Fat/water separation in k‐space with real‐valued estimates and its combination with POCS

Abstract

Purpose: To develop reconstruction methods for improved image quality of chemical shift displacement-corrected fat/water imaging combined with partial Fourier acquisition. Theory: Fat/water separation in k-space enables correction of chemical shift displacement. Modeling fat and water as real-valued rather than complex improves the conditionality of the inverse problem. This advantage becomes essential for k-space separation. In this work, it was described how to perform regularized fat/water imaging with real estimates in k-space, and how fat/water imaging can be combined with partial Fourier reconstruction using Projection Onto Convex Sets (POCS). Methods: The reconstruction methods were demonstrated on chemical shift encoded gradient echo and fast spin echo data from volunteers, acquired at 1.5 T and 3 T. Both fully sampled and partial Fourier acquisitions were made. Data was retrospectively rejected from the fully sampled dataset to evaluate POCS and homodyne reconstruction. Results: Fat/water separation in k-space eliminated chemical shift displacement, while real-valued estimates considerably reduced the noise amplification compared to complex estimates. POCS reconstruction could recover high spatial frequency information in the fat and water images with lower reconstruction error than homodyne. Partial Fourier in the readout direction enabled more flexible choice of gradient echo imaging parameters, in particular image resolution. Conclusion: Chemical shift displacement-corrected fat/water imaging can be performed with regularization and real-valued estimates to improve image quality by reducing ill-conditioning of the inverse problem in k-space. Fat/water imaging can be combined with POCS, which offers improved image quality over homodyne reconstruction. K E Y W O R D S artifact correction, chemical shift imaging, Dixon, fat-water separation, partial Fourier, POCS 654 | BERGLUND Et aL.

RARE two‐point Dixon with dual bandwidths

“…Given the estimated off‐resonance and for each sample, real‐valued estimates of X were calculated using weighted least‐squares 20 : where the + and * superscript denotes the pseudoinverse and conjugate operator, respectively. The NSA of the species is calculated from the signal variance divided by the variance of the estimate ρ ∈ { W , F } of interest: …”

“…A non‐uniform k‐space regularization was applied to achieve a leveled noise power spectrum in the fat and water estimates 19 . The real‐valued constraint allows partial Fourier data to be separated in a modified projection onto convex sets (POCS) technique, 20 enforcing k‐space symmetry in the estimates rather than in the signal domain. This also improves the conditionality of k‐space points with insufficient CSE by including the conjugate sample pair.…”

“…Here, we introduce a dual readout bandwidths (dBW‐RARE) approach that aims to achieve high readout resolution, completely remove dead times and avoid redundant CSE while still complying with CPMG conditions. The problem of choosing optimal readout waveforms to minimize the noise in the estimates is also investigated, where a modified POCS technique was used to allow an additional degree of freedom 20 . We demonstrate the technique for and PD weighted images on 1.5T and 3T systems.…”

Chemical shift encoding using asymmetric readout waveforms

“…By modeling x as real‐valued in image space, fat and water estimation enforces conjugate symmetry in k‐space 20 . Although Equation is exactly determined (two complex measurements , , and four real‐valued estimates, ϕ , ψ , W , and F ), estimating and demodulating only low resolution ϕ ‐ and ψ ‐maps leaves an over‐determined inverse problem.…”

“…17 By modeling x as real-valued in image space, fat and water estimation enforces conjugate symmetry in k-space. 20 Although Equation (4) is exactly determined (two complex measurements S 1 , S 2 , and four real-valued estimates, ϕ, ψ, W, and F), estimating and demodulating only low resolution ϕand ψ-maps leaves an over-determined inverse problem. The additional degrees of freedom can be used to take advantage of the known differences in signal uncertainties Λ for an improved estimation: Here,  () is the Fourier transform operator, and S , = Se −i(̂+̂t) denotes an acquired echo after image space demodulation with the estimated ̂ and ̂.…”