Maximum likelihood signal estimation in phase contrast magnitude MR images A. J. den Dekkera , J. Sijbersa , M. Verhoyeb and D. Van Dycka Department of Physicsa and Biologyb , University of Antwerp, Belgium. Proceedings of SPIE Medical Imaging, Vol. 3338, p. 408-415, (1998) ABSTRACT When conventional techniques are employed in the quantitative analysis of phase contrast magnitude Magnetic Resonance (MR) data, the results obtained are biased. The bias is due to the contributions from inherent random noise. To remove this bias, knowledge of the actual shape of the data probability density function becomes essential. In the present work, the full knowledge of the probability distribution of the phase contrast magnitude MR data is exploited for optimal estimation of the underlying signal. Keywords: Flow imaging, phase contrast imaging, magnitude MR images, Maximum Likelihood estimation
1. INTRODUCTION Phase Contrast Magnetic Resonance (PCMR) imaging is widely used to detect flow. It is an enhancement procedure by which signal contributions from stationary spins are suppressed through posterior image processing.1 PCMR images can be computed in several ways.2 One way is by acquiring a complex valued base image S+ with a flow encoding gradient applied in for example the positive X direction followed by an identical acquisition of S− where the flow encoding gradient is applied in the opposite direction. The complex base images S+ and S− are assumed ± 2 to be corrupted by Gaussian, zero mean noise n± r and ni with variance σn , where r and i denote the real and imaginary image component, respectively. The goal of phase contrast image formation is to visualize the phase difference ∆Φ associated with flow. The complex base images are given by: S+ −
S
+ = Cej(Φ0 +∆Φ) + n+ r + jni j(Φ0 −∆Φ)
n− r
(1)
jn− i
= Ce + + (2) √ where Φ0 is an unknown phase offset and j = −1. In Eq. (1-2) it is assumed that the voxel magnitude C is identical in both base images. The same flow encoding strategy can be employed to acquire two additional pairs of complex valued MR images, S± Y and S± Z , where the flow encoding gradients are applied in the other two orthogonal directions Y and Z, respectively. Then, from the six complex valued MR images, the phase contrast magnitude image SCD can be computed via complex difference calculation: q − 2 − 2 − 2 kSCD k = S+ + S+ + S+ (3) X − SX Y − SY Z − SZ It is now easy to show that the PCMR pixel variable, denoted by M , can be written as: v uK uX M =t s2k k=1
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(4)
with K denoting twice the number of orthogonal Cartesian directions in which flow is encoded. The set {sk } are independent Gaussian variables with variance 2σn2 . Without loss of generality the mean values {ak } of {sk } can be written as: 2Ck sin(∆Φk ) odd ak = for k values. (5) 0 even Hence, the deterministic signal component of the PCMR pixel variable is given by: v uK uX a2k A=t
(6)
k=1
The real and imaginary pixel variables of each complex image are known to be Gaussian distributed. As PCMR data are derived from the square root of the sum of the squares of these Gaussian distributed variables, which is a non-linear transformation, the PCMR data are no longer expected to be Gaussian distributed. It has been shown that, when PCMR data are being used in quantitative analysis as an estimate of the underlying flow-related signal component magnitude, results are biased. The bias is due to the contributions from inherent random noise which is not Gaussian distributed. Since the bias is not merely an additive component, it cannot be just subtracted out. To remove this bias, knowledge of the actual shape of the data probability density function (PDF) becomes essential. This PDF was recently deduced by Andersen and Kirsch.1 In the present work, the full knowledge of the PDF of the phase contrast MR data is exploited for optimal estimation of the underlying signal. It has to be noticed that, although this paper focuses on complex difference processed images, the estimation techniques derived in the remainder of this work can, under certain conditions, also be applied to images obtained by phase difference processing. This has to do with the fact that for both methods in the end one has to estimate the underlying signal component from magnitude images of which the pixel variable M can be described by Eq. (4). The only difference is that for phase difference processing the dimension or number of degrees of freedom K directly equals the number of orthogonal Cartesian directions in which flow is encoded, whereas for complex difference processing the dimension K is twice this physical dimension.1
