Theory

In this subsection, we outline the theory for the estimation of the image SNR. In the description of the method, we assume the MR imaging process to be stationary; i.e., the statistical properties of two images, acquired at different times, are equal [68]. We assume that an experimental MR image i, defined on an N x N square lattice, consists of a deterministic signal s, corrupted by additive, uncorrelated noise n with zero mean ( $\langle n \rangle =0$). The signal s includes possible blurring, caused by the system point spread function:

$$i(\vec r) = s(\vec r) + n(\vec r)\quad ,$$ (7.3)

where $\vec r$ denotes the image point coordinates. As a definition of the SNR, the ratio of the signal standard deviation to the noise standard deviation is chosen:

$${\rm SNR} = {\sigma_s \over \sigma_n}\quad .$$ (7.4)

The SNR, as defined above, cannot be determined exactly from one experimental acquisition only. However, it has been shown [67] that in case of uncorrelated, additive noise, two consequent acquisitions i₁ and i₂

$$i_1(\vec r) s(\vec r) + n_1(\vec r)$$ = (7.5)

$$i_2(\vec r) s(\vec r) + n_2(\vec r)$$ = (7.6)

can be used to estimate the SNR. The cross-correlation function (CCF) of the two images becomes:

$$i_1\otimes i_2 = s\otimes s + n_1 \otimes s + s \otimes n_2 + n_1 \otimes n_2\quad .$$ (7.7)

Since the noise is uncorrelated, one has

$$E\left[n_1 \otimes s\right ] = E\left [s \otimes n_2\right ] = E\left [n_1 \otimes n_2 \right ] = 0\quad ,$$ (7.8)

so that

$$E\left [i_1\otimes i_2\right ] = s\otimes s\quad ,$$ (7.9)

i.e., the CCF of the two images is equal to the auto correlation function (ACF) of the signal. This observation is utilized in the cross-correlation coefficient (CCC), which is defined as:

$$\rho(\vec r) = {i_1 (\vec r) \otimes i_2 (\vec r) - \langle... ...\rangle \langle i_2 \rangle \over \sigma_1 \sigma_2}\quad ,$$ (7.10)

where $\langle i_1 \rangle$, $\langle i_2 \rangle$, $\sigma_1$, and $\sigma_2$ are the mean and standard deviation of the two MR images i₁ and i₂, respectively. The SNR can be computed from the maximum of the CCC. When the two acquisitions are perfectly registered (no shift of the sample has occurred) this maximum occurs in the center of the CCC:

$$\rho_m = {\langle i_1 i_2\rangle - \langle i_1\rangle\langl... ... \left[\langle i_2^2\rangle - \langle i_2\rangle ^2\right]}}$$ (7.11)

or using Eq. (7.9):

$$\rho_m = {\langle s^2 \rangle - {\langle s \rangle}^2 \ove... ...{\langle s\rangle}^2 + \langle n_2^2\rangle\right]}}\quad ,$$ (7.12)

which finally results in the following simple expression:

$$\rho_m = {\widehat{\sigma_s^2} \over \widehat{\sigma_s^2} + \widehat{\sigma_n^2}}\quad .$$ (7.13)

From this it is easy to derive the expression for the SNR estimate:

$$\widehat{{\rm SNR}} = \sqrt{{\rho_m \over 1- \rho_m}}\quad .$$ (7.14)

Notice that the subtraction of $\langle i_1\rangle\langle i_2\rangle$ in the numerator of Eq. (7.11) along with the denominator make the SNR estimator (7.14) insensitive to differences in scaling constants between the two MR images.

The calculation of $\rho_m$ can be completely performed in the Fourier domain. Indeed, using Parseval's theorem, one obtains from Eq. (7.11):

$$\rho_m = {\langle I_1 I_2^*\rangle - I_1(\vec 0) I_2(\vec 0... ...] \left[\langle I_2^2\rangle - I_2^2(\vec 0)\right]}}\quad ,$$ (7.15)

where I and I^* are the complex raw MR data and its complex conjugate, respectively. The vector $\vec 0$ represents the center of the CCC. Eq. (7.15) allows the SNR of the MR image to be calculated directly from the raw MR data. In this way, the SNR can be predicted before the Fourier transformation takes place.

The method can be applied, provided the images are perfectly registered. If the acquisitions are not perfectly registered, the maximum of the CCF will in general decrease, which leads to an underestimation of the SNR. However, the CCF maximum will not be affected if the images differ from a uniform translational shift and hence the SNR estimation is still valid. Uniform geometrical registration can easily be performed by examining the position of the CCF maximum. In this way, subpixel registration can even be achieved by for example bilinear interpolation.