1 / 25

Matching Shapes with a Reference Point

Matching Shapes with a Reference Point. Volkan Çardak. Problem. Given two figures A,B Determine how much they “resemble each other” Figure = union of finitely many points and line segments in two dimensions or triangles in three dimensions. Measure for “resemblance”.

quincy
Download Presentation

Matching Shapes with a Reference Point

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Matching Shapes with a Reference Point Volkan Çardak

  2. Problem • Given two figures A,B • Determine how much they “resemble each other” • Figure = union of finitely many points and line segments in two dimensions or triangles in three dimensions

  3. Measure for “resemblance” • Hausdorff-metric δH • Recall : For A, B element of Cd let → δH (A,B) := max min d2 (a,b) a Є A b Є B • Hausdorff-distance between two sets A and B → → δH (A,B):= max {δH (A,B), δH (B,A) }

  4. Measure for “resemblance” • This can be done in O((n+m) log(n+m)) [ABB] • But more natural: Figures A,B are not fixed but can be moved • So a need for transformations • Translation • Rigid motion • Similarity

  5. So what to determine? • min δH( A,T(B) ) T Є TR • Does it make any difference if we exchange the sets A and B in this problem?

  6. Finding optimal matches in 2d • In [ABB] an algorithm of O( (nm log(nm) log*(nm) ) for TR = translation along one fixed direction • In [AST] one of O( (nm)2 log3(nm) ) for TR = arbitrary translations • In [CGHKKK] an O( (nm)3log2(nm) ) for TR = arbitrary rigid motions • Latter two not applicable

  7. Optimal vs Approximation • Try not to find an optimal solution but an approximation to the optimal one by simpler algorithms • Let’s assume optimal match yields the Hausdorff-distance δ • These simpler algortihms will find a matching T such that δH ( A,T(B) ) ≤ αδ constant α>1 • Approximate matching with loss factor α

  8. How to do the approximation? • The idea is to use “reference points” for the approximation algortihms • By using reference points with the approximation algorithms more favourable run time • Before we take a look at the defintion let’s take a look at a simple reference point

  9. Reference Point example • Only applicable for translations • Reference point rA= (xAmin ,yAmin ) • The points rA and rB that are assigned to sets A,B have the property that when B is transformed to match A optimally, then the distance of the transformed rB to rA is also bounded by a constant factor times the Hausdorff distance of the matching. • figure

  10. Reference Point example • With an optimal match then the following holds: | xAmin – xB’min | ≤δ & | yAmin – yB’min | ≤δ (B’ is translated image of B) • rA is as reference point for A of quality √2

  11. Reference Point example • Instead of finding optimal translation of B to match A we just match rB to rA to obtain an image B” of B. • δH ( A,B” ) ≤ δH ( A,B’ ) + δH ( B’ ,B” ) ≤ (√2+ 1) δ • So the Hausdorff-distance of the match found by the reference points is at most by a factor √2 + 1 ≈ 2.4 worse than the optimal one

  12. Optimal vs Approximation (again) • The approximation algorithm has a lineair running time, since you only need to determine the reference points • Best known algorithm for finding the optimal match O( (mn)2 log3(mn) ) (recap from slide 6) • Which approximation algorithms, later in this presentation

  13. Reference Point (definition) • So far we have only seen an example of a reference point for translations • Definition: Let TR be a set of transformations on Rd . A mapping r: Cd → Rd is called a reference point with respect to TR iff • r is equivariant with respect to TR, i.e., for all A,B ЄCd and TЄTR we have r ( T(A) ) = T ( r(A) ) and • There exists some constant c ≥ 0 such that if for all A,B ЄCd , d2 ( r(A), r(B) ) ≤ c δH(A,B)

  14. Approximation algorithms • Based on the existence of a reference point for TR there are the following 3 algorithms for approximately optimal matchings where TR stand for the set of translations, rigig motions and similiraty transformations.

  15. Approximation algorithms • Algorithm T • Compute r(A) and r(B) and translate B by r(A) - r(B) (so that r(B) is mapped onto r(A) ). Let B’ be the image of B. • Output B’ as the approximately optimalsolution ( tohether with the Hausdorff-distance δH(A,B’) )

  16. Approximation algorithms • Algorithm R • As in Algorithm T. • Find an optimal matching of A and B’ under rotations of B’ around r(A) . • Output the solution B” and the Hausdorff-distance δH(A,B”)

  17. Approximation algorithms • Algorithm S • As in Algorithm T. • Determine the diameters d(A) and d(B) and scale B’ by α:= d(A) / d(B) around the center r(A) . • as step 2 in algorithm R with the scaled image of B’. • as step 3 in algorithm R.

  18. Loss factors of the algorithms • These algorithms are simpler than the ones for finding the optimal solutions, since after step 1 the matchings are restricted to ones leaving the reference point invariant → in d dimensions this eliminates d degrees of freedom. • Algorithm T finds approximately optimal matching for translations with loss factor α = c+1 • Algorithm R finds approximately optimal matching for rigid motions with loss factor α = c+1 • Algorithm S finds approximately optimal matching for similarity transformations with loss factor α = c+3

  19. Runtimes algorithms • Algorithm T : lineair • Algorithm R : problem is how to find the optimal matching under rotations. This can be done in O( nmlog(nm) log*(nm) ) in the plane [ABB]. So this is also the overall runtime • Algorithm S : also O( nmlog(nm) log*(nm) )

  20. Reference point: Steiner Point • We have seen a reference point which holds for translations. • Now let’s take a look at the Steiner Point, which is a reference point which for translations, rigid motions, scaling.

  21. Reference point: Steiner Point • Definition: Bd is the d-dimensional unit ball and Sd-1 its boundary, the (n-1)- dimensional unit sphere in Rd. Let A be a convex body (convex and compact subset) in Rd. The support functionhA : Rd → R of A is given by: hA (u) = max ‹a,u› aЄA The Steiner point s(A) of A is defined as s(A) = d ∫ hA (u) u dω(u) ──── Sd-1 Vol (Sd-1 )

  22. Reference point: Steiner Point • The Steiner point is a reference point for similarity transformations in arbitrary dimension d≥2. Quality is χd . 2d Vol(Bd-1 ) χd =______________ Vol(Sd-1 )

  23. Reference point: Steiner Point • For a polygon the Steiner point can be computed easily. It is the weighted average of the vertices where each vertex is weighted by its exterior angle divided by 2 π. n • s(P) = 1/(2 π) ∑ φi ∙ vi i=1 • This can be done in lineair time

  24. Reference point: Steiner Point

  25. References • [AAR] O.Aichholzer, H.Alt and G.Rote, Matching shapes with a Reference Point, 2001 • [ABB] H.Alt, B.Behrends and J.Blömer, Approximate matching of polygonal shapes, Proceedings 7th Annual Symposium on Computational Geometry, 1991, 186-193 • [AST] P.J. Agarwal, M.Sharir and S.Toledo, Applications of parametric searching in geometric optimization, Proc. 3rd ACM-SIAM Sympos.Dicrete Algorithms, 1992, 72-82 • [CGKKK] P.P.Chew, M.T.Goodrich, D.P. Huttenlocher, K.Kedem, J.M. Kleinberg. D.Kravets, Geometric pattern matching under Euclidean motion, Proc.5th Canadian Conference on Computational Geometry, 1993, 151-156

More Related