Corner Detection Algorithms for Digital Images in Last Three Decades ...

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Corner Detection Algorithms for Digital Images in Last Three Decades AMBAR DUTTA, AVIJIT KAR

AND

B N CHATTERJI

ABSTRACT Corner detection is an important step in many computer vision applications. A large number of corner detection algorithms were already developed. An exhaustive review of the existing corner detection algorithms is, therefore, invaluable for the researchers working in this area. In the present literature, we have found a few reviews on this area. Most reviewers classified corner detectors into two categories – boundary-based and intensity-based. Some reviewers classified the algorithms into two groups – template-based and geometry-based. Though we follow the first classification, still we feel that there is a requirement of further subdivision. So, we subdivide each category into further two subcategories – spatial-domain and transform-domain. In this paper, we have reviewed a total of 114 corner detection algorithms developed from 1977 to 2006, classified them in each of the four categories and found that most of the works have been done in spatial domain only as compared to transform domain approaches.

1. INTRODUCTION Corner detection in images is an important step in many tasks in machine vision, including scene analysis, motion and structure from motion analysis, image registration, image matching, object recognition etc. A corner is an image point with high contrast along all the directions. Hence, it is well distinguished from neighboring points. A corner detection algorithm must satisfy the following criteria to be useful for feature point matching: (i) consistency, (ii) accuracy, and (iii) speed. The performance of corner detection algorithms is affected by the attributes, viz., corner angle, corner arm length, corner adjacency, corner sharpness, gray level distribution and noise level. Vision researchers have proposed a considerable number of corner detection algorithms. A Rosenfeld et al [1], A Heyden et al [2], Z Zheng et al [3], C Schmid et al [4], P I Rockett [5] and F Mokhtarian et al [6, 7] provided good literature survey on the existing corner detection algorithms. Corner detection algorithms can be broadly divided into two classes – boundary based and intensity based. Boundary based methods rely on extraction of contours of regions either by edge detection or by segmenting the image into regions followed by boundary finding. These methods, thus, largely depend on a segmentation process. On the other hand, the

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intensity based methods estimate a measure to detect corners directly from the gray values of the original images without a prior segmentation. Each of these two above categories can further be subdivided into spatial domain and frequency domain methods. Spatial domain methods directly operate on the pixel values of the image. In transform domain methods, the image is first transformed to some other domain, which is then passed through a suitable filter and finally the filtered information is mapped back to the spatial domain with the help of an inverse transform operation. The remainder of the paper is structured as follows. In section 2, we discuss different performance measures of corner detectors. In the subsequent sections (section 3-6), we present a literature survey of 114 corner detection algorithms under different categories of corner detection algorithms. Finally, we conclude in section 7.

2.

PERFORMANCE MEASURES FOR CORNER DETECTORS

The exact number of corners in the image, the number of corners truly detected, the number of corners missed and the number of extra corners detected play very important role for the comparison of corner detection algorithms. To evaluate the performance of corner detection I E T E

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Corner Detection Algorithms

Boundary based

Spatial Domain

Transform Domain

Intensity based

Spatial Domain

Transform Domain

Fig 1 Classification of corner detection algorithms

algorithms, a few performance measures had already been proposed in the literature [6, 7, 108]. We have also proposed three performance measures – Detection Gradient, False Positive Ratio and False Negative Ratio, comparable with the existing measures, which are defined below. | NA – ND | + | NM + NF | DG = ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ NA FPR =

NF ⎯⎯⎯ NA

FNR =

NM ⎯⎯⎯ NA

where NA, ND, NM and NF denote the total number of corners present in the image, the number of true corners detected, the number of missed (false negative) corners and number of extraneous (false positive) corners detected in the image, respectively. In the best case, the value of DG is zero, which signifies that all the corners are correctly detected without any missed or false corners; the same is true for FPR and FNR.

3.

