Effects of polar sampling in k‐space

Abstract

Magnetic resonance imaging allows numerous k-space sampling schemes such as cartesian, polar, spherical, and other non-rectilinear trajectories. Non-rectilinear MR acquisitions permit fast scan times and can suppress motion artifacts. Still, these sampling schemes may adversely affect the image characteristics due to aliasing. Here, the Fourier aliasing effects of uniform polar sampling, i.e., equally spaced radial and azimuthal samples, are explained from the principal point spread function (PSF). The principal PSF is determined by assuming equally spaced concentric ring samples in k-space. The radial effects such as replication, smearing, truncation artifacts, and sampling requirements, are characterized based on the PSF.

Density‐weighted concentric circle trajectories for high resolution brain magnetic resonance spectroscopic imaging at 7T

“…Apparently, e‐CONCEPT without density compensation results in bad localization, while DW‐CONCEPT does not. Also the polar sampling artifact due to the Jinc ‐shaped PSF 29 is clearly visible, contrary to DW‐CONCEPT because of oversampling. ePE clearly shows significant Gibbs ringing artifacts, which can be reduced by retrospective filtering.…”

“…(2) evaluates to 125.2%, see Appendix A. For e‐CONCEPT the density is given by ${\mathrm{normal\rho }}_{\mathrm{e}-\text{CONCEPT}}=d_1/k$, where d 1 is a normalization constant, and k is the k‐space radius 29. Using that, Eq.…”

“…Assuming further a continuous k‐space, the k‐space area assigned to a k‐space point can be described by the Jacobian determinant J 29, 30, 31 of the coordinate transformation $(k_{\mathrm{normalx}},k_{\mathrm{normaly}})=(K(k)\cos k_{\varphi },K(k)\sin k_{\varphi })$resulting in $J=K\frac{dK}{dk}$. K ( k ) describes an arbitrary radii distribution of the circles and translates from an equidistant set of k‐space radii k , to a non‐uniform radii distribution K ( k ).…”

Basis function cross‐correlations for Robust k‐space sample density compensation, with application to the design of radiofrequency excitations

“…Equation [5] states that each Fourier basis function f (j ) (r) corresponding to each sample along the traced trajectory can be analyzed into two components: one parallel to f (j) (r), and a second component, g (r), that is orthogonal to it. This is the same concept as used in the Gram-Schmidt orthogonalization procedure (27).…”

“…the samples uniformly and unit spaced, the weights are identified as the determinant of the Jacobian matrix for the coordinate transformation (4,5). This correspondence of weights with a differential area element has in turn given rise to heuristic approaches, which determine weights based on some measure of area assigned to each sample (2), such as the Voronoi tessellation method (6).…”