Removal of Intensity Bias in Magnitude Spin-Echo MRI Images by Nonlinear Diffusion Filtering

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1 Reoval of Intensity Bias in Magnitude Spin-Echo MRI Iages by Nonlinear Diffusion Filtering Alexei A. Sasonov *, Chris R. Johnson Scientific Coputing and Iaging Institute, University of Utah, 50 S Central Capus Drive, Roo 3490, Salt Lake City, UT, 84112, USA ABSTRACT MRI data analysis is routinely done on the agnitude part of coplex iages. While both real and iaginary iage channels contain Gaussian noise, agnitude MRI data are characterized by Rice distribution. However, conventional filtering ethods often assue iage noise to be zero ean and Gaussian distributed. Estiation of an underlying iage using agnitude data produces biased result. The bias ay lead to significant iage errors, especially in areas of low signal-to-noise ratio (SNR). The incorporation of the Rice PDF into a noise filtering procedure can significantly coplicate the ethod both algorithically and coputationally. In this paper, we deonstrate that inherent iage phase soothness of spin-echo MRI iages could be utilized for separate filtering of real and iaginary coplex iage channels to achieve unbiased iage denoising. The concept is deonstrated with a novel nonlinear diffusion filtering schee developed for coplex iage filtering. In our proposed ethod, the separate diffusion processes are coupled through cobined diffusion coefficients deterined fro the iage agnitude. The new ethod has been validated with siulated and real MRI data. The new ethod has provided efficient denoising and bias reoval in conventional and black-blood angiography MRI iages obtained using fast spin echo acquisition protocols. Keywords: MRI, anisotropic diffusion filtering, vector-valued diffusion, fast spin echo, Rice distribution, intensity bias 1. INTRODUCTION MRI iages are coplex-valued and contain zero-ean uncorrelated Gaussian noise in both real and iaginary iage parts. Such noise could be efficiently eliinated using filtering procedures based on spatial averaging. In practice, iage analysis is often perfored on agnitude data, as the iage agnitude is not susceptible to variations characteristic for iage phase 1. Magnitude data are characterized by Rice probability density function (PDF) 2. Though spatial averaging produces biased results for Rice distributed data, the estiation ethods based on Gaussian PDF are still preferred, as the inclusion of the Rice PDF into a filtering procedure usually coplicates it both algorithically and coputationally 3. While for high SNR areas the bias is not significant, it becoes ore pronounced for low SNR regions. This systeatic error ight present a serious proble in studies where contrast-to-noise ratio (CNR) in low SNR areas is essential (i.e. black-blood angiography 4 ), or in quantitative evaluations of T1/T2 aps 5. One way to avoid the biased results is to filter real and iaginary parts separately. This would eliinate the Gaussian noise and produce unbiased estiation of the coplex iage channels and, consequently, the iage agnitude. However, while MRI iage agnitude generally satisfies ergodicity assuption 6, nonsooth iage phase variations generally introduce rough odulations into both real and iaginary iage channels breaking the assuption. Aside fro other acquisition approaches, MRI acquisition techniques based on spin-echo phenoenon 1 create iages with sooth and slowly varying iage phase. This is due to the refocusing properties of the spin-echo effect. The phase soothness is used in partial Fourier ethods 7 that use low-frequency phase estiates in reconstructing a reduced set of asyetrically sapled k-space data. The phase introduces sooth odulations in the intensities of the coplex iage channels that could be handled by any nonlinear filtering approaches 8, 9 to realize the idea of separate channel filtering. * sasonov@sci.utah.edu; phone ; fax ; Scientific Coputing and Iaging Institute, University of Utah, 50 S Central Capus Drive, Roo 3490, Salt Lake City, Utah 84112