BOUNDARY BASED DOMAIN METHODS

SPATIAL

Freeman and Davis [8] and Rutkowski and Rosenfeld [1] detected corners by using chaincode values. Medioni and Yasumoto [9] used B-

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Splines to develop a technique for corner detection and curve representation. Beus and Tiu [10] developed an algorithm based on chain-coded plane curves that eliminates the detection of spurious corners using a maximum cut-off value to determine the length of the forward and backward arms of each point. Davies [11] proposed corner detector based on generalized Hough transform. Ogowa [12] computed a symmetry measure at every point on a digital curve and then extracted corners at the local maxima of the measure. Rangarajan, Shah and Brackle [13] proposed an optimal gray level corner detector based on Canny’s optimal one-dimensional edge detector [14]. Rattarangsi and Chin [15] presented scale-based corner detection algorithm on planar curves. Mehrotra, Nichani and Ranganathan [16] described two methods for corner detection, one was based on the first directional derivative of Gaussian and the other was based on the second directional derivative of Gaussian. The detector also computed corner angle and orientation. Bell and Pau [17] developed a corner detector based on Freeman’s chain-code. Liu and Tsai [18] proposed a method based on the principle of preserving gray and mass moments. Cooper, Venkatesh and Kitchen [19] used two different approaches to detect corners in an image – one, by using dissimilarity along the contour direction to detect curves in the image contour, and the other by estimating image curvature along contour direction. Orange and Greon [20] proposed segmentation model for the detection of corners

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based on a geometrical model. Xie, Sudhakar and Zhuang [21] presented a cost minimization approach to corner detection in which they associated a corner with different cost factors capturing desirable characteristics of a corner. Pei and Horng [22] extracted corners from curvature local maxima of the shape resulting from the nest moving average of the original image. Seeger and Seeger [23] detected corners from a gray-level image by a real-time parameter-free algorithm. Sugimoto and Tomita [24] proposed an algorithm for the detection of various feature points (corner, inflection and transition). Zhang, Haralick and Ramesh [25] described a maximum a posteriori (MAP) probability technique for corner detection. Pikaz and Dinstein [26] proposed an approach based on a simple decomposition of the curve into the minimal number of concave and convex sections to detect feature points and to smooth noisy digital curves. Sohn, Alexander, Kim and Snyder [27] presented a constraint regularization approach to detect corners that overcome the problem of determining the unique smoothing factor. Koplowitz and Plante [28] proposed a scheme for chain-coded curves by measuring the number of links to either side of a point that can produce the largest digital straight line. Ray and Ray [29] presented a corner detection algorithm using an iterative Gaussian convolution with constant window size. Dias, Kassim and Srinivasan [30] presented an artificial neural network based corner detection algorithm in a 2-D image. Wang, Wu, Huang and Wang [31] proposed a modified contour tracking and efficient corner detection algorithm using bending value. Lai, Paul and Wu [32] presented an edge-corner detection algorithm, called Cellular Vectorization Method. Sheu and Hu [33] proposed a rotationally invariant two-phase scheme for corner detection where the candidate corners are detected first and then these corners are reinvestigated for the global trend. Arrebola, Camacho, Bandera and Sandoval developed techniques which are based on local [34] and circular [35] histograms of the contour chain code. Ji and Haralick [36] applied statistical techniques to detect corners from chain-encoded digital arcs. Ray and Ray [37] used a discrete scale-space kernel to detect corners on a digital arc. Tsai [38] proposed a boundary based corner detection algorithm by developing two artificial neural networks, one for detecting corners with high curvature, and the other for detecting points of Vol. 25, No. 3, May-June’08

tangency and inflection. Zheng and Zhao [39] implemented a parallel algorithm for detecting dominant points on multiple digital curves. Shilat, Werman and Gdalyahu [40] presented a method for the detection of corners along ridges/troughs and local minima points. Mokhtarian and Soumela [41] presented a corner detection algorithm based curvature scale space representation in which first the edges are extracted from the original image using Canny edge detector [14], and then corners are found from the edge image where there is a local maxima of absolute curvature. Sohn, Kim and Alexander [42] presented a mean field annealing approach to boundary smoothing for curvature estimation to improve the capability of detecting corners. Li and Chen [43] described a corner detection algorithm on planar curves as a fuzzy classification problem containing three stages – evaluation, classification and location. Luo, Cross and Hancock [44] described a corner detection algorithm based on the topographic analysis of a vector potential image representation. Tsai, Hou and Su [45] presented a quantitative measure of corners based on the smaller eigen value of the covariance matrix of boundary points over a small region of support. Guest and Fairhurst [46] described a clustering approach to corner point analysis in hand-drawn images. Ludtke, Luo, Hancock and Wilson [47] proposed a corner detection algorithm using mixture model of edge orientation. Oh and Chien [48] described a corner detection algorithm by combining the generalized symmetry transform (GST) operator with the parametric corner equation. Shen and Wang [49] proposed a fast corner detector using modified Hough transform. Ray and Pandyan [50] presented an adaptive corner detector for planar curves. Marji and Siy [51] proposed an algorithm for detecting dominant points and polygonal approximation of digitized closed curves. Wu [52] proposed an efficient method for dominant point detection using adaptive bending value. Urdiales, Trazegnies, Bandera and Sandoval [53] presented a fast algorithm for corner detection defined at different scales by estimating the curvature of a contour in a local adaptive way. Banerjee, Kundu and Mitra [54] presented a support vector machine based algorithm for corner detection. Guru, Dinesh and Nagabhushan [55] presented a fast and efficient corner detection algorithm by introducing a new measure “cornerity index” for the quantification of the prominence of a corner point. Arrebola and I E T E

