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An Algorithm for Associating the Features of Two Images / G. L. Scott, H. C. Longuet-Higgins A direct method for stereo correspondence based on singular value decomposition / M. Pilu. CSE 291 Seminar Presentation Andrew Cosand ECE CVRR. Outline. Correspondence Problem Examples Discrepancy

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Cse 291 seminar presentation andrew cosand ece cvrr

An Algorithm for Associating the Features of Two Images / G. L. Scott, H. C. Longuet-HigginsA direct method for stereo correspondence based on singular value decomposition / M. Pilu

CSE 291 Seminar Presentation

Andrew Cosand

ECE CVRR


Outline
Outline L. Scott, H. C. Longuet-Higgins

Correspondence Problem

  • Examples

  • Discrepancy

    S&L-H Solution

  • Distance Measure

  • Singular Value Decomposition

  • Relation to Kernel Trick

    Pilu’s Contribution


Correspondence problem
Correspondence Problem L. Scott, H. C. Longuet-Higgins

Which features in image A correspond to features in image B?


Correspondence problem1
Correspondence Problem L. Scott, H. C. Longuet-Higgins

This task is trivial for humans, but difficult for computers.


Correspondence problem2
Correspondence Problem L. Scott, H. C. Longuet-Higgins

  • Used for stereo image pairs & motion images.

  • Feature correspondence should exhibit Similarity, Proximity and Exclusivity.

  • Complexity is combinatorial with number of features to compare.


Stereo imaging
Stereo Imaging L. Scott, H. C. Longuet-Higgins

Trinocular camera captures 3 images, horizontally and vertically offset.


Stereo imaging1
Stereo Imaging L. Scott, H. C. Longuet-Higgins

Feature correspondence is used to extract depth information from stereo images

  • Distances between cameras are known.

  • Distances between the same feature in different images is determined.

  • Distance from cameras to actual object can be calculated.


Motion tracking
Motion Tracking L. Scott, H. C. Longuet-Higgins

Corresponding features are tracked through sequential images to determine object or camera motion.

Compound Motion

Object Motion Only


Local vs global
Local vs. Global L. Scott, H. C. Longuet-Higgins


Discrepancy
Discrepancy L. Scott, H. C. Longuet-Higgins

Small scale discrepancy constrains corresponding features to be close together.

  • Slow object movement, slight camera motion, narrow baseline stereo

    Large scale discrepancy allows widely separated features.

  • Fast object movement, large camera motion, wide baseline stereo


Ternus
Ternus L. Scott, H. C. Longuet-Higgins


Ternus1
Ternus L. Scott, H. C. Longuet-Higgins


Ternus2
Ternus L. Scott, H. C. Longuet-Higgins


Achieving good global correspondence
Achieving Good Global Correspondence L. Scott, H. C. Longuet-Higgins

Requires relationships between points

  • The inner product of x,y coordinates yields a deficient feature space. (Also location biased)

  • Gaussian weighted distance better captures the spatial relationships between points (location and proximity).

  • S&LH provides superior sphered (decorrelated) relationship.

  • Pilu adds similarity relationship.


Scott longuet higgins
Scott & Longuet-Higgins L. Scott, H. C. Longuet-Higgins

Define a distance metric between features

Gij=e(-rij2/22)

Create matrix of relationships for all possible feature pairs

G11

Gij


S lh distance measure
S&LH Distance Measure L. Scott, H. C. Longuet-Higgins

Gaussian Weighted

  •  scales distance weighting (discrepancy)

  • Analytic with respect to distance, coordinates

  • Decreases monotonically with distance

  • Positive Definite for identical images


Positive definite matrices
Positive Definite Matrices L. Scott, H. C. Longuet-Higgins

  • Comparing identical feature sets yields a symmetric positive definite matrix.

  • Symmetric gets us real eigenvalues.

  • Positive definite has positive eigenvalues (which means real square roots).

  • G = UUT = QQT => Q = U1/2

Matrix Factors

Real

Inner

Product


Singular value decomposition
Singular Value Decomposition L. Scott, H. C. Longuet-Higgins

SVD factors a matrix into the product of two orthogonal matrices and a diagonal matrix of singular values (eigenvalues).

G = TDU, G is m-by-n,

  • T is orthogonal, m-by-m

  • D is diag(1, 2, … p), m-by-n, p=min{m,n}

  • U is orthogonal, n-by-n


Scott longuet higgins1
Scott & Longuet-Higgins L. Scott, H. C. Longuet-Higgins

Use Singular Value Decomposition on matrix. G = TDU


Scott longuet higgins2
Scott & Longuet-Higgins L. Scott, H. C. Longuet-Higgins

Set diagonal elements of D to 1, ‘recover’ relationship matrix.

P = TIU = TU

Eliminating singular matrix rescales data in feature space, essentially sphereing it.


Scott longuet higgins3
Scott & Longuet-Higgins L. Scott, H. C. Longuet-Higgins

Largest feature in row and column indicates mutual best match (correspondence)


Relation to kernel trick
Relation to Kernel Trick L. Scott, H. C. Longuet-Higgins

Gaussian Distance is essentially a kernel

  • Relates to a dot product in infinite dimensionial space.

  • This gives a richer feature space with useful relationships between features.

  • This is why the SVD works here.


Pilu s improvement
Pilu’s Improvement L. Scott, H. C. Longuet-Higgins

  • Rogue features don’t correspond to anything, complicating the process.

  • S&LH only deals with proximity and exclusivity.

  • Similarity constraint can eliminate rogue features, which shouldn’t be similar to anything.


Pilu s improvement1
Pilu’s Improvement L. Scott, H. C. Longuet-Higgins

Modify relationship metric to include gray-level correlation.

Gij = (e-(Cij – 1)2/22) e(-rij2/22)

Gij = ((Cij+1)/2) e(-rij2/22)

  • Adds similarity to feature space (kernel operation).

  • Rogue features can be eliminated because they are not similar to anything.


Results
Results L. Scott, H. C. Longuet-Higgins

  • Achieves globally better feature matches rather than locally good matches.

  • Resistant to rogue points.


Summary
Summary L. Scott, H. C. Longuet-Higgins

  • S&LH essentially maps input to a rich, high dimensional feature space using kernel trick, then uses SVD to determine matches.

  • Pilu improves kernel to achieve better feature space.

  • Combination works well.


References
References L. Scott, H. C. Longuet-Higgins

This presentation drew material from the following sources

  • S. Belonge, Notes on Spectral Correspondence

  • M. Pilu, A direct method for stereo correspondence based on singular value decomposition

    • variants

  • G. L. Scott, H. C. Longuet-Higgins, An Algorithm for Associating the Features of Two Images


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