Corner Detection with Covariance Propagation

Corner Detection with Covariance Propagation Qiang Ji and Robert M. Haralick Intelligent Systems Laboratory Department of Electrical Engineering, FT-10 University of Washington Seattle, WA 98 195 Abstract proposed for detecting comer points including maximum curvature, deflection angle, maximum deviatior,, and total This paper presents a statistical approach for detect- fitting errors [1][5][6][11]. A major problem with existing ing corners from chain encoded digital arcs. An arc point approaches is that the employed criterion is not tied to a sta- is declared as a corner if the estimated parameters of the tistical analysis, therefore rendering existing methods sus- two fitted lines of the two arc segments immediately to the ceptible to noise. To overcome this, we present a statistical right and left of the arc point are statistically sign9cantly approach for corner detection. Here, the corner criterion different. The corner detection algorithm consists of two is treated as a random variable and is subject to perturba- steps: corner detection and optimization. While corner de- tion. Given an arc segment, a comer is defined to be a point tection involves statistically identifying the most likely cor- where two underlying line segments meet and form a ver- ner points along an arc sequence, corner optimization deals tex, whose included angle is statistically larger than an angle with improving the locational errors of the detected corners. threshold. The comer detection procedure involves sliding The major contributions of this research include devel- a context window of specified length over the arc sequence, oping a method for analytically estimating the covariance performing a least square line fitting to the arc points lo- matrix of the fitted line parameters and developing a hy- cated immediately to the left and right of the center of the pothesis test statistic to statistically test the difference be- window, estimating the parameters of the fitted lines, and tween the parameters of two fitted lines. Performance eval- their covariance matrices, and finally performing a hypoth- uation study showed that the algorithm is robust and ac- esis test to test the statistical difference between the param- curate for complex images. It has an average misdetection eters of the two fitted lines. If the difference is significant, rate of 2.5% and false alarm rate of 2.2% for the complex the arc point located in the center of the window is declared RADIUS images. This paper discusses the theory and per- as a comer point. Finally, a comer optimization procedure formance characterization of the proposed corner detector. is performed to improve the locational errors of the detected comers. The major contributions of this research include developing a method for analytically estimating the covariance 1. Introduction matrix of the fitted line parameters and developing a hypothesis test statistic to statistically test the difference be- Comers have long been important two dimensional fea- tween the parameters of two fitted lines. This paper is ar- ranged as follows. In section 2, we state the problem and tures for computer vision research. They have been used ex- tensively for matching, pattern recognition, and data com- present the associated noise and comer models. Section 3 discusses in detail the theoretical aspects of the corner de- pression. Various algorithms have been developed for detector. The tecting comers. Comer detection algorithms can be roughly performance evaluation of the comer detector is covered in sections 4 and The paper ends in section 6 grouped into two categories: one is based on the detection 5. directly from the underlying grayscale images; the other is with a discussion and summary of the proposed approach. based on the digital arcs, resulting from edge detection and linking. Various techniques have been developed for comer 2. Problem statement detection from digital arc sequences. The basis for these techniques is to identify the locations of the endpoints of A comer point represents a discontinuity in the curva- each maximal line segment. Different criteria have been ture of a curve. The location of the discontinuity can be 1063-6919/97 $10.00 0 1997 IEEE 362 approximated by the intersection of two straight lines that underlie the arc segments to the right and left of the corner point, Perturbation to the points on the ideal underlying H~ : e12< eo H~ : e122 eo (3) lines gives rise to the observed arc segments. This section is where 812 repfesents the population mean of random vari- concerned with the definition of the perturbation model and able 612. the comer model. The hypothesis testing identifies the most likely comer point along an arc sequence. Specifically, Jiven a signifi- 2.1. Perturbation model