Corner Detection Using Laplacian of Gaussian Operator. Salvatore Tabbone. CRIN CNRS/INRIA Lorraine. Campus scientique, B.P. 239

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1 Corner Detection Using Laplacian of Gaussian Operator Salvatore Tabbone CRIN CNRS/INRIA Lorraine Campus scientique, B.P Vand uvre-les-nanc CEDEX France Abstract This paper describes a fast corner detection algorithm making use of the Laplacian of Gaussian operator. We propose a general corner model and analze its behavior in scale space. The stud shows that the response of the operator has a stable elliptic etremum which alwas lies inside the corner. Using a multiscale representation limited to two scales and the sole laplacian of Gaussian operator, we build an accurate corner detector with low cost. Promising eperimental results show the ecienc of the detector. 1 Introduction Corners are etremel useful features. The are of great use to solve correspondence problems in optical ow, in structure from motion [, 16, ] and in stereovision [5]. Various methods have been proposed to detect corners and the essentiall divide in two classes. On the one hand, the image is segmented in digital curves before the etraction of maima of curvature [1, 9, 1, 19]. On the other hand, the detector directl operates on gra level images [13, 16, 3]. In latter approaches, corners are usuall dened as points where the rate of change of the gradient direction is maimal. Deriche and Giraudon [7, 1] have shown that these detectors (second class) suer from a delocalization problem, more precisel the etracted corner does not correspond to a real corner in the gra level image. To overcome this problem, the combine two properties: 1) the rotationnall invariant operator DET [] (I I?I ) gives rise to an elliptic maimum inside the corner which moves in scale space on the bisector, ) the laplacian of a corner has a zerocrossing whose position is stable in scale space. To detect corners the compute two images of DET at two dierent scales and match each maimum of the rst image with its correspondent in the other image. This wa, the corresponding zero-crossing is searched on the line passing through these two points. Guidicci [11] presents an algorithm using dierential geometr to dene the characteristics of a corner (i.e., amplitude of the wedge, aperture angle and smoothness of the wedge). Recentl, researchers [1, 17] introduced a new method based on measuring the cornerness of points owing to some rst dierential geometr tools. Other methods which consider a corner as the intersection point between two half contours can be found in [15, 1]. In this paper, we propose a corner detector of the second tpe. A general corner model is proposed and its behavior is analzed. The stud shows that the convolution of the corner and the laplacian of Gaussian operator gives rise to an elliptic point which is alwas inside the corner. To investigate these properties, we propose a detector with low time conpleit and good localization. Section presents a mathematical model of corner and stud its behavior in scale space. Properties of this behavior are investigated in Section 3. Eperimental results are given in Section and a discussion on the ecienc of the detector follows. Analsis of a Scale Space Corner Model In this section, we consider a general corner model and analze its behavior in scale space. More precisel, we show that near a corner, the response of the laplacian of Gaussian 1 has an elliptic etremum (local maimum or local minimum) which is alwas inside the corner, and that the etremum moves in scale space on the bisector of the corner. 1 LOG in what follows

