BOUNDARY BASED CORNER DETECTION AND LOCALIZATION USING NEW ‘CORNERITY’ INDEX: A ROBUST APPROACH D.S Guru, R. Dinesh* and P. Nagabhushan Department of studies in Computer Science University of Mysore, Manasagangotri, Mysore -570 006, INDIA

Abstract In this paper, a novel boundary based corner detection algorithm is proposed. The proposed algorithm is computationally fast and efficient. The proposed method computes an expected point for every point on a boundary curve. An expected point corresponding to a point on the boundary curve is defined to be the geometrical centroid of the symmetrical boundary segment of size 2k+1, for some integer k>0, within the neighborhood of the point in consideration. A new ‘cornerity’ index for a point on the boundary curve is defined to be the distance between the point and its corresponding expected point. The larger the cornerity index, the stronger is the evidence that the boundary point is a corner point. A set of rules is worked out to guide the process of locating true corner points. The conducted experiments establish that the proposed approach is invariant to image transformations viz., rotation, translation and scaling.

Keywords:

Object recognition, Boundary curve, Region of support, Curvature estimation, Cornerity index, Corner detection, Corner localization.

1. Introduction Corner detection in an image is an important aspect of computer vision. Since information about a shape is concentrated at the corners and corners practically prove to be descriptive primitives in shape representation and image interpretation, corner detection is useful to many vision applications including object recognition. Corners on a curve arise where two adjacent segments which are relatively found to be straightline segments, intersect. That is, the corner points are found at locations where nature of curve changes significantly and abruptly. Many algorithms have been developed for detecting corners on the boundary curve of an object. Freeman and Davis (1977) designed a scheme that detects local curvature maxima points as corner points. The algorithm consists of scanning the chain code of the curve with a moving line segment to connect the end points of a sequence of links. As the line segment moves from one chain code to the next, the angular differences * Author for Correspondence: e-mail: [email protected]

between the successive segment positions are used as a measure of local curvature. The procedure is parallel and needs more than one input parameter to decide the size of the region of support (Ray and Ray, 1992). Anderson and Bezdek (1984) devised an algorithm that defines tangential deflection and curvature of discrete curve on the basis of the geometrical and statistical properties associated with the eigenvalues and eigenvectors of sample covariance matrices. The corners are located by using the criterion of excessive cumulative tangential deflection between successive points. Sankar and Sharma (1978) have designed an iterative procedure for detecting dominant points with maximum global curvature based on the local curvature of each point with respect to its immediate neighbors. The procedure is parallel and needs no input parameter. However, it fails to operate successfully on curves consisting of features of multiple size (Ray and Ray 1992). As an improved method, Ray and Ray (1992) have proposed an algorithm that detects a dominant point based on the angle between the k-vector and the l-vector of the point of interest. However, the determination of k-vector and l-vector at every point, particularly at the points on a straight-line of longer length is time consuming. Other interesting algorithms were also reported; Harlick and Shapiro (1992) used a facet model, Rattarangsi and Chin (1992) used a coarse-to-fine tree parsing technique, Lee et al (1993) employed wavelet transformation. Zhang and Zhao (1997) proposed a parallel algorithm based on an analysis of morphological residues and corner characteristics. Numerous boundary based algorithms that are generally easy to implement and computationally fast were also proposed. However, they may suffer from poor detection due to the instability of discrete curvature measurement when the curve is rotated in different orientations (Tsai, 1997). Tsai (1997) measures the curvature by using neural networks to recognize included angles at boundary points. Sohn et al (1998) proposed a method of boundary smoothing for curvature estimation using a

