Elsevier

Ultramicroscopy

Volume 149, February 2015, Pages 64-73
Ultramicroscopy

Atomic resolution tomography reconstruction of tilt series based on a GPU accelerated hybrid input–output algorithm using polar Fourier transform

https://doi.org/10.1016/j.ultramic.2014.10.005Get rights and content

Highlights

  • Three-dimensional atomic structural reconstruction demonstrated using simulated diffraction data in a tilt series.

  • Using diffraction patterns allows alignment free tomography reconstruction.

  • This method can be applied to general image based tomography by using the power spectra of the images as input.

  • Reconstruction is based on an iterative transformation algorithm (ITA) using polar fast Fourier transform.

  • The ITA algorithm is accelerated using graphics processing unit (GPU) for competitive high performance.

Abstract

Advances in diffraction and transmission electron microscopy (TEM) have greatly improved the prospect of three-dimensional (3D) structure reconstruction from two-dimensional (2D) images or diffraction patterns recorded in a tilt series at atomic resolution. Here, we report a new graphics processing unit (GPU) accelerated iterative transformation algorithm (ITA) based on polar fast Fourier transform for reconstructing 3D structure from 2D diffraction patterns. The algorithm also applies to image tilt series by calculating diffraction patterns from the recorded images using the projection-slice theorem. A gold icosahedral nanoparticle of 309 atoms is used as the model to test the feasibility, performance and robustness of the developed algorithm using simulations. Atomic resolution in 3D is achieved for the 309 atoms Au nanoparticle using 75 diffraction patterns covering 150° rotation. The capability demonstrated here provides an opportunity to uncover the 3D structure of small objects of nanometers in size by electron diffraction.

Introduction

Three-dimensional reconstruction at atomic resolution is a major challenge in materials structure characterization. Recent advances in electron optics have enabled a direct determination of atomic structure at sub-ångstrom resolution in 2D projection [1], [2], [3], [4]. However, imaging atoms inside, and at the surface of, a nanoparticle or determining the structure of a 3D defect requires information of the 3D atomic structure. Tomographic reconstruction at atomic resolution provides a possible way forward to 3D structure determination for any objects.

At atomic resolution, several groups have reported 3D reconstruction using Z-contrast images obtained in a scanning transmission electron microscope (STEM) equipped with an annular dark field (ADF) detector. Aert et al. reported a successful reconstruction of the 3D atomic structure of Ag precipitates in the Al matrix using discrete tomography [5]. This method only requires electron images recorded in a few zone-axis projections, but a prior knowledge of the structure is necessary for the reconstruction. The other approach is to detect atomic position in 3D using the STEM depth sectioning method [6], [7], [8] or using a combination of quantitative STEM and multi-slice simulation [9]. The resolution of these methods, however, is limited along the beam direction by the probe elongation effect resulting from the small electron beam convergence angle and by electron multiple scattering along atomic columns [6], [10]. Recently, Jianwei Miao׳s group at UCLA demonstrated the 3D reconstruction of an Au nanoparticle at 2.4 Å resolution by using tomographic reconstruction based on the so-called equal-sloped fast Fourier Transform [11]. The reconstruction is based on the Z-contrast image data recorded in a tilt series from −72.6° to 72.6° in equal-slope increment. This method has been further applied to image dislocations in an Au nanoparticle in 3D [12].

Here we report a new method for 3D tomographic reconstruction based on the hybrid input–output (HIO) algorithm developed by Fienup [13] and polar Fourier fast Transform (FFT). Only diffraction data obtained in a tilt series are used for reconstruction. Because information recorded in a diffraction pattern is limited only by scattering, the algorithm reported here has the potential to achieve the highest resolution. Another major application of this method is in tomography where the projected object functions, instead of diffraction patterns, are recorded directly in real space. Using the projection-slice theorem, Fourier transform (FT) of the projected object function gives the diffraction pattern with information up to the image resolution. By using only amplitudes of FT, our method does not require alignment of projected object functions, which has to be done at the precision of Å for atomic resolution tomography and thus provides a major advantage over regular tomographic methods.

