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# Iccv95 curvature

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Talk given by Gabriel Taubin at ICCV'95

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### Iccv95 curvature

1. 1. ESTIMATING THE TENSOR OF CURVATURE OF ASURFACE FROM A POLYHEDRAL APPROXIMATION GABRIEL TAUBIN IBM T.J.Watson Research Center ICCV95 June 1995
2. 2. THE PROBLEMHOW TO ESTIMATE PRINCIPAL CURVATURES AND PRINCIPALDIRECTIONS OF A SURFACE AT THE VERTICES OF APOLYHEDRAL APPROXIMATION (USUALLY A TRIANGULATION),MINIMIZING THE COMPUTATIONAL COST
3. 3. THE TENSOR OF CURVATURE OF A SURFACE (1) THE TENSOR OF CURVATURE OF THE SURFACE S IS THE MAP p 7! p THAT ASSIGNS EACH POINT p OF S TO THE FUNCTION THAT MEASURES THE DIRECTIONAL CURVATURE p(T ) OF S AT p IN THE DIRECTION OF THE UNIT LENGTH VECTOR T , TANGENT TO S AT p. THE DIRECTIONAL CURVATURE FUNCTION p( ) IS A QUADRATIC FORM.*
4. 4. THE TENSOR OF CURVATURE OF A SURFACE (2)ALTERNATIVELY, FOR EACH POINT p OF S ,THERE IS A 3 3 SYMMETRIC MATRIX Kp, SUCH THAT p (T ) = T Kp T tFOR EACH TANGENT VECTOR T TO S AT p, AND N t Kp N = 0FOR THE NORMAL VECTOR N TO S AT p.NOTE THAT N IS AN EIGENVECTOR OF KpWITH ASSOCIATED EIGENVALUE 0.THE PRINCIPAL DIRECTIONS , fT1 T2g, OF S AT pARE THE OTHER TWO EIGENVECTORS.THE PRINCIPAL CURVATURES , f 1 2g, p pARE THE CORRESPONDING EIGENVALUES.
5. 5. HOW TO ESTIMATE PRINCIPAL CURVATURES AND DIRECTIONS1. ESTIMATE THE MATRIX Kp2. COMPUTE THE EIGENVALUES AND EIGENVECTORS OF Kp
6. 6. HOW WE ESTIMATE THE MATRIX KpWE ESTIMATE THE MATRIX Mp = 21 Z + p(T ) T T t d ;INSTEAD, WHERE T = cos( ) T1 + sin( ) T2AND fT1 T2g ARE THE PRINCIPAL DIRECTIONS OF S AT p.NOTE THAT fT1 T2g CAN BE REPLACED BY ANY PAIR OFORTHONORMAL TANGENT VECTORS, WITHOUT CHANGING THEVALUEOF Mp.ALSO NOTE THAT SUCH A PAIR OF VECTORS CAN BE OBTAINEDFROM THE NORMAL VECTOR N TO S AT p.SO, Mp IS ONLY FUNCTION OF THE NORMAL VECTOR N , AND OFTHE DIRECTIONAL CURVATURES p(T ) ALONG TANGENT DIREC-TIONS.
7. 7. WHY WE ESTIMATE THE MATRIX Mp ?BECAUSE IT HAS THE SAME EIGENVECTORS AS Kp.N IS AN EIGENVECTOR OF MpWITH ASSOCIATED EIGENVALUE 0.THE PRINCIPAL DIRECTIONS , fT1 T2g, OF S AT p ARE THE OTHER TWO EIGENVECTORS OF Mp, BECAUSE T1 Mp T2 = 0 tTHE CORRESPONDING EIGENVALUES, fm1 m2g, p pARE LINEAR FUNCTIONS OF THE PRINCIPAL CURVATURES m1 = T1 Mp T1 = 3 1 + 1 2 p t 8 p 8 p m2 = T2 Mp T2 = 1 1 + 3 2 p t 8 p 8 pTHE PRINCIPAL CURVATURES CAN BE COMPUTED AS FOLLOWS 1 p = 3mp ; m2 1 p 2 p = 3mp p2 ; m1
8. 8. HOW WE ESTIMATE DIRECTIONAL CURVATURES p(T )IF q IS ANOTHER POINT ON THE SURFACE, CLOSE TO p BUT DIFFER-ENT FROM p, AND T IS THE UNIT LENGTH NORMALIZED?PROJECTIONOF THE VECTOR q ; p ONTO THE TANGENT PLANE hN i , THE DIREC-TIONAL CURVATURE CAN BE APPROXIMATED AS FOLLOWS 2N t(q ; p) : p (T ) kq ; pk