2. METHODS 2.1. The generalized Rice distribution As was clarified in Section 1, phase contrast MR data are deduced from a number of complex valued MR images. The ith PCMR magnitude data point is given by: v uK uX Mi = t s2k,i (7) k=1
where si,k denotes the real or the imaginary component of the ith complex valued data point of one of the base images S± . Data from PCMR imaging are distributed according to what one might call a generalized Rice distribution1 : pM (M |A) =
M σ2
M A
K2 −1
M 2 + A2 MA K exp − I (M ) 2 −1 2σ 2 σ2
(8)
with σ 2 the noise variance. The unit step function (.) is being used to indicate that the expression for the density pM (M ) is valid for non-negative values of the magnitude M only. Fig. 1 shows the PDF for K = 2, 4 and 6 and for signal-to-noise ratio’s equal to 0 and 3, where the signal-to-noise ratio (SNR) is defined as: SNR =
A σ
(9)
The first and second order moments of the generalized Rician distribution pM are given by: √ Γ [(K + 1)/2] 1 K A2 , ; − E [M ] = 2σ F − 1 1 Γ(K/2) 2 2 2σ 2 2 E M = Kσ 2 + A2
(10) (11)
respectively. Here Γ denotes the Gamma function and 1 F1 is the confluent hypergeometric function of the first kind.
2.2. Noise estimation Techniques to estimate the noise variance from phase contrast MR data are simple extensions of noise estimation techniques commonly applied to conventional magnitude MR data (K = 2). Possible ways to estimate σ 2 from N background data points are (cfr. Henkelman et al.3 ): c2 = σ
N 1 X 2 M KN i=1 i
(12)
or from two realizations of the same PCMR image (cfr. Sijbers et al.4 ). Usually, much more data points are available for estimation of σ 2 than for the estimation of the locally constant signal. Hence, in the following we assume this value to be known. In Section 2.4 we briefly comment on simultaneous estimation of noise and signal.
2.3. Signal estimation In the following, it is assumed that an unknown deterministic signal component A needs to estimated from a number of PCMR pixel values of a region Ω where the signal component is assumed to be constant. Below, three signal estimation methods are discussed, two of which are based on spatial averaging, whereas one is based on Maximum Likelihood estimation. 2.3.1. Mean estimator The most intuitive way of estimating the unknown signal component is through simple averaging of pixel values in the region Ω. Without a priori knowledge of the proper data PDF, this action would be justified as it is the optimal (i.e., Maximum Likelihood) estimation procedure in case the data would be corrupted by Gaussian noise which is the most general type of noise: N 1 X Mi (13) Aˆm = N i=1 The variance of this ’mean estimator’ is given by: 1 Var Aˆm = E M 2 − E[M ]2 N
(14)
However, as PCMR data are not Gaussian distributed, it is clear that a huge bias would be introduced in case the signal is estimated by averaging pixel values. The bias, relative to the true signal component A, is in general defined by: E[A] ˆ − A Relative bias = (15) × 100% A ˆ denotes the expectation value of the signal estimator A. ˆ For the mean estimator (13) the expectation where E[A] value E[Aˆm ] is given by E[M ] because the average operator is an unbiased estimator of the expectation value. Hence, the relative bias of Aˆm can be computed from Eq. (10). Note that it follows from Eq. (10) and Eq.(15) that the relative bias can be written solely in terms of the SNR and is independent of the number of averaged pixel values N . Fig. 2 shows the relative bias of Aˆm as a function of the SNR for various values of K. From the figure, it is clear that the bias increases rapidly with decreasing SNR. Also, the bias increases with increasing number of flow encoding directions.
