BlackNUFFT: Modular customizable black box hybrid parallelization of type 3 NUFFT in 3D

Abstract

Many applications benefit from an efficient Discrete Fourier Transform (DFT) between arbitrarily spaced points. The Non Uniform Fast Fourier Transform reduces the computational cost of such operation from $O\left(N^2\right)$ to $O(NlogN)$ exploiting gridding algorithms and a standard Fast Fourier Transform on an equi-spaced grid. The parallelization of the NUFFT of type 3 (between arbitrary points in space and frequency) still poses some challenges: we present a novel and flexible hybrid parallelization in a MPI-multithreaded environment exploiting existing HPC libraries on modern architectures. To ensure the reliability of the developed library, we exploit continuous integration strategies using Travis CI. We present performance analyses to prove the effectiveness of our implementation, possible extensions to the existing library, and an application of NUFFT type 3 to MRI image processing.

Program summary

Program Title: BlackNUFFT

Program Files doi: http://dx.doi.org/10.17632/vxfj6x2p8x.1

Licensing provisions: LGPL

Programming language: C++

External routines/libraries: deal.II , FFTW, PFFT

Nature of problem: Provide a modular and extensible implementation of a parallel Non Uniform Fast Fourier Transform of type 3.

Solution method: Use of hybrid shared distributed memory paradigm to achieve high level of efficiency. We exploit existing HPC library following best practices in scientific computing (as continuous integration via TravisCI) to reach higher complexities and guarantee the accuracy of the solution proposed.

Introduction

The standard Fast Fourier algorithm relies on a distribution of the points on a regular equispaced grid. However, many applications benefit from an accurate and reliable Fourier transform between $N$ arbitrarily spaced points. Many image reconstruction techniques, such as Magnetic Resonance Imaging, are based on a non Cartesian grid in the frequency domain [1], [2], [3]. In [4], [5] the authors propose accelerated convolution techniques based on the discrete Fourier Transform between arbitrary points in both space and frequency domains. The computational cost of the standard discrete transform increases quadratically, quickly becoming unbearable even on modern computational platforms.

A solution to this problem is the Non Uniform Fast Fourier Transform (NUFFT) developed by Dutt and Rokhlin [6]. The authors provide a deep theoretical analysis to approximate the Fourier transform using the classical Fast Fourier Transform algorithm. Another description of such algorithm has been addressed by Leslie Greengard and June-Yub Lee in [7]. In the literature there are 3 different types of NUFFT: type 1 operating between arbitrary points in space and a regular grid in frequency, type 2 correlating a regular grid in space and arbitrary points in frequency, and type 3 dealing with arbitrary points in both spaces.

The key idea of NUFFT is to transfer the non equi-spaced data on a uniform grid to be used with standard FFT algorithms. In [7] the authors use fast convolutions with a Gaussian function to create a uniform grid where standard Fast Fourier Transform algorithms can be used. This is a key step to retrieve the solution in an affordable time, since Gaussian convolutions can be efficiently computed by means of Fast Gaussian Gridding, reducing the overall computational time. Another possible solution is the usage of Kaiser–Bessel window or the min–max interpolator. In recent years the latter has been efficiently implemented in the library PNFFT [8]. In [9] the authors propose a parallelization of type 1 NUFFT based on the P3DFFT [10]. Type 3 NUFFT, developed in [7], has been successfully applied in fast numerical convolutions [4], and has applications in many fields of engineering. Several parallelizations have been performed for such an algorithm: in [11], [12] highly scalable optimization of the library on modern multicore systems are proposed, in [13] the authors use the hardware to accelerate the NUFFT, while in [14] the multicore architecture of GPUs is used to tune and optimize NUFFT.

While the previous implementations have proved to be effective, they are extremely hardware specific and optimized. Moreover, their modification is far from trivial from the user’s perspective. We propose an OpenSOURCE flexible parallelization strategy of the type 3 NUFFT algorithm based on existing High Performance Computing libraries: we call such a library BlackNUFFT [15]. We use Intel Threading Building Blocks (TBB) for shared memory parallelism [16], [17], together with the standard Message Passing Interface (MPI) for distributed parallelism. High modularity is a key aspect of our implementation, allowing the user to plug-in new algorithms for each aspect of NUFFT. We use a black box approach for FFT on the fine grid, making it possible to interchange back-end libraries for the FFT algorithm. We provide a first implementation that uses by default FFTW [18] as back-end FFT. This choice is mainly due to the extensive documentation and high reliability of such a library. We also propose a second implementation using PFFT [19] as back-end FFT library. In this way we improve the parallel efficiency of our library and we show the advantages of using a black-box modular approach that allows the interchanging of its building blocks. We compare the presented BlackNUFFT with the library developed by Lee and Greengard which is freely available under GPL license [20]. We use existing implementations of distributed vectors provided by existing OpenSOURCE libraries. In particular we refer to the implementation of the deal.II library [21], [22], [23], which provides both high performance on modern architectures and the possibility of easily switching between 32 and 64 bits indexing. This capability is a keystone to reach higher computational complexity. We present a performance analysis considering multithread and multiprocessor environments for the current implementation of the library. BlackNUFFT is available under LGPL license on GitHub [15].

The work is organized as follows: in Section 2 we collect some considerations on theoretical aspects of the NUFFT [7], [6] and we describe the actual structure of the algorithm. In Section 3 we exploit shared memory parallelism to achieve a first multicore parallel analysis. In Section 4 we combine shared and distributed memory paradigms to reach higher levels of scalability. Section 5 describes the coupling with the parallel pruned library PFFT [19]. In Section 6 we briefly describe the application of BlackNUFFT to the reconstruction of MRI images. Section 7 describes possible procedures to expand BlackNUFFT capabilities.