Title: Minimal Surfaces for Stereo
1Minimal Surfaces for Stereo
- Chris Buehler, Steven J. Gortler,
- Michael F. Cohen, Leonard McMillan
- MIT, Harvard
- Microsoft Research, MIT
2Motivation
- Optimization based stereo over greed based
- No early commitment
- Enforce interactions each pixel sees unique item
- Penalize interactions non-smoothness
3Stereo by Optimization
- Early algorithms dynamic programming
- (Baker 81, Belumeur Mumford 92)
- Dont generalize beyond 2 camera, single scanline
4Stereo by Optimization
- Recent Algorithms iterative a-expansion
- ( Kolmogorov Zabih 01)
- very general
- NP-Complete
- Local opt found quickly in practice
- Recent algorithms MIN-CUT
- (Roy Cox 96, Ishikawa Geiger 98)
- Polynomial time global optimum
- New interpretation to such methods
5Contributions
- Stereo as a discrete minimal surface problem
- Algorithms Polynomial time globally optimal
surface - Using MIN-CUT (Sullivan 90)
- Build from shortest path
- Applications to stereo vision
- Rederive previous MIN-CUT stereo approaches
- New 3-camera stereo formulation (Ayache 88)
6Planar Graph Shortest Path
- Given an embedded planar graph
- faces, edges, vertices
7Planar Graph Shortest Path
- A non negative cost on each edge
57
8Planar Graph Shortest Path
- Two boundary points on the exterior of the
complex
9Planar Graph Shortest Path
- Find minimal curve (collection of edges) with
given boundary
10Planar Graph For stereo
11Algorithms
- Classic Dijkstras
- Works even for non-planar graphs
- Wacky use duality
- But this will generalize to higher dimension
12Duality
13Duality
- face vertex
- edge cross edge
- - same cost
57
14Duality
15Duality
16Cuts
- Cuts of dual graph partitions of dual verts
- Cost sum of dual edges spanning the partition
- MIN-CUT can be found in polynomial time
17Cuts
- Claim Primalization of MIN-CUT will be shortest
path
18Why this works
- Cuts of dual graph partitions of dual verts
19Why this works
- Partition of dual verts partition of primal
faces
20Why this works
- Partition of primal faces primal path
21Why this works
- Cuts in dual correspond to paths in primal
- MIN-CUT in dual corresponds to shortest path in
primal
22Same idea works for surfaces!
23Increasing the dimension
Planar graph verts, edges, faces cost on
edges boundary 2 points on exterior sol min
path
Spacial compex verts, edges, faces, cells
cost on faces boundary loop on exterior
sol min surface
24Increasing the dimension
Planar graph verts, edges, faces boundary
2 points on exterior sol min path
Spacial compex verts, edges, faces, cells
cost on faces boundary loop on exterior
sol min surface
25Increasing the dimension
Planar graph verts, edges, faces boundary
2 points on exterior sol min path
Spacial compex verts, edges, faces, cells
cost on faces boundary loop on exterior
sol min surface
26Dual construction for min surf
- face vertex
- edge cross edge
- cell vertex
- face cross edge
MIN-CUT primalizes to min surf
27Checkpoint
- Solve for minimal paths and surfaces
- MIN-CUT on dual graph
- Apply these algorithms to stereo vision
28Flatland Stereo
Geometric interpretation of Cox et al. 96
pixel
Camera Left
Camera Right
29Flatland Stereo
Geometric interpretation of Cox et al. 96
pixel
Camera Left
Camera Right
30Flatland Stereo
Cost unmatched/discontinuity, ß
Camera Left
Camera Right
31Flatland Stereo
Cost correspondence quality
Camera Left
Camera Right
32Flatland Stereo
33Flatland Stereo
Uniqueness monotonicity solution is
directed path
34Flatland Stereo
Note unmatched pixels also function as
discontinuities
Occlusion, discontinuity
Match
35Flatland to Fatland
Camera Left
Camera Right
36Flatland to Fatland
Camera Left
Camera Right
372 cameras, 3d
382 cameras, 3d
39One Cuboid Among Many
Solve for minimal surface
40Geometric interpretation IG98
41Three Camera
Rectification (Ayache 88)
42Three Camera
43Three Camera
44Three Camera
45Three Camera
46One cuboid
47Dual graph of one cuboid
48One Cuboid Among Many
Solve for minimal surface
49More divisions of middle cell
50More expressive decomposition
51Complexity
- Vertices and edges 20 n d
- n pixels per image
- d max disparity
- Time complexity O((nd)2 log(nd))
- About 1 min
52Results
LL image
RC
KZ01
MS
53LL image
RC
KZ01
MS
54Future
- Application of MS to n cameras
- Monotonicity/oriented manifold enforces more than
uniqueness - see Kolmogorov Zabih (today 1100am)
- Other applications of MS