To demonstrate a freely-available software tool for utilizing the polyhedral Fourier transform to simulate physiologically relevant phantoms.Analytical Fourier transforms (FT) enable accurate arbitrary k-space sampling of digital phantoms. Most digital phantoms with analytical FTs are comprised of simple shapes (e.g. cubes and ellipsoids) whose analytical FT expressions are known. The standard is the 2D and 3D Shepp-Logan phantom, which utilizes ellipses and ellipsoids.¹ Recently, a closed form expression for the analytical FT of polyhedral meshes was presented, enabling simulations of physiologically relevant shapes with uniform intensity,² or non-uniform intensities.³ Here, we present and demonstrate a computationally efficient MATLAB (The Mathworks, Natick, MA) implementation of the polyhedral FT, polyFT.m, targeted to perform MRI acquisition and reconstruction simulations using physiologically relevant shapes.

1. Implementation

Two implementations are incorporated into polyFT.m. The first one uses only MATLAB and is highly optimized, while the second one includes mex files (compiled C files callable from MATLAB) to improve computational speed. As mex files require an operating system (OS) dependent compiler, the cross-platform MATLAB-only implementation is designed for ease of use without requiring the capacity to compile mex files.

The equations of the polyhedral FT include several layers of summation, ² making the computations amenable to parallelization. Hence, both implementations are designed to utilize the Parallel Computing Toolbox (PCT) of MATLAB, taking advantage of multiple CPUs. If the PCT is not available, polyFT.m utilizes standard for-loops.

2. Functionality

PolyFT.m enables the computation of analytical FT of phantoms with uniform or non-uniform intensity. A closed surface mesh is required for the phantom with uniform intensity. The mesh can include polygonal faces with varying number of edges; however, a constant number of edges results in faster computational speed. A tetrahedral volume mesh is required for the phantom with non-uniform intensities, and intensities are individually assigned to each tetrahedron comprising the mesh. Smaller tetrahedrons lead to emulation of smoother intensity variations though increase computation time.

3. Demonstrations

PolyFT.m is packaged with additional demos that compute the FTs of simple polyhedra, such as cubes, ellipsoids, and dodecahedrons, and a realistic brain surface mesh (Figures 1 and 2).⁴ Additionally, a demo comparing 3D Cartesian and stack-of-stars sampling is also provided. ⁸ (Figure 3) PolyFT.m can also be used in simulations involving shapes with non-uniform intensities such as coils sensitivity profiles and enables analytical evaluation of both acquisition and reconstruction algorithms. Thus, a simple demo utilizing a simulated 8-channel sensitivity map ⁵ and Generalized Partially Parallel Acquisitions ^{6, 7} (GRAPPA) is provided as well. (Figure 4).

4. Evaluation of Computational Time

The computation time of mex implementation and the MATLAB-only implementation were compared using release 2015b on a MacBook Pro with four 2.7 GHz Intel Core i7 CPUs and 16 GB DDR3 memory. K-space points and faces were varied and total computation time and computation time per k-point per face were recorded. Three-run averages are reported.

Table 1 shows the computational time of two implementations. Both display similar computational speed (~ 90 ns / k-point / triangular face) though the MATLAB-only implementation was slower when larger meshes with more faces were used or when the number of k-space points increased.

PolyFT.m can provide a platform for the evaluation of the performance of advanced acquisition strategies and reconstruction algorithms. The use of physiologically relevant digital phantoms can make development more efficient by avoiding the expenses of physical experiments.

The polyhedral FT is computationally intensive. Hence, significant effort was devoted to optimizing performance across all OSs. By incorporating parallelization, large simulations involving complex meshes can be performed efficiently

A highly efficient MATLAB implementation of the analytical polyhedral Fourier transforms, polyFT.m, is presented. PolyFT.m allows users to get Fourier transforms of polyhedral phantoms with uniform or non-uniform intensity enabling the digital evaluation of arbitrary k-space sampling trajectories and advanced reconstruction algorithms. The package is now available through the Mathworks user community and GitHub (https://github.com/Shuo-Han/polyFT).