1 / 14

Optimal Multiresolution Polygonal Approximation Algorithm

A detailed study on an efficient multi-resolution polygonal approximation algorithm for minimizing segment numbers and controlling deviation within error tolerances. The algorithm is based on a heuristic split method and a weighted shortest path approach in a feasibility graph. The algorithm's complexity is O(RN3). Test results on shape "Britain" with N=10,109 show promising efficiency.

jrohde
Download Presentation

Optimal Multiresolution Polygonal Approximation Algorithm

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. Alexander Kolesnikov and Pasi Fränti DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF JOENSUU JOENSUU, FINLAND Optimal multiresolution polygonal approximation

  2. Min-# polygonal approximation =0.1 Input curve P Approximation curve Q N=11,000 vertices M=117 vertices

  3. Multi-resolution approximation 3=0.005 2=0.02 1=0.1 Input curve P Q1 Q2 Q3 N M1 M2 M3 1 > 2 > 3 M1 < M2 < M3 < N

  4. Multiresolution: The main principle Vertices of approximation curve Qrfor coarse resolution level rare subset of vertices of curveQr+1for the fine resolution level (r+1): Qr  Qr+1 Qr+1 Qr                r r+1

  5. Problem formulation Find such Rmultiresolution polygonal curvesQ1,Q2,…,QR so the total number of segments M is minimal, and maximum deviationfor the curves is less than the given error tolerancies r: subject toD(Qr)  r , r=1, 2, …, R, here approximation error D(Qr) is maximal deviation of vertices of input curve P from the approximation lines of curve Qr

  6. Heuristic split algorithm(Ramer-Douglas-Peucker) DO Add approximation node to curveQr-1 to split the segment with maximum deviation UNTIL D(Qr)<r,.

  7. Single-resolution min-# algorithm 1. Construct feasibility graphs G=G()on vertices of input curve P for the given tolerance . Edge e(i,j)G() if deviation d(i,j) < . 2. Find the shortest path in the graph G(): M min. • • • • • • • • • • GraphG() Shortest path • •

  8. Proposed algorithm 1. Construct R feasibility graphs G1, G2, …, GR for the given error tolerances 1, 2, ..., R. 2. Find the weighted shortest path in the graph G1. • Weight w(i,j) of edge e(i,j) in graph G1: w(i,j) =1+weight of the Shortest Path in sub-graph H2(i,j)G2embeddedinto the edge e(i,j). • Weights of the edges of sub-graphs H2(i,j) are defind in analogous way.

  9. Embedded sub-graph • Edge e(i,j)}G1defines embedded subgraph H2(i,j) of the graph G2 e(i,j)G1 i j H(i,j,2)G2 i j

  10. Example of weight calculation e(i,j) G1 H2(i,j)G2 H3(i,k1)G3 H2(k2,j)G3 k2 j k1 i 1 > 2 > 3 w(i,j)=1+[(1+2)+(1+3)+(1+3)]

  11. Pseudo-code of recursive algorithm • (M) = MR_min_num( r, n1, n2 : integer) • FOR n = n1 TO n2 DO • C(n) • FOR j = n-i TO n1 DO • IF e(j,n)  G(P,r) THEN • IF r = R THEN • Weight1; • ELSE • Weight1+ MR_min_num( r+1, j, n); • ENDIF • IF C(j) + Weight < C(n) THEN • C(n)C(j) + Weight; • A(n)j; • ENDIF • ENDIF • ENDFOR • ENDFOR • MC(n2) • RETURN M

  12. Sequential algorithm • All shortest paths in all graphs are calculated from • high resolution level to low one (bottom-up) • Then the final solution is constructed (top-down). • Complexity of the algorithm: O(RN3)

  13. Efficiency G=100*Mopt/M % Test shape ”Britain”, N=10,109 Error tolerances: =0.1, 0.05, 0.02, 0.01, 0.005, 0.002. R=6 R=6

  14. Conclusions • Optimal algorithm for multiresolution min-# problem is introduced • The algorithm is based on weighted shortestpath algorithm in graph • Time complexity of the algorithm is O(RN3).

More Related