Existing methods – main ideas Normal partition Various flattening methods Disadvantages

Existing methods – main ideas Normal partition Various flattening methods Disadvantages www.phwiki.com

Existing methods – main ideas Normal partition Various flattening methods Disadvantages

Henneman, Todd, Contributing Writer has reference to this Academic Journal, PHwiki organized this Journal Bounded-distortion Piecewise Mesh Parameterization O. Sorkine, D. Cohen-Or, R. Goldenthal, D. Lischinski IEEE Visualization 2002 July 2002 Existing methods – main ideas Partition/cut the mesh in preprocess Interactive user input Normals bucketing Region growing from feature curves Flatten each patch by energy minimization Convex mapping Harmonic mapping Con as long as mal mapping Normal partition Doesn’t detect developable structures Prone to self-intersection problems

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Various flattening methods Floater 97, Lévy in addition to Mallet 98 Maillot et al. 93, Eck et al. 95, Lee et al. 98 S in addition to er et al. 01, Gu et al. 02 Haker et al. 00, Sheffer 02, Lévy et al. 02 Zigelman et al. 02 Bennis et al. 91 Disadvantages A-priori partition sets lower bound on the distortion cannot comply with preset upper bound on the distortion. If the distortion is too high, need to subdivide the partition in addition to recompute the parameterization. Most of the methods cannot prevent triangle flips in addition to global self-intersections (overlaps). High computational cost ( as long as non-linear optimizations). Our contribution Parameterization with bounded distortion Simultaneous partition in addition to parameterization Valid parameterization – no self-intersections

Algorithm overview Greedy algorithm: grow one patch at a time, until no more vertices can be added. At each step, attempts to flatten the “best” vertex adjacent to the current patch. The distortion of each mesh triangle is guaranteed to be below specified threshold. Algorithm overview Select r in addition to om seed triangle, flatten it. Maintain a priority queue of the vertices adjacent to the current patch. Flatten triangles adjacent to current patch: At each step, take the best vertex off the queue Check as long as self-intersections Stop when no triangles can be added to the patch, in addition to start a new one. The 3D surface

The 3D surface The planar patch The 3D surface The planar patch The 3D surface The planar patch

The 3D surface The planar patch The 3D surface The planar patch The 3D surface The planar patch

The 3D surface The planar patch The 3D surface The planar patch The 3D surface The planar patch

The 3D surface The planar patch The 3D surface The planar patch The 3D surface The planar patch

The 3D surface The planar patch The distortion metric In 2D In 3D S p1 p2 p3 q1 q2 q3 T T’ The distortion metric The Jacobi matrix of S is 3×2 real matrix: The singular values of J are:

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The distortion metric The distortion metric The values max in addition to min are the maximal stretching in addition to shrinking caused to a unit-length vector by the mapping S. We want to equally “punish” stretch in addition to shrink, there as long as e we define: D(T, T’) = max{max , 1/ min } Any other reasonable metric can be used! Flattening a single vertex

Flattening a single vertex The obtained position v can be used as initial guess as long as a relaxation procedure that finds better position v’ ( as long as example, minimizes the average distortion of triangles incident to the vertex). However, it is slower, in addition to the initial guess per as long as ms well in practice. Vertex grade components Maximal distortion caused to the triangles flattened with the vertex. If it’s greater than the threshold, the grade is set to zero in addition to the vertex can’t be flattened in the current patch! The ratio between patch area in addition to squared perimeter (to create round patches with small boundary length). Crease angles or other segmentation in as long as mation. More criteria Checking self-intersections Local self-intersection – triangle flipping

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