Explicit mathematical expression for the Cramér-Rao lower bound for experimental design and parameter estimation in parallel imaging
Mustapha Bouhrara1 and Richard G. Spencer1

1National Institutes of Health, Baltimore, MD, United States


The Cramér-Rao lower bound (CRLB) is widely used in the design of magnetic resonance (MR) experiments for parameter estimation. Previous work has considered only Gaussian or Rician noise distributions in this calculation. However, the noise distribution for multiple-coil acquisitions, such as in parallel imaging, obeys the noncentral χ-distribution under many circumstances. Here, we present the general mathematical expression for the CRLB calculation for parameter estimation from multiple-coil acquisitions. Our results indicate that the CRLB calculation must account for the noncentral χ-distribution of noise in multi-coil acquisitions, especially in the low-to-moderate signal-to-noise ratio (SNR) regime.


The Cramér-Rao lower bound (CRLB) defines the best achievable precision of any unbiased estimator for a deterministic parameter, given a data model (1-2), and is widely used in the design of MR experiments for parameter estimation (3-6). The purpose of this work is to present the explicit mathematical expression for the CRLB in multiple-coil acquisitions, such as parallel imaging.


Noncentral χ-distribution

The probability density function (PDF) for signal intensity within a given voxel of a magnitude image reconstructed from multiple-coil acquisitions is given by (7):

$$P_χ (M,A,σ,m)=\frac{A^{1-m}}{σ^2}\ M^m\ exp(-\frac{M^2+A^2}{2σ^2})\ I_{m-1}(\frac{MA}{σ^2}),\ \ \ \ [1]$$

where A is the magnitude of the underlying noise-free signal, M is the magnitude of the observed signal, m is the number of coils, σ2 is the noise variance, and Im is the modified mth order Bessel function of the first kind.

CRLB for the noncentral χ-distribution

Calculation of the CRLB requires inversion of the Fisher matrix (1-6). For N measured data values fit to a signal model defined by a parameter vector β, the Fisher matrix elements for the noncentral χ-distribution are:

$$F_{ij}=-E[\frac{∂}{∂β_i}\frac{∂}{∂β_j}\ log⁡(L_χ (M,A,σ,m))]=-E[\frac{∂}{∂β_i}\frac{∂}{∂β_j}\log⁡(\prod_{n=1}^NP_χ (M_n,A_n,σ,m))],\ \ \ [2]$$

where E stands for expectation value, and Lχ is the likelihood function. The CRLB for the standard deviation (SD) of an unbiased parameter βi is:

$$SD(β_i )=\sqrt{F_{ii}^{-1}}.\ \ \ \ [3]$$

After some algebra, Eq.2 reduces to

$$F_{ij}=\frac{1}{σ^2} \sum_{n=1}^N\frac{∂A_n}{∂β_i}\frac{∂A_n}{∂β_j}\ R_n,\ \ \ [4]$$


$$R_n=-\frac{A_n^2}{σ^2}+\frac{A_n^{1-m}}{σ^4} \int_{0}^{∞}M_n^{m+2}\ \frac{I_m^2(z_n)}{I_{m-1}(z_n)}\ exp(-\frac{M_n^2+A_n^2}{2σ^2})\ dM\ \ \ [5]$$

and must be calculated numerically.


PDF and Rn as a function of SNR and number of coils

The PDF of voxel intensity, given by Eq.1, was calculated for different values of m as a function of M/σ and for A=1, 40 and 80. Results were also obtained using a Gaussian PDF, appropriate for single coil acquisition at high SNR. Note that m=1 represents the conventional Rician noise model. In addition, the factor given by Eq.5 was calculated as a function of SNR for different values of m.

CRLB of diffusion kurtosis signal model

We consider a model, $$$A(β,b_n )=A_0\ exp(-b_n D_{app}+(b_n^2 D_{app}^2 K_{app})⁄6)$$$, describing diffusion kurtosis as a function of b-value and parameter set β=(A0, Dapp, Kapp), where Dapp and Kapp are the apparent diffusion coefficient and kurtosis, and A0 is the weighted signal obtained at b= 0 s/mm2. We evaluated the minimum SDs (i.e. CRLB) in the estimation of Dapp and Kapp as a function of SNR and for different m. Four b-values of 0, 1000, 2000, and 3000 s/mm2 were considered.

Monte Carlo (MC) simulation

MC simulations were used to assess the SDs in the estimation of Dapp and Kapp as a function of SNR. This approach mimics actual experimental implementation, as opposed to the calculated CRLB, which is a theoretical bound. Input parameters were identical to those used above, with analysis restricted to m=32. For each SNR, 10000 noisy signals were created and reconstructed using the sum-of-squares method (7) and fit to the expectation value of the noncentral χ-distribution (7-8).


Fig.1a shows that while at high SNR the noncentral χ-distribution approaches a Gaussian, at low-to-moderate SNR it deviates substantially from the Gaussian distribution. This departure is more pronounced for larger values of m. Fig.1b shows that for A>>σ (high SNR), the value of Rn approaches 1, in which case the Fisher matrix for the noncentral χ-distribution becomes identical to that of the Gaussian distribution for any m. At low-to-moderate SNR, the deviation from Rn=1 increases as m increases; we see that at low-to-moderate SNR, the Fisher matrix for the noncentral χ-distribution cannot be approximated by the Gaussian distribution.

Fig.2 shows that for all m, the SDs in the estimation of Kapp and Dapp decrease roughly exponentially with increasing SNR. Furthermore, the SDs increase with increasing m, indicating that the Gaussian approximation in the CRLB calculation is overly optimistic, especially at low-to-moderate SNR. At high SNRs, SDs in derived parameter estimates from both Gaussian- and noncentral χ-distributions converge, as expected, to the same values.

Fig.3 shows that the minimal theoretical SDs as defined by the CRLB can be achieved at moderate-to-high SNR. Most importantly, the CRLB assuming a noncentral χ-distribution for noise provided a closer match to the MC simulations as compared to the Gaussian CRLB results, as expected.


We have provided a general expression for the Fisher matrix, as used in the CRLB calculation, for imaging data described by noncentral χ-distributions, and demonstrated that the Gaussian assumption in the CRLB calculations may not hold in the setting of low-to-moderate SNR regime. These results have immediate application to experimental design for parallel imaging.


This work was supported by the Intramural Research Program of the National Institute on Aging of the National Institutes of Health.


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Fig. 1a. The noncentral χ- (Eq. 1) and Gaussian- probability density functions (PDF) as a function of M/σ for different number of coils, m. Results are shown for three different values of A of 1, 30 and 80, corresponding to low, moderate, and high SNR, respectively. For all cases, σ was fixed to 1. Fig. 1b. Correction factor, Rn, of the Fisher matrix given by Eq. 5, calculated as a function of SNR, and for different numbers of coils (i.e. m).

Fig. 2. The CRLB of the standard deviation (SD) in the estimation of Kapp and Dapp as a function of SNR. Results are obtained for different number of coils, m, and for the case of Gaussian distribution. Analysis was performed assuming four b-values of 0, 1000, 2000, 3000 s/mm2. The input parameter values for Dapp and Kapp were fixed to 1 μm2/ms and 1, respectively.

Fig. 3. Simulation results showing dispersion, that is, the standard deviation (SD) in the estimation of Kapp and Dapp as a function of SNR. The CRLB results for the SD in the estimation of Kapp and Dapp for both the noncentral χ- and Gaussian-distributions are presented. Results are obtained for m = 32. The input parameter values for Dapp and Kapp were fixed to 1 μm2/ms and 1, respectively.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)