Lec10 alignment
- 2. Image alignment: Motivation Panorama stitching Recognition of object instances
- 3. Image alignment: Challenges Small degree of overlap Occlusion, clutter
- 4. Image alignment • Two broad approaches: • Direct (pixel-based) alignment – Search for alignment where most pixels agree • Feature-based alignment – Search for alignment where extracted features agree – Can be verified using pixel-based alignment
- 5. Alignment as fitting • Previous lectures: fitting a model to features in one image ∑ i i M x ) , ( residual Find model M that minimizes M xi
- 6. Alignment as fitting • Previous lectures: fitting a model to features in one image • Alignment: fitting a model to a transformation between pairs of features (matches) in two images ∑ i i M x ) , ( residual ∑ ′ i i i x x T ) ), ( ( residual Find model M that minimizes Find transformation T that minimizes M xi T xi xi '
- 8. Feature-based alignment outline • Extract features
- 9. Feature-based alignment outline • Extract features • Compute putative matches
- 10. Feature-based alignment outline • Extract features • Compute putative matches • Loop: • Hypothesize transformation T (small group of putative matches that are related by T)
- 11. Feature-based alignment outline • Extract features • Compute putative matches • Loop: • Hypothesize transformation T (small group of putative matches that are related by T) • Verify transformation (search for other matches consistent with T)
- 12. Feature-based alignment outline • Extract features • Compute putative matches • Loop: • Hypothesize transformation T (small group of putative matches that are related by T) • Verify transformation (search for other matches consistent with T)
- 13. 2D transformation models • Similarity (translation, scale, rotation) • Affine • Projective (homography)
- 14. Let’s start with affine transformations • Simple fitting procedure (linear least squares) • Approximates viewpoint changes for roughly planar objects and roughly orthographic cameras • Can be used to initialize fitting for more complex models
- 15. Fitting an affine transformation • Assume we know the correspondences, how do we get the transformation? ) , ( i i y x ′ ′ ) , ( i i y x ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ′ ′ 2 1 4 3 2 1 t t y x m m m m y x i i i i ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ′ ′ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ L L L L i i i i i i y x t t m m m m y x y x 2 1 4 3 2 1 1 0 0 0 0 1 0 0
- 16. Fitting an affine transformation • Linear system with six unknowns • Each match gives us two linearly independent equations: need at least three to solve for the transformation parameters ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ′ ′ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ L L L L i i i i i i y x t t m m m m y x y x 2 1 4 3 2 1 1 0 0 0 0 1 0 0
- 17. What if we don’t know the correspondences? ?
- 18. What if we don’t know the correspondences? • Need to compare feature descriptors of local patches surrounding interest points ( ) ( ) = ? feature descriptor feature descriptor ?
- 19. Feature descriptors • Assuming the patches are already normalized (i.e., the local effect of the geometric transformation is factored out), how do we compute their similarity? • Want invariance to intensity changes, noise, perceptually insignificant changes of the pixel pattern
- 20. • Simplest descriptor: vector of raw intensity values • How to compare two such vectors? • Sum of squared differences (SSD) – Not invariant to intensity change • Normalized correlation – Invariant to affine intensity change Feature descriptors ( ) ∑ − = i i i v u v u 2 ) , SSD( ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − = ∑ ∑ ∑ j j j j i i i v v u u v v u u v u 2 2 ) ( ) ( ) )( ( ) , ( ρ
- 21. Feature descriptors • Disadvantage of patches as descriptors: • Small shifts can affect matching score a lot • Solution: histograms 0 2 π
- 22. • Descriptor computation: • Divide patch into 4x4 sub-patches • Compute histogram of gradient orientations (8 reference angles) inside each sub-patch • Resulting descriptor: 4x4x8 = 128 dimensions Feature descriptors: SIFT David G. Lowe. "Distinctive image features from scale-invariant keypoints.” IJCV 60 (2), pp. 91-110, 2004.
- 23. • Descriptor computation: • Divide patch into 4x4 sub-patches • Compute histogram of gradient orientations (8 reference angles) inside each sub-patch • Resulting descriptor: 4x4x8 = 128 dimensions • Advantage over raw vectors of pixel values • Gradients less sensitive to illumination change • “Subdivide and disorder” strategy achieves robustness to small shifts, but still preserves some spatial information Feature descriptors: SIFT David G. Lowe. "Distinctive image features from scale-invariant keypoints.” IJCV 60 (2), pp. 91-110, 2004.
