Temporal and spatio-temporal interpolation

One of the first technique proposed to reconstruct dynamic MRI data from partial (k, t)-space samples is the sliding window method. The most basic version of sliding window is the zeroth-order hold technique [69] which simply estimates the missing k-space samples at time frame t with the latest data point at time frame t − 1 from the same k-space location.

A more sophisticated approach was proposed in 1999 called unaliasing by Fourier- encoding the overlaps using the temporal dimension (UNFOLD) [70]. This technique

Sub-Nyquist dynamic MRI

uses a lattice undersampling scheme that produces aliasing artefacts that can be eas- ily removed with simple filtering in the temporal Fourier domain.

In 2003, Tsao et al. [71] proposed an approach called broad-use linear acquisi- tion speed-up technique (k-t BLAST). This method uses a variable-density sampling scheme, which is a combination of acquired random k-space lines (called undersam- pled dataset) with sampling of central part of the k-space (called training dataset). These datasets are converted into the temporal Fourier domain, and respectively contain aliasing artefacts and a low-spatial resolution estimate. k-t BLAST uses the training dataset to guide the reconstruction and removes artefacts in the undersam- pled dataset.

3.4.2 Compressed sensing

In 2006, Lustig et al. [72] proposed k-t SPARSE, a compressed sensing method for dynamic MR imaging that uses a sparsifying transform adapted for dynamic MR imaging, random undersampling and `1 norm reconstruction. k-t SPARSE solves

the convex optimisation problem min

x kΨxk1 s.t. kEx − yk2≤ , (3.48)

where Ψx represents the sparse signal with Ψ being a sparsifying transform, E is the MRI encoding operator modelling both the random sub-Nyquist sampling pro- cess and Fourier transform and y is the stacked (k, t)-space measurements vector. The sparsifying transform Ψ represents in k-t SPARSE a Fourier transform in the temporal direction and a wavelet transform in the spatial direction. The temporal Fourier transform is further discussed in section 3.4.5. Formulation (3.48) is of Mo- rozov type as shown in (3.7) with the constraint representing the data fidelity term. A nonlinear conjugate gradient descent algorithm with backtracking line search is used to solve the unconstrained (Lagrangian) version of problem (3.48). In this method described in Ref. [33], the absolute values of the `1 norm are approximated

by smooth (differentiable) functions.

Later, Jung et al. proposed k-t FOCUSS [73–75] which is based on the focal underdetermined system solver (FOCUSS) [76], a general estimation method to find localised energy solution from limited data that employs successive quadratic opti- misation to obtain sparse solutions. More specifically, k-t FOCUSS first estimates a low-resolution version of the (y-f )-space signal and performs a FOCUSS recon- struction to recover it. The (y-f )-space signal is the dynamic MRI signal in the temporal Fourier domain (see section 3.4.5) which proves to be sparse. Hence, k-t FOCUSS addresses the CS dynamic MRI problem (`1 minimisation) by recovering

Linear inverse problems

optimisation technique. Consider the equation

y = Ex (3.49)

where y and x represent respectively the undersampled (k, t)-space measurement vector and the sparse (y-f )-space image, and E = AFyFt is the MRI encoding

operator modelling the random sub-Nyquist sampling (A), the Fourier transform along the y direction (Fy) and the temporal Fourier transform (Ft). The solution

of Eq. (3.49) is not unique, and the minimum norm solution is unlikely to give a sparse reconstruction. Standard CS dynamic MRI methods solve

min

x kxk1 s.t. ky − Exk2 ≤ . (3.50)

Instead, consider the weighted minimum norm problem find x = Wq,

where q : min

q kqk2 s.t. ky − EWqk2≤ ,

(3.51)

and where W is a weighting matrix. In its Lagrangian (unconstrained) form, the problem can also be written

min

q ky − EWqk 2

2+ λkqk2 (3.52)

which has the following closed form solution

q? = WHEH(EWWHEH+ λI)−1y. (3.53) Hence, the solution of problem (3.51) is

x?= Wq?= ΛEH(EΛEH+ λI)−1y (3.54) where Λ = WWH. Now consider x =ex + Wq wherex represents a low-resolutione estimate of the (y-f )-space, then k-t FOCUSS solves a slightly modified version of problem (3.52),

min

q ky − Ex − EWqke

2

2+ λkqk2 (3.55)

which has the closed form solution x? =ex + ΛE

H_(EΛEH_{+ λI)}−1

(y − Eex). (3.56) Hence, at each iteration, k-t FOCUSS essentially updates the weighting matrix Wk

Sub-Nyquist dynamic MRI according to Wk=        |xk−1₁ |p _{0 · · · 0} 0 |xk−1₂ |p _{· · ·} .._. .. . . .. ... 0 0 · · · |xk−1_N |p        , 1/2 ≤ p ≤ 1, (3.57) where xk−1= [xk−1₁ , xk−1₂ , . . . , xk−1_N ]> (3.58) is the N -dimensional (y-f )-space vector estimate at k − 1, and then computes xk according to Eq. (3.56). For a weighting matrix power factor of p = 1/2 in W, the authors of k-t FOCUSS showed that the FOCUSS solution is asymptotically equivalent to the `1 minimisation. One of the major advantage of k-t FOCUSS over

k-t SPARSE is that it is computationally much more efficient since it uses quadratic (`2 norm) optimisation technique.

An improvement over k-t SPARSE called k-t group SPARSE was proposed by Usman et al. [77] in 2011, which exploits the fact that the sparse coefficients in the temporal Fourier domain typically form a group structure. An overview of some of these previously described methods can be found in the review paper by Tsao and Kozerke [69].