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Sandoval [56] detected corners by means of the hierarchical computation of a multi-resolution structure. Sarfraj, Rasheed and Muzaffer [57] presented a simple, robust and efficient real-time corner detection algorithm based on the change of sign of slope of the curve along the contour. Olague and Hernandez [58] proposed an accurate and flexible model-based multi-corner detector. Poyato, Garcia, Carnicer and Cuevas [59] described an efficient algorithm for detecting dominant points on the boundaries of digital planar curves. Sobania and Evans [60] proposed a morphological corner detector using paired triangular structuring element. The detector operated on the boundaries of a segmented image in a binary format and detected only interior or concave corners. Mokhtarian and Mohanna [7] presented a nice literature survey on the existing corner detection algorithms, provided two performance measure, accuracy and consistency, for corner detection algorithms and gave an enhanced version of original CSS corner detection algorithm [41], which worked on multiple scales also.

4.

BOUNDARY BASED TRANSFORM DOMAIN METHODS

Chen, Lee and Sun [61-63] proposed multiscale gray-level corner detection algorithms based on the modulus and orientation information of the wavelet transform which are used to detect edges and localize corners respectively. They also provided a multi-scale corner detection algorithm based on the wavelet transform of contour orientation. Kohlmann [64] derived a feature detection algorithm with 2-D intensity changes using 2-D Hilbert transform. Quddus and Falmy [65, 66] proposed a wavelet-based corner detection algorithm on planar curves. Peng, Zhou and Ding [67] proposed a boundary-based corner detection using wavelet transform. Gao, Sattar, Quddus and Venkateswarlu [68] proposed a multi-scale corner detection algorithm based on continuous wavelet transform on contour images. Yeh [69] proposed a robust, rotation- and scale-invariant wavelet-based corner detection algorithm on circular arcs by using the eigen vectors of the covariance matrices. Sun, Tang and You [70] proposed a wavelet-based corner detection algorithm by estimating the curvature of a contour in an adaptive way. Vol. 25, No. 3, May-June’08

5.

INTENSITY BASED DOMAIN METHODS

SPATIAL

Moravec [71] gave the concept of “points of interest” as points where a significantly high intensity variation occurs in every direction. Beaudet [72] presented a determinant operator, which has large values near the corners. Kitchen and Rosenfeld [73] were the first to apply the differential operators that consists of first and second order partial derivatives of the image to detect corners. They proposed a cornerness measure based on the product of the gradient magnitude and the change of the gradient direction along an edge contour. The local maxima of the measure isolated corner points. The detector is very sensitive to noise. Wu and Rosenfeld [74] detected corner points by a filter projection method. Zuniga and Haralick [75] proposed a facet model based corner detector. Paler, Foglein, Illingworth and Kittler [76] extracted corners from the local distribution of gray level values. Harris and Stephens [77] estimated the measure of local autocorrelation using first-order derivatives that is suggested by performing an analytic expansion of the Moravec [71] operator. At each pixel location a 2X2 autocorrelation matrix is computed and if both the eigen values are large, the pixel is considered to be a corner. The algorithm is computationally expensive. Forstner and Gulch [78] used the same cornerness measure as Harris and Stephens [76], but their algorithm has higher computational complexity. Tomasi and Kanade [79] derived the same equation by analyzing the optical flow equation presented by Lucas and Kanade [80]. Noble [81] explained the method of estimation of image curvature by Harris and characterized twodimensional surface features. Deriche and Giraudon [82] detected corners using a scale space based approach by combining useful properties from Laplacian and Beaudet’s cornerness measure. Shi and Tomasi [83] proposed a method for feature selection, a tracking algorithm based on a model of affine image changes, and a technique for monitoring features during tracking. Wang and Brady [84, 85] presented a simple and accurate corner detector based on the cornerness measurement of the total surface curvature. Cui and Lawrence [86] proposed scale-space consistent algorithm to detect corners in binary images based on corner attributes. Lee and Bien [87] formulated a pattern classification problem to I E T E

