MSR Image based Reality Project

1
MSR Image based Reality Project
http//research.microsoft.com/larryz/videoviewint
erpolation.htm

2
The visibility problem
Which points are visible in which images?
3
Volumetric stereo
Scene Volume V
Input Images (Calibrated)
Goal Determine occupancy, color of points in V
4
Discrete formulation Voxel Coloring
Discretized Scene Volume
Input Images (Calibrated)
Goal Assign RGBA values to voxels in
V photo-consistent with images
5
Complexity and computability
Discretized Scene Volume
3
N voxels C colors
6
Issues

• Theoretical Questions
• Identify class of all photo-consistent scenes
• Practical Questions
• How do we compute photo-consistent models?

7
Voxel coloring solutions

• 1. C2 (shape from silhouettes)
• Volume intersection Baumgart 1974
• For more info Rapid octree construction from
  image sequences. R. Szeliski, CVGIP Image
  Understanding, 58(1)23-32, July 1993. (this
  paper is apparently not available online) or
• W. Matusik, C. Buehler, R. Raskar, L. McMillan,
  and S. J. Gortler, Image-Based Visual Hulls,
  SIGGRAPH 2000 ( pdf 1.6 MB )
• 2. C unconstrained, viewpoint constraints
• Voxel coloring algorithm Seitz Dyer 97
• 3. General Case
• Space carving Kutulakos Seitz 98

8
Reconstruction from Silhouettes (C 2)
Binary Images

• Approach
• Backproject each silhouette
• Intersect backprojected volumes

9
Volume intersection

• Reconstruction Contains the True Scene
• But is generally not the same
• In the limit (all views) get visual hull
• Complement of all lines that dont intersect S

10
Voxel algorithm for volume intersection

• Color voxel black if on silhouette in every image
• for M images, N3 voxels
• Dont have to search 2N3 possible scenes!

O( ? ),
11
Properties of Volume Intersection

• Pros
• Easy to implement, fast
• Accelerated via octrees Szeliski 1993 or
  interval techniques Matusik 2000
• Cons
• No concavities
• Reconstruction is not photo-consistent
• Requires identification of silhouettes

12
Voxel Coloring Solutions

• 1. C2 (silhouettes)
• Volume intersection Baumgart 1974
• 2. C unconstrained, viewpoint constraints
• Voxel coloring algorithm Seitz Dyer 97
• For more info http//www.cs.washington.edu/homes
  /seitz/papers/ijcv99.pdf
• 3. General Case
• Space carving Kutulakos Seitz 98

13
Voxel Coloring Approach
Visibility Problem in which images is each
voxel visible?
14
Depth Ordering visit occluders first!
Scene Traversal
Condition depth order is the same for all input
views
15
Panoramic Depth Ordering

• Cameras oriented in many different directions
• Planar depth ordering does not apply

16
Panoramic Depth Ordering
Layers radiate outwards from cameras
17
Panoramic Layering
Layers radiate outwards from cameras
18
Panoramic Layering
Layers radiate outwards from cameras
19
Compatible Camera Configurations

• Depth-Order Constraint
• Scene outside convex hull of camera centers

20
Calibrated Image Acquisition
Selected Dinosaur Images

• Calibrated Turntable
• 360 rotation (21 images)

Selected Flower Images
21
Voxel Coloring Results (Video)
Dinosaur Reconstruction 72 K voxels colored 7.6
M voxels tested 7 min. to compute on a 250MHz
SGI
Flower Reconstruction 70 K voxels colored 7.6 M
voxels tested 7 min. to compute on a 250MHz SGI
22
Limitations of Depth Ordering

• A view-independent depth order may not exist

p
q

• Need more powerful general-case algorithms
• Unconstrained camera positions
• Unconstrained scene geometry/topology

23
Voxel Coloring Solutions

• 1. C2 (silhouettes)
• Volume intersection Baumgart 1974
• 2. C unconstrained, viewpoint constraints
• Voxel coloring algorithm Seitz Dyer 97
• 3. General Case
• Space carving Kutulakos Seitz 98
• For more info http//www.cs.washington.edu/homes
  /seitz/papers/kutu-ijcv00.pdf

