Slide a window along the input arc sequence S. Least-squares estimate. σ 2. σ Estimate 1. Statistically test the difference between θ 1 and θ 2

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1 Corner Detection 2D Image Features Corners are important two dimensional features. Two dimensional image features are interesting local structures. They include junctions of dierent types Slide 3 They can concisely represent object shapes, e therefore playing an important role in matching, like 'Y', 'T', 'X', and 'L'. Much of the work on 2D pattern recognition, robotics, and mensuration. features focuses on junction 'L', aks, corners. Previous Research Corner Corner detection from the underlying gray scale e2 images. Slide 4 Corners are the intersections of two edges of suciently dierent orientations. Corner detection from binary edge images (digital arcs).

2 Corner Detection from Gray Scale Image I Corners are located in the region with large intensity variations in every direction. Let I c and I r be image gradients in horizontal and vertical directions, we can dened a matrix C as in two directions Slide 7 e5 2 P I 2 c 4 P Ic I r P Ic I r P I 2 r 3 C = 5 where the sums are taken over a small neighborhood Compute the eigenvalue of C, and 2. If the minimum of the two eigenvalues is larger than a Slide 8 e6 threshold, the the point is declared as a corner. It is good for detecting corners with orthogonal edges.

3 in four directions Corner Detection from Gray Scale Image II compute local autocorrelation in four directions Fit the image intensities of a small neighborhood with and take the local lowest result as the measure of Slide e9 interest. Image response at each point is a local quadratic or cubic facet surface. Look for saddle thresholded to determine whether the point is a points by calculating image Gaussian curvature (the product of two principle curvatures, see appendix A 5). corner or not. Corner Detection from Gray Scale Image II Assume corners are formed by two edges of suciently Corner Detection from Gray Scale Image II dierent orientations, in a N N neighborhood, Saddle points are points with zero gradient, and a max in Slide 2 compute the direction of each point and then construct 0 one direction but a min in the other. Dierent methods are a histogram of edgel orientations. A pixel point is used to detect saddle points. declared as a corner if two distinctive peaks are observed in the histogram.

4 ^y n jn = :::Ng, statistically A Corner Detection from Digital Arcs Criteria maximum curvature. Slide 5 3 deection angle. maximum deviation. total tting errors. A Statistical Approach Corner Detection from Digital Arcs Problem Statement Objective Given an arc sequence Given a list of connected edge points (a Slide 6 0 ^x n 4 digital arc) resulted from an edge detection, S = f identify arc points that partition the digital arc determine the arc points along S that are most into maximum arc subsequences. likely corner points.

5 Analytically compute 2 and 2 2, the variances of ^ and ^ 2. Approach Overview (cont'd) Slide a window along the input arc sequence S. Estimate ^ and ^ 2, the orientations of the two arc subsequences located to the right and left of the window center, via least-squares line tting. Slide 8 Overview Approach Input arc sequence S Perform a hypothesis test to statistically test the Slide a window along S dierence between ^ and ^ 2. Least-squares estimate θ θ and 2 σ Estimate 2 σ 2 and 2 Slide 7 Statistically test the difference between θ and θ 2 P-value > α? Yes Not a corner No A corner Details of the Proposed Approach Noise and Corner Models Slide 9 Covariance Propagation Hypothesis Testing

6 y n ^ y n H 0 : 2 < 0 H : 2 0 n = ::: N A ^ 2,and 0 a threshold. Noise Model Given an observed arc sequence S = f(^x n ^y n ) t jn = :::Ng, it is assumed that (^x n ^y n ) result from random perturbations to the ideal points (x n y n ), lying on a line x n cos + y n sin ; =0, through the following noise model: Slide ^ x n 0 x n = A 0 n cos + A sin where n are iid as N(0, 2 ). Xcos ^θ + Ysin^θ = ^ ρ Σ θ ^ ρ^, m th point Moving window Corner Model digital arc segment Xcos^θ + Ysin^θ 2 2 = ^ρ 2 Σ^ θ ^ 2, ρ 2 Covariance Propagation Problem statement Slide 23 2 Analytically estimate, the covariance matrix of least-squares estimate ^ =(^ ^) t,in terms of the input covariance matrix X. where ^ 2 = j^ ; ^ 2 j, 2 is the population mean of

7 X g 2 ( X)= F g(x ) = ); ( ( )tx X : +p x2 + y2 ; 2 if y cos x sin k = N NX NX k n (k n ; k ) 2 Geometric Interpretation of k Covariance Propagation (cont.) Covariance Propagation (cont.) From Haralick's covariance propagation theory, dene Dene 8 < F ( ^ ^X) = k = NX n= (^x n cos^ +^y n sin ^ ; ^) 2 x 2 + y 2 ; 2 ; p otherwise 24 and Slide 26 and n= then g(x ) 2 k = n= X g(x ) ( g(x ) )[( ); ] t X 25 Slide 27

8 k 2 k + 2 k 2 k N H 0 : 2 < 0 H : 2 0 ^ 2 2 ^ 2 +^ 2 2 T 2 2 Y Hypothesis Testing (x,y) k where 0 is an angular threshold and 2 is the population mean of RV ^ 2 = j^ ; ^ 2 j. 28 Slide 30 Let X T = Under null hypothesis Covariance Propagation (cont.) 29 X k = k 2 k A if P (T ) <, then a corner else not a corner. Slide 3

9 32 Corner Detection Example 33