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Annular Dark Field Scanning TEM ADF STEM 3 4 5 6 and Energy Filtered TEM EF. TEM 7 8 The main requirement is that the acquired projection images depend monotonically. on some physical property of the sample integrated along a set of parallel lines. Most electron tomography users do not deal with reconstruction algorithms directly Rather. these methods are implemented in a software package that provides the user with an interface to. set certain reconstruction parameters while performing the computations internally A range of. software packages exist for reconstruction in electron tomography each with their own advan. tages and drawbacks and their own user base Some microscope vendors offer a software pack. age along with the microscope such as the Inspect3D software of FEI the Digital Micrograph. software from the Gatan Company and the Hitachi Tomography plugin 9 Within the academic. community several free packages have been developed including IMOD 10 11 EFTET J 7. Protomo 12 UCSF tomography 13 14 TomoJ 15 TOM Toolbox 16 and TxBR 17. Key advantages of using established software packages for electron tomography are the reli. ability implied by the software s proven track record and the user friendliness that such packages. can provide On the other hand the implemented reconstruction tools typically provide only. limited flexibility to the user of the software Here we mention three such limitations i the. set of reconstruction algorithms that can be used is limited typically only offering one or two. alternatives e g WBP and SIRT ii the software assumes a fixed geometrical setup for the. experiment e g single axis tilting or dual axis tilting with an angle of 90 degrees between the. two series without the flexibility to change the experiment iii the computational efficiency of. the implemented algorithms is sometimes limited requiring a long time to reconstruct a large. For a routine user of electron tomography the functionality offered by the established soft. ware packages is often sufficient However for a technique developer either on the experimental. or on the algorithmic side the limitations imposed by using a fixed software package can be. an obstacle in experimenting with new concepts and ideas At present no software platform is. available that suits the development of advanced efficient algorithms capable of dealing with. various geometries and constraints, The All Scale Tomographic Reconstruction Antwerp ASTRA Toolbox is a software platform. developed at the University of Antwerp Belgium and at the Centrum Wiskunde Informatica. CWI Amsterdam The Netherlands to address the need for a fast flexible development plat. form for tomography algorithms 18 19 It provides a set of building blocks that can deal with. various geometrical setups and incorporate a variety of constraints in an efficient manner The. toolbox is accessible through MATLAB and Python providing a powerful platform for algorithm. prototyping and is available as open source software under a GPLv3 license 20. Due to its flexible nature the ASTRA Toolbox is suitable for addressing a wide range of. computational problems in many tomographic applications such as medical CT biomedical 21. or industrial micro CT synchrotron tomography 22 and electron tomography 23 It offers full. 3D flexibility for modelling misalignments and multi directional tilt series and allows to perform. parallelized computations using such complex geometrical setups on Graphics Processing Units. GPUs Through its integration in MATLAB and Python advanced numerical code such as. regularized reconstruction algorithms e g total variation minimization TV min 24 can be. directly applied to large experimental datasets, In this article we provide an overview of the ASTRA Toolbox and its design We demonstrate. the flexibility of the toolbox by constructing a complex reconstruction algorithm for a dual tilt. geometry with just a few lines of MATLAB code We investigate the resulting reconstruction. quality and the computational efficiency of the toolbox by a series of experiments based on. experimental dual axis tilt series,2 Basic concepts and framework design. We start by describing the architecture of the ASTRA Toolbox and its key components In. ternally the software consists of three layers Fig 1 i The first low level layer provides effi. ciently implemented algorithm building blocks such as projection and backprojection operators. that are GPU accelerated using NVIDIA CUDA ii The second middle level C layer con. tributes a range of algorithms such as reconstruction algorithms that make use of these building. blocks iii The third top level presents an easy to use interface of these algorithms and building. blocks to the end user Two options are provided a MATLAB interface implemented using the. MATLAB MEX framework and a Python interface Both offer the same features and design. philosophy and differ only slightly in their syntax In the remainder of this work we will focus. on the conceptual structure as experienced by a user running MATLAB scripts. user scripts users,layer 3 MATLAB Python,mex interface interface. layer 2 C algorithms Toolbox,building blocks,CPU GPU hardware.

Figure 1 Schematic overview of the ASTRA Toolbox design. Three main concepts are involved in using the toolbox Fig 2 i the projection and volume. data Section 2 1 ii the spatial geometry of the experimental setup Section 2 2 and iii. the algorithms to be executed on the data Section 2 3. projection,data projection,geometry forward and algorithms. backprojection e g reconstructions,volume geometry. Figure 2 Schematic overview of the basic ASTRA Toolbox objects and their relations. 2 1 Projection and volume data, Data objects are used to store projection or volume data within the toolbox Typically input. projection data is first loaded into the MATLAB environment in the form of a double precision. matrix However for the ASTRA algorithms to use this data it has to be accessible from the. lower CPU or GPU layer of the toolbox Simple functions are provided to copy data from the. interface to the toolbox and to create an empty dataset in system memory Once this is done. each data object is referred to by a unique identifier or handle much like how file I O is handled. in MATLAB Later on the user can use this identifier to use the data in an algorithm or to copy. the data from system memory back into the MATLAB environment. 2 2 Spatial geometry of the setup, Each data object is linked to its corresponding volume geometry or projection geometry. specifying the setup of the scanning system The volume geometry describes the pixel or voxel. grid on which the object is represented This volume has the shape of a rectangle or box centred. around the origin The projection geometry describes the source and detector setup relative to. the volume geometry The ASTRA Toolbox supports different types of geometry parallel fan. beam and cone beam used in different types of tomography For electron tomography a parallel. beam projection geometry is the most relevant In this geometry the position and orientation. of the detector and the electron beam can be fully specified in 3D for each projection direction. separately, Figure 3 Scheme depicting the projection geometry of a single projection direction.