2 In this paper, we deonstrate this idea using a novel nonlinear filtering schee based on an anisotropic diffusion approach 10. The anisotropic diffusion filter is an iterative denoising technique used for MRI iage enhanceent 11. The technique provides efficient reoval of noise in hoogeneous tissue areas while retaining iage structures. Our ethod uses the inherent phase soothness property of spin-echo iages to fulfill the separate filtering of coplex iage parts. The separate diffusion processes are coupled through cobined diffusion coefficients found using the iage agnitude. In the next sections, we first give an overview of concepts related to the subject of the paper, naely, the PDF of MRI agnitude data, anisotropic diffusion filtering in Perona-Malik forulation 10, and properties of spin-echo iages. We then describe the new ethod and provide ipleentation details. We present results of testing the ethod on siulated, phanto and patient brain data. Finally, we discuss the advantages and liitations of the new technique. 2. THEORY 2.1 PDF of MRI iage agnitude MRI data are collected in the spatial-frequency doain, or k-space. A coon way to reconstruct data sapled on the Cartesian grid is to transfor the data into the spatial doain using a fast Fourier transfor (FFT). Resulting iage is coplex-valued, and both real and iaginary parts are described by a Gaussian PDF. The iage agnitude is coputed as M I I 2 2 = Re( ) + I( ), (1) where Re(I) and I(I) are real and iaginary parts of iage I correspondingly. The nonlinear agnitude operation leads to the Rice PDF of the iage agnitude 2, 12 : 2 2 M + µ µ σ M M pm M = e I M σ σ 2 2 ( µσ, ) ε( ), (2) where µ represents the true iage aplitude, σ is the standard deviation (STD) of the data, I 0 (.) is the 0 th order odified Bessel function of the first kind, and ε(.) is a unit step function indicating that the PDF is valid only for non-negative values of M. If ore than two Gaussian variables are used to calculate iage agnitude, then a generalized Rice PDF 2 should be used instead. The Rice PDF for several values of SNR defined as µ/σ is plotted in Figure 1. For high SNR (SNR>3), the Rice PDF approxiates the Gaussian PDF, and for low SNR (SNR<3) it approxiates a Rayleigh distribution: M 2 2σ pm ( M σ) = e ε 2 ( M) (3) σ The Rayleigh distribution characterizes saples in air background areas where SNR=0. It is coon practice to estiate the iage noise STD using relations M and σbg = σ (4) µ = σ, (5) bg where σ bg and µ bg is the STD and the ean of saples in air background areas. The bias relative to the aplitude µ vs. SNR is depicted in Figure 2. As could be appreciated fro the plot, the bias is low SNR areas is coparable with iage aplitude. There could be significant degradation of local CNR defined as the difference in SNR between two tissue types. CNR is one of the ost iportant characteristics of diagnostic quality of MRI iages. Accuracy of disease diagnosis is strongly dependent on the difference in signal strength between noral and pathological tissue types.

3 Figure 1: Rice PDF for several values of SNR. Figure 2: Relative bias of spatial averaging signal estiation vs. SNR. The bias is independent of the nuber of averaged pixel values and copletely deterined by iage SNR. 2.2 Anisotropic diffusion filtering A nice property of anisotropic diffusion filtering is that it preserves iportant iage structures while reoving noise. This is achieved by soothing the iage along object boundaries and preventing diffusion across the boundaries. Anisotropic diffusion filtering is equivalent to solving the following partial differential equation with respect to iage function Irt: (, ) Irt (, ) = t ( g( I, k) I( r, t) ), (6) where g( I, k) is a onotonically decreasing diffusivity function that depends on the value of the local iage gradient I and conductance paraeter k, and t is an artificial tie paraeter. In this work, we used an exponential diffusivity function 10 2 g( I, k) = exp( ( I / k) ) (7) The paraeter k sets a threshold between iage gradients to be soothed and the ones to be preserved. The choice of the conductance paraeter is crucial for the filter perforance. In practice, a good choice for MRI data is σ k 2σ (8) The paraeter σ could be estiated using analysis of air background saples (Eqs. [4, 5]). In practice, any iages such as color iages and ultiple contrast MRI data are vector-valued functions. A siple way to extend the anisotropic diffusion to vector-valued iages is to siply solve Eq. [6] separately for each iage channel. However, this approach does not work very well when diffusion coefficients are different for each iage coponent 14. Flexibility of the forulation based on the anisotropic diffusion (Eq. [6]) offers various opportunities to control the nonlinear filtering through a choice of diffusion coefficients. Most anisotropic diffusion schees use coon diffusion coefficients for all iage coponents. This allows preservation of correlating and contrasting effects aong ultiple iage channels 11, 15. This approach has been taken in application of the anisotropic diffusion filtering to ulti-echo MRI data 11, where diffusion coefficients were calculated using a cobined iage gradient easure.

