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

S&L-H Solution– Distance Measure– Singular Value Decomposition– Relation to Kernel Trick

Pilu’s Contribution

Correspondence Problem

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

Correspondence Problem

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

Correspondence Problem

• 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

Trinocular camera captures 3 images, horizontally and vertically offset.

Stereo Imaging

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

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

Object Motion Only Compound Motion

Local vs. Global

Discrepancy

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

Ternus

Achieving Good Global Correspondence

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

Define a distance metric between featuresGij=e(-r

ij2/22)

Create matrix of relationships for all possible feature pairs

G11

Gij

S&LH Distance Measure

Gaussian Weighted scales distance weighting (discrepancy)– Analytic with respect to distance, coordinates– Decreases monotonically with distance– Positive Definite for identical images

Positive Definite Matrices

• 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 = UUT = QQT => Q = U1/2

Matrix Factors Inner Product

Real

Singular Value Decomposition

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-Higgins

Use Singular Value Decomposition on matrix. G = TDU

Scott & 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-Higgins

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

Relation to Kernel Trick

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

• 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 Improvement

Modify relationship metric to include gray-level correlation.

Gij = (e-(Cij – 1)2/22) e(-rij2/22)

Gij = ((Cij+1) /2) e(-rij2/22)

– Adds similarity to feature space (kernel operation).– Rogue features can be eliminated because they are not

similar to anything.

Results

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

• Resistant to rogue points.

Summary

• 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

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