CSE 291 Seminar Presentation Andrew Cosand ECE CVRR 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 S&L-H Solution Pilu’s Contribution 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 features Gij=e(-rij2/22) 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 = UUT = QQT => Q = U1/2 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)2/2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 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