Imaging, contrast and noise
MR physics: Imaging, contrast and noise
The signal detected from an anatomical region is the sum of the signals from dipoles all over the region. In order to form an image, the dipoles must somehow be spatially resolved. This is usually a two-step process: (i) exciting the magnetization into the transverse plane over a spatially restricted region and (ii) encoding spatial location of the signal during data acquisition. For medical applications, the image must reflect clinically relevant anatomy and physiology. One way of achieving this is by manipulating image contrast with appropriate data acquisition parameters. Ultimately, image quality is constrained by noise, which enters MRI in an interesting way. The basic principles of image formation, contrast manipulation, and SNR are introduced in the following sections. In each section, the role of signal processing is highlighted.
Spatially selective excitation
The usual goal in spatially selective excitation is to tip magnetization in a thin spatial slice or section, say of thickness 
along the z axis, into the transverse plane. Conceptually, this is accomplished by first causing the Larmor frequency to vary linearly in one spatial dimension 
and then, while holding the field constant, applying an RF excitation pulse crafted to contain significant energy only over a limited range of temporal frequencies (BW) corresponding to the Larmor frequencies in the slice.
To a first approximation, the amplitude of the component at each frequency in the excitation signal determines the flip angle of the protons resonating at that frequency. As a result, if the temporal Fourier transform of the pulse has a rectangular distribution about 
, a rectangular distribution of spins around 
is tipped away from the z axis over a spatial extent 
. For small tip angles (generally, 
), we can solve the Bloch equations explicitly to get the spatial distribution of 
following an RF pulse 
(t) in the presence of a magnetic field gradient of amplitude 
:
assuming that all the magnetization initially lies along the z axis. Under these conditions, a rectangular slice profile is achieved if 
).
Signal processing methods have been applied to improve the accuracy and to extend the applications of selective excitation. For one-dimensional excitation, the sharpness of the edge and the degree of suppression in the stopband have been improved by taking into account the non-linear nature of the Bloch equation [6,7]. More recently, the Shinnar-LeRoux algorithm has been developed to transform one-dimensional pulse design into a finite-impulse-response digital filter design problem, where many standard digital signal processing tools can be applied [8]. Selective excitation has also been extended to multiple spatial dimensions through linear systems theory under the small-tip Fourier approximation [9]. The two-dimensional case is the dual of spatial encoding, which is described in the following section.
Image formation through spatial frequency encoding
The imaging equation
Once one has isolated a volume of interest using selective excitation, the volume can be imaged by manipulating the precession frequency (determined by the Larmor relation --- Eq. 2), and hence the phase of 
[4]. For example, introduce a linear magnetic field gradient, 
in the x direction so that 
; each dipole now contributes signal at a frequency proportional to its x-axis coordinate. In principle, by performing a Fourier transform on the received signal, one can determine 
as a function of x. An equivalent point of view follows from observing that each dipole contributes signal with a phase that depends linearly on its x-axis coordinate and time. Thus the signal as a whole samples the spatial Fourier transform of the image along the 
spatial frequency axis, with the sampled location moving along this axis linearly with time.
A more general viewpoint can be developed mathematically from the Bloch equation. For simplicity, the problem is limited to producing a spatial map in two dimensions, x and y. In practical situations, this is achieved using spatially selective excitation (sec. iii-A) where only protons in a thin slice at 
, are tipped into the transverse plane so that 
[3]. Let the magnetic field after excitation be 
. Assume 
is relatively constant during data acquisition (i.e. acquisition duration 
T1,T2,T2*). Let the time at the center of the acquisition be 
. During acquisition,
The signal received, 
, is the integral of this signal over the xy plane. Letting 
and 
,
If this signal is demodulated by 
then the resulting baseband signal, 
, is the two-dimensional spatial Fourier transform of 
[10] at spatial frequency coordinates 
and 
. One chooses 
and 
so that, over the full data acquisition, the 2D frequency domain is adequately sampled and the desired image can be reconstructed as the inverse Fourier transform of the acquired data.
Image characteristics: Sampling issues
In general, the 
frequency domain, referred to as k-space, cannot be sampled completely after a single excitation. This is because of a variety of physical limitations (finite relaxation time of the dipoles and SNR limitations) and technical limitations (slew rate limits on 
and 
). Thus, k-space is sampled in a sequence of n excitation-acquisition cycles with repetition time 
.
The most popular method of sampling k-space is referred to as two-dimensional Fourier transform (2DFT) or spin-warp imaging [11]. During each acquisition, this method samples the signal along a line in k-space corresponding to 
some constant. The subsequent n-1 a
