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Corners+Ransac

# Corners+Ransac - Corner Detection Basic idea Find points...

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Computer Vision : CISC 4/689 Corner Detection Basic idea: Find points where two edges meet—i.e., high gradient in two directions “Cornerness” is undefined at a single pixel, because there’s only one gradient per point Look at the gradient behavior over a small window Categories image windows based on gradient statistics Constant : Little or no brightness change Edge : Strong brightness change in single direction Flow : Parallel stripes Corner/spot : Strong brightness changes in orthogonal directions

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Computer Vision : CISC 4/689 Corner Detection: Analyzing Gradient Covariance Intuitively, in corner windows both I x and I y should be high Can’t just set a threshold on them directly, because we want rotational invariance Analyze distribution of gradient components over a window to differentiate between types from previous slide: The two eigenvectors and eigenvalues ¸ 1 , ¸ 2 of C (Matlab: eig(C) ) encode the predominant directions and magnitudes of the gradient, respectively, within the window Corners are thus where min(¸ 1 , ¸ 2 ) is over a threshold courtesy of Wolfram
Computer Vision : CISC 4/689 Contents Harris Corner Detector Description Analysis Detectors Rotation invariant Scale invariant Affine invariant Descriptors Rotation invariant Scale invariant Affine invariant

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Computer Vision : CISC 4/689 Harris Detector: Mathematics [ ] 2 , ( , ) ( , ) ( , ) ( , ) x y E u v w x y I x u y v I x y = + + - Change of intensity for the shift [ u,v ]: Intensity Shifted  intensity Window  function or Window function w(x,y) = Gaussian 1 in window, 0 outside
Computer Vision : CISC 4/689 Harris Detector: Mathematics [ ] ( , ) , u E u v u v M v   2245     For small shifts [ u,v ] we have a bilinear approximation: 2 2 , ( , ) x x y x y x y y I I I M w x y I I I = where M is a 2 × 2 matrix computed from image derivatives:

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Computer Vision : CISC 4/689 Harris Detector: Mathematics [ ] ( , ) , u E u v u v M v   2245     Intensity change in shifting window: eigenvalue analysis λ 1 , λ 2 – eigenvalues of M ( λ max ) -1/2 ( λ min ) -1/2 Ellipse E(u,v) = const If we try every possible orientation n , the max. change in intensity is λ 2
Computer Vision : CISC 4/689 Harris Detector: Mathematics λ 1 λ 2 “Corner” λ 1 and λ 2 are large, λ 1 ~ λ 2 ; E increases in all directions λ 1 and λ 2 are small; E is almost constant in all directions “Edge” λ 1 >> λ 2 “Edge” λ 2 >> λ 1 “Flat” region Classification of image points using eigenvalues of M :

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Computer Vision : CISC 4/689 Harris Detector: Mathematics Measure of corner response: ( 29 2 det trace R M k M = - 1 2 1 2 det trace M M λ λ λ λ = = + ( k – empirical constant, k = 0.04-0.06)
Computer Vision : CISC 4/689 Harris Detector: Mathematics λ 1 λ 2 “Corner” “Edge” “Edge” “Flat” R depends only on eigenvalues of M R is large for a corner R is negative with large magnitude for an edge | R | is small for a flat region R > 0 R < 0 R < 0 |R| small

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Computer Vision : CISC 4/689 Harris Detector
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• Spring '08
• AMINI
• Computer vision, Scale-invariant feature transform, Blob detection, corner detection, Interest point detection, Harris corner detector

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