4 a b Figure 3: Iage phase for (a) FSE and (b) gradient echo (GRE) scans of the sae phanto object. The phases are shown in the liits of object ask obtained by thresholding iage agnitude. Note sooth behavior of FSE iage phase and rapid phase variations on edges of the object for GRE iage phase. 2.3 Phase properties of spin-echo MRI iages MRI iages should be real valued, if the ideal experient conditions are et. However, in practice, the iages are coplex. Phase variations in the iages occur due to the various factors such as a noncentered sapling window, patient-to-patient variations in coil loading, inhoogeneity in RF pulse spin-flipping angle, eddy currents, ain field inhoogeneities, susceptibility, and cheical shift effects 1. The first three factors create global, slowly varying phase changes, while the last three are responsible for fast phase transitions. Spin-echo effect used to for MRI signal eliinates the spin dephasing that is due to ain field, susceptibility and cheical shift effects and prevents the corresponding phase variations in MRI iage. As a result, iage phase for spin-echo acquisitions is described by a sooth slowly varying function (Fig. 3a). For gradient-echo-based (GRE) sequences, the behavior of the phase is ore coplicated (Fig. 3b). In this work, we ake use of the phase soothness to separate real and coplex iage channel filtering. Due to this property, real and iaginary parts of coplex spin-echo MRI iages are well described by a piecewise slowly varying odel (Fig. 4). The phase soothness property is used in partial Fourier reconstruction 7 of spin-echo data that akes use of Heritian syetry of the Fourier transfor. In this approach, the k-space is sapled asyetrically to reduce the a b c Figure 4: Magnitude (a) of a phanto iage, and plots of iage agnitude (b) and real and iaginary iage parts (c) using the profile indicated in (a).

5 aount of data needed for iage reconstruction and, consequently, achieve the scan tie reduction. The syetrically sapled k-space center is used to obtain low-frequency estiate of iage phase. The issing data are then filled in using phase correction algoriths, for exaple, POCS-based reconstruction 16. The algoriths are known to produce artifacts in areas where true iage phase significantly deviates fro its low-resolution phase estiate. These variations are due to agnetic susceptibility changes on air-tissue boundaries and blood flow effects METHODOLOGY The ethod we present in the paper is a particular case of a vector-valued diffusion 15. The iage channels are soothed separately. At the sae tie, the sae diffusion coefficients are utilized to ensure the consistency of the separate diffusion processes. To calculate the coefficients, diffusion stopping gradients are found fro the iage agnitude. The ain rationale for using this schee is that the iage agnitude provides a ore reliable representation of tissue structures than the real and iaginary iage channels in separate. The corresponding diffusion equation to be solved has the for of Eq. [6]. However, the iage function Irt (,) is now coplex-valued, and the diffusivity function depends on the gradient value of its agnitude. Discretizing Eq. [6] with an explicit discretization schee 17, and taking into account the iage coplexity and the diffusivity function odification, we coe to the following nuerical schee for diffusive soothing of coplex iage. Starting with an initial guess I (0), we proceed in a fixed nuber of iterations as follows: ( t+ 1) () t () t () t = + n n n η ( ) ( t+ 1) () t () t () t = + n n n η ( ) ( t+ 1) ( t+ 1) ( t+ 1) = Re( ) + *I( ) Re( I ) Re( I ) t Re( I ) g( I, k) I( I ) I( I ) t I( I ) g( I, k) (9) I I i I Here, η () is the discretization neighborhood of pixel, and tie step t establishes the diffusion rate. The iage function gradients are approxiated by the nearest neighbor differences: () t () t () t n n I = I I () t () t () t n n () t () t () t n n Re( I ) = Re( I ) Re( I ) I( I ) = I( I ) I( I ), (10) Initial Coplex Iage n 3 n 1 I n1 I n3 I n2 n 2 Calculate Diffusion Coefficients On Iage Magnitude I n4 Diffuse Real Iage Channel Diffuse Iaginary Iage Channel n 4 Cobine into New Coplex Iage Figure 5: Four-point discretization for nuerical evaluation of iage gradients. Figure 6: Flowchart of anisotropic diffusion on coplex iage.

