1. Linearizing (assuming small (u,v)): Brightness Constancy Equation: The Brightness Constraint Where:),(),(yxJyxII t  Each pixel provides 1 equation in 2 unknowns (u,v). Insufficient info. Another constraint: Global Motion Model Constraint

2. Camera induced image motion = + Independent motions = 3D Camera motion + 3D Scene structure + Independent motions The 2D/3D Dichotomy Image motion = 2D techniques 3D techniques Singularities in “2D scenes” Do not model “3D scenes”  Requires prior model selection

3. Global Motion Models 2D Models: Affine Quadratic Homography (Planar projective transform) 3D Models: Rotation, Translation, 1/Depth Instantaneous camera motion models Essential/Fundamental Matrix Plane+Parallax  Relevant when camera is translating, scene is near, with depth variations.  Relevant for: *Airborne video (distant scene) * Remote Surveillance (distant scene) * Camera on tripod (pure Zoom/Rotation) * 2D Models are easier to estimate than 3D models (much fewer unknowns  numerically more stable). * 2D models provide dense correspondences.

4. Example: Affine Motion Substituting into the B.C. Equation: Each pixel provides 1 linear constraint in 6 global unknowns (minimum 6 pixels necessary) Least Square Minimization (over all pixels): Every pixel contributes  Confidence-weighted regression

5. Example: Affine Motion Differentiating w.r.t. a 1, …, a 6 and equating to zero  6 linear equations in 6 unknowns: Summation is over all the pixels in the image!

6. image I image J JwJw warp refine + Pyramid of image JPyramid of image I image I image J Coarse-to-Fine Estimation u=10 pixels u=5 pixels u=2.5 pixels u=1.25 pixels ==> small u and v... Parameter propagation:

7. Quadratic – instantaneous approximation to planar motion Other 2D Motion Models Projective – exact planar motion (Homography H)

8. Panoramic Mosaic Image Original video clip Generated Mosaic image Alignment accuracy (between a pair of frames): error < 0.1 pixel

9. Original Outliers Original Synthesized Video Removal

10. ORIGINAL ENHANCED Video Enhancement

11. Direct Methods: Methods for motion and/or shape estimation, which recover the unknown parameters directly from measurable image quantities at each pixel in the image. Minimization step: Direct methods: Error measure based on dense measurable image quantities (Confidence-weighted regression; Exploits all available information) Feature-based methods: Error measure based on distances of a sparse set of distinct feature matches (SIFT, HOG,...)

12. Benefits of Direct Methods High subpixel accuracy. Simultaneously estimate matches + transformation  Do not need distinct features for image alignment: Strong locking property.

13. Limitations Limited search range (up to ~10% of the image size). Brightness constancy assumption.

14. Video Indexing and Editing

15. The 2D/3D Dichotomy Image motion = Camera induced motion = + Independent motions = Camera motion + Scene structure + Independent motions 2D techniques 3D techniques Singularities in “2D scenes” Do not model “3D scenes” Source of dichotomy: Camera-centric models (R,T,Z)

16. The Plane+Parallax Decomposition Original SequencePlane-Stabilized Sequence The residual parallax lies on a radial (epipolar) field: epipole

17. Benefits of the P+P Decomposition Eliminates effects of rotation Eliminates changes in camera calibration parameters / zoom Camera parameters: Need to estimate only the epipole. (i.e., 2 unknowns) Image displacements: Constrained to lie on radial lines (i.e., reduces to a 1D search problem)  A result of aligning an existing structure in the image. 1. Reduces the search space:

18. Remove global component which dilutes information ! Translation or pure rotation ??? Benefits of the P+P Decomposition 2. Scene-Centered Representation: Focus on relevant portion of info

19. Benefits of the P+P Decomposition 2. Scene-Centered Representation: Shape = Fluctuations relative to a planar surface in the scene STAB_RUG SEQ

20. - fewer bits, progressive encoding Benefits of the P+P Decomposition 2. Scene-Centered Representation: Shape = Fluctuations relative to a planar surface in the scene Height vs. Depth (e.g., obstacle avoidance) A compact representation global (100) component local [-3..+3] component total distance [97..103] camera center scene Appropriate units for shape

21. Start with 2D estimation (homography). 3D info builds on top of 2D info. 3. Stratified 2D-3D Representation: Avoids a-priori model selection. Benefits of the P+P Decomposition

22. Original sequencePlane-aligned sequenceRecovered shape Dense 3D Reconstruction (Plane+Parallax)

23. Original sequence Plane-aligned sequence Recovered shape

24. Original sequence Plane-aligned sequence Recovered shape Dense 3D Reconstruction (Plane+Parallax)

25. Brightness Constancy constraint P+P Correspondence Estimation The intersection of the two line constraints uniquely defines the displacement. 1. Eliminating Aperture Problem Epipolar line epipole p

26. other epipolar line Epipolar line Multi-Frame vs. 2-Frame Estimation The two line constraints are parallel ==> do NOT intersect 1. Eliminating Aperture Problem p another epipole Brightness Constancy constraint The other epipole resolves the ambiguity !