Removal of Impulsive Noise from MRI Images using Quadratic Filter

Removal of Impulsive Noise from MRI Images using Quadratic Filter

Leslie Raju Thomas1 , Gopika Krishnan2 , Ansy Mol R S 3 and Aswany Roy 4

Department of Electronics and Communication, College of Engineering, Karunagappally

Abstract In the past few decades there had been signicant progress in the theory of linear time invariant (LTI) systems which successfully can model many real time systems and phenomena. But most of the natural systems and processes are inherently nonlinear, and there arise the need for nonlinear systems. The power series proposed by Vito Volterra can model mild polynomial nonlinearities. This paper focuses on quadratic volterra lter for removing impulsive noise from MRI images, the acquisition of which happens at extremely noisy conditions. By conventional ltering methods, it is very difcult to remove impulsive noise added with the raw MRI data due to the rapidly changing magnetic eld. Comparisons of performance parameters with conventional systems establish the superiority of quadratic systems in removing impulsive noise.

Keywords MRI, Impulsive Noise, Volterra Series, Quadratic Filter, Singular Value Decomposition.

1. INTRODUCTION

   Magnetic resonance imaging (MRI), is a medical imaging technique used in radiology to investigate the anatomy and function of the body. This imaging technique has a wide range of applications in medical diagnosis such as to nd problems such as tumors, bleeding, injury, blood vessel diseases, or infection. Unlike images generated by digital cameras, images generated by medical imaging systems such as MRI machine are largely corrupted by impulsive noise as the acquisition of image happens under strong and rapidly varying magnetic elds, typically 1.5 Tesla to 2 Tesla. So noise removal is essential for improving the interpretability of information in images and for generating perceptually more pleasing images from the given input images.

   Both linear and nonlinear lters can be used for noise removal. Linear lters are useful in many image processing applications. The obvious advantage of a linear lter is its simplicity in design and implementation. But linear systems fail when it comes to removing impulsive noise added with the raw MRI data due to the rapidly changing magnetic eld. Besides, medical images are formed by complex nonlinear processes and the inherent nonlinearities in them cannot easily be modeled by conventional linear systems. Mild polynomial nonlinearities can be modeled by Volterra series. It is a power series that add quadratic, cubic and higher order components in parallel with the linear term. So this series has the added advantage that existing LTI systems can be augmented by adding parallel polynomial lters to yield improved performance. It is widely observed that much of the

   nonlinear behavior can be modeled with the quadratic term alone. So the present work was to design and implement a quadratic lter that can take into account the inherent nonlinearities in image formation while removing the impulsive noise.

2. DISCRETE VOLTERRA SERIES

   Although the theory of linear systems is very advanced and useful, most of the real life and practical systems are nonlinear, making them difcult for mathematical modeling. Mild polynomial nonlinearities can be modeled by Volterra power series. An Nth order Volterra lter with input vector x[n] and output vector y[n] is realized by [1]

   (1)

   where r indicates the order of nonlinearity, with r=1 implying a linear system, r=2 implying a quadratic system and so forth. The term h0 denotes the output offset when no input is present and the term hr[n1,n2,…nr ] denotes the rth order Volterra kernel. Identication of this kernel for a nonlinear system is the chief issue in designing polynomial systems [2].

   In practical systems, the polynomial nonlinearities are often comprised of the quadratic term alone. So it is proposed that a two dimensional quadratic lter can model and process inherent nonlinearities in medical images, resulting in better noise removal and sharper edges.

3. TWO DIMENSIONAL QUADRATIC VOLTERRA FILTER

   The two dimensional quadratic lter is governed by the equation Eq. 2

   (2)

   Out of the four indices m11 , m12 , m21 , m22 of the kernel H2 , two stem from the quadratic nature of the kernel and the remaining two denote the two dimensions of the signal processed. Eq. 2 is represented in the matrix form as in Eq. 3.

   (3)

   where

   Fig.1. Flow of work

   where each sub-matrix H[i, j] is given by

   (4)

   (5)

   The solution for this minimum error is selected as the lter kernel. As the direct implementation of this kernel is computationally complex, singular value decomposition is done on H2noise to yield an approximate realization . In the third phase, the kernel is tested with known images for ascertaining the improvement in signal to noise ratio and peak signal to noise ratio before applying to noisy MRI images. The noisy MRI image is taken and is subjected to ltering by as implemented for removing impulsive noise.

   The principal issues in Volterra systems are the identication of the kernel and its computationally efcient implementation. Unlike in linear ltering, there are no general design rules for nding H2. Design of two dimensional kernels for specic applications can be done using methods like optimization, bi-impulse response method etc [3].

   The current work uses the optimization of mean square error using Powell method [4] as it yields faster convergence. The second issue is in realizing the kernel with minimum computational complexity [5], [6]. A feasible implementation can be done with an appropriate decomposition of H2 like LU or singular value decomposition (SVD). In this paper quadratic lter is implemented using SVD.

