation [3]. Even though it violates
eiθ
(6)
t
) 2 1 + ( =(I kθ )
and with initial conditions: