Estimation of the noise variance from the background histogram mode of an MR image D. Poot (
[email protected]), J. Sijbers Vision Lab, University of Antwerp, Belgium A. J. den Dekker, R. Bos Delft Center for Systems and Control, Delft University of Technology, The Netherlands
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The noise variance in magnetic resonance (MR) images has always been an important parameter to account for in the processing and analysis of (f)MRI data. Algorithms for segmentation, clustering, restoration, noise reduction, statistical inference etc, highly depend on the noise variance. We focus on noise variance estimation based on the background mode of the histogram of a single MR image in which the pixel grey values are represented by integers.
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RMSE of σ estimators
1 Introduction
8 6 4 2 0 10
2 Methods When in MR data the underlying noiseless signal is zero, as it is in air, the noise corrupted magnitude data is Rayleigh distributed [1]. This Rayleigh distribution appears as the background mode in the histogram of the MR data. To robustly estimate the noise variance from this mode, the left part of the histogram of the image is used. The noise variance σ 2 can be estimated by searching for the value of m for which the histogram attains a maximum
σb = mmax
(1)
Alternatively, σ can be estimated from the background mode of the histogram by maximum likelihood (ML) estimation. If {li } with i = 0, ..., K denotes the set of boundaries of histogram bins, and ni represents the number of observations (counts) within the bin [li−1 , li ], then the ML estimate σbML,K is given by the maximum of the likelihood function or equivalently by "
Ã
2
l − 02
σbML,K = arg minσ ∑Ki=1 ni ln e Ã
l2 − i−12 2σ
∑Ki=1 ni ln e
2σ
2
−e
l2 − i2 2σ
−e
l − K2 2σ
! −
!# (2)
The ML estimator can be computed for each number of bins. A procedure has been constructed that automatically selects b . This procedure the number of bins that provides the best σ uses a χ 2 test to determine the quality of the fit on the histogram, as well as some heuristic rules (mainly) to prevent the selection of too few bins.
ML estimator Maximum of histogram
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40 σ
50
60
70
Figure 1: Performance comparison of the Maximum and the ML estimator of the noise variance in terms of root mean square error (RMSE) as function of the σ used in the simulation. Average of 250 simulations.
3 Results and conclusion To test the performance of the estimators, different levels of noise are added to a noise free image [2]. In figure 1 the root mean square error (RMSE) of the estimators is compared as a function of the simulated σ . The ML estimator, in which the number of bins is automatically selected, has the best performance (i.e. lowest RMSE). Acknowledgments The research of D. Poot is funded by the I.W.T. (Institute for Science and Technology - Belgium). J. Sijbers is a postdoctoral fellow of the F.W.O. (Fund for Scientific Research - Flanders, Belgium). References [1] A. J. den Dekker and J. Sijbers, Advanced Image Processing in Magnetic Resonance Imaging, chapter Estimation of signal and noise from MR data, pp. 85–143, CRC Press, 2005. [2] C.A. Cocosco, V. Kollokian, R.K.-S. Kwan, and A.C. Evans, “Brainweb: Online interface to a 3D MRI simulated brain database; http://www.bic.mni.mcgill.ca/brainweb/ ,” NeuroImage, vol. 5, no. 4, pp. S425, 1997.
