Selection of arbitrary 3D Cartesian sampling patterns for support-contrained MRI, parallel MRI, and dynamic MRI can be heuristical, and g-factor calculations require a computationally expensive simulation. To provide theoretical guidance and a method to optimize 3D Cartesian sampling, a novel concept of a differential distribution is introduced to represent a distribution of pairwise differences between sample locations, and is related to point-spread-functions. Its relationship to noise amplification in a generalized sensitivity encoding model and linear reconstruction is then used to efficiently optimize multidimensional k-space sampling. Examples in support-constrained MRI, parallel MRI, and dynamic MRI demonstrate reduced noise amplification
Consider a generalized sensitivity-encoding model, yt,c(k)=st(k)F{∑mSm,t,c(r)xm(r)}+ϵc,t(k),(1) where xm is an image, Sm,t,c are sensitivity functions, F is an N-point discrete Fourier transform operator, st are sampling functions (binary masks), and ϵc,t is Gaussian noise. (1) can represent sensitivity encoding with channels indexed by c with multiple maps (m), and dimensions (t) (e.g. time, echoes), over which sampling can be varied and for which a basis (e.g. for signal-intensity-time curves) is known.
We introduce a novel definition of a differential distribution, which describes the distribution of pairwise differences between sample locations, or equivalently, a circular cross-correlation of sampling patterns: p(Δk,t,t′)=(st⋆s′t)(Δk) The differential distribution describes the sampling geometry and is the Fourier transform of the product of PSFs, the squared magnitude of the PSF in the single-time-point case. Representing (1) as a single linear operator E, the eigenvalues of EHE, λk(EHE), have two properties: 1) their sum is constant and 2) their sum-of-squares can be computed as the inner (dot) product: ∑kλk(EHE)2=⟨w,p⟩,(2) where w=1N2∑c,c′|∑mF{S∗m,t′,c′(x)Sm,t,c(x)}|2 Together, these properties imply that minimizing (2) minimizes the spread in the eigenvalues, the minimum value corresponding to all eigenvalues equal (ideal conditioning). (2) analytically expresses a measure of noise enhancement in terms of sampling geometry, which is naturally linked with coil sensitivity variation by a Fourier transform relationship. Figure 1 shows an example of w for coil sensitivities from a breast coil that prescribes an optimal spacing of samples in ky-kz space, and this is generalized to ky-kz-t sampling where signal-intensity-time curves have a temporal basis of B-splines (Figure 2). The relationship can be exploited as a computational shortcut in a sequential selection of samples chosen to minimize (2). Adding a sample only requires updating the cost associated with sampling locations within a local neighborhood determined by the extent of p.
Three problems were considered: 1) support-constrained imaging, 2) single-time-point parallel imaging, 3) multi-time-point parallel imaging. Sampling patterns minimizing (2) and Poisson-disc sampling patterns were compared using g-factor maps and maps in k-space of the change in (2) associated with adding a sample. Cross-shaped7 and elliptical support regions were considered (Fig. 3A-B). Sensitivity maps from a 16-channel breast coil estimated using ESPIRiT8 were also considered (Fig. 3C), both for single-time-point imaging and multi-time-point imaging where a temporal basis of 5 third order B-splines spanning 20 time frames was assumed (Fig. 3D).
Figure 1: The differential distribution p describes sampling geometry as a distribution of pairwise differences between sample locations. The weighting function w represents coil sensitivity information in all directions. Optimal sampling geometry, or differences between sample locations p, matches w, as minimizing their inner product minimizes the spread in the eigenvalues.