Improving the Performance of the Prony Method Using a Wavelet Domain Filter for MRI Denoising

The Prony methods are used for exponential fitting. We use a variant of the Prony method for abnormal brain tissue detection in sequences of T 2 weighted magnetic resonance images. Here, MR images are considered to be affected only by Rician noise, and a new wavelet domain bilateral filtering process is implemented to reduce the noise in the images. This filter is a modification of Kazubek's algorithm and we use synthetic images to show the ability of the new procedure to suppress noise and compare its performance with respect to the original filter, using quantitative and qualitative criteria. The tissue classification process is illustrated using a real sequence of T 2 MR images, and the filter is applied to each image before using the variant of the Prony method.


Introduction
The main goal of this paper is to detect abnormal tissues in 2 weighted magnetic resonance image sequences. The identification of the distribution of relaxation times in 2 weighted magnetic resonance brain images has been used to classify tissues and as a tool for the segmentation of tumors [1][2][3]. This is possible because the tissue corresponding to a given pixel in a sequence of 2 weighted MRI images is characterized by a relaxation rate and therefore allows for differentiation of the tissues that conform to the region determined by a set of pixels. In this paper, we assume that the intensity at a pixel, in a sequence of 2 weighted MRI brain images, can be approximated by where Δ is the echo time interval. Here, represents the intensity of gray in the pixel, is the number of images in the sequence, and is the number of tissues considered in the pixel, ≥ 2 + 1. The exponents are called the nonlinear parameters and correspond to the relaxation rates associated with the different tissues present in the images, and the coefficients , 1 , . . . , are called the linear parameters and are related to the background noise of the image and the proportions of each one of the tissues in the pixel, respectively. The background noise is the noise which comes from all possible sources, some of which are unknown, and usually corresponds to patient movement and the image acquisition process. Martín-Landrove et al. [3,4] proposed the model and used a variant of Prony's method to find the linear and nonlinear parameters. We have used two methods to compute the parameters in the exponential model (1): the variant of the Prony method already mentioned [3][4][5][6] and the variable projection method of Golub and Pereyra [7][8][9]. Prony-type methods are widely used for estimating a signal frequency component; a discussion of such methods can be found in [10][11][12][13]. On the other hand, the variable projection method has been used in many applications in fields like electrical engineering, medical image processing, chemical sciences, environmental sciences, magnetic resonance applications, and so forth. A comprehensive bibliographic reference can be found in [8], (144 items).
In [5] Paluszny et al., comparing the solution obtained by the variant of the Prony method and the solution given by the variable projection method, it was noted that both methods successfully converge in the case of low level noisy data, and none produces a satisfactory solution for high noise levels; the variable projection method proved to be more robust in the 2 Computational and Mathematical Methods in Medicine presence of higher noise levels but required ten times more computational time to get results comparable to those of the variant of the Prony method.
The exponential fit given by (1) is an ill-conditioned problem, and this means that relatively small changes in the data may result in large variations in the linear and nonlinear parameters; therefore, a filter is required to modify the data before using any of the considered methods. In particular, [6] provides an analysis of the sources of instability for the variant of the Prony method.
In the literature, there are a number of different filters to enhance MRI. We chose to modify the filter designed by Kazubek [14], to get a wavelet domain filter for removing noise on each image, before applying the Prony method, and trying to keep the computation time low.

The Two-Dimensional Wavelet
Transform. Let us consider an image , and let ( , ) be the gray intensity corresponding to the pixel given by coordinates ( , ). The discrete wavelet transform (DWT) is a decomposition of in two functions: where F is a representation of in terms of shifted and dilated versions of a low-pass scaling function ( , ) and F is a representation of in terms of shifted and dilated versions of a prototype bandpass wavelet function ( , ). The coefficients which determine the components F and F are called scaling coefficients and wavelet coefficients, respectively. Once the functions ( , ) and ( , ) have been chosen, the components F and F are uniquely determined by the coefficients; therefore, it is said that the discrete wavelet transform consists of the two sets of coefficients. The scaling coefficients allow for a low resolution reconstruction of the image and the wavelet coefficients deal with the reconstruction of the details. There is a large amount of bibliographic information about the theoretical as well as computational aspects of the DWT; we will give, here, a brief description to point out the relevant issues regarding the design of a filter, in the wavelet domain, for MR images.