9. 9. THE ALGORITHM (1)POLYHEDRAL SURFACE REPRESENTED AS A PAIR OF LISTSS = fV F g, A LIST OF VERTICES V = fvig,AND A LIST OF FACES F = ffig.CURRENT IMPLEMENTATION ONLY FOR TRIANGULAR FACES.THE NORMAL Nvi TO A VERTEX IS ESTIMATED ASTHE WEIGHTED AVERAGE OF THE NORMALS TO THEINCIDENT FACES, WITH WEIGHTS PROPORTIONAL TOSURFACE AREA OF THE FACES,AND NORMALIZED TO UNITP LENGTH. 2 jfk j Nfk Nvi = P fk F i : k fk 2F i jfkj Nfk k
10. 10. THE ALGORITHM (2)THE MATRIX Mvi IS APPROXIMATED BY THE FINITE SUM M~ vi = X wij ij Tij Tij : t 2 vj V iWHERE{ V i IS THE SET OF VERTICES THAT SHARE AN EDGE WITH vi.{ Tij IS THE UNIT LENGTH NORMALIZED PROJECTION OF THE VECTOR vj ; vi ONTO THE TANGENT PLANE hNvi i?.{ ij IS THE APPROXIMATION OF vi (Tij ) 2Nvti (vj ; vi) ij = kvj ; vik{ AND THE WEIGHT wij IS PROPORTIONAL TO THE SUM OF THE SURFACE AREAS OF THE TWO TRIANGLES THAT ARE INCIDENT TO BOTH VERTICES vi AND vj .
11. 11. SURFACE SMOOTHINGIT IS REQUIRED WHEN DEALING WITH REAL DATASEE OUR OTHER PAPER IN THIS CONFERENCE CURVE AND SURFACE SMOOTHING WITHOUT SHRINKAGE PRELIMINARY EXPERIMENTSWE DETERMINE THE ACCURACY OF THE ALGORITHMBY COMPARING ESTIMATED VALUES WITH TRUE VALUES INCASES WHERE GROUND TRUTH IS AVAILABLE.
12. 12. EXPERIMENT 1ERRORS AS A FUNCTION OF INCREASING RESOLUTION A B CPolyhedral approximations of a sphere. A: Icosahedron B: Icosahedron subdivided once. C: Icosahedron subdivided twice.ERROR < 0:2 10;3 IN THE THREE CASES
13. 13. EXPERIMENT 2ERRORS AS A FUNCTION OF INCREASING RESOLUTION A B CPolyhedral approximations of a torus. A B CEstimation errors histograms for the torus. Horizontal axis (error value) goesfrom 0:00 to 0:05, and there are 20 vertical bars. No vertex showed error largerthan 0:05. The vertical axis is scaled to make the maximum of each histogramequal to 1.
14. 14. EXPERIMENT 3ERRORS AS A FUNCTION OF TRIANGULATION(DIFFERENT TRIANGULATIONS OF THE SAME SURFACE) A B CThree iso-surfaces constructed by evaluating the same function f (x y z) = e;((x;1=2)2+y2;1=4)2;(z2=4)2 + e;((x+1=2)2+z2;1=4)2;(y2=4)2 ; 0:1on di erent grids, and then adjusting the position of the vertices to make themlie on the implicit surface de ned by the function. The three grids are of thesame size, but their orientations with respect to the surface di er. Surface Ahas 5 842 vertices, B has 6 198 vertices, and C has 5 763 vertices.
15. 15. EXPERIMENT 3 A B CEstimation error histograms. Horizontal axis (error value) goes from 0:00 to0:10, and there are 100 vertical bars. The vertical axis is scaled to make themaximum of each histogram equal to 1.Only the vertices with error les than 0:10 where considered in the histograms.In the three cases there where a few vertices with error larger than 0:10 (27for surface A, 45 for surface B, and 70 for surface C). These outlier verticeswhere located in the regions of highest curvature, where the edge length shouldhave been shorter to produce a good estimation. The mean error though, wasaround 0:01 in the three cases (0:011 for surface A, 0:009 for surface B, and0:013 for surface C). This can clearly be seen in the histograms.