1124 Toby Sanders and Ilke Arslan


in the images is the small object in the bottom right corner of the second image. This object is not seen in a few of the projections, in particular, looking at the sinograms, it can be seen that the object moves out of the viewing range approximately between angles 25° and 55°. Due to this, our governing equation (2) when taken over a finite domain Ω no longer holds for the second image. This is evidenced in the rightmost plots, where we used (2) to find the center-of- mass coordinates cx by taking the least squares solution to Θcx=b. There is not a good fit for the second image, whereas the first has a perfect fit. From this purely mathematical viewpoint, it would seem


rather hopeless to align based on the governing equation. However, typically the majority of the mass of the projected objects in f is contained within Ω, and it can be the case that the nonzero parts of f are almost entirely within the domain Ωfor each angle. Thus, it was first suggested to use only some of the cross-sections for the alignment, those which were “good” candidates based on the total mass and observed variation of this mass through the projections. In addition, a weighting function was applied uniquely to each projection image to account for the introduction of mass in the edge of the frames.


COMPLETE ROBUST ALIGNMENTMODEL


In this work, we expand further upon the ideas in section 2 for an even more robust alignment technique, which allow us to overcome issues with a changing mass between the projections. As before we should only consider some of the cross-sections in which the observed mass is stable, that is the objects in those cross-sections appear to be in a vacuum. Unfortunately, in some instances it may be difficult to find any cross-section which truly satisfies this property. For example, if the objects in the projections sit on top of some nonuniform carbon grid, then this grid will be transitioning in and out of the frame in every projection. Therefore, it may be completely unreasonable to consider any cross-section really fixed. This is especially true at very differing angular ranges, where completely different sections of the grid may be visible in the projections, making the model for a rigid path of the center-of-mass completely unreliable. However, locally we may be able to trust that at least a few cross-sections of neighboring projections more or less con- tain the same mass and the centers-of-mass between pro- jections at nearby angles should follow a rigid motion. Thus, the strategy we propose is to enforce the rigid movements between projections at nearby angles.


Basic Idea


Let us begin by using a simple example to describe our approach. Suppose we have four projection images Pθ1


with θ1<θ2<θ3<θ4, and say, for example, θ4−θ1≤10°. Then the motion of the center-of-mass based on the governing equation should bemore reliable for Pθ1


ðÞf ; Pθ2 ðÞf ; Pθ3 ðÞf ; Pθ4 ðÞf ; Pθ2 ðÞf , all acquired at nearby angles ðÞf ;


and Pθ3ðÞf , and perhaps some transition of mass makes the centers-of-mass of Pθ1


ment. The same argument holds that Pθ2 Pθ4


ðÞf and Pθ4


x = 1 Mx


y  Pθj


denote the center-of-mass of each projection by tθj


ðfxÞðyÞdy Ω


and define the matrices Θ1 =


2 4


cos θ1 cos θ2 cos θ3


ðÞf can perhaps give us reliable quantities. Therefore, let us Z


ðÞf less reliable for align- ðÞf ; Pθ3


ðÞf ; and


sinθ1 sinθ2 sinθ3


3 5; and Θ2 =


Likewise, we define the vectors cx;1 = cy


; cx;2 = cy x;2


cz


x;1 x;1


and bx;2 =


2 4


cx,2 so that Θ1cx;1 = bx;1 and Θ2cx;2 = bx;2; (4)


for every x, or more appropriately for only the selection of “good” x cross-sections. In addition, the chosen cross- sections for x for (3) and (4) need not be the same, and the measure of “good” cross-sections is dependent upon the three neighboring projections. Finally, since the acquired projections are not aligned and the equations likely do not have good solutions, the job is to determine shifts of the projections so that (3) and (4) do have good solutions. To match this local motion to the global alignment problem, we enforce the condition that the solution which arises from (3) and (4) should make it so that only a single shift is given for Pθ2


do not need to be precisely the same. With this in mind, we proceed by describing the general method in full detail.


ðÞf and Pθ3 ðÞf , although the determined centers-of-mass


General Formulation Suppose we have k misaligned projections images,


Pθ1 ðf Þðx; yÞ; ~ ~ Pθ2 ðf Þðx; yÞ; :::~ Pθk ðf Þðx; yÞ;


where θ1<θ2< … <θk. Moreover, suppose that we have accurately determined the shifts along the tilting axis so that we only need to determine the shifts along the y-axis, and we will write the projections as


Pθ1 ðfxÞðyÞ; ~ ~ Pθ2 ðfxÞðyÞ; :::~ Pθk ðfxÞðyÞ: (3)


tθ2x tθ3x tθ4x


3 5:


Then perhaps independently there exists solution cx,1 and x;2


2 4


cos θ2 sin θ2 cos θ3 sin θ3 cos θ4 sin θ4


; bx;1 = cz


2 4


tθ1x tθ2x tθ3x


3 5;


3 5:


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