Wavelet Domain Filters for MRI.
When we consider a noisy image and its DWT, it is supposed that the noise in the image is determined by the noise associated with the scaling and wavelet coefficients. A wavelet domain filter corresponds to two procedures, one for the noise reduction in the scaling coefficients and the other for dealing with the noise associated with the wavelet coefficients. The filtered image is the result of applying the inverse wavelet transform to the modified coefficients. The differences between various proposals for wavelet domain filters lie in the algorithms to be applied to each one of the two sets of coefficients; see, for example, [14][15][16][17][18][19][20][21][22].
Although the noise in MRI comes from various sources, we consider that the perturbation in magnitude follows a Rice distribution, [15,23]. Kazubek considers that most of the energy used in the image generation corresponds to the scaling coefficients and hence these contribute the most to the perturbation of the image in the presence of noise; therefore, Kazubek proposes to apply a filter to reduce the noise in the scaling coefficients under the hypothesis that the noise is Rician.
According to some authors, numerical experiments show that noise in the wavelet coefficients can be approximated by a Gaussian distribution; for this reason we made a satisfactory noise reduction in these coefficients using a Wiener filter (consult [14,15,21]) in addition to the Kazubek filter in the scaling coefficients.
The arithmetic mean of a random variable which follows a Rician distribution is a biased estimator of the value without noise. The next section shows a brief description of the Rician noise and a formula for reducing the bias between the value without noise and the arithmetic mean. We will use this formula later in the filter, on the scaling coefficients.

Removing the Bias from Rice Distributed Data
In a magnetic resonance image, the gray intensity corresponding to a given pixel is the magnitude of a complex valued function whose real and imaginary components are distributed as Gaussians with the same standard deviation. The different approaches for MRI noise removal can be divided into three classes; some methods act on the real and imaginary components, most of the methods deal with the square of the magnitude, and a third kind of methods acts on the magnitude of the complex image. We are interested in the latter class of methods because our data are in magnitude.
In [14], Kazubek implements a wavelet domain filter which acts on the magnitude of the image and whose relevant issues are the use of the Haar system of orthogonal functions in the wavelet transform and a formula to apply on the scaling coefficients that is used to suppress the bias discussed in Section 2.
Next, we present an analysis of Kazubek's formula for bias removal and propose a modification of his algorithm by applying the bilateral filter to the scaling coefficients [24].
Let 1 and 2 be the mean values of two random variables 1 and 2 , respectively, which are the real and imaginary components of a complex image. If 1 and 2 are distributed as Gaussians with standard deviation , then the probability density function for the magnitude, = √ 2 1 + 2 2 , is a Rice distribution given by where 0 is the modified Bessel function of the first kind and zeroth order and = √ 2 1 + 2 2 is the underlying Computational and Mathematical Methods in Medicine 3 noise-free signal amplitude. Let be the mean value of ; then, where = / and 1 is the modified Bessel function of the first kind and first order [15]. Let = / ; we can get a numerical approximation of the inverse = V −1 ( ) for 0 ≤ ≤ 50.
Bessel functions 0 and 1 satisfy 0 = 1 ; then, and using the power series expansion of functions 0 , 1 , and (Details of the proof can be found in the Appendix.) Therefore, V( ) is a monotonically increasing function in the interval [0, +∞) and V (0) = 0.
Using polynomial approximations of the functions for > √ 15 (consult Abramowitz and Stegun [25]), we get where = 0.999999 + and | | ≤ 5.139(10 −7 ). Then, for large , V( ) behaves as the function On the other hand, we have and a straightforward computation shows that This function is a particular case of function √ 2 + with > 0 and < 0.
The idea to get a numerical approximation of V −1 ( ) consists in considering a function Φ( ) such that function and its derivative satisfy similar asymptotic features of takes positive values and is a monotonically decreasing function in the same interval. Then, there is a unique positive root, 0 , of and if ≥ 0 , then It follows that [ 0 , ∞) is the domain of To get the parameters , , , and , we perform a least square fit to the discretized curve ( ) for = (0.1) and = V( ), = 1, . . . , 500. This problem was solved numerically using the MATLAB function lsqcurvefit. With MATLAB R2008a and taking the initial values  Figure 1(a) shows the function ( ) as a numerical approximation of −1 ( ) and Figure 1(b) shows the quality if the solution given by the MATLAB function lsqcurvefit. In MRI, as in acoustic, electricity, and telecommunications, the signal to noise ratio (SNR) is frequently used to measure, in decibels, the noise impact on a signal. Let and Nowak [15] suggests that a SNR greater than 10 dB corresponds to a noisy image, whose Rice distribution can be approximated by a Gaussian distribution. We will consider a method to suppress the bias in our MR images using the inverse function discussed in this section when SNR is less than 34 dB and a simple wavelet domain filter for the other cases.