6 The tie step t is chosen using analysis of the explicit discretization schee. We used a 4-point discretization approach (Fig. 5). In order to ensure the stability of the discretization, t should be in the range of (0, 0.25]. The diffusion schee is illustrated in Fig. 6. In all filtering experients, the conductance paraeter was chosen based on the iage noise STD. The paraeter was deterined using air background area saples (Eqs. [4, 5]). Tie step t was set equal to The ethod was ipleented in MATLAB 6.1 and run on a standard id-range PC. 4. DATA In order to evaluate filter perforance, we created a synthetic iage (256-by-256 pixels) using the well-known Shepp-Logan phanto as an iage agnitude (Fig. 7a), and a linear 2D function taking values in the range [-π, π] as an iage phase (Fig. 7b). Then, a zero ean Gaussian noise with σ = 0.02 was added to both real and iaginary iage channels. The filter perforance was easured by estiating the standard deviation (STD) of pixel values in the hoogeneous areas of the phanto object. The root-ean-squared (RMS) error taken against the groundtruth iage (Fig. 7a) was used to estiate the filtered iage quality. a b c Figure 7: Test iage. a: iage agnitude, b: iage phase, c: agnitude of noisy iage. The real data were obtained scanning the phanto object and patient head on a 1.5T MR scanner (GE SIGNA, GE Medical Systes, Milwaukee, WI) with a head coil using FSE acquisition protocol. Black-blood angiography data were acquired using 3D FSE protocol for black-blood iaging. 5. RESULTS 5.1 Siulation studies Filtering experients were perfored on both iage agnitude and coplex iages. The results are presented in Fig. 8. The newly developed technique provided significant reduction of RMS error (Fig. 8a) that corresponded to a substantial decrease in iage noise (Fig. 8b). The analogous decrease in noise STD was observed for standard filtering. However, the noise eliination was not accopanied with a noticeable decrease in the RMS error. This could be explained by the fact that filtering the iage agnitude left the intensity bias intact while filtering the coplex iage eliinated the bias (Fig. 8c). In other words, the RMS error in the iage is introduced ostly by the intensity bias. Another iportant observation is that the significant iage iproveent and noise reduction could be reached using only a few iterations (10-20). 5.2 Real data experients Figure 9 shows saple profiles for the iage in Fig. 4a filtered with both standard and our proposed approaches (100 iterations, k=1.75σ). Filtering on the coplex and agnitude iages produces alost identical results in high SNR areas, where bias is sall. This supports the idea that filtering could be safely done separately on real and iaginary parts of the spin-echo iage. However, the results differ drastically for low SNR areas, where the bias is high. The bias for our proposed ethod is significantly reduced.

7 a b c Figure 8: Perforances of anisotropic diffusion filters on synthetic iage. a: plots of RMS vs. iteration nuber; the change in RMS for coplex filtered iage is by factor 14, while that of agnitude filtered is by factor b: plots of iage noise STD vs. iteration nuber The plots for both filtering types are approxiately the sae. c: saple profiles illustrating the filter perforances. Note the bias in low SNR areas for both initial and agnitude filtered iages. Figure 10 presents an exaple of filtering T2-weighted brain iages. The filters were applied in 15 iterations with k=1.75σ. The absolute values of differences aong initial (Fig. 10a), agnitude filtered (Fig. 10b) and coplex filtered (Fig. 10c) iages were used to visualize the filtering results. While both filters efficiently eliinated the noise coponent (Fig. 10d,e), the newly developed filter produced the iage with a reduced bias (Fig. 10f). Figure 11 deonstrates results of the application of the anisotropic diffusion filters to a stack of forty 2D blackblood angiography iages (15 iterations, k=σ). The brain tissues were segented and asked before creating a Miniu Intensity Projection (MIP) iage, coonly used to visualize black-blood angiography data. While both agnitude filtering and coplex filtering reduce iage noise, the MIP iage for coplex filtering has better iage contrast. A prior application of our filtering ethod to the iage stack resulted in iproved black-blood data visualization. Figure 9: Filtering the phanto data. Plots of initial, agnitude filtered and coplex filtered iages corresponding to the profile shown in Fig. 4a are given.

