Tobias Wech^{1}

This presentation will provide an introduction to the k-space formalism. MRI will be approximated as a linear and stationary system and the point spread function as well as the modulation transfer function will be introduced as descriptive tools. Finally, the sampling-theorem of Nyquist and Shannon will be discussed with respect to classical MRI and newer techniques like compressed sensing or MR-fingerprinting.

MR physicists and engineers, pulse sequence developers and clinicians who want to deepen their understanding of MRI.

Understanding:

- The k-space formalism in MRI
- The point spread function and the modulation transfer function
- Characteristics of linear and stationary systems
- The sampling theorem of Nyquist and Shannon

The data acquisition in MRI can be elegantly described as a measurement process in the so-called k-space $$$K(k)$$$. This raw data domain is reciprocal to the image space, such that an inverse Fourier transform needs to be applied to reconstruct the spin density $$$\rho(r)$$$:

$$\rho(r) \propto \int_{-\infty}^{\infty} K(k) \cdot e^{ikr} dk \quad [1] $$

In classical spin-warp sampling, the continuous k-space signal is measured at discrete positions and up to a maximum coordinate $$$k_{max}$$$, yielding an image $$$I(r)$$$:

$$I(r) \propto \int_{-\infty}^{\infty} box(k) \cdot III(k) \cdot K(k) \cdot e^{ikr} dk \quad [2]$$

The $$$box$$$ function equals 1 from $$$-k_{max}$$$ to $$$+k_{max}$$$ and is zero elsewhere; $$$III$$$ represents the dirac comb, which features periodic dirac delta peaks with a distance of $$$\Delta k$$$.

Expression $$$[2]$$$ can be re-written as a convolution:

$$I(r) \propto \int_{-\infty}^{\infty} box(k) \cdot III(k) \cdot K(k) \cdot e^{ikr} dk \propto \hat{F}(box(k)) \circledast \hat{F}(III(k)) \circledast \rho(r), \quad [3]$$

where $$$ \hat{F}$$$ reresents the inverse Fourier transform. As $$$\hat{F}(III(k))$$$ is again a dirac comb, with periodic dirac delta functions and an inter-pulse distance of $$$2 \pi / \Delta k$$$, the imaged object is also repeated with this spatial frequency. In order to avoid foldings, a sufficiently small $$$\Delta k \lt 2 \pi / l$$$ needs to be used, where $$$l$$$ represents the extent of the object in the sampled dimension. This latter inequality is also known as "Nyquist-Shannon sampling theorem". Conversely, a field-of-view (FOV) can be assigned to the imaging process:

$$FOV = 2 \pi / \Delta k \quad [4]$$

The first term on the right hand side of [3] represents the inverse Fourier transform of the $$$box$$$-function, which yields a $$$sinc$$$-function:

$$\hat{F}(box(k)) = sinc(k_{max} \cdot r) = \frac{sin(\pi \cdot k_{max} \cdot r)}{\pi \cdot k_{max} \cdot r}$$

The discretely sampled object is convolved with this function, such that the width of the main-lobe of the $$$sinc$$$ - which soleley depends on $$$k_{max}$$$ - defines the resolution of the system.

The point spread function (PSF) represents a general and abstract tool to describe the imaging process $$$T$$$: Based on measurements, an image $$$I(r)$$$ of a real object $$$O(r)$$$ is created according to the following equation:

$$I(r) = T(O(r)) \quad [5]$$

If the transformation $$$T$$$ is linear:

$$\sum_i T(\lambda_i O_i) = \sum_i \lambda_i T(O_i), \quad [6]$$

it can be fully described by its Green-function $$$G$$$:

$$I(r) = \int G(r,\tilde{r}) O(\tilde{r}) d \tilde{r}. \quad [7]$$

$$$G$$$ thereby describes the transformation of a point source $$$\delta(r-\tilde{r})$$$ at position $$$\tilde{r}$$$. If T is also stationary (or equivalently: shift-invariant), $$$G$$$ simplifies to $$$G(r,\tilde{r}) = G(r - \tilde{r}) $$$ and is typically called PSF in this form. Equation [7] can now be formulated as

$$I(r) = \int PSF(r - \tilde{r}) O(\tilde{r}) d \tilde{r} = (PSF \circledast O ) (r), \quad [8]$$

such that the imaging process represents a convolution of the object with this PSF.

As classical MRI can be considered as linear and stationary in good approximation, the PSF serves as an excellent tool to describe its characteristics.

Haacke EM, Brown RW, Thompson MR, Venkatesan R. MagneticResonance Imaging: Physical Principles and Sequence Design. Wiley-Liss, 1999.

Twieg DB. The k-trajectory formulation of the NMR imaging process with applicationsin analysis and synthesis of imaging methods. Medical Physics 10(5):610-21, 1983

Steckner MC, Drost DJ, Prato FS. Computing the modulationtransfer function of a magnetic resonance imager. Medical Physics 21(3):483-489, 1994

Edelstein WA, Hutchison JMS, Johnson G, Redpath T. Spin warp NMR imaging and applications to human whole-body imaging. Physics in Medicine and Biology 25(4):751-756, 1980