Fully three-dimensional reconstruction from data collected on concentric cubes in Fourier space: implementation and a sample application to MRI

Optical computed tomography for spatially isotropic four-dimensional imaging of live single cells

Quantitative three-dimensional (3D) computed tomography (CT) imaging of living single cells enables orientation-independent morphometric analysis of the intricacies of cellular physiology. Since its invention, x-ray CT has become indispensable in the clinic for diagnostic and prognostic purposes due to its quantitative absorption-based imaging in true 3D that allows objects of interest to be viewed and measured from any orientation. However, x-ray CT has not been useful at the level of single cells because there is insufficient contrast to form an image. Recently, optical CT has been developed successfully for fixed cells, but this technology called Cell-CT is incompatible with live-cell imaging due to the use of stains, such as hematoxylin, that are not compatible with cell viability. We present a novel development of optical CT for quantitative, multispectral functional 4D (three spatial + one spectral dimension) imaging of living single cells. The method applied to immune system cells offers truly isotropic 3D spatial resolution and enables time-resolved imaging studies of cells suspended in aqueous medium. Using live-cell optical CT, we found a heterogeneous response to mitochondrial fission inhibition in mouse macrophages and differential basal remodeling of small (0.1 to 1 fl) and large (1 to 20 fl) nuclear and mitochondrial structures on a 20- to 30-s time scale in human myelogenous leukemia cells. Because of its robust 3D measurement capabilities, live-cell optical CT represents a powerful new tool in the biomedical research field.

Three-dimensional reconstruction from cone-beam data in O(N3logN) time

We have used direct Fourier techniques to modify and implement the 3D reconstruction method from cone-beam projections proposed by Grangeat. In this way we manage to decrease the computational complexity from O(N4) to O(N3 log N). Just as Grangeat's original method is exact in the mathematical sense, so is our method, provided a complete set of projection data is acquired. Also in accordance with Grangeat, our algorithm consists of two distinct phases: phase 1, from cone-beam data to derivatives of Radon data; phase 2, from derivatives of Radon data to reconstructed 3D object. In phase 1 we use the direct Fourier method in reverse to obtain line integrals in the detector plane. In phase 2 the 2D linogram method is employed for reconstruction of vertical and horizontal planes in the Radon space.

A dual approach to linogram imaging for MRI

An alternate scheme for linogram image reconstruction, which is more logical from the viewpoint of its applicability to MRI data, is presented here. As a result, an intermediate step for this method, the direct Fourier method (DFM) gives the same results as the earlier developed general reconstruction algorithm labeled as the linogram method (LM). However, the two differ in the pathways taken to the solution. As a result, an intermediate step for DFM corresponds directly with the geometry of linogram data collected for MRI, in contrast to the LM reconstruction. The two reconstruction methods are delineated within the context of MRI data reconstruction, and applied to reconstruct images from linogram spin-echo data of a physical phantom, obtained on a clinical 1.5 T scanner.

Characterization of and correction for artifacts in linogram MRI

Chemical shift differences, field inhomogeneity, and gradient nonlinearity result in artifacts in magnetic resonance imaging. Three artifacts are characterized for linogram imaging and it is shown that, based on computer simulations and theory, linogram MRI behaves similarly to 2DFT. A correction technique similar to a scheme for 2DFT imaging based on the Dixon technique and coordinate transform methods is proposed. The algorithm is applied to correct for field inhomogeneity and gradient nonlinearity-induced artifacts in both simulations and images of a clinical phantom. The results show good correlation with the theory. It is concluded that linogram imaging offers certain attractive features of both 2DFT and PR imaging techniques, and is a potentially viable alternative to PR imaging in the presence of field inhomogeneity.

VOIR: a volumetric image reconstruction algorithm based on Fourier techniques for inversion of the 3-D Radon transform

A novel volumetric image reconstruction algorithm known as VOIR is presented for inversion of the 3-D Radon transform or its radial derivative. The algorithm is a direct implementation of the projection slice theorem for plane integrals. It generalizes one of the most successful methods in 2-D Fourier image reconstruction involving concentric-square rasters to 3-D; in VOIR, the spectral data, which is calculated by fast Fourier techniques, lie on concentric cubes and are interpolated by a bilinear method on the sides of these concentric cubes. The algorithm has great computational advantages over filtered-backprojection algorithms; for images of side dimension N, the numerical complexity of VOIR is O(N(3) log N) instead of O(N (4)) for backprojection techniques. An evaluation of the image processing performance is reported by comparison of reconstructed images from simulated cone-beam scans of a contrast and resolution test object. The image processing performance is also characterized by an analysis of the edge response from the reconstructed images.