# An Algorithm for Associating the Features of Two Images

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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
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.
Scott & Longuet-Higgins
Define a distance metric
between features
Gij=e(-r 2/22)
ij

Create matrix of           G11

relationships for all
Gij
possible feature pairs
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 = UUT = QQT => Q = U1/2
Matrix Factors   Inner        Real
Product
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)2/22)   e (-rij2/22)

Gij = ((Cij+1) /2) e(-r 2/22)ij

– 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

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