Today I will take a look at the basics of parallel magnetic resonance imaging (MRI) from a frames perspective. I know of only one publication  discussing frame-based encoding for MRI nevertheless I think that the mathematics of frames provides a solid foundation for understanding parallel imaging. In my opinion all practical or currently used methods of doing parallel imaging should be understood in terms of how well they can approximate a frame-based image reconstruction.

Back in the day there was but one coil element in a receiver designed for magnetic resonance imaging of the head. Modern commercial MRI scanners are now equipped with head-imaging receivers comprised of an array of 8 to 32 coil elements. These extra elements are primarily used for two purposes: (1) To improve signal-to-noise ratio (SNR), or (2) To decrease the time required to collect the imaging data (accelerated imaging). The term parallel imaging is usually associated with the second purpose but I will use the term to define imaging methods which simply make use of a receive array for either purpose (with or without acceleration). So what is a frame and what does this mathematical concept have to do with MRI?

Today I will focus on 2D multislice imaging since this is the workhorse sequence for functional MRI (fMRI). If you are familiar with the mathematics of conventional (non-parallel) 2D imaging then you are aware that: (1) First the magnetization in a particular 2D “slice” is induced to precess about the ${z}$-axis (the main static field being in the ${z}$-direction) by an applied radio-frequency magnetic field (the transmit or excitation field) and (2) Subsequently applied time-varying linear gradients in the z-component of the magnetic field are responsible for generating a set of encoding functions that span the imaging space. These encoding functions of conventional MRI are that of a Fourier basis.

Parallel imaging uses a set of encoding functions which are in part due to the individual receive coil elements used to detect the signal generated by the precessing magnetization. There are presently two parallel imaging methods by which 2D imaging may be accelerated. One method, which I will refer to as the phase-encoded frame method, accomplishes this acceleration by using the focal sensitivity of the individual receive coils to eliminate the need to acquire some of the analysis coefficents corresponding to the elements of the conventional Fourier imaging basis. Another method, which I will refer to as the slice-encoded frame method, uses the focal sensitivity of the receive coils to generate encoding functions in the slice direction ${z}$. In the literature it appears that this second method is most often referred to as multiband imaging.

Today’s post will address the phase-encoded frame method of parallel imaging leaving the discussion of slice-encoded frame method for another time. The phase-encoded frame uses a set of encoding functions generated by a product of some subset of the usual Fourier basis with a set of window functions ${g_n^*(x,y)}$ (where ${n = 1, \ldots, N_c}$ and ${N_c}$ is the number of receive coil elements in the array) that are related to the receive field of the coil elements of the receiver array. This set of encoding functions may comprise a frame. A frame is set of functions that spans the imaging space and can be used to perform stable analysis (data acquisition) and synthesis (image reconstruction) of the image data.

The data associated with the Fourier-like term of the phase-encoded frame may be sampled in different temporal orders and this temporal order establishes a trajectory through a spatial frequency space universally (MRI universe that is) referred to as ${k}$-space. In practice different temporal orderings can change the nature of imaging artifacts in the presence of perturbations to the ideal imaging situation. In this post I will not choose a particular trajectory. So I am in effect assuming an ideal imaging experiment with none of the many nasty perturbations that may occur in practical MRI. In future posts I will return to the topic of trajectories (particularly that of the 2D multislice imaging workhorse of fMRI – echo planar imaging) and the artifacts associated with them.

So without further ado …

The Phase-Encoded Frame

So, what is a frame? A few definitions and properties should be enough to get us started but please see Ole Christensens’s book  for a rigorous introduction to frames.

Frame Definition: A set of functions ${{\mathcal F}= \lbrace f_n \rbrace}$ in a Hilbert space ${\mathcal H}$ is called a frame if there exists ${A>0, B<\infty}$ so that, for all ${f}$ in ${\mathcal H}$ $\displaystyle A \| f \|^2 \le \sum_n |\langle f, f_n \rangle|^2 \le B \| f \|^2 \ \ \ \ \ (1)$

The ${A}$ and ${B}$ are called the frame bounds. If ${A=B}$ the frame is said to be a tight frame. There will be more to say about frame bounds and tight frames in later posts.

Frame Operator Definition: To every frame ${{\mathcal F}}$ there corresponds an operator ${S}$, known as the frame operator, from ${\mathcal H}$ onto itself defined by $\displaystyle S f = \sum_n \langle f, f_n \rangle f_n \ \ \ \ \ (2)$

for all ${f \in \mathcal H}$.

Synthesis Property: If ${{\mathcal F}}$ constitutes a frame for all ${f}$ in ${\mathcal H}$ then $\displaystyle f = \sum_{n'=-\infty}^\infty \langle \rho, f_{n'} \rangle \hat{f}_{n'} \ \ \ \ \ (3)$

where the set of functions ${\lbrace \hat{f}_n \rbrace}$ is called the dual frame to frame ${{\mathcal F}}$. It is important to note that there is not a unique dual frame corresponding to a given frame. Depending upon the application this nonuniquess may come in handy.

