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# CSE 291 Seminar Presentation Andrew Cosand ECE CVRR - PowerPoint PPT Presentation

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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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 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 L. Scott, H. C. Longuet-Higgins

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

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

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

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 L. Scott, H. C. Longuet-Higgins

Trinocular camera captures 3 images, horizontally and vertically offset.

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 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 L. Scott, H. C. Longuet-Higgins

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 L. Scott, H. C. Longuet-Higgins

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

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

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.

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

Use Singular Value Decomposition on matrix. G = TDU

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

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

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 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 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 L. Scott, H. C. Longuet-Higgins

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

• Resistant to rogue points.

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 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