Image formation

In order to obtain an MR image, physical hardware is needed to encode spatial information into the MR signal, and a mathematical tool is needed to decode the measured signals. Spatial information is encoded through application of slice selecting, frequency encoding, and/or phase encoding magnetic field gradients. During the time these gradients are applied, the precession frequencies of the voxel magnetizations are modulated, depending on their spatial coordinates $\vec r$:

$$\omega(\vec r) = \gamma (B_0 + \vec G. \vec r)\quad ,$$ (1.13)

with $\vec G$ denoting the magnetic field gradient vector. The MR signal received is the total signal coming from all voxel magnetizations excited by the RF pulse. Ignoring the precession due to the main magnetic induction B₀, we have:

$$S\left(t\right) = \int\!\!\int\limits\!\!\int \beta(\vec r)\exp\left(i\gamma\vec G.\vec r t\right) d\vec r\quad ,$$ (1.14)

where $d\vec r$ is used to represent volume integration. In Eq. (1.14), $\beta(\vec r)$ denotes the magnetic resonance coefficient, representing the spin density, weighted by for example T₁ and T₂ relaxation:

$$\beta(\vec r) = \rho(\vec r) f(T_1(\vec r),T_2(\vec r))\quad ,$$ (1.15)

where the function f(.) is determined by the scanning pulse sequence.
An inverse discrete Fourier transform is used to decode the acquired MR signals. Indeed, Eq. (1.14) has the form of a Fourier transformation. To make this more obvious, Mansfield introduced the concept of a reciprocal space vector $\vec k$, given by [7]:

$$\vec k = {1\over 2\pi}\gamma\vec G t\quad .$$ (1.16)

For this reason, the Fourier space, in which MR signals are acquired, is often referred to as the K-space. Fourier reconstruction is the most commonly used reconstruction method in MR imaging that results in a final complex valued image given by:

$$\beta\left(\vec r\right) = \int\!\!\int\!\!\int S(\vec k)\exp\left(-i\vec k.\vec r\right) d\vec k\quad .$$ (1.17)

Hence an MR image is a spatial representation of the magnetization distribution.