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Minimal Surfaces for Stereo. Chris Buehler, Steven J. Gortler, Michael F. Cohen, Leonard McMillan MIT, Harvard Microsoft Research, MIT. Motivation. Optimization based stereo over greed based No early commitment Enforce interactions: each pixel sees unique item

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Minimal surfaces for stereo

Minimal Surfaces for Stereo

Chris Buehler, Steven J. Gortler,

Michael F. Cohen, Leonard McMillan

MIT, Harvard

Microsoft Research, MIT


Motivation
Motivation

  • Optimization based stereo over greed based

    • No early commitment

    • Enforce interactions: each pixel sees unique item

    • Penalize interactions: non-smoothness


Stereo by optimization
Stereo by Optimization

  • Early algorithms: dynamic programming

    • (Baker ‘81, Belumeur & Mumford ‘92…)

    • Don’t generalize beyond 2 camera, single scanline


Stereo by optimization1
Stereo 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


Contributions
Contributions

  • 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)


Planar graph shortest path
Planar Graph Shortest Path

  • Given: an embedded planar graph

    • faces, edges, vertices


Planar graph shortest path1
Planar Graph Shortest Path

  • A non negative cost on each edge

57


Planar graph shortest path2
Planar Graph Shortest Path

  • Two boundary points on the exterior of the complex


Planar graph shortest path3
Planar Graph Shortest Path

  • Find minimal curve: (collection of edges) with given boundary


Planar graph for stereo

Selected Match

Selected Occlusion

Camera Left

Camera Right

Planar Graph For stereo


Algorithms
Algorithms

  • Classic: Dijkstra’s

    • Works even for non-planar graphs

  • Wacky: use duality

    • But this will generalize to higher dimension



Minimal surfaces for stereo

Duality

  • face vertex

  • edgecross edge

    • - same cost

57


Minimal surfaces for stereo

Duality

  • Split exterior


Minimal surfaces for stereo

Sink

Source

Source

Duality

  • Add source and sink


Minimal surfaces for stereo

Sink

Source

Cuts

  • Cuts of dual graph = partitions of dual verts

  • Cost = sum of dual edges spanning the partition

  • MIN-CUT can be found in polynomial time


Minimal surfaces for stereo

Sink

Source

Cuts

  • Claim: Primalization of MIN-CUT will be shortest path


Why this works

Sink

Sink

Source

Source

Why this works

  • Cuts of dual graph = partitions of dual verts


Why this works1

Sink

Sink

Source

Source

Why this works

  • Partition of dual verts = partition of primal faces


Why this works2

Sink

Sink

Sink

Source

Source

Source

Why this works

  • Partition of primal faces = primal path


Why this works3

Sink

Sink

Sink

Source

Source

Source

Why this works

  • Cuts in dual correspond to paths in primal

  • MIN-CUT in dual corresponds to shortest path in primal



Increasing the dimension
Increasing 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


Increasing the dimension1
Increasing 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


Increasing the dimension2
Increasing 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


Minimal surfaces for stereo

Sink

Source

Dual construction for min surf

  • face vertex

  • edgecross edge

  • cell vertex

  • face cross edge

MIN-CUT primalizes to min surf


Checkpoint
Checkpoint

  • Solve for minimal paths and surfaces

    • MIN-CUT on dual graph

  • Apply these algorithms to stereo vision


Flatland stereo
Flatland Stereo

Geometric interpretation of

Cox et al. 96

pixel

Camera Left

Camera Right


Flatland stereo1
Flatland Stereo

Geometric interpretation of

Cox et al. 96

pixel

Camera Left

Camera Right


Flatland stereo2
Flatland Stereo

Cost: unmatched/discontinuity, β

Camera Left

Camera Right


Flatland stereo3
Flatland Stereo

Cost: correspondence quality

Camera Left

Camera Right


Flatland stereo4

Camera Left

Camera Right

Flatland Stereo


Flatland stereo5

Match

Unmatched

Camera Left

Camera Right

Flatland Stereo

Uniqueness & monotonicity solution is directed path


Flatland stereo6

Camera Left

Camera Right

Flatland Stereo

Note: unmatched pixels also function as discontinuities

Occlusion,

discontinuity

Match


Flatland to fatland
Flatland to Fatland

Camera Left

Camera Right


Flatland to fatland1
Flatland to Fatland

Camera Left

Camera Right




One cuboid among many
One Cuboid Among Many

Solve for minimal surface



Three camera
Three Camera

Rectification

(Ayache ‘88)








One cuboid among many1
One Cuboid Among Many

Solve for minimal surface




Complexity
Complexity

  • Vertices and edges: 20 n d

    • n: pixels per image

    • d: max disparity

  • Time complexity O((nd)2 log(nd))

  • About 1 min


Results
Results

LL image

RC

KZ01

MS


Minimal surfaces for stereo

LL image

RC

KZ01

MS


Future
Future

  • Application of MS to n cameras

    • Monotonicity/oriented manifold enforces more than uniqueness

    • see Kolmogorov & Zabih (today 11:00am)

  • Other applications of MS