We use the HIO algorithm originally proposed by Fienup to retrieve the missing phase in the diffraction data. This algorithm has been extensively applied in X-ray and electron diffractive imaging [14], [15], [16], [17], [18], [19]. In the HIO algorithm, phase is retrieved to reconstruct the object function via iterations between the real (image) space and the Fourier space with modifications made in each space using constraints imposed by the measured diffraction intensities and support for the object [19]. The approach is general without the need for prior information about the structure except the object support.

Iteration in HIO requires forward and backward fast Fourier transformation (FFT). However, conventional FFT uses equispaced rectilinear sampling, which cannot be directly extended to tomography reconstruction of diffraction data recorded in a tilt series. To overcome this issue, we use polar FFT as implemented in the general category of non-equispaced FFT (NFFT) methods. The motivation here is to avoid resampling in the diffraction space and its related issues [20]. The basic concept of NFFT computation is to use conventional FFT in a Cartesian grid, while the Fourier frequencies are oversampled and a window function is used to interpolate between equally and non-equally sampled space. The inverse FT is obtained through a least square minimization [21]. In the 3D HIO algorithm using polar FFT, the object function is sampled in a Cartesian grid, while the Fourier space is sampled in the cylindrical grid, where in the xy plane the sampling is in the polar grid with points equally spaced on concentric circles (Fig. 1(a)).

For 3D reconstruction, computational cost increases dramatically in order to achieve high resolution. Furthermore, because of oversampling, polar FFT requires a significant increase in the data size, which makes it challenging to implement the HIO algorithm not only in the Fourier space but also in the object space, while maintaining the computational efficiency. We have sought to overcome this challenge using GPU-based acceleration [22], [23], taking advantage of its massively parallel processing architecture.

In what follows, we first describe the methodology of a 3D HIO algorithm based on polar FFT. This is followed by a description of its implementation on GPU. The last section describes a numerical test of our algorithm by reconstructing the 3D structure of an Au icosahedron with 309 atoms using the calculated projected potentials. We analyze the computation performance of our GPU-based algorithm by simulating the noise and missing data wedge in possible experimental situations. Based on the simulation results, we demonstrate the requirements for data sampling, the resolution and the limitation of the method.

Section snippets

Cylindrical Fourier transform (CFT)

Reconstruction of the 3D object using the HIO algorithm is performed in both the Fourier and object spaces. A set of 2D diffraction patterns are recorded in a tilt series. The diffraction patterns are combined in 3D by taking the rotation axis as the z-axis in the cylindrical coordinate as illustrated in Fig. 1 (a). For each xy plane of constant z, the data is sampled in the polar coordinate of k=(k,θ) with kn=nΔk and θm=mΔθ. Along z, the sampling is performed in equal space with zl=lΔz.

GPU implementation for 3D reconstruction algorithm and Its performance

We implemented our algorithm on a NVIDIA GPU. To support parallel computing on GPU, NVIDIA designed the Compute Unified Device Architecture (CUDA) as a parallel computing platform and programming model [27]. A general description about GPU computation based on CUDA can be found in the Refs. [28], [29].

In our 3D HIO algorithm, the computation of 3D inverse CFT, modification in the object space (f(ρCn,ρCn)), 3D forward CFT and modification in Fourier space (F(FC) andC(|FoDP|,|FC|)), follow

Experimental simulations

To test the above 3D reconstruction algorithm in a simulated experimental situation, we designed a work flow as shown in Fig. 5. The process consists two parts: data preparation, and reconstruction using iterative transformations. In data preparation, an icosahedral nanocrystal consisting of 309 Au atoms was first built. To acquire tilting series 2D images, the nanocrystal is rotated along z axis to angles from 0° to 179° and projected potential is calculated along each direction. For

Summary and discussions

We have described a 3D reconstruction algorithm based on the cylindrical Fourier transform computed by non-equispaced fast Fourier transform. The algorithm is accelerated and implemented using CUDA on GPU. The algorithm was used to test 3D atomic resolution reconstruction for a 309 atoms Au nanoparticle using diffraction patterns calculated from the simulated projected images as input.

Results using simulated test data show that the algorithm is capable of reconstructing the 3D structure at

Acknowledgment

This work described here is supported by the U.S. Department of Energy, Basic Energy Science, under Award no. DEFG02-01ER45923 and National Natural Science Foundation of China, Award no. 61139002.

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