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deterministic approximation of simulated annealing. The methods (Beus and Tiu 1987; Anderson and Bezdek 1984; Liu and Srinath 1990) generally work reliably for polygonal objects as the vertices of a polygon have large curvature values and the points elsewhere on the boundary have approximately zero curvature. However they may detect many spurious corners for the objects involving circular arcs of varying radii. Points on a circular arc of a small radius generally have high curvature and may have curvature values larger than that of the intersection point of two straight line segments in the discrete domain. Using a curvature measure, detection of two or more superfluous corners on a digital arc is not uncommon. The detection and localization become poor and unstable when the objects of curved shapes can be changed in scales and rotated in arbitrary orientations. Instead of using the curvature measures for corner detection, a new corner detection method based on the eigenvalues of the covariance matrix of data points on a curve segment was proposed by Tsai et al (1999). The method computes the eigenvalues associated with the covariance matrix of the data points over a small region of support at every point on the boundary. A point was said to be corner if the corresponding small eigenvalue exceeds a predefined threshold value and has a local maxima. It fails in avoiding spurious corners on circular arcs under different rotations and scale changes. In addition, the algorithm requires considerable amount of time to compute eigenvalues at every point on the boundary. In specific, it requires (10k+2) n, 5n, (6k+5) n, 6n and n additions, subtractions, multiplications, divisions and square root computations respectively for a boundary of size n with the region of support of size 2k+1. Chang and Horng (1994) have proposed an algorithm to detect corner points using nest moving average filter. The nest moving average filter is used as a low pass filter, which attenuates high frequency components. A low pass filter suppresses corners and results in a smooth curve. Therefore, corner points have a larger amount of shift than any other point and can easily be detected by a low pass filter, just by computing the difference between the original boundary and the one processed with the nest moving average filter. But this method requires two parameters to decide the size of two moving average filters, and in addition, the boundary should be represented in a parametric form. The method requires 2n(M+m+2), n, 2n, 2n and n additions, subtractions, multiplications, divisions and absolute operations respectively for a boundary of size n with two filters of size 2M+1 and 2m+1. This method suffers from poor localization of corner

points and is very much sensitive to quantization error curve which is unavoidable if the boundary curves undergo transformations. In this paper, a boundary based corner detection algorithm is proposed. The proposed algorithm is computationally fast and efficient. The proposed method computes an expected point for every point on a boundary curve. An expected point corresponding to a point on the boundary curve is defined to be the geometrical centroid of the symmetrical boundary segment of size 2k+1 within the neighborhood of the point in consideration. A new cornerity index for a point on the boundary curve is defined to be the distance between the point and its corresponding expected point. The larger the cornerity index, the stronger is the evidence that the boundary point is a corner point. The points within the vicinity of actual corner points will have relatively larger value for cornerity index than other points, and among such a cluster of points, the point which bear local maximum cornerity index value is selected as the corner point. A non-parametric approach is suggested for localization of true corner points. The approach requires (4k+8) n, 2n, 2n, 2n and n additions, subtractions, multiplications, divisions and square root computations respectively for a boundary of size n with the region of support of size 2k+1. The results of the experiments conducted demonstrate that the proposed method is invariant to image transformations viz., rotation, translation and scaling. The rest of the paper is organized as follows. Section 2 introduces the new measure of curvature based on the expected point of a curve segment and then discusses its usage as a quantitative measure of corners. Section 3 presents the experimental results with a comparative study. The discussion is given in section 4 and section 5 follows with conclusion. 2. The Proposed method for corner detection In this section, a measure for the prominence of a corner is derived from the statistical and geometrical properties associated with the mean values of the X and Y coordinates of a set of data points on a boundary curve over a region of support. Let the sequence of n digital points describe a closed boundary curve B of an object. B = { pi = (xi, yi), ∀ i = 1,2,…,n } Here pi is a neighbor of p(i+1) mod n and (xi, yi) are the Cartesian coordinates of pi. Let Sk(pi), for some integer k > 0, denote a small curve segment (of B), the mid point of which is pi and is called the region of support for the point pi.

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i.e., Sk(pi) = { pj  j = i-k,…., i ,…i+k }. The expected point pie of the segment Sk(pi) is given by, pie = (xie, yie) where xie and yie are the respective mean values of the X and Y coordinates of the points in Sk(pi) and are given by i+k

x ie =

1 2k + 1

y ie =

i+k 1 yj ¦ 2 k + 1 j =i − k

¦x

j =i − k

and

j

Replacement of each boundary point by its expected point will smoothen the boundary curve. Thus, the expected point corresponding to a corner point will have a larger shift when compared to other points on the boundary curve. Therefore, the cornerity index of pi is defined to be the Euclidean distance d between the points pi and its expected point pie and is given by

d = ( xi − xie ) 2 + ( y i − y ie ) 2 The cornerity index indicates the prominence of a corner point. The larger the value of the cornerity index of a boundary point, the stronger is the evidence that the boundary point is a corner.

