OBJECTS RECOGNITION BY MEANS OF PROJECTIVE INVARIANTS CONSIDERING CORNER-POINTS.

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1 OBJECTS RECOGNTON BY EANS OF PROJECTVE NVARANTS CONSDERNG CORNER-PONTS. Vicente,.A., Gil,P. *, Reinoso,O., Torres,F. * Department of Engineering, Division of Sstems Engineering and Automatic. iguel Hernandez Universit, Avda. del Ferrocarril, s/n Elche Spain suni@umh.es, o.reinoso@umh.es, *Department of Phsics, Sstems Engineering and Theor of the Signal. Universit of Alicante, Crtra. San Vicente, s/n 8 Alicante Spain pgil@disc.ua.es, medina@disc.ua.es ABSTRACT This paper presents an object recognition technique based on projective geometr for industrial pieces that satisf geometric properties. First at all, we consider some methods of corner detection which are useful for the etraction of interest points in digital images. For object recognition b means of projective invariants, an ecessive number of points to be processed supposes a greater compleit of the algorithm We present a method that allows to reduce the points etracted b different corner detection techniques, based on the elimination of non-significant points, using the estimation of the straight lines that contain those points. Secondl, these groups of points are then used to build projective invariants which allow us to distinguish one object from another. Eperiments with different pieces and real images in gre-scale show the validit of this approach. Kewords: object recognition, corner detector, projective invariants, projective geometr.. NTRODUCTON Object recognition is an essential part of an highlevel robotic sstem. n the last ears, there has been variet of approaches to tackle the problem of object recognition. However, there is not a general technique that allows to recognize an tpe of objects, independentl of its intrinsic properties (color, shape, teture, size,...). Eisting recognition techniques can be classified b the wa objects are represented in the model data base or b the tpe of features use. On the first kind we can find geometric representations (silhouettes, superquadrics, algebraic surfaces, comple representation,..) and appearance-base representations, where wide variet of techniques eist, differing in which image information is used and the how the data is stored, but in all, the representations are learned from the images. B the tpe of features used, we can find techniques based on global features (such as area, compactness,..) or on local features (line segments, ). These approaches are clearl limited to a certain applications, and perhaps the snthesis of sets of these techniques might resolve the problem of object recognition. On the other hand, the projective geometric provides new tools that allow a generic recognition of the object, through the emploment of geometric invariants of points, straight lines or conics eisting in the object. [Bandlow98], [Tarel]. The remainder of the paper is organized as follows. n section we review the corner detection techniques emploed to detect significant points on the image. n section we present a method that allows to eliminate those points previousl detected that correspond with false corner points. n following sections we present some concepts of the projective invariant theor. Based on these concepts in section we show the procedure to construct the projective invariants with the selected points and the algorithm

2 proposed to recognized the objects in the image. Finall in section 6 we show some eperiments carried out with the proposed procedure.. CORNER DETECTON Vision-based object recognition begins with an etraction of global or local features from the object image. n our recognition sstem the features etracted are the corners from the image object. We can define a corner as the image points belonging to a contour where the contour presents a local maimum curvature or as the intersection of two o more contours. [Deriche9]. Kitchen proposed a measure of cornerness based on the changed of gradient direction along an edge contour multiplied b the local gradient magnitude (Eq.), the maimum values of k show the possible corners [Kitchen8]. k + + () Noble given a theoretical formulation for the corner detection problem using differential geometr [Noble88]. There are a lot of corner detection techniques. ost of them can be classified into two groups: the first group are based in operations over an object image with a pre-etraction of edges, and the other one consists of approaches that work directl at the gra scale level. n the eperiments presented in following sections, we used classical techniques as the Beaudet, Kitchen, Noble and Harris detectors, and one more recent, the SUSAN detector which provided better results for the pieces we have considered. (a) n [Beaudet78] was proposed a rotationall invariant operator called DET. This operator is obtained using a second order Talor s epansion of the intensit surface (,): DET () The corner detection is based on the thresholding of the absolute value of the etrema of this operator. DET can be estimated as the Hessian determinant, H, which is related to the product of the principal curvatures k min k má, called the Gaussian Curvature: H () DET kmin kma + + () For a piel (,): if k min k má, the piel is a parabolic point; if k min k má >, the piel is an elliptic point and if k min k má <, the piel is a hperbolic point. DET and the Gaussian Curvature have the same sign because the denominator of Eq. is alwas positive. Near a corner, DET gives a positive and negative response on both sides of the edge. (b) (d) (c) (e) (a) Original. (b) Beaudet. (c) Kitchen. (d) Noble. (e) SUSAN. Figure The corner detector SUSAN (Smallest Univalue Segment Assimilating Nucleus) [Smith97] measures the certaint that a piel of the image be a corner as of its area USAN, that is defined like the total number of piels in a neighbourhood that the have similar values of intensit, in certain degree, to