2 .1 Notations Let g denote the one-dimensional Gaussian lter: g(x) = 1 p e? X : where?r F (; ) =?[F (; ) + F (; )] =?u g(u ) (v ) + u g(u) (v); () Following Berzins[3], we work in a coordinate sstem where the unit measure is equal to the scale factor of the lter. To do this, let: = X ; = Y ; Z () = g(t)dt;?1 denote the error function, G denote the two-dimensional Gaussian G(; ) = g()g(); and U denote the step function: U () = if 1 elsewhere :. A General Corner Model A general unit amplitude corner with aperture angle? can be modelled b (see Fig. 1): I(; ) = U (tan()? ) U (? tan( )): The ltered image is given b the convolution theorem: F (; ) = (G I)(; ) = =R 1?1 R 1?1 G(; ) U (tan()(? )? + )U (?? tan( )(? ))dd; =R g() (? tan( )(? ))d? R?1?1 g() (? tan()(? ))d: (1) The rst and second derivatives in and are given b: F (; ) =? sin( )g(u )(v ) + sin()g(u)(v); F (; ) = cos( )g(u )(v )? cos()g(u)(v); F (; ) = sin( )g(u ) (sin( )u (v )? cos( )g(v ))? sin()g(u) (sin()u(v)? cos()g(v)); F (; ) = cos( )g(u ) (cos( )u (v )+ sin( )g(v ))? cos()g(u) (cos()u(v) + sin()g(v)); u = sin()? cos(); v = cos() + sin(); u = sin( )? cos( ); v = cos( ) + sin( ): Figure a illustrates the surfaces obtained b a 3D plot of Eq. for a aperture. We nd a local maimum in all directions inside the corner. As shown in Figure b, the maimum is eactl located on the bisector of the corner and its position depends on the scale factor. An increasing scale (more smoothing) or a sharper angle pushes the local maimum awa from the corner on the bisector. For a right angle, its position is approimatel X = Y = 1:15 and for a sharper angle with aperture, X = :76; Y = 1:31. The point on the opposite side of the zero-crossing is onl minimal in the bisector direction. Using dierential geometr techniques, we have found that this point is hperbolic. Intuitivel, we can eplain this behavior. The local maimum is an accumulation point where two maima of the zero-crossings located on each side of the corner meet. As the scale increases, the slope of the zero-crossing decreases and the accumulation point is pushed awa from the corner. The discussion is the same for a light corner on a dark background. In this case, we have found a local minimum inside the corner and a maimum onl in the bisector direction on the opposite side of the corner. Therefore, the sign of the etremum gives information about the corner contrast. In terms of differential geometr, we can eplain this behavior. The mean curvature H [1] for a surface z = F (; ) is dened b: H F + F + FF + F F? F F F = (1 + F + F ; ) 3 which can be rewritten as H = 1 g (r F n? rf ); where g = (1 + F + F ) 3, n? is the second directional derivative in the direction orthogonal to the gradient. For the smooth corner model (Eq. 1), the n? is zero all along the contour (at the corner, the gradient direction is the same as the corner F bisector one i.e. (;) = tan( + F (;) )), and the mean curvature and the laplacian of F have the same sign

3 because the denominator g is alwas positive. In this case we have: H = k 1 + k ' 1 g r F; where k 1 and k are the maimum and the minimum principal curvatures. Reasoning in terms of dierential geometr (see [] for a review), H has an elliptic on one side of the corner and a hperbolic point on the other side as does the LOG (see Figs. c, d). To summarize, the LOG creates an etremum in the neighborhood of a corner which lies inside the corner on the bisecting line and is stable in scale space. The position of the etremum depends on the corner aperture and the degree of smoothing. Furthermore, its sign gives information about the corner contrast. 3 Etracting Corners 3.1 Algorithm We have shown in the previous section that the position of the etremum is stable. It moves in scale space on the bisector of the corner and never disappears when the scale decreases. The algorithm is roughl the same as in [6]. It consists of the following four steps: compute two images LAP 1 and LAP of LOG responses at two dierent scales 1 <. All convolutions are done in 1D, using the propert of separabilit of the LOG, detect and threshold the local etrema in LAP. We use two thresholds. The rst eliminates false etrema resulting from noise. Usuall, these points have ver low laplacian values. The second computes the cornerness of etrem At each etremum, we compute the cornerness (determinant of the Hessian DET) and we t a quadratic function to have subpiel position, nd for each etremal point P ( ; ) in LAP an etremum P 1 ( 1 ; 1 ) in LAP 1. The search is done in a 9 9 neighborhood centered at the position P. The candidate point is the one having the same sign as P and the higher value of cornerness, search on the line (P ; P 1 ) the corresponding zero-crossing. For the separated LOG convolution, the total number of multiplications required is: M (N? M + 1) ; where the image size is N N and the mask size is M = p. For the sake of comparaison, the total number of multiplications to compute DET of a gra level image is: (6 M + ) (N? M + 1) : The total number of multiplications, after running at the two scales, is roughl 35% less for the LOG operator. Eperimental Results Several eperiments with snthetic and real images were conducted to test the ecienc of the proposed corner detection techniques. All the etracted corners are indicated b the sign and are superimposed on the gra level images. Figure 3a illustrates the results of our approach on an indoor scene. We have shown in Section 3 that the sign of the elliptic point gives information about the corner contrast. We have investigated this interesting propert in Figs. 3b and 3c. Almost all corners have been detected perfectl. 5 Conclusions Using onl a LOG operator, we can detect both corners and edges in a gra level image. To etract corners, we have shown that this operator gives rise to a stable elliptic point which is alwas inside the corner on the bisector. Furthermore, the main advantages of this operator compared to others are the low time and space compleities (rotationnall invariant) and the nice scaling propert (never creates new zero-crossing when the scale increases []). Further work will be devoted to analzing the interactions between two adjacent corners. The have in their neighborhood two close elliptic points which merge at high scale. To recover these points we would like to develop a multi-scale corner detector using a coarse-to-ne strateg. The idea is to detect elliptic points at high scale and to localize corners at low scale. Acknowledgements The author thanks Slvain Petitjean and Djemel Ziou for man helpful comments. References [1] H. Asada and M. Brad. The Curvature Primal Sketch. IEEE Transactions on PAMI, (1):1, 196.