it is not necessarily zero. It is also observed that a true corner point will have significantly high cornerity index which is relatively larger when compared to that of its neighbors. Moreover, the experimental study has also revealed that although boundary points on a smooth curve segment with a small radius of curvature bear significantly high cornerity index, the relative differences of their cornerity indices are negligibly small. These experimental observations have been revealed in Fig1. Fig-1(b) shows the graph of the cornerity index versus the boundary points of the boundary curve shown in Fig-1(a). Based on these observations the following rules are worked out to guide the process of true corner point localization. (i) Select those boundary points which bear significantly large cornerity index by eliminating the boundary points which lie on straight line segments bearing negligibly small cornerity index value. (ii) Since all points on a smooth curve segment are in general associated with almost same cornerity index, and actual corner points bear cornerity index larger than that of their neighbors, it is suggested to select the set of connected points such that the variations in their cornerity indices are considerably large. This rule helps in selecting only the set of points with in the vicinity of actual corner points by eliminating the points on smooth curve segments. (iii) Select the points which bear local maximum cornerity index as true corner points. It could be noticed that these rules do not require any priori knowledge in locating true corner points.

(a)

(b) Fig-1: (a) A boundary curve; (b) Graph of cornerity index versus boundary points of (a) Though, theoretically the cornerity index of a boundary point lying on a straight line segment is expected to be zero, it is experimentally observed that

3. Experimental results In order to reveal the accuracy and superiority of the proposed model in pragmatic scenario, we have conducted an extensive experimentation on various shapes with different scaling factors and in different orientations. The set of shapes considered for experimentation includes shapes with smooth curve segments of different radii of curvature, curvilinear segments, straight line segments and also the boundary curves of real objects. For each shape, 36 different samples are generated, out of which 24 are rotated versions, 6 are scaled versions and the remaining 6 are scaled and rotated versions. Out of shapes considered, we present here, only a few shapes (that are considered by most of researchers) along with the obtained results. Fig-2 and Fig-3 respectively show the set of considered shapes and the results obtained due to the application of the

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proposed methodology. The detected corner points are marked by the symbol “O”. The proposed method is then employed on each sample of every shape to detect corner points. The detected corner points are categorized into actual true corner points, non-corner points detected as corner points (false corner points) and corner points undetected. Out of many shapes considered, the Table-1, summarizes the details about only six shapes (shown in Fig-2) which are generally considered by many researchers. In order to explore the consistency of the proposed method with the human vision system we compared the results obtained by the proposed model with the results provided by a panel of human observers. 10 people were given the curves shown in Fig-2 and were asked to determine, according to them, corner points. They were left free to choose as many points as they perceive. The results obtained by the proposed method are validated against the results given by the panel of human experts and is understood that our method is highly consistent with the human vision system (see Table-1).

Fig-2: A few shapes considered for experimentation

Furthermore, to reveal the superiority of our model over other models we have also made a comparative study. For the purpose of this study we have considered the methods proposed by Tasi et. al., (1999) and Chang et. al., (1994). Same set of shapes are considered for the comparative study and the parameter values which determine the region of support are kept constant throughout the experimentation. Fig-4 and Fig-5 respectively show the results due to the Tasi’s (1999) and Chang’s (1994) methods. Table-1 also summarizes these results. It can be observed from the Table-1 that the method proposed by Tsai et. al.,(1999), most of the times, detect all corner points as corner points, but it detects some spurious points as corner points on a smooth curve segment of a small radius of curvature. It is also evident from the Table-1 that the method proposed by Chang et. al., (1994) fails to detect even some true corner points in addition to identifying some non-corner points as corner points. On the other hand, our method is capable of detecting all corner points irrespective of shape features and it does not detect even a single spurious corner point.

Fig-3: Results of the proposed method

Fig-4: Results of the Tsai’s (1999) method

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requires two parameters to decide the value of M and m. (iv)

The proposed method has good localization capability in locating true corner points.

(v)

The proposed method is invariant to image transformations viz., translation, rotation and scaling.

(vi)

The proposed method is highly consistent with the human vision system.

5. Conclusion Fig-5: Results of the Chang et. al., (1994) method

4. Discussions In this work, we compared our method with the method proposed by Tasi et. al., (1999) and the method proposed by Chang et. al., (1994). Compared to these methods our method offers the following advantages: (i)

Unlike other two methods, the proposed method locates corner points based on the local properties of the curve segment and does not make use of any parameter to decide the threshold value for the selection and localization of corner points.

(ii)

Unlike other methods, the proposed method works well even if the size of the region of support is varied in the range [5..12], because while locating corner points, the proposed method does not require any parameter to locate corner points, which is not the case with the other methods, where their performances entirely depend on the size of the region of support.