3 the value of intensit of the piel that is considered. The detector SUSAN situates a circular mask around the piel respected as the nucleus, calculates the area USAN, decides if the piel is a corner reducing the size of the area USAN of a specific threshold (normall the half or less than the total area USAN), and eliminates the false corners b means of a suppression of not most maimum. To compare these corner detectors techniques in Fig. we show the results obtained using a snthetic image. As we can see, SUSAN corner detector offers best results over objects with ver defined boundaries. The objects we are used in our eperiments have this propert.. ELNATON OF NON-SGNFCANT PONTS. Afterwards, the points that belong to each of the straight lines (horizontal, vertical or inclined) are ordered based on their values as coordinates and/or coordinates. Once these points have been ordered, we eliminate all the points that belong to the straight line but are not the ends. Thus, we eliminate all the inner points. During the process of eliminating the false corner points, we ma find points that satisf more than one equation of the straight line. Such points are internal points of the straight line r i which can be the ends of other straight lines r j. For such cases, a previous pre-processing is required before its elimination as inner points. An eample of the elimination of points is shown in Fig.. Often, the corner detectors analsed recognize certain points as corners which do not have the characteristic to be corner-points (false corner points). We consider a corner-point an point at which two or more edges converge. Similarl, an edge is considered as an of the borders of object s contour which separates two planes of different surfaces or which indicates an abrupt change of luminance. n this section a method is proposed which allows to eliminate false corner-points. This method is based on the knowledge of the possible straight lines present in the input image. There are a lot of procedures that allow us to determine the straight line that best approimates a set of points (border detection, Hough transform). For each one of the possible corner-points detected P j (,), a set of point lists lr i is constructed. This set indicates whether the points belong to each one of the different straight lines detected r i. n order to determine whether a point belongs to a straight line, we verif if the point satisfies the straight-line equations allowing a threshold, ε. The selection of the threshold is important, so if ε is too great, we eliminate all the points in the image. According to the tpe of straight lines, we have differents sets: U Pj / j ai j + bi ± ε lri U Pj / i j ± ε U Pj / i j ± ε () where lr i is the i-set formed b all the corner points that belong to a straight line r i, and a i and b i are the straight line coefficients obtained b the Hough transform. (a) (b) (c) (a) Original. (b) SUSAN. (c) Elimination Non-corners. Figure

4 . BASC THEORY OF PROJECTVE NVARANT. Projective geometr is based on the geometr inherent inside a central transformation (camera model). n transformations more usual, as Euclidean or Similarit transformations eist invariants properties well known, as the length and the area in the Euclidean or the ratio of lengths and the angles in the Similarit transformation. n Projective transformation there also are invariants, the most known is the ratio of ratio of lengths (or cross-ratio). n this section, we review some aspects of projective invariants built with points from the images. [Hartle]. Given five general D points P i, i,.., of an object, and p i, i,.., which correspond to points of the image plane, respectivel, two projective invariants on a plane are defined as follows, [Song]: (6) where ijk represents the determinant of a matri, which is composed of the coordinates of three points p i,p j,p k of an image plane and the corresponding points of an object plane (coordinates of the world), respectivel. We rewrite Eq. as follows: (7) Nevertheless, the representations of these invariants, are sensitive to changes. Therefore, the value of an invariant associated to a set of points depends frequentl on the order in which these points are considered during their computation. This is wh J is introduced as a more stable invariant when there is a permutation of points: p p p p p p p p p p p p p p p p p p p p p p p p J ( J, J, J, J, J ) (8) For an object D, a point is represented in homogeneous coordinates as P (P,P,P,), and for an image, p(p,p,). λ λ λ J ( λ ), J ( λ ), J, J, J λ λ λ λ λ [ λ ] J with λ 6 6λ + 9λ 8λ + 9λ 6λ + λ 6 λ + λ λ + λ λ + (9) where λ can fulfil an of the following relations, for λ and λ. λ ( p, p, p, p, p ) λ (,,, ) p p p p, p () n this wa, the J invariant (Eq.9) obtained b means of the procedure is independent of the order of the points chosen to calculate it. Projective invariants pla an important role in the method proposed in following section to recognize the objects in the image. Due to five points in a plane of an object (P, P, P, P, P ) determine the invariant, and the projection of these points of the object over the image plane (p, p, p, p, p ) determine the same invariant, if we stud the invariants generated through the corner points of the image we will know the object with those invariants.. ETHODOLOGY. n this section, we eplain how to recognize pieces using projective invariants of these objects. The main idea consist of to calculate off line some projective invariants of the model of the objects that we have to recognize on line. These projective invariants constitute the data base of the reference invariants. Then, we calculate all the possible projective invariants with the corner detected in the image and matching them with the data base of the reference invariants previousl calculated. From the input images, corner points are etracted (Fig.). The corners can be obtained using different corner operators or corner detectors (see section ). We should mention that some of these cornerdetectors detect small piel-regions as corners. That is to sa, the detected corners are formed b several piels and it is necessar the compute the centres of gravit to reduce such regions to one single piel. Afterwards, we construct candidate sets of five points and for each combination (set of five points) the invariant J i is computated. As the number n of combinations C (Eq.) is ver high, the computational cost is increased eponentiall. This factor forces the reduction of the number of possible combinations to avoid an overload of the sstem. One