8 a b c d e f Figure 10: Brain iage filtering. a: initial noisy iage agnitude; b: filtered iage agnitude; c: agnitude iage after application of the proposed filter; d, e: absolute difference of (a) against (b) and (c) correspondingly; f: absolute value of difference of (b) against (c). Note that the difference (f) illustrates the bias eliinated using the proposed ethod. Figure 11: Black blood angiography iage filtering. Fro left to right: initial noisy, agnitude filtered and coplex filtered iages. Fro top to botto: single slice, MIP on the stack of segented iages, ROI of the MIP iage.

9 6. DISCUSSION Anisotropic diffusion filtering assues that the underlying iage is piecewise constant. If the filter is applied to regions of constant slopes with a large nuber of iterations, it ay split the regions into piecewise constant plateaus. This effect is called a staircasing artifact 14. Though both real and iaginary parts of the iage deviate fro this odel, we did not observe the described artifacts in our experients. We attribute the behavior of the filter to the fact that the phase is slowly varying enough not to ake the iage channels deviate significantly fro this odel. The standard tie for filtering a 256-by-256 iage on id-range PC was about 0.5 sec per iteration. We did not observe any significant filtering artifacts due to breaking the phase soothness assuption of spin-echo MRI data. The possible explanation is that the phase variations due to blood flow and susceptibility changes usually correlate with iage structures such as vessels and tissue-air boundaries. Hence, the artifacts could exhibit theselves as local edge degradation. The application of the ethod to other kinds of MRI data such as data acquired with gradient echo protocols ay be probleatic, as the phase soothness property is not retained in such data. Hence, the application of our proposed ethod is liited to spin-echo data. Further developents ay include extending the schee to filtering the iage derivatives that would allow application of the filter to iages that strongly deviate fro a piecewise constant odel. This ight be particularly useful for iages acquired with surface coils 18 that are odulated by rapidly varying sensitivity function. 7. CONCLUSIONS Intensity bias in the MRI iage agnitude could present a serious proble in MRI data analysis. The reoval of the bias by using filtering approaches that take into account the Rician PDF of the data is typically a nuerically and algorithically coplicated procedure. Instead, we suggested using standard spatial averaging techniques that are efficient for the eliination of Gaussian noise for filtering spin-echo MRI data. The phase soothness property allows separate filtering of both real and iaginary iage parts using short scale spatial averaging. We proposed a new ethod for unbiased nonlinear diffusion filtering of spin-echo MRI data. The new ethod was tested on a wide range of both phanto and patient data obtained using FSE acquisition protocols. Our ethod provided efficient iage denoising and siultaneously reoved an intensity bias in the iage agnitude. At the sae tie, the new ethod did a good job at spatial resolution preservation enhancing iage edges. The ethod is coputationally efficient and algorithically siple. AKNOLEDGEMENTS This work was supported by NIH BISTI grant 1P20HL The authors thank Eugene Kholovski for his assistance with iaging experients and Ross Whitaker for his valuable coents. REFERENCES 1. E.M. Haacke, R.W. Brown, M.R. Thopson, R. Venkatesan, Magnetic Resonance Iaging: Physical Principles and Sequence Design, 914 p., John Wiley & Sons, New York, H. Gudbjartsson, S. Patz, The Rician Distribution of Noisy MRI Data, Magn Reson Med, 34, pp , J. Sijbers, A.J. dendekker, A. VanderLinden, M. Verhoye, D. VanDyck, Adaptive Anisotropic Noise Filtering of Magnitude MRI Data, Magn Reson Iaging, 17, pp , R.R. Edelan, D. Chien, D. Ki, Fast Selective Black-Blood MR Iaging, Radiology, 181, pp , J. Liu, A. Nieinen, J.L. Koenig, Calculation of T1, T2 and Proton Spin Density in Nuclear Magnetic Resonance Iaging, J Magn Reson, 85, pp , Y. Wang, T. Lei, Statistical Analysis of MR Iaging and its Applications in Iage Modeling, Proceedings of the IEEE International Conference on Iage Processing and Neural Networks, 1, pp , 1994.

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