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<<Break-Curve And Surface-Tangent Matching >> click
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<Pair Matches>
Matches of Pairs of Sherds.
There are seven figures of break-curves on black backgrounds and five figures of break-curves on white backgrounds. The first seven are for correct matches, and the last five are for incorrect matches. A match is based on 5 equispaced points along a break-curve, starting at a vertex, and an associated outer-surface normal vector at the location of each of the 5 points. Successive points are a Euclidean distance d from each other, and the sequence of 5 points extends over only a portion of the break-curve between successive vertices on the curve. Hence, only a portion of the break-curve common to a pair of matching sherds is used in this alignment shown. The vertices were marked by a person, but in practice most of them can be estimated automatically. Hence, one sherd in a pair is held fixed and a best Euclidean transformation is estimated for the other (3 translation parameters, 3 rotation parameters). This is a linear-least squares problem (an eigen value/vector problem). The piece of break-curve used for the match is indicated by a pair of numbers, one on each side of the break-curve and located at the vertices used. In the 12 figures, a "1" and a "1'" are the pairs of numbers.
For incorrect matches illustrated in Figs. 8-12, the least-squares error is small -- the 5 points and associated surface normals match well. However, there are at least two ways to detect these matches as being incorrect. First, For Figs 8,9,11, portions of the sherds overlap, and this can be detected quickly. Second, for Figs. 10,12 each of the matched pairs will not have an accurate profile curve, i.e., there will usually be a sizable fitting error in estimating a profile curve for the pair. Note that in Fig.10 the two break curves match very well even though the match is not a correct one.
(fig.1) sherd 7 vs 11 (fig.2) sherd 3 vs 13 (fig.3) sherd 1 vs 5 (fig.4) sherd 4 vs 8
(fig.5) sherd 5 vs 9 (fig.6) sherd 10 vs 9 (fig.7) sherd 2 vs 8 (fig.8) sherd 10 vs 5
(fig.9) sherd 10 vs 5 (fig.10) sherd 5 vs 4 (fig.11) sherd 1 vs 10 (fig.12) sherd 3 vs 7
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<Triplet Matches>
Matches of Triplets of Sherds.
Two examples of correctly-matched triplets are shown. With each there are three views of the assembled sherds in order that one be able to see how good the assembly is. In these cases one sherd is held fixed and each of the other two is transformed, which means that two transformations must be estimated -- 12 paramters. These paramters are estimated simultaneously using all the matching data in a single cost function. Hence, the estimate of a pair of transformations is based on the matching of 15 pairs of points and associated pairs of normals. Unfortunately, in this case the estimation is nonlinear. Note that the matching is not perfect, especially for the bottom of the pot in the second triplet. The matching can be improved by using data at the other vertices as well, by automatically adjusting the locations of the first points at the vertices better, and, more generally, by using points along the entire common portions of the break-curves and using denser sets of points and dynamic local curve-scaling during the point matching. The approach extends immediately to 4 or more groups of sherds, where k-1 transformations must be estimated simultaneously for optimally matching k sherds.
sherd 1-5-9
sherd 8-6-2
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