2.3.2. Modified RMS estimator An easy way to reduce the bias is by exploiting the second moment of the generalized Rice distribution as was given in Eq. (11). Indeed, an unbiased estimator of A2 is given by: N X c2 = 1 A M 2 − Kσ 2 N i=1 i
(16)
c2 can be computed to yield: The PDF of A c2 = N pAb2 A 2σ 2
c2 + Kσ 2 A A2
! N K−2 4
c2 + Kσ 2 + A2 A exp −N 2σ 2
!
q c2 + Kσ 2 N A A c2 + Kσ 2 (17) A I N K −1 2 σ2
From the unbiased estimator of A2 , given in Eq. (16), a modified root mean square (RMS) estimator of A would be given by: v u N u1 X ˆ Arms = t M 2 − Kσ 2 (18) N i=1 i However, root extraction is a non-linear operation which makes the estimator Aˆrms biased. Also, the modified RMS estimator is only appropriate whenever the argument of the square root operator is non-negative. A possible, at c2 is first sight quite arbitrary, solution to this problem is to artificially put the estimator Aˆrms to zero whenever A negative: ( p c2 if A c2 ≥ 0 A (19) Aˆrms = c2 < 0 0 if A The PDF of Aˆrms is then given by: pAˆrms
Z0 ˆ Arms = pAb2 (x) xdx δ Aˆrms + −Kσ 2
N Aˆrms σ2
Aˆ2rms + Kσ 2 A2
! N K−2 4
Aˆ2 + Kσ 2 + A2 exp −N rms 2σ 2
!
q Aˆ2rms + Kσ 2 N A Aˆrms (20) I N K −1 2 σ2
where δ(.) denotes the Dirac Delta function. Notice that the first term of Eq. (20) vanishes for high SNR. The bias of the modified estimator (19) can be computed from E[Aˆrms ] =
Z∞ pAˆrms (x) xdx
(21)
0
Fig. 3 shows the bias for various values of K and N . In general, the bias of the modified estiator is significantly smaller compared to the bias of the mean estimator. It should be noticed that the bias of the mean estimator turns out to be positively valued whereas the bias of the modified RMS estimator has a negative sign. That, however can not be observed from Fig. 3 or Fig. 2 due to the absolute value operator in Eq. (15). For both estimators the bias increases with increasing number of flow encoding directions. However, in contrast to the mean estimator (13), the bias of the modified estimator (19) decreases with increasing N .
2.3.3. Maximum Likelihood estimator In this section, the ML approach is clarified for the estimation of the unknown signal parameter A from a set of N independent magnitude PCMR data points {Mi }. The proposed technique consists of maximizing the joint PDF, also referred to as the likelihood function, of N generalized Rician distributed data points with respect to A. The likelihood function of N independent magnitude data points {Mi } is given by: L({Mi }|A) =
N Y
pM (Mi |A)
(22)
i=1
Then the Maximum Likelihood (ML) estimate of the PCMR signal A is the global maximum of L, or equivalently the maximum of log(L), with respect to A: n o AˆML = arg max (log L) (23) A
Leaving only the terms that depend on the variable A, we have explicitely: log(L) ∼ −N
N K N A2 X Mi A − 1 log A − + log I K −1 2 2 2σ 2 σ2 i=1
(24)
It can be shown that log(L) has only one maximum for positive A values. Hence the computational requirements c2 , given by Eq. (16) is are very low. It can also be shown that the ML estimator yields the value zero whenever A ˆ negative. This observation makes the modification of Arms as described by Eq. (19) less arbitrary.
2.4. Simultaneous signal and noise estimation If the value of the noise variance σ 2 is not known a priori, the signal A and variance σ 2 have to be estimated simultaneously from the N available data points by maximizing the log-likelihood function with respect to A and σ2 : c 2 ˆ {AML , σ ML } = arg max (log L) (25) A,σ 2
Although optimization of a two-dimensional function is non-trivial, computational requirements were observed to be limited due to the smooth shape of the log-likelihood function.
