2. W HAT IS IT ? Point cloud - Set of vertices in a 3D coordinate system. These vertices are usually defined by X, Y, and Z coordinates, and typically are intended to be representative of the external surface of an object. It is a point-based rendering, i.e., the 3D models are represented by points (point cloud) and not by triangles, as they are commonly represented.

3. M OTIVATION Evolution of point acquisition’s devices. Best efficiency when compared with rendering based on triangles or another primitive that needs information about connectivity of points. High quality (with efficiency) of rendering when the point cloud is dense. Hardware implementation is possible.

4. P IPELINE 3D Acquisition Spherical Flipping Convex Hull Point Selection Rendering

5. 3D ACQUISITION (S CANNER ) 3D Scanning - A scan is taken at some viewpoint. View planning - A decision is made about where to position the scanner relative to the object in order to perform the next scan. Registration - After all scans have been acquired, a global registration algorithm may be used to simultaneously minimize misalignment errors over all pairs of overlapping scans. Merging - The aligned scans frequently contain significant regions in which many scans overlap. Merging these logically separate scans into a single model both reduces storage and averages away some of the scanning noise. Post-processing - Depending on the final use for the model, the output of the merging step may undergo further processing, including noise or outlier reduction, automated filling of small remaining holes, and curvature-adaptive resampling or decimation.

6. 3D ACQUISITION (S CANNER ) Triangulation-based Triangulation-based 3D scanners find the positions of points on a surface by computing corresponding pixels from two viewpoints. The correspondence defines a pair of rays in space, and the intersection of the rays determines a 3D position.

7. 3D ACQUISITION (S CANNER ) Passive Stereo Active Stereo Structured Light Light Stripe Multiview Triangulation and Structure from Motion Pulsed and Modulated Time of Flight Shape from Shading and Photometric Stereo

8. 3D ACQUISITION (P HOTOGRAPHY )

10. S PHERICAL F LIPPING Consider a D-dimensional sphere with radius R, centered at the origin (C), and constrained to include all the points in P. Spherical flipping reflects a point ϵ P with respect to the sphere (spherical mirror) by applying the following equation: where: = point inside the sphere R = radius of sphere

11. S PHERICAL F LIPPING Intuitively, spherical flipping reflects every point internal to the sphere along the ray from C to to its image outside the sphere. Figure 2 - Spherical flipping (in red) of a 2D curve (in blue) using a sphere (in green) centered at the view point (in magenta). Figure 1Figure 2

12. C ONVEX H ULL The convex hull of a set Q of points is the smallest conve x polygon P for which each point Q is either on the boun dary of P or in its interior.

13. C ONVEX H ULL (3D) 3 rd Party library

14. C ONVEX H ULL Denote by the transformed point cloud of P: Calculate the convex hull of U {C}, i.e., the set that contains the transformed point cloud AND the center of the sphere.

15. A point is marked visible from C if its inverted point lies on the convex hull of U {C}. P OINT S ELECTION

16. B EFORE

17. A FTER

18. R EFERENCES Point-Based Graphics (The Morgan Kaufmann Series in Computer Graphics) Markus Gross, Hanspeter Pfister Direct Visibility of Point Sets Sagi Katz, Ayellet Tal, Ronen Basri Mesh segmentation using feature point and core extraction Sagi Katz, George Leifman, Ayellet Tal Introduction to Algorithms Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein

19. P OINT S ELECTION (P ROVE ) The shape of L for different values of β (in degrees), where = (10;0), R = 30. Result of spherical flipping

20. L and the X-axis define the empty region associated with. “”How much”” is visible, depends on the size of the region. For every given point, there exist two special points on either side of P, and ϵ P. The region bounded between the curves in the “L equation”, from through and from through, is the largest possible empty region. P OINT S ELECTION (P ROVE )

21. The empty gray region between Lj and Lk, as defined by the values of βj + βk. From “L Equation”, it can be deduced that the largest region corresponds to the smallest β. This means that βj and βk that correspond to the largest possible empty region, are the smallest possible for. Note that βj and βk can be extracted from “L Equation”. For to be visible, the sum of βj and βk should satisfy βj + βk = const (i.e., a large empty region is associated with )

22. P OINT S ELECTION (P ROVE )