Surface Reconstruction from Unorganized Points

1. Surface Reconstruction from Unorganized Points Hugues Hoppe et. al SIGGRAPH�92 Min Chen Qingdi Liu

2. Outline Introduction Algorithm Overview Results Discussion

3. Problem Statement Input an unorganized set of points assumed to lie on or near an unknown manifold M Output a simplicial surface that approximates M

4. Features a robust solution to the unifying general surface reconstruction problem with relatively few assumptions about the set of points X and underlying surface U

5. Algorithm Overview Two main sampling assumptions X is a ?-dense, ?-noise sample of U Features of U that are too small are not recoverable An approximation of ?+? is a user-specified parameter for our program

6. Stage 1: Define Signed Distance Function Tangent Plane Estimation each data point x --> an oriented tangent plane ( oi, ni ) center oi --> centroid of Nbhd( xi) normal ni --> principal component analysis( SVD ) on covariance matrix of Nbhd(xi)

7. Consistent Tangent Plane Orientation Set up Riemannian Graph G = ( O, E ) cost W( i, j ) = 1 - | Ni ? Nj | Traversing an Euclidean Minimum Spanning Tree of G

8. Signed Distance Function f f(p) = disti(p) = ( p - oi )?ni Undefined distance value for boundary identification. Zero set Z(f) is our estimate for M

9. Stage 2: Contour Tracing Variation of marching cube algorithm( Geoff Wyvill, 1986 ) use hash tables to avoid repeated calculation from seed cube to visit only appropriate cubes Deal with cubes with undefined distance Degenerate zero case -- perturbing to a small positive value

10. Results

15. Problem with disconnected surface For surfaces with real holes close by, it tries to connect them.