(iii)

The proposed method is computationally efficient and it requires (4k+8) n, 2n, 2n, 2n and n additions, subtractions, multiplications, divisions and square root computations respectively for a boundary of size n, whereas Tsai’s (1999) method requires (10k+2) n, 5n, (6k+5) n, 6n and n additions, subtractions, multiplications, divisions and square root computations respectively for a boundary of size n and the method proposed by Chang et. al., (1994) requires 2n(M+m+2), n, 2n, 2n and n additions, subtractions, multiplications, divisions and absolute computation respectively for a boundary of size n with two filters of size 2M+1 and 2m+1. In addition, the method proposed by Chang et. al., (1994)

A new measure called cornerity index is introduced in this paper to quantify the prominence of a corner point. The expected point for every point on the boundary of an object is computed to be the geometrical centroid of the data points chosen in a small region of support. The cornerity index of a point is defined to be the distance between the point and its corresponding expected point. A set of rules is worked out to guide the process of locating true corners. The proposed method, being computationally efficient is invariant to image transformations. Several experiments are conducted to corroborate the efficacy of the proposed method on various shapes with different scales and in different orientations. References Anderson. I.M and I.C.Bezdek, (1984). Curvature and tangential deflection of discrete arcs; a theory based on the commutator of scatter matrix pairs and its application to vertex detection in planar shape data. IEEE Trans. Pattern Anal Machine Intell, 6, 27-40. Beus. H.L and S.S.H Tiu, (1987). An improved corner detection algorithm based on chain coded plane curves. Pattern Recognition, 20, 291-296. Chang S.P, J.H. Horng, (1994). Corner point detection using nest moving average. Pattern Recognition, 27(11), pp. 1533-1537. Freeman H and L. S. Davis, (1977). A corner finding algorithm for chain coded curves. IEEE Trans. Comput. 26, 297-303. Haralick R and L. Shapiro, (1992). Computer Vision Vol I, pp 410-419. Addison-Wesley, Reading, Massacusetts. Lee. J.S, Y.N.Sun, C.H. Chen and T. C. Tsai, (1993).Wavelet based corner detection. Pattern recognition, 26(6), 853-865.

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Liu , H.C and M.D. Srinath., (1990). Corner detection from chain code. Pattern Recognition, 23, 51-68. Rattarangsi. A and R. Chin, (1992). Scale based detection of corners of planar curves. IEEE. Trans. Pattern Anal. Mach. Intell. 14, 430-448. Ray. B. K and K.S. Ray, (1992). An algorithm for detection of dominant points and polygonal approximation of digitized curves. Pattern Recognition Letters, 13, 849-856. Sankar P.V and C.V. Sharma, (1978). A parallel procedure for the detection of dominant points on digital curve. Computer Graphics and Image Processing, 7, 403-412. Sohn. K, J.H. Kim and W.E. Alexander, (1998). A mean field annealing approach to robust corner detection. IEEE Trans. Systems Man Cyber. Part B, 28, 82-90. Tsai. D.M., (1997). Boundary based corner detection using neural networks. Pattern Recognition, 30(1), 85-97. Tsai. D.M, H. T. Hou, and H. J. Su, (1999), Boundary-based corner detection using eigenvalues of covariance matrices. Pattern Recognition Letters, 20, 31-40. Zhang. X and D. Zhao, (1997). A parallel algorithm for detecting dominant points on multiple digital curves. Pattern Recognition

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Table 1: Results of the experiments conducted.

Number of Samples Average Boundary Length Total Boundary Length Expected number of Corner points on each sample by human experts Total number of corner points expected due to all samples by human experts Total number of corner points detected Number of true corner points detected Number of false corner Proposed points detected method Number of undetected corner points Total number of corner points detected Number of true corner Nest Moving points detected Number of false corner Average points detected Number of undetected corner points Total number of corner points detected Number of true corner Eigenvalue Based corner points detected detection Number of false corner scheme points detected Number of undetected corner points

shape 1 36 477.14 17177 4

shape 2 36 368.62 13270 11

shape 3 36 274.45 9880 2

shape 4 36 575.56 17266 8

shape 5 36 411.84 14826 16

shape 6 36 725.31 26111 5

144

396

72

288

576

180

144

396

72

288

576

180

144

396

72

288

576

180

0

0

0

0

0

0

0

0

0

0

0

0

712

458

314

720

788

446

144

370

72

280

540

180

568

88

242

440

248

266

0

26

0

8

36

0

162

496

72

680

576

396

144

390

72

280

576

180

18

100

0

400

0

216

0

6

0

0

0

0

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