Implementing a New Wavelet Domain
Bilateral Filter for MRI 4 (2) is a linear combination of functions that belong to a specific family of orthogonal functions; each scaling coefficient depends on three parameters: , the level of decomposition in the discrete wavelet transform, and the pair ( 1 , 2 ) which characterizes the support of the corresponding orthogonal function. Let ( , ) be the function in (2); if we apply the discrete wavelet transform using the Haar system of orthogonal functions, then the scaling coefficients are given by Details can be found in [26].

Wiener Filter for the Wavelet Coefficients.
As the scaling coefficients, the wavelet coefficients depend on three parameters ,ℎ 1 ,ℎ 2 , 1 ≤ ≤ . For simplicity, let us write = ,ℎ 1 ,ℎ 2 and suppose that the noise in the wavelet coefficients may be approximated by a Gaussian distribution [14]. Then, we attenuate the contribution of this coefficient bŷ According to Nowak, we assume that the noise wavelet coefficient is an unbiased estimator of the value of the wavelet coefficient in the noise-free case and denote its mean by = [ ]. The filter weight is chosen by minimizing [( − ) 2 ]. In fact, if 2 is the variance of the wavelet coefficient , then therefore is defined by Let 2 * be an estimate of 2 ; then, Usually the parameter 2 * is approximated by 2 , where ≥ 1 is a new parameter to be chosen for each image and is Computational and Mathematical Methods in Medicine 5 as defined in Section 2. In previous works [15,18,21], the value = 2 has proven to be convenient. Hence, the filter weight in (20) is defined by

Bilateral Filtering.
The bilateral filter is a nonlinear filter for images, proposed by Tomasi and Manduchi [24], whereby contours are successfully recovered from noisy images. Let x ∈ be a pixel in image ; the bilateral filter takes a sum with weights on the pixels in a local neighborhood of x, x . These weights depend on the spatial distance and the intensity in each pixel in the neighborhood. The response of the bilateral filter in the pixel is given bỹ where The performance of the bilateral filter depends on the choice of , and the size of the neighborhood x . Let x be a pixel located close to an edge which separates two regions of the image. When the pixel y is in the same region as x, (x, y) is close to one; on the other hand, when y is in the other region, then (x, y) is close to zero if the intensities (x) and (y) differ greatly and then the filter tends to preserve the pixel intensity due to the effect of the (x, y) components.
In our proposal the bilateral filter is applied to the twodimensional array consisting of the scaling coefficients of the wavelet transform (k) = ( 1 , 2 ) = , 1 , 2 . We follow Anand and Sahambi [20] and choose the neighborhood x to be a 15 × 15 matrix centered at x and the parameters = 5 and = 1.5 , where is the standard deviation given in Section 3.

Algorithm for MRI Denoising
(1) Compute an approximation of the variance, 2 , choosing a rectangular 1 × 2 neighborhood , in the background of the image [15]: (2) Perform a level = 3 wavelet decomposition using the Haar system of orthogonal functions.
(3) Perform a bias correction of the level 3 scale coefficients using the inverse function considered in Section 3.
From (19) we see that is an approximation to the expected value given in (4) so that is an approximation of the value defined in Section 2. Using (4) and (17), we get Now we compute the new scaling coefficients , 1 , 2 by (4) Apply the bilateral filter to the two-dimensional array ( 1 , 2 ), given by the new scaling coefficients: ( 1 , 2 ) = , 1 , 2 .
(5) Obtain a provisional denoised image, , computing the inverse wavelet transform using the new scaling coefficients and the unchanged wavelet coefficients.
(6) Perform a level = 4 wavelet decomposition of using the Daubechie system with four vanishing moments.
(8) Get a denoised image by computing the inverse wavelet transform on the set of coefficients given by the scaling coefficients in (6) and the new wavelet coefficients in (7).

