Geometry and Matching of Curves - University of Washington
Interest Operators Find interesting pieces of the image Multiple possible uses image matching stereo pairs tracking in videos creating panoramas object recognition 1 Goal: Local invariant photometric descriptors
() local descriptor Local : robust to occlusion/clutter + no segmentation Photometric : distinctive Invariant : to image transformations + illumination2 changes History Matching Matching based on correlation alone Matching based on line segments Not very discriminating (why?)
Solution : matching with interest points & correlation [ A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry, Z. Zhang, R. Deriche, O. Faugeras and Q. Luong, Artificial Intelligence 1995 ] 3 Approach Extraction of interest points with the Harris detector Comparison of points with cross-correlation Verification with the fundamental matrix (later in the course) 4
Harris detector Interest points extracted with Harris (~ 500 points) 5 Cross-correlation matching Initial matches (188 pairs) 6 Global constraints Robust estimation of the fundamental matrix
99 inliers 89 outliers 7 Summary of the approach Very good results in the presence of occlusion and clutter local information
discriminant greyvalue information robust estimation of the global relation between images for limited view point changes Solution for more general view point changes wide baseline matching (different viewpoint, scale and rotation) local invariant descriptors based on greyvalue information 8 History - Recognition Problems : occlusion, clutter, image transformations, distinctiveness Solution : recognition with local photometric invariants [ Local greyvalue invariants for image retrieval,
C. Schmid and R. Mohr, PAMI 1997 ] 9 Approach () local descriptor 1) Extraction of interest points (characteristic locations) 2) Computation of local descriptors 3) Determining correspondences 4) Selection of similar images
10 Interest points Geometric features repeatable under transformations 2D characteristics of the signal high informational content Comparison of different detectors [Schmid98] Harris detector 11 Harris detector
Based on the idea of auto-correlation Important difference in all directions => interest point 12 Background: Moravec Corner Detector take a window w in the image shift it in four directions w (1,0), (0,1), (1,1), (-1,1) compute a difference for each compute the min difference at
each pixel local maxima in the min image are the corners E(x,y) = w(u,v) |I(x+u,y+v) I(u,v)|2 u,v in w 13 Shortcomings of Moravec Operator Only tries 4 shifts. Wed like to consider all shifts. Uses a discrete rectangular window. Wed like to use a smooth circular (or later elliptical) window. Uses a simple min function. Wed like to characterize variation with respect to direction.
Result: Harris Operator 14 Harris detector Auto-correlation function for a point ( x, y ) and a shift (x, y ) f ( x, y ) 2 ( I ( x , y
) I ( x x , y y )) k k
k k ( xk , y k )W Discrete shifts can be avoided with the auto-correlation matrix what is this? x with I ( xk x, yk y ) I ( xk , yk ) ( I x ( xk , yk ) I y ( xk , yk )) y x
f ( x, y ) I x ( xk , yk ) I y ( xk , yk ) ( xk , y k )W y 15 2 16 Harris detector 2 ( I (
x , y )) x k k x y ( xk , yk )W I x ( xk , y k ) I y ( xk , y k ) ( xk , yk )W I ( x , y ) I ( x , y ) x
( I ( x , y )) y x ( xk , yk )W k k y k
k 2 y k k ( xk , y k )W Auto-correlation matrix M
17 Harris detection Auto-correlation matrix captures the structure of the local neighborhood measure based on eigenvalues of M which form a rotationally invariant descriptor. 2 strong eigenvalues => interest point 1 strong eigenvalue => contour 0 eigenvalue => uniform region Interest point detection threshold on the eigenvalues
local maximum for localization 18 Some Details from the Harris Paper Let and be the two eigenvalues Tr(M) = + Det(M) =
Response R = Det(M) k Tr(M) R is positive for corners, - for edges, and small for flat regions Select corner pixels that are 8-way local maxima Trace and determinant are easy to compute. 19 Determining correspondences () ?
= () Vector comparison using a distance measure What are some suitable distance measures? 20 Some Matching Results 21 Summary of the approach
Very good results in the presence of occlusion and clutter local information discriminant greyvalue information invariance to image rotation and illumination Not invariance to scale and affine changes Solution for more general view point changes local invariant descriptors to scale and rotation extraction of invariant points and regions 22
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