5 constraint that must be added is not to calculate the invariants of combinations that have three or more collinear points. n! C n ( n )!! () The J i calculated are compared with the models of the pieces, defined as Jref i. (See Fig. ). between all reference and the image invariants), but the average of the four minimum errors n Fig., we illustrate the algorithm developed to recognize the objects in the image through its projective invariants. corners detection Each of these vectors of J-invariants (calculated as combinations of points) is compared to a vector or several vectors of reference J-invariants. The vectors of reference J-invariants identif each one of the models stored in the database. As such, for each piece stored in the database we have a set of reference J-invariants. possible combinations selection of candidate corners generation of corners combinations START combination n? es input image no -points combination corner detection model of object no suitable combination? set of canditate corners es compute J-invariants compute reference J- invariants computation of J-invariants reference J-invariants matching. search and compare END no Object Recognition Sstem. Figure minimum error? es The degree of similarit is measured b the mean squared error. The minimum error will provide the correct matching, and it allows us to state which piece-model of the database is represented in the image captured. Due to in an image several corner piels would be detected, a great number of invariants would be generated. Some of these invariants would be ver similar to other of the different models. So, to determine the model of the piece with the better match with the object in the image it is necessar taking into account not onl the best match (minimum error object identification Recognition process detailed. Figure

6 6. EXPERENTS ON REAL AGES. The proposed recognition technique has been evaluated on real images from different pieces. The sstem has eight different models or pieces. Some of the pieces are shown on Fig., all of them have an side with or more corners. /Error RECOGNTON BETWEEN ODELS model A model B model C model D Different eperiments have been carried out, b changing the size of the set of the model J- invariants (Jref i.). On Fig. 6 we can see the error obtained during the matching with the J i and Jref i using just a set of Jref i. Fig.6 shows the results achieved after comparing images with the reference invariants of models. To compare these invariants we have chosen the average of the minimum errors between ever image invariant and the reference invariants of each model. We have represented the inverse of this average error. Fig. 7 show the same eperiment but we have taking into account the average of the minimum 8 errors (8 Jref i ), comparing again the images with the reference invariants of models. Adding a noise equivalent to the displacement of the corner detected approimatel piels, does not introduce error in the recognition. n Fig. 8 we illustrate the results achieved considering an error in the corner detection method. As we can see the images evaluated allow perfectl to separate the model A and B and recognize model A as the object in the image. Finall in Fig.9 we show the results achieved when we compare four images of two different models (image and from object B, and image and from object D) with the reference invariants of all possible models. /Error 6 8 mages /Error Figure 6 RECOGNTON model A model B 6 8 mages 9 Figure 7 RECOGNTON ADDNG NOSE model A model B (a) (b) Two objects from the set. Figure 6 8 Nois images Figure 8

7 /Error ODEL SEPARATON image image image image [Song] Bong Seop Song, l Dong Yun, Sang Uk Lee: A target recognition technique emploing geometric invariants. Pattern Recognition,, pp. -,. [Tarel] Tarel, J.P, Cooper, D.B.: The Comple Representation of Algebraic Curves and ts Simple Eploitation for Pose Estimation and nvariant Recognition. EEE Transactions on Pattern Analsis and achine ntelligence, Vol., No 7, pp ,. A B C D E F G H odels Figure 9 7. CONCLUSONS. n this paper, an object recognition sstem through projective invariants has been presented. The algorithm proposed allows to recognised the object in the image comparing the J-invariant of several points in the image with the J-invariants of ever model previousl calculated. Furthermore, the method emploed allows us to partiall reduce the false points detected, and so reducing the search space for the calculated J-invariants. Eperiments show that just a small set of J-invariants can identif the object, reducing the computational cost. REFERENCES [Bandlow98] Bandlow,T, Hauck,A, Einsele,T, Färber,G.: Recognising Objects b their Silhouette. macs Conf. on Comp. Eng. in Sstems Appl., pp , 998. [Beaudet78] Beaudet,P.R.: Rotational nvariant mage Operators. nt. Conf. Pattern Recognition, pp.79-8, 978. [Deriche9]Deriche,R, Giraudon,G.: A Computational Approach for Corner and Verte Detection. nternational Journal of Computer Vision, :, pp.-, 99. [Hartle] Hartle,R, Zisserman, A.: ultiple View Geometr in Computer Vision, Cambridge Universit Press,. [Kitchen8] Kitchen,L, Rosenfeld, A.: Gra Level Corner Detection. Pattern Recognition Letters, pp.9-, 98. [Noble88] Noble, J.A.: Finding Corners. mage and Vision Computing, Vol. 6, pp.-8, 988. [Smith97] Smith, S., Brad, J..: SUSAN: A New Approach to Low Level mage Processing. nternational Journal of Computer Vision, pp. -78, 997.