3. EXPERIMENTS AND DISCUSSION It is already clear from Fig. 2 and 3 that the accuracy of the modified RMS estimator described in Subsection 2.3.2 is an order of magnitude better than that of the mean estimator (13) though a significant bias still remains. As to compare the modified RMS estimator with the ML estimator described in Subsection 2.3.3, a simulation experiment was set up. Thereby, K data points with deterministic signal component ak were polluted with Gaussian zero mean noise with equal variance after which a magnitude data point M was computed according to Eq. (7). The same procedure was repeated N times as to obtain N generalized Rician distributed magnitude data points. The deterministic signal component A of M , given in Eq. (6), was then estimated, once using the modified RMS estimator and once using the ML estimator. The data generation process and posterior signal estimation was repeated 105 times as a function of the SNR, after which the average value and the 95% confidence interval was computed. Fig. 4 shows the signal estimation results for K = 6 and N = 8 as a function of the SNR. When the c2 values was larger than 5%, the modified RMS estimator was regarded to be percentage for obtaining negative A inappropriate. The SNR levels for this to occur was observed to be smaller than 1.5. For this reason, the modified RMS estimator was compared to the ML estimator only for SNR values higher than 1.5. From Fig. 4 it is clear that the ML estimator is slightly but significantly more accurate than the modified RMS estimator. A similar behavior of the performance of the estimators was observed for all combinations of K = 2, 4, and 6 and N = 4, 8, and 16. The results also show that the precision, i.e., the standard deviation, of both estimators is approximately equal. It can also be seen that at high SNR the ML estimator cannot be distinguished from an unbiased estimator.
4. CONCLUSIONS In this paper, three methods were discussed for the estimation of the underlying signal component associated with flow from a number of generalized Rician distributed phase contrast magnitude MR data points. It was shown that the Maximum Likelihood estimator outperforms conventional estimators in terms of accuracy. The proposed Maximum Likelihood estimation technique can be applied to a region of interest as well as on a pixel-by-pixel basis. Also, the method is valid for any number of directions in which flow is encoded.
REFERENCES 1. A. H. Andersen and J. E. Kirsch, “Analysis of noise in phase contrast MR imaging,” Medical Physics 23(6), pp. 857–869, 1996. 2. N. J. Pelc, M. A. Bernstein, A. Shimakawa, and G. H. Glover, “Encoding strategies for three-direction phasecontrast MR imaging of flow,” Journal of Magnetic Resonance Imaging 1(4), pp. 405–413, 1991. 3. R. M. Henkelman, “Measurement of signal intensities in the presence of noise in MR images,” Medical Physics 12(2), pp. 232–233, 1985. 4. J. Sijbers, A. J. den Dekker, D. Van Dyck, and E. Raman, “Estimation of signal and noise from Rician distributed data,” in Proceedings of the International Conference on Signal Processing and Communications, pp. 140–142, (Gran Canaria, Canary Islands, Spain), February 1998.
0.7 K=2 K=4 K=6 0.6
Generalized Rician PDF
0.5
0.4
0.3
0.2
0.1
0 0
0.5
1
1.5
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2.5 M
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(a) SNR = 0 0.45 K=2 K=4 K=6
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Generalized Rician PDF
0.35
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0.25
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0.1
0.05
0 0
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M
(b) SNR = 3 Figure 1: Probability density function of PCMR data for K = 2, 4 and 6.
7
100
Relative bias (%)
K=2 K=4 K=6
10
1 1.5
2
2.5
3
3.5
4
4.5
5
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Figure 2. Relative bias of a simple spatial average estimator for K = 2, 4 and 6. The relative bias is independent of the number of averaged pixels N .
10
Relative bias (%)
K=6 ; N = 4 K=4 ; N = 4 K=2 ; N = 4 K=6 ; N = 8 K=4 ; N = 8 K=2 ; N = 8 K=6 ; N = 16 K=4 ; N = 16 K=2 ; N = 16
1
0.1 1.5
2
2.5
3
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SNR
Figure 3: Relative bias of the estimator Aˆrms for K = 2, 4, 6 and N = 4, 8, 16. The true value is A = 100.
101
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99
A estimate
98
97
True signal value Modified RMS estimate Expectation value of the modified RMS estimator ML estimate
96
95
94
93 2
2.5
3
3.5
4
4.5
5
SNR
Figure 4. Comparison of the modified RMS estimator Aˆrms with the ML estimator AˆML for K = 6 and N = 8. Each time, the average value of 105 signal estimates is given, along with the 95% confidence interval. The true value is A = 100. The expectation value of Aˆrms according to Eq. (21) has also been shown.
