The properties of SIRT, TVM, and DART for 3D imaging of tubular domains in nanocomposite thin-films and sections
Introduction
Electron tomography (ET) has nowadays become a standard tool for materials research to reveal the three dimensional (3D) morphology of specimens with nanometer resolution [1], [2], [3], [4]. Knowledge of a materials 3D morphology is critical to understanding the material properties, such as, e.g., the efficiency of polymer photovoltaic cells [5], [6]. ET consists of three basic steps [2]. First, a series of 2D projections from multiple directions of the object of interest is acquired in a transmission electron microscope (TEM). This is conventionally done by tilting the object over a large angular range at small tilt increments. Second, the series of projections are aligned with respect to a common origin and tilt axis, correcting for unavoidable displacements during data acquisition. Third, the electron tomogram (3D intensity map) of the object is reconstructed from the tilt series by numerical algorithms.
ET is mainly used for qualitative studies where a 3D visualization of different nanostructures is required. Nevertheless a strong trend towards obtaining 3D quantitative information from electron tomograms is ongoing that is currently often hampered by the low reconstruction quality [7]. Several challenges impede obtaining high-quality reconstructions [8], [9], [10]. The most significant challenge is that the tilt range for ET is limited (often <±80°) because of the sample, the sample holder or the microscope stage [11]. As a consequence, the limited tilt range leads to a missing angular range of information referred to as “missing wedge” [2], [12]. The reconstruction quality is significantly affected by the missing wedge, i.e. elongation in the reconstruction along the beam direction [13]. Another key challenge is the low contrast and low signal-to-noise ratio (SNR) that can be obtained in the projection images, especially for beam sensitive materials [2], [8], [14]. The reason of low contrast is that, e.g., polymers and composites thereof, mainly consist of light elements with small differences in density or composition. In addition, polymers are often very sensitive to electron irradiation and will be shrunk or even bubble during data acquisition if one goes beyond a tolerable cumulative electron dose [15], [16], [17]. Therefore, in order to avoid radiation damage and to preserve the structure of the object, only a limited number of electrons can be used throughout data acquisition, thus, leading to low SNR in the projections of the tilt-series.
To overcome the aforementioned challenges, various methods have been proposed in the literature [1], [3], [8]. Instead of a single tilt axis, dual tilt axis has been used during acquisition, thereby reducing the missing wedge to a “missing pyramid” [18], [19]. Nevertheless, the problem of missing information still persists. The ideal tilt range ±90° can be reached by fabricating the material of interest into a needle shaped specimen. By mounting the sample in a special specimen holder, an angular tilt range of 360 degrees can be achieved thus completely removing the missing wedge [20], [21]. However, fabrication of a needle-shaped specimen requires materials that are mechanically stable enough which hampers the application to many polymer composites. Finally, to enhance the reconstruction quality, different angular sampling procedures and reconstruction schemes have been suggested and implemented [22], [23], [24].
Commonly used reconstruction methods are the weighted backprojection method (WBP) and iterative methods such as simultaneous iterative reconstruction technique (SIRT) [13], [25], [26], [27]. The WBP method has been one of the most widely used algorithms [13], [25]. Apart from high computational efficiency, the major advantage of WBP is that the outcome of the reconstruction is thoroughly determined by the experimental data as all steps in the algorithm are linear [13], [28]. However, the main disadvantage of WBP is that the reconstruction quality is sensitive to the limited tilt range [28]. In contrast, the SIRT method generates reconstructions yielding good visual quality from fewer projections and even from noisy data [26], [29]. Therefore, SIRT gradually becomes an increasingly popular reconstruction method used in electron tomography, although it is computationally more expensive. Recently, more advanced reconstruction algorithms have been proposed, such as the discrete algebraic reconstruction technique (DART) [22] and the total variation minimization based reconstruction technique (TVM) [23]. Using prior knowledge of the specimen such as that the specimen contains only a limited number of phases, i.e., a discrete number of gray levels, aids in solving the ill-posed inversion (reconstruction) problem. For example, the DART algorithm actually segments (binarizes) the reconstruction during the iterative process. DART reconstructions are directly quantifiable and have been successfully applied for the 3D characterization of catalytic CuO nanoparticles and zeolite materials [30], [31]. The TVM method is developed based on compressed sensing. It incorporates the prior knowledge that the boundary of the specimen is sparse in the reconstruction [23]. Using this algorithm, the elongation artifacts and noise in the reconstruction are reduced and the sharpness of the edges is significantly improved. The advantages of the TVM method have demonstrated by the 3D reconstruction of FeO nanoparticles [32] and PbSe/CdSe core-shell nanoparticles [23].
The performance of above advanced reconstruction algorithms have been mainly studied based on experimental data [22], [33] which generally lack the ground truth of the specimens. Nevertheless, some studies have incorporated known phantoms or models into the performance comparison, but these models were based on data that are obtained with a very high SNR in the projections [32], [34], [35]. Hence, no study has included effects of limited electron dose, which we consider the most crucial physical limit to 3D imaging of beam sensitive materials. Thus for beam sensitive materials, detailed studies on the performance of reconstruction algorithms in combination with limited electron dose and varying tilt schemes are lacking.
The aim of this work is therefore to fill this gap by evaluating the fidelity of SIRT, TVM, and DART reconstructions in dependence of a limited total electron dose and a variety of possible acquisition schemes. A comparison to WBP will be presented as a reference. As model structure we focus on the large range of functional nanocomposites composed of tubular domains in a matrix with slab-geometry, i.e. thin-films or thin-sections. Our approach is built on simulating the entire workflow of the bright-field ET from projections, over experimental imaging and recording effects to 3D reconstructions, finalized by a qualitative and quantitative comparison of the initial model and the reconstruction. The reconstruction fidelity is first assessed by image quality. Subsequently, local and global edge profiles and edge spread functions are employed to quantify the resolution in the reconstructions. Moreover, we evaluate the resolvable separation between particles (connectivity and percolation) which is a key question for many functional composites such as photovoltaic bulk heterojunctions or conductive CNT/polymer nanocomposites.
Section snippets
Materials and methods
In this section, we first briefly introduce SIRT, TVM and DART, and then present the simulation approach and evaluation methods [36].
Results
In the following, we present visual inspection (3.1), evaluation based on global edge profiles (3.2), local edge profiles (3.3), and gap profiles (3.4). The projections were also reconstructed using WBP as reference for quantitative analysis. Further details on the WBP parameters can be found in the Supplementary information, Section 1.3. For the purpose of visualization, all the reconstructions are normalized to the same mean density. The normalization method is presented in the Supplementary
Discussion and conclusion
Although the SIRT, TVM and DART reconstruction methods have been compared before in the literature [22], [23], [32], [33], these studies are generally based on experimental data, lacking a detailed knowledge of the ground truth. In this work, we have simulated the electron tomography workflow based on a well-defined model of tubular domains in a matrix with slab geometry. The model was chosen as it closely resembles a large range of functional materials, such as a P3HT/PCBM bulk heterojunction
Acknowledgments
The authors thank prof. M. Dijkstra and Dr. R. Ni (Utrecht University, the Netherlands) for providing the random packing code, and Maarten Wirix for characterization of CCD noise. This research forms part of the research program of the Dutch Polymer Institute (DPI), projects #615. This work is supported by the Research Foundation Flanders (FWO Vlaanderen) through a Ph. D. research grant to B.G.
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