The k‐trajectory formulation of the NMR imaging process with applications in analysis and synthesis of imaging methods

Abstract

The fundamental operations of nuclear magnetic resonance (NMR) imaging can be formulated, for a large number of methods, as sampling the object distribution in the Fourier spatial-frequency domain, followed by processing the digitized data (often simply by Fourier transformation) to produce a digital image. In these methods, which include reconstruction from projections, Fourier imaging, spin-warp imaging, and echo-planar imaging, controllable gradient fields determine the points in the spatial-frequency domain which are sampled at any given time during the acquisition of data (the free induction decay, or FID). The detailed time dependence of the resulting trajectory of sample points (the k trajectory) determines the relative weight and accuracy with which image information at each spatial frequency is measured, establishing theoretical limitations on image quality achievable with a given imaging method. We demonstrate here that these considerations may be used to compare the theoretical capabilities of NMR imaging methods, and to derive new imaging methods with optimal theoretical imaging properties.

Noquist: Reduced field‐of‐view imaging by direct Fourier inversion

“…In conventional reconstruction, an expression analogous to Eq. [22] can be formulated for each individual temporal frame:…”

Improved k‐space trajectory measurement with signal shifting

“…If the slice is off-centered (x i 0), Eq. [1] can be written as the product of the Fourier transform of the centered slice and a phase term dependent on the slice location x i : S i ͑k x ͑t͒͒ ϭ ˆ͑k x ͑t͒͒e Ϫj2 kx,(t)xi [2] If 1 (t) and 2 (t) are the phases of the complex signals acquired from two slices 1 and 2, the effective k-space coordinate k x (t) can then be obtained, as shown in Ref. 6, using:…”

“…These signals are acquired during the application of the gradient waveform to be calibrated, while the other channels are switched off. If we neglect relaxation effects, the MR signal S i (k x (t)) can be written as S i ͑k x ͑t͒͒ ϭ ͵ Ϫϱ ϩϱ i ͑x͒e Ϫj͑2 kx,͑t͒xϩ ͑x,t͒͒ dx, [1] where i (x) represents the transverse magnetization density at location x modulated by the excitation profile cor-responding to the slice at position x i , where k x (t) is the k-space coordinate to be determined, and (x,t) is the phase accumulated by temporal and nonlinear spatial B 0 field variations. With well-shimmed, relatively thin and close slices, we may neglect the nonlinear spatial variations of B 0 and (x,t) becomes (t).…”

“…In this case, Eq. [1] represents a Fourier transform relation between signal S i and the "effective magnetization density" i (x)e -j (t) . In practice, the phase term (t) can be neglected in the following calculations (6).…”

“…MR images are obtained from discrete samples acquired in the spatial frequency domain, or k-space (1). If the sampled k-space coordinates deviate from the assumed ones, the resulting image presents artifacts, such as image rotation or inhomogeneous intensity distribution (2,3).…”