An interactive computer graphic system for 3-D stereoscopic reconstruction from serial sections: Analysis of metastatic growth

Extending Unbiased Stereology of Brain Ultrastructure to Three-dimensional Volumes

“…1,[20][21][22] Transformation parameters are calculated automatically from a set of point correspondences given by the user. This method is less labor intensive than manual adjustments (see User Interface, below) and will work for any choice of basis functions.…”

“…21 Fortunately, serial EM of neuropil has intrinsic fiducials in the form of cross-sectioned microtubules, mitochondria, and other small, cylindric organelles. The centers of these structures can be used to form a set of point correspondences for alignment.…”

On the choice of a cost function for the reconstruction of surfaces by triangulation between contours

“…For example, one can decide which of two alternative triangles to use at each step on the way around the contours by choosing the triangle with the shorter diagonal [4935,1003]. In terms of the graph-theoretic formulation, this is equivalent to using a cost function equal to triangle perimeter (which is non-negative), but making a final decision about direction at each step along the path rather than looking at all the possibilities and making a globally optimal decision.…”

“…One possibility [4935,1003] is to choose the two vertices which are closest together, that is, to choose i0 and j0 such that Ai0 and Bj0 are closer together than any other pair Ai and Bj. (One can either look at all mn pairs of points before deciding which two points are closest, or, with some risk, arbitrarily choose to use the point on S A which was used as starting point when triangulating between it and the previous contour, and only look at the n distances between it and the points on S B .)…”

“…The resulting triangulation is in fact the same as that of Figure 2. This cost function is related to the shorter-diagonal criterion used by Christiansen & Sederberg [4935] and Chawla [1003]: the two are equivalent for an algorithm making local decisions, but they may be slightly different for a global optimization. Comparing lengths of diagonals obviously requires less computation than comparing perimeters, since one needs to consider only the leading edge of each triangle rather than all three edges.…”