- 24. Feature matching ? • Generating putative matches: for each patch in one image, find a short list of patches in the other image that could match it based solely on appearance
- 25. Feature matching • Generating putative matches: for each patch in one image, find a short list of patches in the other image that could match it based solely on appearance • Exhaustive search – For each feature in one image, compute the distance to all features in the other image and find the “closest” ones (threshold or fixed number of top matches) • Fast approximate nearest neighbor search – Hierarchical spatial data structures (kd-trees, vocabulary trees) – Hashing
- 26. Feature space outlier rejection • How can we tell which putative matches are more reliable? • Heuristic: compare distance of nearest neighbor to that of second nearest neighbor • Ratio will be high for features that are not distinctive • Threshold of 0.8 provides good separation David G. Lowe. "Distinctive image features from scale-invariant keypoints.” IJCV 60 (2), pp. 91-110, 2004.
- 27. Reading David G. Lowe. "Distinctive image features from scale- invariant keypoints.” IJCV 60 (2), pp. 91-110, 2004.
- 28. Dealing with outliers • The set of putative matches contains a very high percentage of outliers • Heuristics for feature-space outlier rejection • Geometric fitting strategies: • RANSAC • Incremental alignment • Hough transform • Hashing
- 29. Strategy 1: RANSAC RANSAC loop: 1. Randomly select a seed group of matches 2. Compute transformation from seed group 3. Find inliers to this transformation 4. If the number of inliers is sufficiently large, re-compute least-squares estimate of transformation on all of the inliers Keep the transformation with the largest number of inliers
- 30. RANSAC example: Translation Putative matches
- 31. RANSAC example: Translation Select one match, count inliers
- 32. RANSAC example: Translation Select one match, count inliers
- 33. RANSAC example: Translation Select translation with the most inliers
- 34. Problem with RANSAC • In many practical situations, the percentage of outliers (incorrect putative matches) is often very high (90% or above) • Alternative strategy: restrict search space by using strong locality constraints on seed groups and inliers • Incremental alignment
- 35. Strategy 2: Incremental alignment • Take advantage of strong locality constraints: only pick close-by matches to start with, and gradually add more matches in the same neighborhood S. Lazebnik, C. Schmid and J. Ponce, “Semi-local affine parts for object recognition,” BMVC 2004.
- 36. Strategy 2: Incremental alignment • Take advantage of strong locality constraints: only pick close-by matches to start with, and gradually add more matches in the same neighborhood
- 37. Strategy 2: Incremental alignment • Take advantage of strong locality constraints: only pick close-by matches to start with, and gradually add more matches in the same neighborhood
- 38. Strategy 2: Incremental alignment • Take advantage of strong locality constraints: only pick close-by matches to start with, and gradually add more matches in the same neighborhood
- 39. A Strategy 2: Incremental alignment • Take advantage of strong locality constraints: only pick close-by matches to start with, and gradually add more matches in the same neighborhood
- 40. Strategy 3: Hough transform • Recall: Generalized Hough transform B. Leibe, A. Leonardis, and B. Schiele, Combined Object Categorization and Segmentation with an Implicit Shape Model, ECCV Workshop on Statistical Learning in Computer Vision 2004 model visual codeword with displacement vectors test image
- 41. Strategy 3: Hough transform • Suppose our features are adapted to scale and rotation • Then a single feature match provides an alignment hypothesis (translation, scale, orientation) David G. Lowe. "Distinctive image features from scale-invariant keypoints.” IJCV 60 (2), pp. 91-110, 2004. model
- 42. Strategy 3: Hough transform • Suppose our features are adapted to scale and rotation • Then a single feature match provides an alignment hypothesis (translation, scale, orientation) • Of course, a hypothesis obtained from a single match is unreliable • Solution: let each match vote for its hypothesis in a Hough space with very coarse bins David G. Lowe. "Distinctive image features from scale-invariant keypoints.” IJCV 60 (2), pp. 91-110, 2004. model
- 43. Hough transform details (D. Lowe’s system) • Training phase: For each model feature, record 2D location, scale, and orientation of model (relative to normalized feature frame) • Test phase: Let each match between a test and a model feature vote in a 4D Hough space • Use broad bin sizes of 30 degrees for orientation, a factor of 2 for scale, and 0.25 times image size for location • Vote for two closest bins in each dimension • Find all bins with at least three votes and perform geometric verification • Estimate least squares affine transformation • Use stricter thresholds on transformation residual • Search for additional features that agree with the alignment David G. Lowe. "Distinctive image features from scale-invariant keypoints.” IJCV 60 (2), pp. 91-110, 2004.
