# Noise in MRI (magnitude) data

Standard imaging techniques acquire complex k-space data, then an image is calculated as the magnitude of the Fourier transform of the k-space data.  The noise in the magnitude images ((R. M. Henkelman. Measurement of signal intensities in the presence of noise in MRI images.  Med. Phys., 12(2):232–233, 1985))  has a Rician distribution, based on the assumption that the noise on the real and imaginary channels is Gaussian.  The probability density function for a Rice distribution is defined ((E. W. Weisstein. Rice distribution. Eric Weisstein’s World of Mathematics, 2000. http://mathworld.wolfram.com/RiceDistribution.html. )) as:

$latex P_{\mathrm{Rice}}(x) = \frac{x}{\sigma^2}\exp\left(-\frac{\mu^2+x^2}{2\sigma^2} \right) I_0\left(\frac{\mu x }{\sigma^2} \right)$

\label{eq:rician_pdf}

where $latex \sigma$ is the “standard deviation” of the function, $latex \mu$ is the mean, $latex x$ is the parameter, and $latex I_0\left(x\right)$ is a modified Bessel function of the first kind.  The two interesting implications of the Rice
distribution is what happens as a function of $latex \mu/\sigma$.

Rice distribution as a function of $latex \mu/\sigma$.  When $latex \mu/\sigma$ is zero, for example in the black area of an MRI image, then the noise is Rayleigh distributed.  When $latex \mu/\sigma$ is large, for example in the object of an MRI image, then the noise approximates a Gaussian distribution.

## Limiting case as $latex \mu -> 0$

As $latex \mu/\sigma \to 0$, the Rician distribution becomes a Rayleigh distribution.  The Bessel function $latex I_0(\mu) = 1$ when $latex \mu = 0$ and $latex \exp\left(-\frac{\mu^2+ x^2}{2 \sigma^2} \right) = \exp\left(-\frac{x^2}{2 \sigma^2}\right)$ when$latex \mu = 0$. Therefore,

$latex P_{\mathrm{Rice}}(x) &=& \frac{x}{\sigma^2} \exp\left(-\frac{x^2}{2\sigma^2} \right) \\ &=& P_{\mathrm{Rayleigh}}(x)$

\label{eq:rayleigh_pdf}

## Limiting case as $latex \mu$ is large

As $latex \mu/\sigma \to \infty$, the Rician distribution becomes similar to a Gaussian distribution.  The asymptotic expansion ((James A. Roberts. http://people.eecs.ku.edu/~jroberts/private/Appendix A.pdf))  of the Bessel function $latex I_0(x)$ when $latex x$ is large is:

$latex I_0(x) \sim \frac{e^x}{\sqrt{2 \pi x}} \left[ 1 + \frac{1}{8x} + \frac{1\cdot 9}{2!(8x)^2} + \frac{1\cdot 9 \cdot 25}{3!(8x)^3} + \cdots \right]$
\label{eq:bessel_expansion}

then, because $latex x \approx \mu$ the scaling factor in front of the exponential can be simplified leaving:

$latex P(x) & = & \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left[-\frac{\left(x-\mu\right)^2}{2\sigma^2}\right] \\& = & P_{\mathrm{Gaussian}}(x)$

which is the probability function for Gaussian distribution.

All simulations, in the following chapters, used Rician noise.  The Rician noise was created as $latex y_e(t_i) = \sqrt{ \left[y(t_i) + e_1 \right]^2 + e_2^2 }$, where $latex y$ is the true signal, and $latex e_1$ and $latex e_2$ are random numbers from a Gaussian distribution with zero mean and standard deviation $latex \sigma$.  The standard deviation, $latex \sigma$, for the Gaussian distribution was defined as $latex y(t_1) / {\rm SNR}$.