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detect gray-level corners in real-time algorithm using fuzzy logic. Diaz, Domingo and Ayala [88] detected two-dimensional feature point from a grayscale image using the application of statistical techniques. Smith and Brady [89] proposed SUSAN corner detector using a concept that each image point is associated with it a local area of similar brightness. If the brightness of each pixel within a mask is compared with the brightness of that mask’s nucleus, then an area of the mask can be defined which has the same (or similar) brightness as the nucleus. This area of the mask is known as the “USAN”, an acronym for “Univalue Segment Assimilating Nucleus”. The value of USAN gets smaller on both sides of an edge and becomes even smaller on each side of a corner. The local minima of the USAN map represent corners in the image. Kautsky, Zitova, Flusser and Peters [90, 91] proposed a two-stage method for the detecting the feature points in which all possible candidates are found first and then the desirable number of resulting significant points is selected among them by maximizing the weight function. Laganiere [92] proposed a morphological corner detector using an asymmetric closing operator. Lin, Chu and Hsueh [93] also used mathematical morphological operators for corner detection with simple integer computation. Stammberger, Michaelis, Reiser and Englmeier [94] proposed a hierarchical approach to corner detection in which the whole image is convolved with only one kernel. Trajkovic and Hedley [95] proposed a fast corner detection algorithm using a multigrid approach to reduce the computational complexity and to improve the quality of the detected corners. Chabat, Yang and Hansell [96] used an operator to detect the true location and orientation of corners. Lv and Zhou [97] modified the cornerness measure of KitchenRosenfeld detector [73] based on the perception of human vision system to make the detector effective in variable illumination scenes. Zheng, Wang and Toeh [2] presented a gradient direction corner detector, derived from the Plessey corner detector, which is based on the measure of the gradient module of image gradient direction. Sojka [98] presented a reliable, robust corner detection algorithm in digital images. Ruzon and Tomasi [99] used both a region model based on the distributions of pixel colors and an edge model for the detection of corners in textured color images. Basak and Mahata [100] proposed a connectionist model to detect corners in binary and gray images. Vol. 25, No. 3, May-June’08

Deschenes and Ziou [101] presented an approach to detect the line junctions in gray images. Alvarez, Cuenca and Mazorra [102] proposed a morphological corner detection algorithm to estimate corners with sub-pixel accuracy and tested the detector’s accuracy in the problem of multiple camera calibration. Wurtz and Laurens [103] presented a corner detection algorithm in color images through a multi-scale combination of end-stopped cortical cells. Telle and Aldon [104] proposed a interest point detector for color images based on non-linear filtering of the image. Gao, Yu, Sattar and Venkateswarlu [105] proposed an improved Plessey corner detector, worked in scalespace domain, for gray level images using multiscale analysis. Bae, Kweon and Yoo [106] used two oriented cross operators, COP (crosses as ordered pair) to extract low-level image features, viz., corners. Elias and Laganiere [107] proposed an approach based on a data structure similar to pyramid, but of circular levels, to determine the location and orientation of corners. Golightly and Jones [108] presented an algorithm for corner detection and matching for visual tracking of power line inspection and proposed two performance measures – detection rate and error rate – for the corner detection algorithms. Mikolajczyk and Schmid [109] presented two techniques for corner detection invariant to scale and affine transformations. Alkaabi and Deravi [110] presented a fast corner detection algorithm based on pruning candidate corners. Zhou, Liut and Cai [111] improved SUSAN corner detector [89] by presenting a robust and efficient corner detection algorithm which is capable of detecting the features in different contrast images automatically through self-adjust multi-threshold. Kenney, Zuliani and Manjunath [112] presented an axiomatic approach to corner detection. Vincent and Laganiere [113] proposed a new feature point detector which first performs a simple segmentation based on the intensity values found in the vicinity of each considered point, and then, it tries to fit a simple wedge corner model to the resulting segmented area. Cooke and Whatmough [114] proposed two ways – one by using genetic algorithm and another by using supervised classification techniques – towards the application of learning algorithms in corner detection. Pei and Ding [115] proposed a corner detection algorithm based on determining orientations of the adaptive vertical and tangent axes and observing the variations of brightness I E T E

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along the positive and negative directions of these axes. Rosten and Drummond [116] used machine learning for fast and high quality corner detection.

6.

