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Spatial encoding and image reconstruction 

Spatial encoding and image reconstruction
Chapter:
Spatial encoding and image reconstruction
Author(s):

Sebastian Kozerke

, Redha Boubertakh

, and Marc Miquel

DOI:
10.1093/med/9780198779735.003.0003
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date: 08 March 2021

Slice selection

Excitation produces transverse magnetization which, in turn, can be detected using a receiver antenna or coil until it has decayed to zero due to relaxation, as discussed in Section 1, Chapter 2. Without any further provision, excitation would flip magnetization of the entire sample, and hence the sum of all transverse magnetization components of the sample is detected. Consequently, no spatial resolution is obtained and, accordingly, the process does not yield any spatial selectivity.

In order to selectively excite a slab or slice of the sample, let us recall Spatial encoding and image reconstruction Fig. 1.2.3. The energy difference Δ‎E is proportional to the Larmor frequency ω‎0, which, in turn, depends on B0, as recapitulated in Spatial encoding and image reconstruction Fig. 1.3.1. Spin transitions or excitation can only occur if RF energy on the right, i.e. the Larmor frequency, is applied. If the frequency of the excitation RF pulse has an offset relative to the Larmor frequency, magnetization is not manipulated. This insight is the basis for slice-selective excitation.

Fig. 1.3.1 The energy difference Δ‎E between low- and high-energy states is proportional to the Larmor frequency ω‎0 and consequently proportional to B0. Excitation occurs only if the sample is irradiated with radiofrequency energy at the Larmor frequency ω‎0.

Fig. 1.3.1
The energy difference Δ‎E between low- and high-energy states is proportional to the Larmor frequency ω‎0 and consequently proportional to B0. Excitation occurs only if the sample is irradiated with radiofrequency energy at the Larmor frequency ω‎0.

In Section 1, Chapter 1, magnetic field gradient coils have been mentioned. These coils allow adding a spatially dependent offset to the static magnetic field B0. To this end, the Larmor frequency ω‎0 becomes a function of spatial positon. In analogy to a piano, this process encodes spatial position into frequency, as illustrated in Spatial encoding and image reconstruction Fig. 1.3.2. Accordingly, the position left and the position right are discriminated by their frequency (the tone ‘C’ and its octave). To excite the spins in the left and right positions, RF energy of different frequencies is required in the presence of a gradient field Gz.

Fig. 1.3.2 A magnetic gradient field Gz causes the Larmor frequency ω‎ to change linearly with position z. Accordingly, spin transitions, i.e. excitation, can only occur for spins along z which receive radiofrequency energy at the exact right Larmor frequency ω‎(z). All other spins will not be excited.

Fig. 1.3.2
A magnetic gradient field Gz causes the Larmor frequency ω‎ to change linearly with position z. Accordingly, spin transitions, i.e. excitation, can only occur for spins along z which receive radiofrequency energy at the exact right Larmor frequency ω‎(z). All other spins will not be excited.

To accomplish excitation of a slab or slice, the RF pulse needs to be designed such that it contains the Larmor frequencies Δ‎ω‎ corresponding to the slab or slice thickness Δ‎z, as prescribed by the user and as outlined in Spatial encoding and image reconstruction Fig. 1.3.3. It follows that for a thicker slab or slice, the gradient needs to be made weaker, while for a thinner slab or slice, the gradient should be made stronger for a given RF pulse. A slab or slice can be prescribed along any orientation in space by activating the appropriate combination of the three available gradient coils, as shown in Spatial encoding and image reconstruction Fig. 1.3.4.

Fig. 1.3.3 To select a slab or slice of thickness Δ‎z, a magnetic gradient field Gz is required which linearly increases the Larmor frequency, depending on position z. In order to excite all spins within Δ‎z, the radiofrequency (RF) pulse needs to contain all frequencies within Δ‎ω‎ which yields the RF pulse shape as shown.

Fig. 1.3.3
To select a slab or slice of thickness Δ‎z, a magnetic gradient field Gz is required which linearly increases the Larmor frequency, depending on position z. In order to excite all spins within Δ‎z, the radiofrequency (RF) pulse needs to contain all frequencies within Δ‎ω‎ which yields the RF pulse shape as shown.

Fig. 1.3.4 The MR scanner contains three gradient coils which produce magnetic gradient fields along the x, y, and z axes of the scanner. Arbitrary slab or slices are selected by switching the appropriate combination of gradient coils.

Fig. 1.3.4
The MR scanner contains three gradient coils which produce magnetic gradient fields along the x, y, and z axes of the scanner. Arbitrary slab or slices are selected by switching the appropriate combination of gradient coils.

Spatial encoding

Once a slab or slice is selectively excited, the information within the slab or slice needs to be spatially resolved. Here the same concept as outlined above is used to encode spatial position into the Larmor frequency. This time, however, spatial encoding happens after excitation (and not during excitation as for slice selection) using the so-called frequency encoding and phase encoding.

Let us consider the simplified object, as shown in Spatial encoding and image reconstruction Video 1.3.1. By using a magnetic gradient field along the x-axis, the Larmor frequency for all spins perpendicular to the gradient axis varies linearly along x. Accordingly, the rotation speed of the transverse magnetization differs, depending on position, and hence the process is referred to as frequency encoding.