cant level a,the P-value of each observed 42is computed and compared with a. If the P-value 5 a,the null hypoth- Given an observed sequence of ordered points from a esis is rejected and the arc point being considered (&, &) line arc segment, S = { ($n, gn)l, = 1,.. ,N}, where N is a comer point. Figure 1 illustratively shows the comer is the number of points on the arc segment, the perturba- model just described. In the section to follow, we describe tion model assumes that (On, dn) result from random per- the theoretical derivations that lead to the solution to the turbations to the ideal points (xn, gn), n = 1,. ,N, con- above hypothesis testing problem. strained to be on the line xncos8+ynsin8-p=0, n=l,..., N; I, rMovingwindow where 8 and p are the parameters of the underlying line that \ gives rise to the observed arc segment. It is further assumed that the random perturbations are independently and iden- tically Gaussian distributed in the direction perpendicular to the underlying line. Analytically, the perturbation model can be expressed as follows: digiia arc where n = 1,. , N and tnare independently and identi- Figure 1. An example of the corner model cally distributed as N(0, a2). 2.2. Corner model 3. Theory for the proposed approach For a piecewise linear approximation of a curve, comer In this section, we detail the theoretical aspects of the points are the end points of each line segment. Thus, an end developed algorithm. Specifically, we describe least-square point is a comer point if the underlying two line segments line fitting, covariance propagation, and hypothesis testing. immediately to the right and left of the point meet and form a vertex, whose included angle is statistically larger than a 3.1. Least-square line fitting given angle threshold. A comer is defined as follows. Given an observed sequence of ordered points from an To estimate the line parameters for each arc segment, we arc segment, S = { ( ) In = 1,. , N} and a point perform a least square line fitting to the arc points. The least-square fitting can be formulated as follows: (&, &) along the arc segment, the arc point divides the Assume phts (Pn,in),n = I.." lie on an arc seg- arc segment S into two sub-segments SI and SZ , where ment S, resulting from perturbation of ideal points (xn, Vn) locating on the line 2, cos t9 + yn sin 8 - p = 0. Perturba- tion to each point follow the perturbation model in equation IC + 1,. ,N}. Let 81 and 62 be the estimated orientations (1). of the two lines that fit to SIand S2,and i12be the included To estimate the best fitting line parameters b and j using angle between the lines, is then defined as the least square method, we need to minimize the sum of squared residual errors: Given an included angle threshold eo, the comer detection problem may be formulated as a hypothesis testing problem as follows: (4) 363 where 5 is called design matrix, 6 the parameter matrix, Then based on [21, CAe, the covariance matrix of the esti- and S = atfithe scatter matrix. fi and 0 are defined as mated line parameters 6 can be computed from: where& = cos8andp = sin8 As a result, we need to minimize DtSh subject to b2 + where & and $& are evaluated at ideal line parameter 8 j2= 1. Introducing the Lagrange multiplier A, the func- and points X, and CAX represents the input perturbation. tion to be minimized can be expressed as From eq(9), we can easily obtain E2 = && - A($& - 1) (5) where C is referred to as constraint matrix and is defined as c=(i ; g) and, Taking partial derivatives of e2 w.r.t. 6 and A, and setting them to zeros yields the simultaneousequations ag2Nx2 $&-A& = 0 (6) ax @& = 1 (7) This system is readily solved by considering generalized eigenvectors of eq(6). 2N x2 3.2. Covariance propagation For the given perturbation model in equation (l),the input covariance matrix CAX is given by The random perturbation on ideal X=(zn, points yn), d ... 0 0 n = 1, ...,N,lyingonlinesncosO+ynsinO-p = 0, 0 d ... 0 yields observed arc points 2 = (Zn,fjn), n = 1,. , N. The use of 2 for estimating line parameter 0 = (3 yields 6 = ( ) , a least square estimate of 0.ne per- where turbation accompanying 2 induces a correspondingpertur- cos2 e sin 8 cos e bation on 6. In this section, we will analytically estimate d= ( sinecos8 sin20 A@,the perturbation of Q, expressed in its covariance ma- ) trix CA@,in terms of the covariance matrix cAX of 2. Define Based on the covariance propagation theory [21, the scaler criterion function F that needs to be minimized can be defined as N and ~(6,2)= C(2,cosB + insin8 - j12 n=l Define g as follows: 364 After algebraic operations and simplifications,we obtain me distribution of the test statistic under null hypothesis is a non-central Chi-squared with two degrees of freedom.

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