Validation of the MRI Filter Applied to Simulated Images
We compare the performances of the modified and the original Kazubek's filter using a synthetic image generated with MATLAB, see Figure 2, and introduce the noisy images by adding Rician noise to the synthetic image. The Rician noise was generated as where ( , ) is the true signal and 1 and 2 are random numbers from a Gaussian distribution with mean zero and standard deviation ; five levels of were used = The structural similarity index in a region Ω is estimated as where and are the means over Ω of and , respectively, and are the corresponding variances, , is the covariance of and , and the constants 1 and 2 are given by √ 1 = 0.01 * 255 = 2.55 and √ 2 = 0.03 * 255 = 7.65; see [27]. If the images are divided in regions, Ω 1 , . . . , Ω , the SSIM for the two images ( , ) and ( , ) is defined by The resultant SSIM index is a value between −1 and 1, and value 1 is only reachable in the case of two identical data sets. Quantities SNR, PSNR, RMSE, and MAE are currently used in computer and medical sciences to compare two images; however, a good quantitative performance could be not consistent, in some cases, with visual perception. SSIM has become a more convenient way to compare two images, when PSNR and RMSE show indistinguishable results for the perception of the human eye.
Tables 1, 2, 3, 4, and 5 show the errors between the noisy images and the filtered images using the quantities SNR, PSNR, RMSE, MAE and SSIM, respectively. The first row in each table corresponds to the noisy image and various values, the second row corresponds to the filtered image using Kazubek's filter, and the third row corresponds to the filtered image using the modified filter. For each level, our modified Kazubek's algorithm shows an improved performance with respect to the original algorithm using all the criteria to measure the error. The performance of both filters is nearly indistinguishable in the cases = 1,2 and we get better filtered images as the noise level increases to = 5, 8, and 12, see Figure 3.

Numerical Results
To see the effect introduced by the filter on a real magnetic resonance image, as a part of a tissue segmentation process, we consider the set of images referenced in [3]. In this case, we have a set of eight echoes (magnetic resonance images of the same axial section) from a sequence of T2-MRI, with an echo time of 44 milliseconds. According to [2,3] we can see a part of the brain affected by tumoral tissues (ROI1) and another region showing no visible impairment (ROI2); regions ROI1 and ROI2 are called the region of lesion and region of control, respectively. Figure 4 shows the echo corresponding to = 6, from the set of images. For each pixel in a region the variant of Prony method is applied, to find the linear and nonlinear parameters defined in (1) with = 8 and Δ = 0.044; this process is performed for = 1, 2, and 3, and we choose the ones minimizing the residual. In Table 6, we show the computation times of the classification process, using our implementation of the variant of the Prony method.
For each region of interest, the frequency diagram is computed as follows: we divided the interval [0, 30] into 100 bins of equal length; = [ , +1 ], 1 ≤ ≤ 100, since the brain tissues relaxation rates belong to this interval, as we can see in [2]. For the pixels in a specific region and a selected subinterval , we add the linear parameters whose corresponding nonlinear parameters belong to , and this process is done for 1 ≤ ≤ 100, to get a diagram of frequencies. Each frequency diagram is normalized to obtain a probability density function.
In Figure 5(a), we see the solution obtained in [3], which was calculated by the variant of the Prony method. Figure 5(b) shows the results using our implementation of the variant of the Prony method, in red the probability density corresponding to the control region, and in blue the corresponding to the region of lesion. The differences in the shape of these graphics are caused because our implementation of the method is not the same as that reported in [3]. Figure 5(c) shows the result of applying the MRI filter designed in this paper, on each one of the eight images, before running the variant of the Prony method. The filtering process produces a curve with less dispersion for the region ROI2 and a softer edge in the curve corresponding to region ROI1.

Conclusion
This paper takes as its starting point the wavelet domain filter for MRI developed by Kazubek. We improved the original filter introducing the bilateral filter in the wavelet domain. The resulting filter exhibits a better performance and requires only a small increase in computational time of the whole tissue classification process. The new filter is applied to real brain MRI in a process of brain tissue classification proposed by Paluszny et al. [3], getting different results to those previously reported, which lead to an improvement in tissue identification with 2 relaxation techniques. Our results are validated with the main quality criteria for image denoising.  Figure 5: Probability densities corresponding to ROI1 and ROI2, in blue and red, respectively. (a) Graphic reported in [3]. (b) Solution obtained by Prony's method without a filtering process. (c) Solution obtained when we apply the wavelet domain filter to the images before using the Prony method.