INTENSITY BASED TRANSFORM DOMAIN METHODS

Quddus and Falmy [117] proposed a corner detection algorithm based on scale interaction model using Gabor filter suitable for both binary and gray level images with variable background. Pedersini, Pozzoli, Sarti and Tubaro [118] proposed a wavelet-based multi-resolution corner detection algorithm based on the analysis of multi-scale Laplacian’s profile. Quddus and Gabbouj [119] proposed a wavelet-based corner detection algorithm using singular value decomposition (SVD). Gao, Sattar and Venkateswarlu [120] presented a multi-scale corner detection algorithm on gray images using Gabor wavelets.

found that most of the work has been carried out on boundary (binary) and gray-scale images. So, there is huge scope of work in the field of corner detection that will directly operate on the color images as well. In addition, any boundary-based corner detection algorithm consists of finding boundary from an image, followed by following and closing the boundary, which always requires some manual intervention resulting in high computational time. 7. REFERENCES 1.

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7. CONCLUSIONS In this paper, we provide a literature survey on existing corner detection algorithms developed in the last three decades starting from 1977 which include a total of 114 algorithms. We then classify these algorithms into two main categories – boundary-based and intensity-based methods, which, in turn, are again subdivided into spatial domain and transform domain methods. After this classification, we have observed that out of these 114 algorithms, 54 algorithms belong to boundarybased spatial domain, 10 belong to boundary-based transform domain, 46 belong to intensity-based spatial domain and only 4 to intensity-based transform domain. But since transform domain methods are intensely used in various applications in image analysis and processing and comparatively much less work has been carried out in transform domain, there is a tremendous scope of work in the field of corner detection in transform domain. Moreover, all transform domain methods discussed in this survey were based on wavelet transform and since there are many other transforms available, e.g. Fourier, Hartley, WalshHadamard, Haar, Slant, Karhunen-Loeve etc, so efficiency of these transforms may be investigated and noted. Though the majority of work in this field has been carried out in the spatial domain, still there is scope more work in this area. We have also

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Authors Ambar Dutta born on 3rd December, 1977, did his BSc (Honors) in Mathematics from Presidency College, Kolkata in 1999 and MCA from Jadavpur University, Kolkata in 2002. He is working as a Lecturer in the department of Computer Science and Engineering, Birla Institute of Technology, Mesra, Kolkata Extension Centre. He is pursuing his PhD from Jadavpur University, Kolkata in the area of image processing (corner detection and matching). His research interest includes Image Processing, Pattern Recognition, Data Mining and Soft Computing. Address: Department of Computer Science and Engineering, Birla Institute of Technology, Mesra, Kolkata Extension Centre, Southend Conclave, 1582, Rajdanga Main Road, Kolkata 700 107, India. email: *

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Avijit Kar did his MSc and PhD in 1980 and 1984 respectively from IIT Kharagpur. He is currently a professor in the department of Computer Science and Engineering in Jadavpur University, Kolkata. He has supervised several PhD theses and is actively involved in many R & D activities and IT related consultancy for Government of India and the private sector. His research interest includes biomedical imaging as well as SAR imaging. He is also into computer systems reliability. He is involved in a large number of industry sponsored development projects. Address: Department of Computer Science and Engineering, Jadavpur University, Kolkata 700 032, India. email: *

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B N Chatterji born on 10th November, 1942, obtained BTech (Hons) (1965) and PhD (1970) in Electronics and Electrical Communication Engineering of IIT, Kharagpur. He did Post Doctoral work at University of Erlangen-Nurenberg, Germany during 1972-73. Worked with Telerad Pvt Ltd, Bombay (1965), Central Electronics Research Institute, Pilani (1966) and IIT, Kharagpur as faculty member during 1967-2005. He was Professor during 1980-2005, Head of the Department during 1987-1991, Dean Academic Affairs during 1994-1997 and Member of Board of Governors of IIT, Kharagpur during 1998-2000. He has published about 150 journal papers, 200 conference papers and four books. He was Chairman of four International Conferences and ten National Conferences. He has coordinated 25 short-term courses and was the chief investigator of 24 Sponsored Projects. He is the Fellow/Life Member/Member of eight Professional Societies. He has received ten National Awards on the basis of his Academic/Research contributions. His areas of interests are Pattern Recognition, Image Processing, Signal Processing, Parallel Processing and Control Systems. Address: Department of Electronics Communication Engineering, B P Poddar Institute of Management and Technology, 137, VIP Road, Kolkata 700 052, India. email:

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