Corner detection based on gradient correlation matrices of planar curves

Abstract

An efficient and novel technique is developed for detecting and localizing corners of planar curves. This paper discusses the gradient feature distribution of planar curves and constructs gradient correlation matrices (GCMs) over the region of support (ROS) of these planar curves. It is shown that the eigen-structure and determinant of the GCMs encode the geometric features of these curves, such as curvature features and the dominant points. The determinant of the GCMs is shown to have a strong corner response, and is used as a “cornerness” measure of planar curves. A comprehensive performance evaluation of the proposed detector is performed, using the ACU and localization error criteria. Experimental results demonstrate that the GCM detector has a strong corner position response, along with a high detection rate and good localization performance.

Introduction

Corners are important features in images, and are frequently used for scene analysis, stereo matching, robot navigation, stitching of panoramic photographs, and object tracking. There are many competing algorithms for detecting corners in images. Since the pioneering work of Förstner [1], and Harris and Stephens [2], the structure tensor of image gradients, denoted by P, has become popular for corner detection. For example, Rohr [4], [5] developed a rotationally invariant corner detector based solely on the determinant of P. Tomasi and Kanade [6], and Shi and Tomasi [7] proposed corner detectors based on the smallest eigenvalue of P. Recently, Kenney and Manjunath [8] defined a reciprocal value of the smallest eigenvalue of P as a “cornerness” measure according to condition number theory. Kenney, Zuliani, and Manjunath [3] discussed and reviewed five detectors, and presented an axiomatic approach to corner detection based on the structure tensor of image gradients. Structure tensors have been used rather successfully to find corners in images in these detectors, and applications in invariant region detection or shape detection [23], [24], [25] have also been reported. It should be noted that these structure tensors are calculated from image intensities.

Corner detectors using boundary based methods [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] have been proposed as an alternative to intensity based methods. Tsai, Hou, and Su [9] proposed a boundary based corner detector using the eigenvalues of covariance matrices of contour coordinate points over the region of support (ROS). This approach requires that the radius of the ROS is large enough to suit the statistical characteristics of Tsai's detector. If the radius is too large however, the responses of adjacent corners may interfere with each other causing the detector to miss small features; while if the radius is too small, the detector may suffer from sensitivity to noise. Rattarangsi and Chin [10], Mokhtarian and Suomela [11], [12], and Mohanna and Mokhtarian [13] proposed a kind of curvature scale space (CSS) technique for corner detection. He and Yung [14] proposed an improved CSS corner detector with an adaptive curvature threshold and a dynamic region of support (ATCSS). Zhong and Liao [26] proposed a direct curvature scale space (DCSS) technique, and presented a corner detector combing CSS and DCSS. Zhang, Lei, and Yang [15] proposed a multi-scale curvature product (MSCP) technique for enhancing the response of corners, while suppressing the noise response. Yeh [20] proposed a curvature estimation method using wavelet transforms of eigenvectors of covariance matrices. These methods based on curvature estimation require calculating higher order derivatives of the smoothed versions of planar curves, and can suffer from sensitivity to noise. Awrangjeb and Lu [27] proposed a robust feature detector integrating multi-scale products, adaptive curvature thresholds, and chord-to-point distance accumulation (CPDA) into the discrete curvature estimation. The CPDA detector does not need to calculate derivatives of the planar curves, and is robust to geometric transformations. This approach may merge nearby features and miss some weak corners due to the larger radius of the ROS required however. Zhang and Wang [28] proposed a corner detector based on the scale evolution difference of contours and a difference of Gaussian filter. To the best of the authors’ knowledge however, no methods using the structure tensor of planar curve gradients for boundary based corner detection have been presented in the literature.

In this paper a novel algorithm for corner detection based on the structure tensor of planar curve gradients is developed. Similar to the well-known Förstner [1], Harris [2] and Rohr [4], [5] detectors, the proposed detector computes the structure tensor of the gradient and seeks corners at the maxima of its determinant. Instead of using the image gradient however, the proposed detector uses the contour's gradient vectors. The gradient correlation matrix (GCM) formulated using Lagrange multipliers only requires calculation of the first derivative of the planar curves. This is advantageous as avoiding the higher order derivatives reduces the effect of noise. A small ROS radius may then be used to improve corner localization and to prevent nearby features from merging. Consequently, the proposed detector offers a high detection rate along with good localization performance. The main contributions and organization of this paper are as follows:

• Firstly, we introduce the motivation for using GCMs by analyzing the gradient feature distribution in the plane spanned by the gradients of planar curves, and formulate the GCM by least squares and the Lagrange multiplier method, where the GCM is viewed as the structure tensor of planar curve gradients.

• Secondly, we analyze the geometric properties of the GCM based on typical corner models [10], and find that the local maxima of the determinant of the GCMs correspond to the positions of dominant (corner) points. Numerical simulations based on the discrete END model are given to illustrate the behavior of the determinant of the GCMs, with experimental results used to validate this theoretical analysis. From this analysis, it follows that the determinant of the GCMs can be used as a “cornerness” measure of planar curves.

• Finally, we perform a comprehensive evaluation of the detection and localization performances of the proposed detector using the ACU [11] and the localization error (LE) [14], [27] criteria. The proposed GCM corner detector has a strong corner position response along with a high detection rate and a good localization performance, in terms of the ACU and LE criteria.