24
Space Carving Algorithm
Image 1
Image N
...

• Space Carving Algorithm

25
Which shape do you get?
V
True Scene

• The Photo Hull is the UNION of all
  photo-consistent scenes in V
• It is a photo-consistent scene reconstruction
• Tightest possible bound on the true scene

26
Space Carving Algorithm

• The Basic Algorithm is Unwieldy
• Complex update procedure
• Alternative Multi-Pass Plane Sweep
• Efficient, can use texture-mapping hardware
• Converges quickly in practice
• Easy to implement

Results
Algorithm
27
Multi-Pass Plane Sweep

• Sweep plane in each of 6 principle directions
• Consider cameras on only one side of plane
• Repeat until convergence

True Scene
Reconstruction
28
Multi-Pass Plane Sweep

• Sweep plane in each of 6 principle directions
• Consider cameras on only one side of plane
• Repeat until convergence

29
Multi-Pass Plane Sweep

• Sweep plane in each of 6 principle directions
• Consider cameras on only one side of plane
• Repeat until convergence

30
Multi-Pass Plane Sweep

• Sweep plane in each of 6 principle directions
• Consider cameras on only one side of plane
• Repeat until convergence

31
Multi-Pass Plane Sweep

• Sweep plane in each of 6 principle directions
• Consider cameras on only one side of plane
• Repeat until convergence

32
Multi-Pass Plane Sweep

• Sweep plane in each of 6 principle directions
• Consider cameras on only one side of plane
• Repeat until convergence

33
Space Carving Results African Violet
Input Image (1 of 45)
Reconstruction
Reconstruction
Reconstruction
34
Space Carving Results Hand
Input Image (1 of 100)
Views of Reconstruction
35
Properties of Space Carving

• Pros
• Voxel coloring version is easy to implement, fast
• Photo-consistent results
• No smoothness prior
• Cons
• Bulging
• No smoothness prior

36
Alternatives to space carving

• Optimizing space carving
• recent surveys
• Slabaugh et al., 2001
• Dyer et al., 2001
• many others...
• Graph cuts
• Kolmogorov Zabih
• Level sets
• introduce smoothness term
• surface represented as an implicit function in 3D
  volume
• optimize by solving PDEs

37
Alternatives to space carving

• Optimizing space carving
• recent surveys
• Slabaugh et al., 2001
• Dyer et al., 2001
• many others...
• Graph cuts
• Kolmogorov Zabih
• Level sets
• introduce smoothness term
• surface represented as an implicit function in 3D
  volume
• optimize by solving PDEs

38
Level sets vs. space carving

• Advantages of level sets
• optimizes consistency with images smoothness
  term
• excellent results for smooth things
• does not require as many images
• Advantages of space carving
• much simpler to implement
• runs faster (orders of magnitude)
• works better for thin structures, discontinuities
• For more info on level set stereo
• Renaud Kerivens page
• http//cermics.enpc.fr/keriven/stereo.html

39
References

• Volume Intersection
• Martin Aggarwal, Volumetric description of
  objects from multiple views, Trans. Pattern
  Analysis and Machine Intelligence, 5(2), 1991,
  pp. 150-158.
• Szeliski, Rapid Octree Construction from Image
  Sequences, Computer Vision, Graphics, and Image
  Processing Image Understanding, 58(1), 1993, pp.
  23-32.
• Matusik, Buehler, Raskar, McMillan, and Gortler ,
  Image-Based Visual Hulls, Proc. SIGGRAPH 2000,
  pp. 369-374.
• Voxel Coloring and Space Carving
• Seitz Dyer, Photorealistic Scene
  Reconstruction by Voxel Coloring, Intl. Journal
  of Computer Vision (IJCV), 1999, 35(2), pp.
  151-173.
• Kutulakos Seitz, A Theory of Shape by Space
  Carving, International Journal of Computer
  Vision, 2000, 38(3), pp. 199-218.
• Recent surveys
• Slabaugh, Culbertson, Malzbender, Schafer, A
  Survey of Volumetric Scene Reconstruction Methods
  from Photographs, Proc. workshop on Volume
  Graphics 2001, pp. 81-100. http//users.ece.gatec
  h.edu/slabaugh/personal/publications/vg01.pdf
• Dyer, Volumetric Scene Reconstruction from
  Multiple Views, Foundations of Image
  Understanding, L. S. Davis, ed., Kluwer, Boston,
  2001, 469-489. ftp//ftp.cs.wisc.edu/computer-vis
  ion/repository/PDF/dyer.2001.fia.pdf

