Corner. Corners are the intersections of two edges of sufficiently different orientations.

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1 2D Image Features Two dimensional image features are interesting local structures. They include junctions of different types like Y, T, X, and L. Much of the work on 2D features focuses on junction L, aks, corners. 1

2 Corner Corners are the intersections of two edges of sufficiently different orientations. 2

3 Corner Detection Corners are important two dimensional features. They can concisely represent object shapes, therefore playing an important role in matching, pattern recognition, robotics, and mensuration. 3

4 Previous Research Corner detection from the underlying gray scale images. Corner detection from binary edge images (digital arcs). 4

5 Corner Detection from Gray Scale Image I Corners are located in the region with large intensity variations. Let I c and I r be image gradients in horizontal and vertical directions, we can defined a matrix C as in two directions C = I 2 c Ic I r Ic I r I 2 r where the sums are taken over a small neighborhood Compute the eigenvalue of C, λ 1 and λ 2.Ifthe minimum of the two eigen values is larger than a 5

6 threshold, the the point is declared as a corner. It is good for detecting corners with orthogonal edges. 6

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8 Corner Detection from Gray Scale Image II Assume corners are formed by two edges of sufficiently different orientations, in a N N neighborhood, compute the direction of each point and then construct a histogram of edgel orientations. A pixel point is declared as a corner if two distinctive peaks are observed in the histogram. 8

9 Corner Detection from Gray Scale Image II Fit the image intensities of a small neighborhood with a local quadratic or cubic facet surface. Look for saddle points by calculating image Gaussian curvature (the product of two principle curvatures, see appendix A 5). 9

10 Corner Detection from Gray Scale Image II Saddle points are points with zero gradient, and a max in one direction but a min in the other. Different methods are used to detect saddle points. 10

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12 Corner Detection from Digital Arcs Objective Given a list of connected edge points (a digital arc) resulted from an edge detection, identify arc points that partition the digital arc into maximum arc subsequences. 12

13 Corner Detection from Digital Arcs Criteria maximum curvature. deflection angle. maximum deviation. total fitting errors. 13

14 Problem Statement A Statistical Approach GivenanarcsequenceS = { ˆx n ŷ n n =1,...N}, statistically determine the arc points along S that are most likely corner points. 14

15 Approach Overview Input arc sequence S Slide a window along S Least-squares estimate θ θ 1 and 2 15 Estimate σ 2 1 and σ 2 2 Statistically test the difference between θ 1 and θ2 P-value > α? Yes Not a corner No A corner

16 Approach Overview (cont d) Slide a window along the input arc sequence S. Estimate ˆθ 1 and ˆθ 2, the orientations of the two arc subsequences located to the right and left of the window center, via least-squares line fitting. Analytically compute σ 2 1 and σ 2 2, the variances of ˆθ 1 and ˆθ 2. Perform a hypothesis test to statistically test the difference between ˆθ 1 and ˆθ 2. 16

17 Details of the Proposed Approach Noise and Corner Models Covariance Propagation Hypothesis Testing 17

18 Noise Model Given an observed arc sequence S = {(ˆx n ŷ n ) t n =1,...N}, itisassumedthat(ˆx n, ŷ n ) result from random perturbations to the ideal points (x n,y n ), lying on a line x n cos θ + y n sin θ ρ =0, through the following noise model: ˆx n ŷ n = x n y n + ξ n where ξ n are iid as N(0, σ 2 ). cos θ sinθ ; n =1,...,N 18

19 Corner Model Xcos θ Ysin θ = ρ 1 Σ θ ^ ρ^ 1, 1 digital arc segment ^ ^ ^ m th point Moving window ^ ^ ^ Xcosθ + Ysin θ = 2 2 ρ 2 Σ ^ θ 2 ρ^, 2 H 0 : θ 12 <θ 0 H 1 : θ 12 θ 0 where ˆθ 12 = ˆθ 1 ˆθ 2, θ 12 is the population mean of ˆθ 12, and θ 0 a threshold. 19

20 Covariance Propagation Problem statement Analytically estimate Σ Θ, the covariance matrix of least-squares estimate ˆΘ =(ˆθ ˆρ) t,intermsofthe input covariance matrix Σ X. 20

21 Covariance Propagation (cont.) From Haralick s covariance propagation theory, define and then F ( ˆΘ, ˆX) = Θ N n=1 (ˆx n cosˆθ +ŷ n sin ˆθ ˆρ) 2 g 2 1 (Θ,X)= F Θ Θ) = ( g(x, Θ ) 1 ( g(x, Θ) ( X g(x, Θ) X )[( g(x, Θ) Θ )t X ) 1 ] t 21

22 Covariance Propagation (cont.) Define k = + x 2 + y 2 ρ 2 if y cos θ x sin θ x 2 + y 2 ρ 2 otherwise and N µ k = 1 N n=1 k n σ 2 k = N n=1 (k n µ k ) 2 22

23 Geometric Interpretation of k Y (x,y) k X 23

24 Covariance Propagation (cont.) Θ = σ2 1 σ 2 k µ k σ 2 k µ k σk µ2 k N σk 2 24

25 Hypothesis Testing H 0 : θ 12 <θ 0 H 1 : θ 12 θ 0 where θ 0 is an angular threshold and θ 12 is the population mean of RV ˆθ 12 = ˆθ 1 ˆθ 2. Let T = ˆθ 2 12 ˆσ 2 θ 1 +ˆσ 2 θ 2 Under null hypothesis T χ 2 2 if P (T ) <α, then a corner else not a corner. 25

26 Corner Detection Example 26