4 [] P.R. Beaudet. Rotationall Invariant Image Operators. In Proceedings of th International Joint Conference on Pattern Recognition, Koto (Japan), pages 57953, 197. [3] V. Berzins. Accurac of Laplacian Edge Detectors. Computer Vision, Graphics and Image Processing, 7:1951, 19. [] P.J. Besl and C. Jain. Invariant Surface Characteristics for 3D Object Recognition in Range Images. In Computer Vision, Graphics and Image Processing, volume 33, pages 33, 196. [5] R. Deriche and O. Faugeras. -D Curve Matching Using High Curvature Points: Application to Stereo. In Proceedings of 1th International Conference on Pattern Recognition, Atlantic Cit, NJ (USA), pages 13, June 199. [6] R. Deriche and G. Giraudon. Accurate Corner Detection: An Analtical Stud. In Proceedings of 3rd International Conference on Computer Vision, Osaka (Japan), pages 667, 199. [7] R. Deriche and G. Giraudon. A Computational Approach for Corner and Verte Detection. International Journal of Computer Vision, 1():111, Februar [] L. Dreschler and H. Nagel. Volumetric Model and 3D Trajector of a Moving Car Derived from Monocular TV Frame Sequences of a Street Scene. Computer Graphics and Image Processing, :199, 19. [9] R. Espelid and I. Jonassen. A Comparison of Splitting Methods for the Identication of Corner-points. Pattern Recognition Letters, 1():793, Februar [1] G. Giraudon and R. Deriche. On Corner and Verte Detection. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, Maui, Hawaii, pages 65655, [11] A. Guiducci. Corner Characterization b Dierential Geometr techniques. Pattern Recognition Letters, (5):31131, December 19. [1] C. Harris and M. Stephens. A Combined Corner and Edge Detector. In Proceedings of th Alve Conference, Cambridge, August 19. [13] L. Kitchen and A. Rosenfeld. Gra-Level Corner Detection. Pattern Recognition Letters, 1():951, December 19. [1] G. Medioni and Y. Yasumoto. Corner Detection and Curve Representation Using Cubic B-Splines. In Proceedings of the IEEE Computer Societ Conference on Robotics and Automation, pages 76769, 196. [15] R. Mehrotra and S. Nichani. Corner Detection. Pattern Recognition, 3(11):13133, 199. [16] H. Nagel. Displacement Vectors from the Second Order Variations in Image Sequences. Computer Vision, Graphics and Image Processing, 1:5117, 193. [17] J.A. Noble. Finding Corners. Image and Vision Computing, 6():111, Ma 19. [1] J.A. Noble. Finding Half Boundaries and Junctions in Images. Image and Vision Computing, 1():193, Ma 199. [19] A. Rattarangsi and R.T. Chin. Scale-Based Detection of Corners of Planar Curves. IEEE Transactions on PAMI, 1():39, April 199. [] M.A. Shah and R. Jain. Detecting Time-Varing Corners. Computer Vision, Graphics and Image Processing, :35355, 19. [1] V. Torre and T.A. Poggio. On Edge Detection. IEEE Transactions on PAMI, ():17163, 196. [] A.P. Yuille and T.A. Poggio. Scaling Theorems for Zero-Crossings. IEEE Transactions on PAMI, (1):155, 196. [3] O.A. Zuniga and R. Haralick. Corner Detection Using the Facet Model. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, Washington DC (USA), pages 337, June I(,)= - b. - ψ I(,) = 1 Figure 1: Corner model. a) D corner model. b) Gaussian output for a right angle ( = =, = ). θ

..15..1..5 -. - - t -6 - - 6 b. -.5.1...6.1.. -.1 - - t -6 - - 6 c. d. -. Figure : a) LOG output for a = angle ( =, = =). b) LOG output in the bisector direction.

5 t b t c. d. -. Figure : a) LOG output for a = angle ( =, = =). b) LOG output in the bisector direction. c) Mean curvature for a = angle ( =, = =) d) Mean curvature in the bisector direction. b. c. Figure 3: a) Corners etracted on an indoor scene. b) Light corners on a dark background. c) Dark corners on a light background

Morphological Corner Detection

Morphological Corner Detection Robert Laganière School of Information Technology and Engineering University of Ottawa Ottawa, Ont. CANADA K1N 6N5 Abstract This paper presents a new operator for corner