- 44. Strategy 4: Hashing • Make each image feature into a low-dimensional “key” that indexes into a table of hypotheses model hash table
- 45. Strategy 4: Hashing • Make each image feature into a low-dimensional “key” that indexes into a table of hypotheses • Given a new test image, compute the hash keys for all features found in that image, access the table, and look for consistent hypotheses model hash table test image
- 46. Strategy 4: Hashing • Make each image feature into a low-dimensional “key” that indexes into a table of hypotheses • Given a new test image, compute the hash keys for all features found in that image, access the table, and look for consistent hypotheses • This can even work when we don’t have any feature descriptors: we can take n-tuples of neighboring features and compute invariant hash codes from their geometric configurations A B C D
- 47. Application: Searching the sky http://www.astrometry.net/
- 48. Beyond affine transformations • Homography: plane projective transformation (transformation taking a quad to another arbitrary quad)
- 49. Homography • The transformation between two views of a planar surface • The transformation between images from two cameras that share the same center
- 50. Fitting a homography • Recall: homogenenous coordinates Converting to homogenenous image coordinates Converting from homogenenous image coordinates
- 51. Fitting a homography • Recall: homogenenous coordinates • Equation for homography: Converting to homogenenous image coordinates Converting from homogenenous image coordinates ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ′ ′ 1 1 33 32 31 23 22 21 13 12 11 y x h h h h h h h h h y x λ
- 52. Fitting a homography • Equation for homography: i T T T i i x h h h x H x ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = = ′ 3 2 1 λ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ′ ′ 1 1 33 32 31 23 22 21 13 12 11 i i i i y x h h h h h h h h h y x λ 0 = × ′ i i x H x ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ′ − ′ ′ − − ′ = × ′ i T i i T i i T i i T i T i T i i i y x x y x h x h x h x h x h x h x H x 1 2 3 1 2 3 0 0 0 0 3 2 1 = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ′ ′ − ′ − ′ − h h h x x x x x x T T i i T i i T i i T T i T i i T i T x y x y 3 equations, only 2 linearly independent 9 entries, 8 degrees of freedom (scale is arbitrary)
- 53. Direct linear transform • H has 8 degrees of freedom (9 parameters, but scale is arbitrary) • One match gives us two linearly independent equations • Four matches needed for a minimal solution (null space of 8x9 matrix) • More than four: homogeneous least squares 0 0 0 0 0 3 2 1 1 1 1 1 1 1 = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ′ − ′ − ′ − ′ − h h h x x x x x x x x T n n T T n T n n T n T T T T T T T x y x y L L L 0 = h A
- 55. Recognizing panoramas M. Brown and D. Lowe, “Recognizing Panoramas,” ICCV 2003. • Given contents of a camera memory card, automatically figure out which pictures go together and stitch them together into panoramas http://www.cs.ubc.ca/~mbrown/panorama/panorama.html
- 56. Issues in alignment-based applications • Choosing the geometric alignment model • Tradeoff between “correctness” and robustness (also, efficiency) • Choosing the descriptor • “Rich” imagery (natural images): high-dimensional patch-based descriptors (e.g., SIFT) • “Impoverished” imagery (e.g., star fields): need to create invariant geometric descriptors from k-tuples of point-based features • Strategy for finding putative matches • Small number of images, one-time computation (e.g., panorama stitching): brute force search • Large database of model images, frequent queries: indexing or hashing • Heuristics for feature-space pruning of putative matches
- 57. Issues in alignment-based applications • Choosing the geometric alignment model • Choosing the descriptor • Strategy for finding putative matches • Hypothesis generation strategy • Relatively large inlier ratio: RANSAC • Small inlier ratio: locality constraints, Hough transform • Hypothesis verification strategy • Size of consensus set, residual tolerance depend on inlier ratio and expected accuracy of the model • Possible refinement of geometric model • Dense verification
- 58. Next time: Single-view geometry