Spherical Parameterization JinKyung Hong Computer Graphics Lab Korea

Spherical Parameterization Jin-Kyung Hong Computer Graphics Lab. Korea University 2004. 09. 23

Goal • The inspired image Jin Kyung, Hong 2 Computer Graphics Lab. @ Korea Univ.

Applications (1/4) • Texture mapping Parameterization of 3 D mesh Jin Kyung, Hong Texture image 3 Texture mapping Computer Graphics Lab. @ Korea Univ.

Applications (2/4) • Morphing Jin Kyung, Hong 4 Computer Graphics Lab. @ Korea Univ.

Applications (3/4) • Remeshing Jin Kyung, Hong 5 Computer Graphics Lab. @ Korea Univ.

Applications (4/4) • Editing Jin Kyung, Hong 6 Computer Graphics Lab. @ Korea Univ.

Parameterization • The planar case (3 D->2 D) magnification triangle mesh Jin Kyung, Hong 7 Computer Graphics Lab. @ Korea Univ.

Parameterization • A kind of parameterization Jin Kyung, Hong 8 Computer Graphics Lab. @ Korea Univ.

Spherical Parameterization • The spherical case (3 D->3 D) Parameterizing triangle mesh spherical triangulation the vertex V: M-O = c (V-O) Jin Kyung, Hong 9 Computer Graphics Lab. @ Korea Univ.

Spherical Parameterization : positions of vertices : i’th row of the matrix Lw : column vectors : the number is vertices in the mesh : auxiliary variables • Key observation Obtaining M (x , y, z) = 0 Jin Kyung, Hong 10 Computer Graphics Lab. @ Korea Univ.

Experiment (1/5) • Example model Jin Kyung, Hong 11 5 -1 -1 0 -1 4 -1 0 -1 -1 0 0 -1 -1 6 -1 0 -1 3 0 0 -1 -1 0 0 4 -1 -1 0 0 -1 -1 0 -1 5 -1 -1 -1 0 -1 -1 6 -1 0 0 -1 -1 3 Computer Graphics Lab. @ Korea Univ.

Experiment (2/5) • Result n = 8; mesh = [ -0. 5 -0. 5 0. 5 ]; Jin Kyung, Hong -0. 5 0. 5 3; 3; 12 Computer Graphics Lab. @ Korea Univ.

Experiment (3/5) • Convex Vertices: 16 Jin Kyung, Hong 13 Computer Graphics Lab. @ Korea Univ.

Experiment (4/5) • Concave Jin Kyung, Hong 14 Computer Graphics Lab. @ Korea Univ.

Experiment (5/5) • Degenerate solutions always exist. Jin Kyung, Hong 15 Computer Graphics Lab. @ Korea Univ.

Examination • Wrong F = [p(: , 1). ^2+p(: , 2). ^2+p(: , 3). ^2 -1; p(: , 4). *p(: , 1)-Lw(: , : )*p(: , 1); p(: , 4). *p(: , 2)-Lw(: , : )*p(: , 2); p(: , 4). *p(: , 3)-Lw(: , : )*p(: , 3); ]; • Right F = [p(: , 1). ^2+p(: , 2). ^2+p(: , 3). ^2 -1; M(: , : )*p(: , 1); M(: , : )*p(: , 2); M(: , : )*p(: , 3); ]; Jin Kyung, Hong 16 Computer Graphics Lab. @ Korea Univ.

Experimental Results (1/8) • Convex Vertices: 16 Jin Kyung, Hong 17 Computer Graphics Lab. @ Korea Univ.

Experimental Results (2/8) • Concave Jin Kyung, Hong 18 Computer Graphics Lab. @ Korea Univ.

Experimental Results (3/8) Jin Kyung, Hong 19 Computer Graphics Lab. @ Korea Univ.

Experimental Results (4/8) Jin Kyung, Hong 20 Computer Graphics Lab. @ Korea Univ.

Experimental Results (5/8) Jin Kyung, Hong 21 Computer Graphics Lab. @ Korea Univ.

Experimental Results (6/8) • Test model vertices: 188 Jin Kyung, Hong 255 22 Computer Graphics Lab. @ Korea Univ.

Experimental Results (7/8) • Bunnyhead Jin Kyung, Hong 23 Computer Graphics Lab. @ Korea Univ.

Experimental Results (8/8) • Bunny Jin Kyung, Hong 24 Computer Graphics Lab. @ Korea Univ.

Questions? Thank you ^^*~ Jin Kyung, Hong 25 Computer Graphics Lab. @ Korea Univ.