1122 Toby Sanders and Ilke Arslan have found this approach ineffective in their own experience,


because a reconstruction from misaligned data generally fits these misaligned projections. Therefore, no refinement was observed in subsequent iterations. Other approaches similar to fiducial marker-based approaches have been proposed, which align by identifying and tracking points or regions of interest within the scene (Brandt et al., 2001; Sorzano et al., 2009). The alignment models within this investigation can


generally be characterized as center-of-mass models (Scott et al., 2012; Sanders et al., 2015). In Scott et al. (2012), the authors pointed out that the 3D center-of-mass of the object of interest should rotate in a circle as the specimen is rotated about a fixed axis. With this inmind, the projections are aligned so that the center-of-mass is posi- tioned at the origin and rotates about a fixed point. This model is ideal for cases when the projected mass within the imaging area is fixed for all angles. In practical terms, it means there must be an isolated set of particles with no additional mass moving in or out of the imaging area throughout the tilt series. In Sanders et al. (2015), a detailed derivation of the


motion of the center-of-mass was developed for each 2D cross-section perpendicular to the axis of rotation. Based off of this derivation, it was determined that a least squares problem could be solved that determined viable paths for each cross-section. This allowed for accurate alignment in the cases when the volume was not necessarily fixed, as the ideal cross-sections can be selected. This ideal cross-section selection procedure is automated within the algorithm. In addition to this, a windowing function was designed to further alleviate any changes in the total mass. The work in this paper further generalizes on the ideas


in Sanders et al. (2015) for a more robust alignment method. Instead of only selecting a subset of cross-sections to use for the alignment model, we select a subset of cross-sections for particular sequences within the tilt series. In other words, the ideal alignment model in Sanders et al. (2015) is viable for any subset of the projection images, and therefore we may apply the alignment models only over angular ranges where the mass is known to be fixed based off of simple statistical estimates. This strategy allows for greater flexibility in the alignment and is thus robust for a larger class of data sets. The complete mathematical development of this


approach is given here. The alignment is carried out on a tilt series of a zeolite cluster with nanoscale platinum particles embedded in the pores. The reconstruction of these nano- particles gives us a close look at the effect and accuracy of the alignment. In addition to this result, as has been pointed out in several other papers (Binev et al., 2012; Leary et al., 2013; Sanders et al., 2017), we highlight the effectiveness of the alignment combined with the proper image reconstruction technique, in this case the total variation (TV) regularization within compressed sensing. In particular, we show that the desired results are not achievable without both the most advanced image reconstruction techniques and the proper


alignment. The associated algorithms are openly available online (Sanders) and Supplementary Material is provided online to reproduce our numerical results.


Supplementary Materials 1


Supplementary Materials 1 can be found online. Please visit journals.cambridge.org/jid_MAM.


PROBLEM DESCRIPTION AND PREVIOUS MODELS


Let a function f(x,y,z) represent the density map of the 3D space on which f is defined, where it is implied that f≥0 everywhere. We define an ideal projection of f at angle θ rotated about the x-axis by


Pθðf Þðx; yÞ = Z Ωz


Qθ = cos θ fx; ðyzÞQθ


- sin θ cos θ ; sin θ


and x,y are in say [0,1]2. The classical tomographic recon- struction problem is then given a set of projections of f at angles say θjk


one would like to reconstruct an accurate approximation of f. A major obstacle in electron tomography is in the uncertainty of the relative position of the projections due to the inherent challenge of having a precise rotational mechanism of the sample at the nanoscale. This results in shifted versions of the ideal projections, which can be expressed by


j = 1 with −90°≤θ1<θ2< … <θk<90°, Pθ fðÞ x; y ðÞ: ~ ðÞ = Pθ fðÞ x - xθ; y - yθ Therefore, the shifts (xθ, yθ) must first be determined for


each projection, or at least relate each projection to some common coordinate system. We note that there could potentially be several other sources of error not accounted for in this misalignment model but could be included later, such as offset tilt axis or inaccurate θ values. Determining the shifts along the x-axis is usually a more


j=1,…, k− 1 based on some optimization model. With such an approach, we can be confident that neighboring projections are relatively well aligned. However, can we trust that Pθk


ðÞf is aligned relative to Pθj ðÞf , for


say or measure for certain if it is? Perhaps visual examination is appropriate, but in our experience this is not suitable.


ðÞf is well aligned with Pθ1 ðÞf , and how can one even


trivial task, as the x-axis is the axis of rotation and one can simply use some conservation of mass approach (Sanders et al., 2015). Aligning along the y-axis can be much more difficult, and traditional techniques are subject to a large accumulative error. For instance, common strategies are carried out so that neighboring images are aligned in sequence, so that Pθj+ 1


ðÞdz; where (1)


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