The lower bounds in equation (1) assures the numerical stability of the synthesis of ${f}$ from the coefficients ${\langle f, f_n \rangle}$. The upper bounds assures that the sequences ${\langle f, f_n \rangle}$ are in ${\ell^2(\mathbb Z)}$, i.e. that ${\sum_j |\langle f, f_n \rangle|^2 < \infty}$ for all ${f \in L^2(\mathbb R)}$.

Frames should be contrasted with bases. A basis is a minimally complete set of functions spanning a vector space. Frames, one the other hand, may be over complete – a basis being a special case of a frame. Now, back to the parallel imaging story …

Assume that there are ${N_c}$ receive coils in the receive array. The signal from the ${n^{th}}$ receiver coil ( ${n=1, \ldots, N_c}$), after Fourier transform in the frequency-encoding (FE) ${x}$-direction, may be written as $\displaystyle s_n(x,m) = \langle \rho, g_{nm} \rangle = \int \rho(x,y) g^*_{nm}(x,y) dy \ \ \ \ \ (4)$

where $\displaystyle g_{nm}(x,y) = g_n^*(x,y) e^{i 2\pi m R\Delta k y} \ \ \ \ \ (5)$

and where ${g_n(x,y)}$ is the receive field of the ${n^{th}}$ coil element of the array, ${R \ge 1}$ is called the reduction factor, and ${m \in \mathbb Z}$. Note that ${\rho(x,y)}$ is assumed to be an image that is compactly supported on ${|y|, |x| < 1/\Delta k}$ and that the phase-encode (PE) direction, ${y}$, will be the direction of possible acceleration. Also note that ${s_n}$ and ${g_n}$ will depend upon ${z}$ as well but I have omitted explicitly writing this dependence. Note that although ${\langle \rho, g_{nm} \rangle}$ is a function of ${x}$ I will often suppress the ${x}$-dependence for the sake of an economy of symbols.

For a suitable ${g_n(x,y)}$, ${\Delta k}$ and ${R}$ the set ${\lbrace g_{nm} \rbrace}$, where ${m \in \mathbb Z}$, may form a frame for images compactly supported on ${|y|, |x| < 1/\Delta k}$ . I will assume that this is the case. (A sufficient condition which the set ${\lbrace g_{nm} \rbrace}$ must satisfy to be a frame on this space will be the topic of a subsequent post). Since ${\lbrace g_{nm} \rbrace}$ is assumed to be a frame we can use the synthesis property of frames given above to write: $\displaystyle \rho = \sum_{m'=-\infty}^\infty \sum_{n'=1}^{N_c} \langle \rho, g_{m', n'} \rangle \hat{g}_{n'm'} \ \ \ \ \ (6)$

where ${\lbrace \hat{g}_{nm} \rbrace}$ is a dual frame corresponding to the frame ${\lbrace g_{nm} \rbrace}$. This is a good place to note once again that there may be more than one dual frame associated with a given frame – the dual frame is not unique. Each dual may have special properties and be advantageous in different imaging scenarios (another topic for future posts?). Also, note that the dual frame of equation (6) depends upon ${x}$ and ${z}$ although I will usually not indicate this explicitly.

Combining equations 4 and 6 we can write $\displaystyle \rho(x,y) = \sum_{m'=-\infty}^\infty \sum_{n'=1}^{N_c} s_{n'}(x,m') \hat{g}_{n'm'} \ \ \ \ \ (7)$

If the dual frame could be found then equation (6) would form the fundamental means of image reconstruction – just measure ${s_{n}(x,m)}$ and perform the summation of equation (6) for a chosen dual frame to obtain ${\rho}$. In practice obtaining the dual frame may be mathematically problematic. Obtaining the dual frame may also be problematic due to an imprecise knowledge of the receive fields ${g_n(x,y)}$ which must be measured or calculated by some means.

There have evolved two main methods used in commercial scanners for reconstructing parallel imaging data. One method goes by the acronym SENSE (Sensitivity Encoding) and the other goes by the acronym GRAPPA (Generalized Autocalibrating Partially Parallel Acquisitions). In appearance the SENSE method is, in its usual mathematical formulation, closer to a frames based method than is GRAPPA. Both methods have strengths and weaknesses.

The topic of my next blog post will be the derivation of autocalibration equations like those used in the GRAPPA method. We will see that the GRAPPA method invokes use of a particular dual frame called the canonical dual frame which is generated by the action of the frame operator ${S}$ upon its frame.

Until then …

 Zhihua Xu and Andrew K. Chan, Encoding With Frames in MRI and Analysis of the Signal-to-Noise Ratio, IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 21, NO. 4, APRIL 2002, p 332-342.

 Ole Christensen, An Introduction to Frames and Riesz Bases, 2003