Video 1.3.1 Frequency encoding maps spatial position along the x-axis to rotation frequency of transverse magnetization, as shown in the rotating frame of reference.

Spatial encoding along the second dimension within the slab or slice is encoded in a similar manner. Instead of detecting the instantaneous rotation speed of transverse magnetization, a phase angle, depending on the position along y, is encoded, as exemplified in Spatial encoding and image reconstruction Video 1.3.2. Accordingly, the encoding process is denoted as phase encoding.

Video 1.3.2 Phase encoding maps spatial position along the y-axis to phase differences of transverse magnetization, as illustrated in the rotating frame of reference.

The k-space concept

Spatial mapping of the object using frequency and phase encoding can be viewed as decomposing the object into sine waves whose amplitudes and phases are recorded. Spatial encoding and image reconstruction Fig. 1.3.5 visualizes the correspondence between the object space and the so-called k-space or measurement space, which records the amplitude and phases of the sine waves. The k-space coordinates are given by frequency encoding and phase encoding according to kx∼Gx*t and ky∼Gy*t with time t. The distance between adjacent k-space locations Δ‎k is given by the inverse of the field of view (FOV). Accordingly, a larger FOV requires smaller steps in k-space. The voxel size or resolution Δ‎x of the object image is given by the FOV divided by the number of k-space points N sampled.

Fig. 1.3.5 Using frequency and phase encoding, any object is decomposed into sine functions with frequency and phase. The amplitudes of these sine functions can be viewed in the so-called k-space representation whose coordinates kx and ky are given by the gradient fields Gx and Gy. The sampling distance Δ‎kx is given by the inverse of the field of view (FOVx), while the spatial resolution Δ‎x is given by the ratio of FOVx divided by the number of k-space points sampled Nx.

Fig. 1.3.5
Using frequency and phase encoding, any object is decomposed into sine functions with frequency and phase. The amplitudes of these sine functions can be viewed in the so-called k-space representation whose coordinates kx and ky are given by the gradient fields Gx and Gy. The sampling distance Δ‎kx is given by the inverse of the field of view (FOVx), while the spatial resolution Δ‎x is given by the ratio of FOVx divided by the number of k-space points sampled Nx.

Simple MR sequence

To record an image, slice-selective excitation, phase encoding, and frequency encoding are played out sequentially in time, as indicated in Spatial encoding and image reconstruction Fig. 1.3.6 with the corresponding k-space view. Upon slice-selective excitation using the slice-select gradient Gz (k-space location ‘A’), the phase-encoding gradient Gy is activated, along with the prewinder of the frequency-encoding gradient Gx. Since the position in k-space is given by the gradient times the time product (k∼G*t), k-space location ‘B’ is reached and the phase-encoding gradient is switched off. Upon activating the frequency-encoding gradient Gx, the MR signal is sampled at Nx points, while traversing from k-space location ‘B’ to ‘C’. The scheme is repeated with different phase-encoding gradient strengths Gy until all Ny lines (= profiles) along ky have been acquired. The duration of each repetition is called the repetition time TR.

Fig. 1.3.6 Simple MR pulse sequence and corresponding k-space trajectory. MR data are sampled and recorded, while the frequency-encoding gradient (Gx) is switched on (B–C). This module is repeated for different phase-encoding gradient strength Gy with repetition time TR.

Fig. 1.3.6
Simple MR pulse sequence and corresponding k-space trajectory. MR data are sampled and recorded, while the frequency-encoding gradient (Gx) is switched on (B–C). This module is repeated for different phase-encoding gradient strength Gy with repetition time TR.

Image reconstruction

Once all (Nx × Ny) MR signals have been collected in k-space, the data can be subjected to a Fourier transform to obtain the image of the object, as illustrated in Spatial encoding and image reconstruction Fig. 1.3.7. The number of MR signals acquired along kx and ky determines the voxel size along x and y, according to Δ‎x = FOVx/Nx and Δ‎y = FOVy/Ny where FOVx and FOVy are the FOVs along the x and y dimensions, respectively. The total scan time is proportional to the number of phase-encoding steps Ny. For the simple MR sequence, the total scan time is determined by Ny*TR where TR denotes the repetition time required to perform an MR experiment with one phase-encoding step.

Fig. 1.3.7 Image reconstruction converts the MR signals acquired in k-space into an image pixel map using the Fourier transform operation on a computer.

Fig. 1.3.7
Image reconstruction converts the MR signals acquired in k-space into an image pixel map using the Fourier transform operation on a computer.

In CMR imaging, the MR pulse sequence is typically synchronized with the R-wave of the electrocardiogram (ECG) and multiple ‘snapshots’ or phases of a cardiac cycle can be imaged by sequentially playing the simple MR sequence before advancing to the next phase-encoding step. Spatial encoding and image reconstruction Fig. 1.3.8 schematically illustrates such a scheme.

Fig. 1.3.8 Schematics of an ECG-triggered multi-phase CMR scan of a cardiac short-axis view.

Fig. 1.3.8
Schematics of an ECG-triggered multi-phase CMR scan of a cardiac short-axis view.