In practice electron microscopes used for tomography contain a stationary source and de. tector and a tilting sample We however define the geometry in a frame of reference where. the sample remains stationary This means that instead of a tilting sample we define a rotating. source and detector setup moving around the stationary sample A 3D parallel beam projec. tion geometry can be regarded as a series of projections each defined by the four following 3D. vectors also refer to Fig 3,The direction of the beam r x ry rz. The centre of the detector plane specified by 3D coordinates d x dy dz For a parallel. beam geometry moving the virtual detector along the direction of the beam does not. have any impact on the computation as projected lines extend indefinitely in that direction. even behind the detector Therefore in the example given here we centre the detector. plane at the origin for all projections, The principal axes of the detector plane typically horizontal and vertical specified as 3D. vectors v x vy vz and u x uy uz The length of these vectors corresponds with the size. of one detector pixel and their direction determines the 3D orientation of the detector By. changing their lengths detector pixels of smaller or larger sizes can be modelled. By specifying the projection geometry in this way not only single axis acquisition schemes. can be modelled but also dual axis or even multi axis schemes Moreover structural sources. of misalignment problems such as a tilt of the detector with respect to the electron beam can be. very accurately modelled and incorporated in all reconstruction algorithms built upon the basic. structure of the ASTRA Toolbox Detector shift for example can be modelled with sub pixel. precision forsaking the need of interpolation on the measured data. Automatic alignment correction of a recorded tilt series is a very difficult problem even. more so for a dual axis tilt series In 25 alignment correction is done by regarding it as an. optimization procedure of all parameters of the projection geometry over a certain objective. function reflecting the reconstruction quality e g the projection difference Such a technique is. only possible when flexible projection geometries are available such as in the ASTRA Toolbox. 2 3 Algorithms, The actual computations are implemented in the algorithm objects At the time of writing. the toolbox provides efficient implementations for the most popular reconstruction algorithms. such as WBP slice by slice SIRT and CGLS a Krylov subspace least squares congruent gra. dients solver analytically equivalent to LSQR 26 27 These algorithms are built upon basic. GPU accelerated projection and backprojection building blocks Configuration of an algorithm. can be done in the MATLAB interface by linking it to the correct data identifiers and by setting. some algorithm specific options e g enabling a minimum constraint in a SIRT reconstruction. Examples of this will be provided in Section 3 It is important to note that the forward pro. jection FP and backprojection BP building blocks itself can also be directly accessed from. within the MATLAB interface This means that a user can easily develop new tomographic al. gorithms or prototypes in which a substantial portion of the computational burden is offloaded. to a much faster GPU card An example of this is provided in Section 3 5 In case GPU cards. are unavailable the ASTRA Toolbox also provides OpenMP accelerated CPU implementations. of these building blocks but only for 2D datasets and slice by slice 3D reconstructions. 3 Concrete example implementing an advanced method for dual axis reconstruction. In this section we demonstrate the key features of the ASTRA Toolbox and how to use. them by constructing an advanced reconstruction algorithm in a step wise manner Firstly in. Section 3 1 we introduce some notation that will be used to describe these advanced iterative. methods In Section 3 2 we describe the common workflow for creating a three dimensional. reconstruction with the ASTRA Toolbox As an example we demonstrate how a dual axis acqui. sition geometry can be defined how projections of a given voxel volume can be computed and. how a reconstruction can be created using the Simultaneous Iterative Reconstruction Technique. SIRT technique, Subsequently we describe how this workflow can be extended by relatively small number. of MATLAB code lines to include more advanced reconstruction methods In Section 3 3. we demonstrate how a recently proposed method for dense particle segmentation 23 called. Partially Discrete Algebraic Reconstruction Technique PDART algorithm can be implemented. with certain image processing operations within MATLAB combined with optimized tomo. graphic reconstruction steps projection and backprojection of the ASTRA Toolbox Next in. Section 3 4 we highlight the fact that the voxel size of the reconstructed volume can be chosen. independently of the detector pixel size which provides the ability to reconstruct a subset of the. volume at lower resolution than the central region of interest leading to improved performance. Finally in Section 3 5 we discuss how the ASTRA building blocks can be used in third party. libraries and scripts such as in an existing Total Variation minimization TVmin script. 3 1 Notation, Let n denote the total number of voxels in the volume and let v Rn denote a vector describ.

ing the voxel values of a certain 3D volume containing the scanned sample Assume a square. detector with t2 the total number of detectors in a single projection With l the total number of. projections the total number of measurements is then m lt2 Let p Rm denote a vector that. contains all tilt series measurements of v We define the matrix W as the projection matrix a. linear operator that describes a forward projection of the scanned object. The matrix W describes how the projection data p depends on the image volume v i e it maps. the volume geometry onto the projection geometry The multiplication of W with a vector v is. called a forward projection FP the multiplication of W T with a vector p is called a backprojec. tion BP Both operations are of crucial importance in tomographic reconstruction as they take. The ASTRA Toolbox a platform for advanced algorithm development in electron tomography Wim van Aarlea 1 Willem Jan Palenstijna b 1 Jan De Beenhouwera Thomas Altantzisc Sara Balsc K Joost Batenburga b d Jan Sijbersa aiMinds Vision Lab University of Antwerp Universiteitsplein 1 B 2610 Wilrijk Belgium bCentrum Wiskunde amp Informatica Science Park 123 NL 1098 XG Amsterdam The

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