40
References

• Other references from this talk
• Multibaseline Stereo Masatoshi Okutomi and
  Takeo Kanade. A multiple-baseline stereo. IEEE
  Trans. on Pattern Analysis and Machine
  Intelligence (PAMI), 15(4), 1993, pp. 353--363.
• Level sets Faugeras Keriven, Variational
  principles, surface evolution, PDE's, level set
  methods and the stereo problem", IEEE Trans. on
  Image Processing, 7(3), 1998, pp. 336-344.
• Mesh based Fua Leclerc, Object-centered
  surface reconstruction Combining multi-image
  stereo and shading", IJCV, 16, 1995, pp. 35-56.
• 3D Room Narayanan, Rander, Kanade,
  Constructing Virtual Worlds Using Dense Stereo,
  Proc. ICCV, 1998, pp. 3-10.
• Graph-based Kolmogorov Zabih, Multi-Camera
  Scene Reconstruction via Graph Cuts, Proc.
  European Conf. on Computer Vision (ECCV), 2002.
• Helmholtz Stereo Zickler, Belhumeur,
  Kriegman, Helmholtz Stereopsis Exploiting
  Reciprocity for Surface Reconstruction, IJCV,
  49(2-3), 2002, pp. 215-227.

41
So far

• Passive Stereo
• Spacetime Stereo
• Multiple View Stereo

42
Next

• Structure from Motion
• Given pixel correspondences,
• how to compute 3D structure and camera motion?

Slides stolen from Prof Yungyu Chuang
43
Epipolar geometry fundamental matrix
44
The epipolar geometry

• What if only C,C,x are known?

45
The epipolar geometry
epipolar geometry demo

• C,C,x,x and X are coplanar

46
The epipolar geometry

• All points on ? project on l and l

47
The epipolar geometry

• Family of planes ? and lines l and l intersect
  at e and e

48
The epipolar geometry
epipolar pole intersection of baseline with
image plane projection of projection center in
other image
epipolar geometry demo

• epipolar plane plane containing baseline
• epipolar line intersection of epipolar plane
  with image

49
The fundamental matrix F
R
C
C
50
The fundamental matrix F
51
The fundamental matrix F
R
C
C
52
The fundamental matrix F
53
The fundamental matrix F
R
C
C
54
The fundamental matrix F

• The fundamental matrix is the algebraic
  representation of epipolar geometry
• The fundamental matrix satisfies the condition
  that for any pair of corresponding points x?x in
  the two images

55
The fundamental matrix F
F is the unique 3x3 rank 2 matrix that satisfies
xTFx0 for all x?x

1. Transpose if F is fundamental matrix for (P,P),
   then FT is fundamental matrix for (P,P)
2. Epipolar lines lFx lFTx
3. Epipoles on all epipolar lines, thus eTFx0, ?x
   ?eTF0, similarly Fe0
4. F has 7 d.o.f. , i.e. 3x3-1(homogeneous)-1(rank2)
5. F maps from a point x to a line lFx (not
   invertible)

56
The fundamental matrix F

• It can be used for
• Simplifies matching
• Allows to detect wrong matches

57
Estimation of F 8-point algorithm

• The fundamental matrix F is defined by

for any pair of matches x and x in two images.

• Let x(u,v,1)T and x(u,v,1)T,

each match gives a linear equation