1. A Volumetric Method for Building Complex Models from Range Images
Brian Curless
Marc Levoy
speaker:Cui Yunpeng
2017/5/16

2. Introduction Reconstruct from scan’s application: Industry Hospital
Art

3. Introduction Input: a set of range images from scanner
Output：a closely range surface of the model

4. Goal: A surface reconstruction algorithm owns such properties
Representation of range uncertainty
Utilization of all range data
Incremental updates
Time and space efficiency
Robustness
No restrictions on topological type
Ability to fill holes in the reconstruction

5. Previous work Two dircetions:
1. unorganized points
①.parametric surfaces
②.implicit function（SDF）
2. exploits the underlying structure of the acquired data
②.implicit function(SDF)
VRIP: use samples of a continuous function to combine structured data

6. Volumetric integration
For a set range images, R1, R2, R3,…,Rn, construct signed distance funchtions(SDF) d1(x),d2(x),d3(x),…d4(x).
Combine these functions to generate cumulative function,D(x).(In fact, combine the volume)
Extract the disired manifold as isosurface, D(x)=0.

7. Volumetric integration
The weight depend on the dot product between each vertex normal and the viewing direction.
Now to prove:
Isosurface of the weighted SDF ＝ Minimization of squared distances between points on the range surfaces and points on the desired surfaces

8. Proof
Assumption: 1.The range sensor is orthographic. 2.The range errors are independently distributed along sensor lines of sight. With the assumption, this is proved in another article:
𝐸 𝑓 = 𝑖=1 𝑁 𝑑 𝑖 2(𝑥,𝑦,𝑧,𝑓) ⅆ𝑥𝑑𝑦𝑑𝑧

9. Proof
x,y,z is postion of surface point, v is directions of sensor, w is weight, s,t is the position of sensor.

10. Volumetric integration
Result:
Incremental calculation:

11. Volumetric integration
Signed distance and weight functions in one dimension and their combine

12. Volumetric integration
Signed distance and weight functions in two dimension and their combine

13. Volumetric integration
Update voxel in three dimension
1. Set all voxel weights to zero
2. Tessellate each range image by constructing triangles
3. Compute a weight at each vertex
4. Cast a ray from the sensor to compute signed distance contribution of each voxel near the surface and intersect it by the triangles mesh
5. The weight is computed by linearly interpolating the weights stored at the intersection triangle’s vertices.

14. Volumetric integration
d is the distance, each voxel has the distance from one range surface

15. Hole filling
Unseen portions of the surface will appear as holes in the reconstruction.
Fill them for continuous meshes and esthetics.

16. Hole filling
Classifying all points in the volume as being in one of three states: unseen, empty, or near the surface.

17. Hole filling
Extension of the algorithm in Update voxel in three dimension
1.Initialize the voxel space to the “unseen” state.
2. Update the voxels near the surface, and mark them as “near surface”.
3.Follow the lines of sight back from the observed surface and mark the corresponding voxels as “empty”. (space carving)
4. Perform an isosurface extraction at the zero-crossing of the signed distance function. Additionally, extract a surface between empty regions and unseen regions.

18. Implementation Software optimzations 1.Run-length encoding
2. Fast volume traversal
3. Fast surface extraction

19. Result Range surface Unorganized point Zippered mesh Volumetric mesh
Scanning a 1.6 mm drill bit from 12 orientations at a 30 degree spacing using traditional optical triangulation methods

20. Result
Zippered surface
Volumetric surface

21. Result Hole filling without backdrop No hole filling
Hole filling with backdrop
Smoothed

22. Result Painted Original Range surface Before hole filling
After hole filling
Hardcopy

23. Result
Statistics for the reconstruction of the dragon and Buddha models, with and without space carving. 

24. Limitation Optical scanning Volumetric algorithm
1.Surface points must be accessible
2.Surface reflectance affects results
Volumetric algorithm
Thin surfaces and sharp corners

25. Future work Carving from video/image silhouettes
Next best view, including backdrops
Large-scale scenes
Surface color acquisition

26. Question
1.Why the weight change in page11? The weight function taper off behind the range points for reasons discussed below. 2.What does the color line mean in page19 (b) figure? It means we get the range surface from different angle view .
taper off behind

27. Question
3.How does it distinguish hole and back-ground when doing hole filling? By input a backdrop, the hole of surface don’t have backdrop’s color. We can see the different in page 21.

28. Question
4.The transition between unseen and empty is discontinuous. How does the paper get a better surface?
In fact, they did post-filtering the mesh after reconstruction using weighted averages of nearest vertex neighbors. It can get a smooth surface.