4. METHODOLOGY

   The methodology of noise removal is as outlined in Fig. 1. The rst phase is the design of the quadratic kernel for noise removal H2noise. There is no general methodology for the design of quadratic systems unlike in the case with LTI systems. Mostly, the methods are heuristic and application dependent but bestows the designer with the power to customize the lter to suit the application. It can often be expressed as one which could be solved by a suitable method of optimization. Here Powell method of optimization is used as described in Sec. V. This step of optimization minimizes the mean square error between a known image and the output of a quadratic lter that receives the noisy version of the image at its input. The optimization is done repeatedly until minimum mean square error yields.

5. DESIGN AND IMPLEMENTATION OF QUADRATIC FILTER

   The design of quadratic lter includes the discovery of its kernel coefficients. The main criteria that followed for the design purpose of lter is the minimization of the difference in desired output and observed output. The important design method based on this criterion is optimization technique. Here we used the Powells conjugate direction method for obtaining H2noise [4], since this algorithm has a fast rate of convergence.

   A synthetic image, X[n1,n2] of 64 × 64 dimension, corrupted by impulsive noise of known variance is simulated. Its noiseless version Yd[n1,n2] of identical dimension is also simulated. The output of the quadratic lter is assumed as:

   (6)

   Let the MSE between y[n1,n2] and yd[n1,n2] is

   (7)

   (8)

   The MSE is minimized to yield an optimum kernel H2noise. The kernel is plotted in Fig. 2.

   Fig.2. Quadratic filter kernel

   A direct implementaton of this kernel as in Eq. 3 is computationally complex. Instead SVD decomposition is performed on to yield an approximation as:

   (9)

   where i are the singular values arranged in a decreasing order and Si are the orthonormal singular vectors. Each 64 × 1 vector can be resized as a 3 × 3 FIR image lter that is equivalent to H(i, j) in Eq. 5.

   The outputs of FIR lters are squared and a weighted sum with i values yields the lter output.

6. EXPERIMENT

   The lter kernel designed in Sec. V is simulated in Python with the help of scipy and pylab modules. The noisy images are imported into Python using the image processing toolbox and subjected to ltering by . The ltered images are compared with those processed by spatial lters like median lter in terms of visual quality. For testing the noise invulnerability of , the experimental set up in Fig. 3 is used. In this, impulsive noise of known variance is added with the images and subjected to ltering. Quantitative measures like signal to noise ratio and peak signal to noise ratio are calculated with reference to the noisy image. Experiment is repeated for conventional lter like median and the results are compared with those of quadratic lter. Conventional lters blur the image while removing noise. Quadratic lter gives a better result.

   Fig.3. Experimental setup for ascertaining the performance parameters

7. RESULTS

   Quadratic lter is designed and implemented with the objective of removing impulsive noise from raw MRI data which is acquired under strong and rapidly changing magnetic elds. The MRI images corrupted by impulsive noise of different noise variances are applied to the quadratic lter implemented by SVD method. The resulting output is compared with those of spatial lter like median lter. It is observed that quadratic lter removes the impulsive noise better than the median lter.

   Fig. 4 show the outputs of various lters for impulsive noise variance =200. The noisy MRI image is as in Fig. 4(b). The output of median lter is shown in Fig. 4(c) and quadratic lter output is in Fig. 4(d). As claimed the output of the quadratic lter is the least noisy. Median lter fails to clean much of the impulsive noise in the input image.

   Fig.4. Output of different filters

   7.1 Signal to Noise Ratio

   The improvement in signal to noise ratio is computed as per the experimental setup in Fig. 3. The SNR is expressed as:

   (10)

   where r[n1,n2] denotes the reference image and t[n1,n2]denotes the test image. N1 × N2 is the size of the images. There is 12 dB improvement in using quadratic lter.

   Besides the improvement in SNR, quadratic lter has the advantage that the edges are not blurred on ltering, ensuring that the periphery of a possible pathological disorder like a tumour remains unambiguous.

8. CONCLUSION

Magnetic resonance imaging (MRI), is a medical imaging technique used to nd problems such as tumors, bleeding, injury, blood vessel diseases, or infection. By conventional ltering methods, it is very difcult to remove impulsive noise added with the raw MRI data due to the rapidly changing magnetic eld.

Conventional nonlinear lters employed for noise removal from images are median lter, mean lter, Gaussian lter etc. Such lters often suffer from poor edge resolution, blurring and poor signal to noise ratio. The paper summarizes the design and implementation of a quadratic noise removal lter based on Volterra series. The design method used for the quadratic kernel is method of optimization. Then it subjected to singular value based decomposition to yield an approximate but computationally simple implementation. The quadratic ltering operation is observed to remove the impulsive noise present in raw MRI data much better than the conventional ltering methods.

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