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
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
     Correspondence Problem
• Used for stereo image pairs & motion
• 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
             Motion Tracking
Corresponding features are tracked through
 sequential images to determine object or
 camera motion.

    Object Motion Only      Compound Motion
Local vs. Global
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
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)
   – Pilu adds similarity relationship.
      Scott & Longuet-Higgins
Define a distance metric
 between features
  Gij=e(-r 2/22)

Create matrix of           G11

  relationships for all
  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
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
  – 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
• Similarity constraint can eliminate rogue
  features, which shouldn’t be similar to
              Pilu’s Improvement
Modify relationship metric to include gray-level

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.
• Achieves globally better feature matches
  rather than locally good matches.
• Resistant to rogue points.
• 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.
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