Surface Reconstruction from Unorganized Points

1. Surface Reconstruction from Unorganized Points Hugues Hoppe Tony DeRose Tom Duchamp John McDonald Werner Stuetzle Presented by NizamovaLiliia 06May 2017

2. Basic definition Visualisation of implicit surfaces - function of three arguments - fixed value of this function - level Surface function It is important to determinate surface function, which describe our unknown surface. There are problems where the surface specified implicitly or table of values defined on an irregular grid

3. Interactive surface sketching • The creation of curves in R2 by tracing the path of a stylus or mouse as the user sketches the desired shape • Surfaces from contours • The data produced by medical application like CT, MRT and other. • Surfaces from range data • The data produced by laser range scanning systems Surface specification

4. Function reconstruction • To determine a function , such that • where given a surface , a set and a set The goal • Surface reconstruction • To determine a surface M (Figure 2f) that approximates an unknown surface M (Figure 2a), using a sample X (Figure 2b) (Figure 1a) Original CSG object (Figure 1b) Sampled points (xi) (n = 4102) (Figure 2f) Final surface after edge collapses

5. Related Work • Implicit reconstruction techniques • Vaughan Pratt. Direct least-squares fitting of algebraic surfaces. Computer Graphics (SIGGRAPH ’87 Proceedings), 21(4):145–152, July 1987. • Shigeru Muraki. Volumetric shape description of range data using “blobby model”. Computer Graphics (SIGGRAPH ’91 Proceedings), 25(4):227–235, July 1991 • Parametric reconstruction techniques • B. C. Vemuri. Representation and Recognition of Objects From Dense Range Maps. PhD thesis, Department of Electrical and Computer Engineering, University of Texas at Austin, 1987. • [27] B. C. Vemuri, A. Mitiche, and J. K. Aggarwal. Curvature- based representation of objects from range data. Image and Vision Computing, 4(2):107–114, 1986.

6. Advantages • Only an unorganized collection of points • Surfaces of arbitrary topology • Not generate spurious surface components

7. Hoppe et al’s Algorithm • Tangent Plane Estimation(PCA) • Consistent tangent plane orientation(MST) • Signed distance function • Contour tracing (Marching Cubes)

8. Tangent Plane Estimation Tangent plane at has center , normal Resulting tangent planes for point clouds

9. Consistent tangent plane orientation Points and tangent planes Graph relating tangent planes Naive propagation of normal Normal propagation using optimized tree Optimized tree

10. Each (center) has a corresponding (vertex in graph) Connect and if and are close Cost on edge is EMST of tangent plane centers Consistent tangent plane orientation Euclidian Minimum Spanning Tree Model this problem as graph optimisation Points and tangent planes Graph relating tangent planes Naive propagation of normal Normal propagation using optimized tree Optimized tree

11. Add edges to EMST: Add edge (i,j) if or are in the K-neighbourhood of other Encode geometric proximity Riemannian Graph over Consistent tangent plane orientation Riemannian Graph

12. Consistent tangent plane orientation Compute orientation from graph Maximise the total cost of the graph The problem is reducible to MAX-CUT

13. Assign cost to edge (i,j) Consistent tangent plane orientation Obtaining good propagation order Traverse Minimum Spanning Tree: tends to propagate along direction of low curvature Traversal order of orientation propagation Oriented tangent planes

14. Signed distance estimation The signed distance from to the tangent plane for is: To allow the surface with boundaries (incomplete data), points far from originals samples are given a distance of "undefined" Estimated signed distance

15. Contour Tracing Contour Tracing is to extract an iso surface from a scalar function use variation of marching cubes algorithm Edges are of length less than density + noise Cubes visited during contouring Output of modified marching cubes

16. Results Contours Range Images Meshes Ray Traced Points Contours Points from 39 planar (horizontal) slices of the CT scan The bust of Spock was reconstructed from points taken from an actual cylindrical range image Points were randomly sampled from a number of existing simplicial surfaces To simulate laser range imaging from multiple view points CSG models were ray traced from multiple eye points

17. Conclusions and Future Work • Conclusions: • Reconstruct a surface in three-dimensional space with or without boundary from a set of unorganized points • Future Work: • Reconstruct manifolds of co-dimension one in spaces of arbitrary dimension

18. Thank you for attention

19. FAQ • Why you assign cost ? • It helps void situation when our unoriented plans are nearly cost. In addition to being non-negative, this assignment has the property that a cost is small • Another questions was the same, but accentuate the word WHY • The order in which the orientation is propagated is important. Because we can obtain something completely different • But the question has another meaning: Why you assign cost not another formula? • I analyse Hoppe’s method. And he uses that formula • Can you explain the meaning of signed distance estimation? • The signed distance f (p) from an arbitrary point p to a known surface M is the distance between p and the closest point z, multiplied by +-1, depending on which side of the surface p lies

20. FAQ • Can you explain why we set distance «underfined»? • If M is known not to have boundaries, this simple rule works well. However, the rule must be extended to accommodate surfaces that might have boundaries. Intuitively, the sample points do not leave holes of radius larger than . If the sample is , the radius of the holes may increase, but by no more than . We therefore conclude that a point z cannot be a point of M if . If the projection z of p onto the closest tangent plane has , we take f(p) to be undefined • Contour tracing. How we can transform from picture 1 to picture 2 on slide 15. Rephrase: How marching cubes algorithm works? • Divide the domain M into cells. • Select cells that intersect with the desired surface • To approximate the surface in one cell. The most optimal method of approximation is triangulation