1. Estimating Visual-Motor Functions CMPUT 610 2001 Martin Jagersand

2. Recall: Visual Servoing Observed features: Motor variables: Local linear model: Visual servoing steps: 1 Solve: 2 Update:

3. Find J Method 1: Test movements along basis Remember: J is unknown m by n matrix Assume movements Finite difference:

4. Find J Method 2: Secant Constraints Constraint along a line: Defines m equations Collect n arbitrary, but different measures y Solve for J

5. Find J Method 3: Recursive Secant Constraints Based on initial J and one measure pair Adjust J s.t. Rank 1 update: Consider rotated coordinates: – Update same as finite difference for n orthogonal moves

6. Spline model of underlying non- linear function Over time acquires several Jacobians J Each J a hyperplane Collection of J’s form a (sparse) piecewise linear spline

7. Jacobian based visual model Assume m>>n All visual change restricted to n freedoms by: 1. Can predict visual change 2. Can also parameterize x visually

8. Affine model Affine basis Image projection of origin: Image basis: e1e1 e2e2 e3e3 O

9. Find affine coordinates Observe (track) y through time Solve an equation system to find q Reprojection: Have q,want y e1e1 e2e2 e3e3 O q

10. Relation Affine – Jacobian image models Rewrite affine model

11. Composite affine and Jacobian model Chain the affine and Jacobian model Represents rigid objects in arbitrary motor frame

12. Transforms Affine-Jacobian model Measurement matrix Affine coordinate equation: