Improved 3D Resolution of Electron Tomograms 1123


Figure 1. Left: test phantom images. Middle: sinogram data of test images. Right: Center-of-mass (COM) determined for each projection angle, and least squares solution for coordinates centers-of-mass for the test images from solving Θcx=b based off of (2). Notice for the first test image the least squares fit is perfect, and that the second test image contains an object near the corner which is not present in all projections, thus making (2) invalid, as demonstrated by the least squares fit.


Again, these small errors between alignment of neighboring projections may be invisible to us, but may accumulate into large errors for projections that are not taken at nearby angles. Therefore, a more rigorous approach that considers a more global analysis of the data. In the development of the approach, we reduce the


dimension of the problem to one particular cross-section x. We denote the cross-sectional function for this fixed x by fx(y, z), and similarly the corresponding projection of the cross-section by Pθ(fx)(y). We denote the total mass in this cross-section by Mx = RR2


of-mass coordinates by cy


x = 1 Mx


cz fxðy; zÞdy dz, and the center-


Z R2


x = 1 Mx


yfxðy; zÞ dy dz and


Z R2


zfxðy; zÞ dy dz: Then as done in Sanders et al. (2015), it is easy to show


the following center-of-mass calculation for the angular projection of fx:1


1 Z Mx R


1In Sanders et al. (2015) there is difference in the sign in (2) due to an alternative definition of Qθ.


y  Pθ ðfxÞðyÞ dy = cy xcosθ + cz xsinθ: (2) some constants cy


Therefore, any projection of fx should satisfy (2) for x and cz


angle. Therefore, if we let 2


Θ =


6 6 6 4


...


cos θ1 sin θ1 cos θ2 sin θ2


...


b = 1 Mx


6 6 6 4


3


cos θk sin θk 2RR y  Pθ1


RR y  Pθ2 RR y  Pθk


7 7 7 5


...


; cx = cy x


x


ðfxÞðyÞdy3 ðfxÞðyÞdy


ðfxÞðyÞdy


7 7 7 5


; and cz


x, which are independent of the


;


then for every x the perfectly aligned projections satisfy


Θcx=b. This equation is fundamental to our approach, and was the primary motivation for original center-of-mass techniques (Sanders et al., 2015). This approach will theo- retically achieve perfect alignment with “perfect” projection data. In addition, the method is robust to random noise. However, the major drawback is that the governing equation (2) is not valid over finite projection domains whenever the volume is not fixed, and at best we can compute the integrals in vector b over some finite interval Ω. This issue is clear to see with the demonstration pre-


sented in Figure 1. Here, the projection data are generated for two nearly identical test images, which can be considered a single cross-section fx of a 3D volume. The subtle difference


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