Technical Note
Estimation of the Noise in Magnitude MR Images

https://doi.org/10.1016/S0730-725X(97)00199-9Get rights and content

Abstract

Magnitude magnetic resonance data are Rician distributed. In this note a new method is proposed to estimate the image noise variance for this type of data distribution. The method is based on a double image acquisition, thereby exploiting the knowledge of the Rice distribution moments.

Introduction

Estimation of the image noise variance (NV) is important for several reasons. Firstly, it provides a measure of the image quality in terms of image detail. Furthermore, knowledge of the NV is useful in the analysis of the magnetic resonance (MR) system: e.g., to test the performance of the MR system itself (receiver coil, preamplifier, etc.). Also, the NV is an important quality measure in functional MR imaging, where signal variations of the order of a few percent need to be detected. Finally, the NV value is often used as input for image processing techniques such as image restoration1, 2 or image filtering.3, 4

Commonly, the image NV is estimated from a single magnitude image; thereby the NV is determined directly from a large uniform signal region or from non-signal regions.5, 6 Although these methods may lead to useful NV estimates, large homogeneous regions are often hard to find, such that only a small amount of data points are available for estimation. Also, back-ground data points sometimes suffer from systematic intensity variations. To cover these disadvantages, methods were developed based on two acquisitions of the same image: the so-called double acquisition methods. Thereby, the amount of noise is for example computed by subtracting two acquisitions of the same object and calculating the standard deviation of the resulting image pixels.[7] Murphy et al.[8] elaborated this technique further and used a parallel rod test object for NV measurements from the signal and non-signal blocks. The double acquisition methods have an advantage over the single image techniques in that they are relatively insensitive to structured noise such as ghosting, ringing and direct current artefacts. However, a strict requirement is the perfect geometrical registration of the images. To overcome this restriction, recently a cross-correlation technique of the two acquisitions was suggested.[9] Besides geometrical registration, another problem may arise: due to small timing errors the raw data from one acquisition may be shifted relative to the other. After Fourier transformation, this results in different phase variations of the complex data such that the above double acquisition NV estimation methods are no longer valid. To overcome this problem, we propose an NV estimation method based on two magnitude MR images.

In magnetic resonance imaging (MRI), the acquired complex data is known to be corrupted by white noise having a Gaussian probability distribution (PD). After inverse Fourier transformation the real and imaginary data is still corrupted with Gaussian noise because of the orthogonality of the Fourier transform. Although all information is present in the real and imaginary images, it is common practice to work with magnitude and phase images instead as they have more physical meaning (proton density, flow, etc.). However, computation of a magnitude image is a non-linear operation which changes the data distribution. It can be shown that the data in a magnitude image is no longer Gaussian but Rician distributed.5, 10 In this note it is demonstrated how the properties of this distribution can be exploited to estimate the image noise variance.

This paper is organized as follows. We first review briefly the properties of the Rice distribution. Then we show how these properties can be exploited to estimate the image noise variance using a double image acquisition. The proposed method is first tested on an artificial image. This was done because in a controlled situation, unforeseen errors, such as a bias, can be detected. Finally, the method is tested on various MR images.

Section snippets

The Rice Distribution

If the real and imaginary data, with mean values AR and AI respectively, are corrupted by zero mean Gaussian, stationary noise with standard deviation σ, it is easy to show that the PDF of the magnitude data will be a Rician distribution (pages 138–139 of[11]), given by: pM(M⊻A)=Mσ2eM2+A22I0(AMσ2) where I0 is the zero order modified Bessel function of the first kind. M denotes the pixel variable of the magnitude image and A is given by: A=AR2 + AI2 Notice that the Rice distribution tends to

Noise Variance Estimation

From Eq. (5)one can determine the noise variance σ2. In MR imaging, a common way to unbiasedly estimate σ2 of a magnitude image is by estimating E[M2] from a spatial average of the squared background data points, where A is known to be zero:6, 10, 12, 13 σ2̂=12〈M2 This approach usually requires user interaction to select the background pixels.

We propose an alternative noise estimation scheme using a double acquisition scheme. When two images are acquired under identical imaging conditions, one

Experiments and Discussion

The performance of the noise variance estimation method was in a first phase tested on an artificial image: the “Lena” image, well known in image processing. This was done because it is a controlled situation in which a possible bias should be revealed if present. The dimensions of the image were 128 × 128. From the Lena image, two independent Rician distributed images with standard deviation σ were generated. The proposed noise estimation method was tested for various values of σ. Simulation

Conclusion

When it comes to estimation of the image noise variance, methods based on a double acquisition are far superior to single image techniques in terms of precision. However, existing double acquisition methods become useless when different phase variations are present in the two images. To overcome this problem, a noise variance estimation method has been proposed based on two magnitude images. Under the condition of geometrical registration, the proposed noise variance estimator has been shown to

References (13)

There are more references available in the full text version of this article.

Cited by (238)

  • Noise estimation in single coil MR images

    2021, Biomedical Engineering Advances
  • An improved nonlocal maximum likelihood estimation method for denoising magnetic resonance images with spatially varying noise levels

    2020, Pattern Recognition Letters
    Citation Excerpt :

    Different methods for estimating noise levels from MR magnitude images are presented in the literature. These estimators can be classified into two kinds, namely, the estimators for Rician model [17–22] and estimators for nc- χ model [18,23–25]. However, most of the aforementioned noise estimators cannot be used as such for pMRI-acquired images because the noise in those images vary spatially.

View all citing articles on Scopus
View full text