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Introduction to Algorithms Contour Stitching

This document explores the problem of contour stitching, focusing on the optimization of triangulations between two closed curves. Given two-dimensional curves with m and n points, it investigates which points correspond to each other and calculates possible triangulations. The solution involves defining spans between curves, using triangles constructed from points on the curves, and adhering to constraints to ensure an acceptable surface topology. It also delves into graph theory concepts to outline how to organize edges and achieve optimal paths with minimal area and aspect ratios.

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Introduction to Algorithms Contour Stitching

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  1. Introduction to Algorithms Contour Stitching CSE 680 Prof. Roger Crawfis

  2. ? ? Contour Stitching • Problem: Given: 2 two-dimensional closed curves Curve #1 has m points Curve #2 has n points Which point(s) does vertex i on curve one correspond to on curve two? How many possible triangulations are there? i

  3. A Solution • Fuchs, et. al. • Optimization problem • 1 stitch consists of: • 2 spans between curves • 1 contour segment • Triangles of {Pi,Qj,Pi+1} or {Qj+1,Pi,Qj} • Consistent normal directions Qj Pi Pi+1

  4. Fuchs, et. al. • Left span • PiQj => go up • Right span (either) • Pi+1Qj => go down • PiQj+1 => go down

  5. Fuchs, et. al. • Constraints • Each contour segment is used once and only once. • If a span appears as a left span, then it must also appear as a right span. • If a span appears as a right span, then it must also appear as a left span.

  6. Fuchs, et. al. • This produces an acceptable surface (from a topological point of view) • No holes • We would like an optimal one in some sense.

  7. Qj+1 Qj Pi Pi+1 Fuchs et. al. • Graph problem • Vertices Vij = span between Pi and Qj • Edges are constructed from a left span to a right span. • Only two valid right spans for a left span.

  8. Fuchs, et. al. • Organize these edges as a grid or matrix. Q ? j i P QjPiQj+1  PiQjPi+1

  9. Fuchs, et. al. • Acceptable graphs • Exactly one vertical arc between Pi and Pi+1 • Exactly one horiz. arc between Qj and Qj+1 • Either • indegree(Vij) = outdegree(Vij) =0 • both > 0 • (if a right, also has to be a left)

  10. Fuchs, et. al. • Claim: • An acceptable graph, S, is weakly connected. • Lemma 2 • Only 0 or 1 vertex of S has an indegree = 2. • E.g., Two cones touching in the center. • All other vertices have indegree=1 • (recall indegree = outdegree)

  11. Fuchs, et. al. • Thereom 1: S is an acceptable surface if and only if: • S has one and only one horizontal arc between adjacent columns. • S has one and only one vertical arc between adjacent rows. • S is Eulerian • closed walk with every arc only once. Q j i P QjPiQj+1  PiQjPi+1

  12. Fuchs, et. al. • Number of arcs is thus m+n. • Many possible solutions still!!! • Associate costs with each edge • Area of resulting triangle • Aspect ratio of resulting triangle

  13. Fuchs, et. al. • Note that Vi0 is in S for some i. • Note that V0j is in S for some j. • With the weights (costs), we can compute the minimum path from a starting node Vi0. • Since we do not know which Vi0, we compute the paths for all of them and take the minimum. • Some cost saving are achievable.

  14. Fuchs, et. al. • Note, that once a search hits the optimal path, it will not cross it, it must follow it.

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