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Computational Geometry and Geometric Shape Matching

Computational Geometry and Geometric Shape Matching. What is Computational Geometry?. Algorithms for geometric objects. Convex Hull. Given a set of pins on a pinboard And a rubber band around them How does the rubber band look when it snaps tight?. Convex Hull.

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Computational Geometry and Geometric Shape Matching

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  1. Computational Geometry and Geometric Shape Matching

  2. What is Computational Geometry? • Algorithms for geometric objects

  3. Convex Hull • Given a set of pins on a pinboard • And a rubber band around them • How does the rubber band look when it snaps tight?

  4. Convex Hull • Given a set of pins on a pinboard • And a rubber band around them • How does the rubber band look when it snaps tight?

  5. Voronoi Diagram • Given all post offices in San Antonio • Find a subdivision of San Antonio into cells such that points in a cell are all closest to one post office

  6. Voronoi Diagram • Given all post offices in San Antonio • Find a subdivision of San Antonio into cells such that points in a cell are all closest to one post office

  7. Security: Art Gallery • Given an art gallery • How many guards do you need to guard the whole gallery? Where should they be located?

  8. Data bases • Given a set of points (data sets) in high dimensional space • Preprocess them such that orthogonal range queries can be answered efficiently.

  9. Geometric Shape Matching • Considergeometric shapesto be composed of a number of basic objects

  10. Geometric Shape Matching • Consider geometric shapesto be composed of a number of basic objects such as points

  11. Geometric Shape Matching • Consider geometric shapesto be composed of a number of basic objects such as points line segments

  12. Geometric Shape Matching • Consider geometric shapes to be composed of a number of basic objects such as points line segments triangles

  13. Geometric Shape Matching • Consider geometric shapes to be composed of a number of basic objects such as points line segments triangles • How similarare two geometric shapes?

  14. Geometric Shape Matching • Consider geometric shapes to be composed of a number of basic objects such as points line segments triangles • How similarare two geometric shapes? • Choice of distance measure • Full or partial matching • Exact or approximate matching • Transformations (translations, rotations, scalings)

  15. Computer-Aided Neurosurgery FU Berlin, Functional Imaging Technologies GmbH and the medical school ‘Benjamin Franklin’ at FU Berlin

  16. Background • Computer assisted neuro surgery (esp. brain tumor surgery)

  17. Background • Computer assisted neuro surgery (esp. brain tumor surgery) • Before Surgery: • Functional MR scan of the brain • 3D model of the brain

  18. Background • Computer assisted neuro surgery (esp. brain tumor surgery) • Before Surgery: • Functional MR scan of the brain • 3D model of the brain During Surgery:

  19. Background • Computer assisted neuro surgery (esp. brain tumor surgery) • Before Surgery: • Functional MR scan of the brain • 3D model of the brain • During Surgery: • Electromagnetic pointing device • Display positions in 3D model

  20. Background • Computer assisted neuro surgery (esp. brain tumor surgery) • Before Surgery: • Functional MR scan of the brain • 3D model of the brain • During Surgery: • Electromagnetic pointing device • Display positions in 3D model Navigation aid mapping positions in the brain to a prerecorded 3D MR image of the brain

  21. Landmark Registration • Set of markers attached to patient’s head 3D model during surgery image world • Small but very noisy point sets • Find nearly rigid motion that maps image markers to world markers

  22. Rigid Point Matching • P={p1,p2,…,pn} Q={q1,q2,…,qm} point sets in R3 P Q

  23. Rigid Point Matching • P={p1,p2,…,pn} Q={q1,q2,…,qm} point sets in R3 P Q • Rigid matching maps edges with same length onto each other

  24. Rigid Point Matching • P={p1,p2,…,pn} Q={q1,q2,…,qm} point sets in R3 P Q • Rigid matching maps edges with same length onto each other • Nearly rigid matching maps edges with similar lengths onto each other

  25. Maintain scorefor each pair indicating the “quality” of matching those two points p1 pn q1 qm • • • • • • qu pi qv pj Scoring Table • Edges with similar lengths indicate a possible matching of and or vice versa • For each pair of similar edges, increase the score of all pairs of involved endpoints

  26. qu pi qv pj Scoring Table • Edges with similar lengths indicate a possible matching of and or vice versa • Maintain scorefor each pair indicating the “quality” of matching those two points p1 pi pj pn q1 qu qv qm • For each pair of similar edges, increase the score of all pairs of involved endpoints

  27. Finding a Transformation • Extract combinatorial matching • from scoring table • Least-Squares Approximation: • Find affine transformation A that minimizes the sum of the squared distances between corresponding points • Test if A is nearly rigid (check determinant, unit vector images, etc.)

  28. Computer-Aided Neurosurgery: Summary • Direct linear algebra approaches were numerically very unstable • Geometric approach of splitting the problem into - finding the combinatorial matching and then - computing the nearly rigid transformation is very easy to implement and proved to be very robust. • The algorithm is integrated into a commercial product and used in practice.

  29. Protein Gel Matching FU Berlin, UofA, German Heart Center Berlin

  30. 2D Gel Electrophoresis • Two-dimensional Gel Electrophoresis (2DE) is • an important method in proteome research • a high resolution technique which is capable to separate thousands of proteins from a tissue sample

  31. 2D Gel Electrophoresis

  32. 2D Gel Electrophoresis • Proteins are concentrated in so called spots of (axis- parallel) elliptic shape

  33. 2D Gel Electrophoresis • Proteins are concentrated in so called spots of (axis- parallel) elliptic shape • Protein analysis by mass spectrometry (expensive)

  34. 2D Gel Electrophoresis

  35. 2D Gel Electrophoresis Gel Matching Protein identification by gel image comparison is faster and not expensive

  36. The Algorithmic Approach Make use of ideas and methods from Computational Geometry: • Spot detection • Assign to each spot the coordinates of its center point and its intensity • Point pattern matching • Consider a gel as a point pattern. Then the problem reduces to a partial approximate point pattern matching.

  37. GPS Curve Location FU Berlin and UofA and UTSA

  38. Finding a Curve in a Map Given: • Ageometric graph G(embedded in R2 with line segments) • Apolygonal curve a Task: Find a path p in G that is the most similar toa

  39. Finding a Curve in a Map Given: • A geometric graph G (embedded in R2 with line segments) • Apolygonal curve a Task: Find a path pin Gthat is the most similar to a

  40. Application: Map Construction • Consider : • A given roadmap, and • a sequence of GPS positions obtained from a person travelling on some of the roads while recording her positioning information using a GPS receiver polygonal curve • Problem: • The noise of the GPS receiver distorts the polygonal curve inherently • Task: • Find the roads in the roadmap that have been traveled

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