Algorithms for the Construction of Digital Convex Fuzzy Hulls

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1 lgorithms for the Construction of Digital Convex Fuzzy Hulls Nebojša M. Ralević Faculty of Engineering, University of Novi Sad Trg Dositeja Obradovića 6, 000 Novi Sad, Serbia Lidija Čomić Faculty of Engineering, University of Novi Sad Trg Dositeja Obradovića 6, 000 Novi Sad, Serbia bstract We consider a digital fuzzy set placed in Oxyplane, with support which is a set of digital points (centroids) in Z. We consider two inds of convexity of a fuzzy set, namely quasi convexity and strong convexity, and we propose two algorithms for the construction of both inds of convex hull of a digital fuzzy set. eywords: fuzzy set, digital fuzzy set, quasiconvexity, convexity, DL-convex hull, digital quasiconvex fuzzy hull, digital convex fuzzy hull. Introduction Convexity is one of the basic properties of sets and shapes. Convex hull H(S) of a set S is defined as the smallest convex set containing S. Convex hull is used as a tool in many different areas. In robotics, for example, it is used in collision detection, while in shape analysis, the difference between the shape and its convex hull is used as a shape descriptor. The problem of the computation of the convex hull has been considered for a long time in many research areas, such as computational geometry, image analysis, and fuzzy set theory. In each of these areas, the problem has specific properties. In computational geometry, a finite set S of points in R n is given, so points have arbitrary real coordinates [6]. In image analysis, points are restricted to have integer coordinates.the notion of the convex hull H(I) of an image I depends on the way a convex set in Z n is defined. Most often, T-convexity, L-convexity, and DL-convexity are used []. In fuzzy set theory, the notion of convexity is also not unique. Most often, quasiconvexity, (strong) convexity, and α-convexity are considered. Moreover, the universal set of a fuzzy set can be either R n or Z n. Here, we consider the problem of the computation of the quasiconvex hull QH() and the (strong) convex hull H() of a fuzzy set, with support supp() which is a digital set, and we propose algorithms for computation of QH() and H() in D. Since a grayscale image I may be considered as a fuzzy set with a discrete domain, our algorithms can be used to find convex hulls of D gray-scale images. Bacground Notions set S in Euclidean space is convex if for any two points P and Q we have: for any λ [0, ], λp + ( λ)q S. The convex hull H(S) of a set S is the intersection of all convex sets that contain S: H(S) = i I{S i S S i, S i is convex}. set S is convex iff H(S) = S. Euclidean plane Oxy (R ) can be digitized using specific regular tessellation; each square is called a pixel. Set D of all pixels (of R ) is called a digital space: D = {[i 0.5, i + 0.5) [j 0.5, j + 0.5) (i, j) Z }. Set R (or region) in the digital space is the union of a set of pixels, and we call it a digital set. The centroid of a pixel is the point of intersection of the two diagonals of the pixel. The union of the centroids of the pixels of R, denoted by R, may be considered as the lattice point or simply a point representation of R. Conversely, R may be called the region representation of R. Clearly, there exists a - correspondence δ between digital space D and set Z, and thus between R and R. We call δ digitalization. Definition Digital straight line segment P P is the set of all pixels that are intersected by Euclidean

2 straight line segment p p, where p, p are centroids (digital points) of pixels P, P. Definition digital region R represented by R is called DL convex if for any two digital points p, p belonging to R, there is a digital straight line segment between P and P whose pixels all belong to R. Definition H DL (R), the DL hull of R is the minimal DL convex set that contains R. n algorithm for the computation of a DL-hull is given in []. Let us present some basic definitions and theorems related to fuzzy sets and their convexity, and extend them on digital fuzzy sets. For more details, see [,, 7]. Definition fuzzy set defined on the universal set X is a set of ordered pairs: = {(x, µ (x)) x X} where µ : X [0, ] is the membership function of in X. Definition 5 The support of a fuzzy set is the set supp() = {x X µ (x) > 0}. Definition 6 n α-cut of a fuzzy set, for α [0, ], is the set: α = {x X µ (x) α }. Definition 7 fuzzy set is a subset of a fuzzy set B, denoted B, iff for all x X. µ (x) µ B (x), Definition 8 fuzzy set of a set R n, given by the membership function µ, is called quasiconvex if µ (λx + ( λ)x ) min{µ (x ), µ (x )} holds for every x, x supp(), and λ [0, ] (see [], [8] ). Theorem fuzzy set is quasiconvex if and only if its α-cuts are empty or convex sets for each α [0, ] (see [5]). Definition 9 fuzzy subset of a set R n, given by the membership function µ, is called convex if µ (λx + ( λ)x ) λµ (x ) + ( λ)µ (x ) holds for every x, x supp(), and λ [0, ]. Theorem fuzzy set is convex if and only if the fuzzy hypograph of, denoted by f.hyp(), is a convex set, where (see [] ). f.hyp() = {(x, t) x R n, t (0, µ (x)]} If is a convex fuzzy set, then for every α [0, ], α-cut of is convex (or empty), but the converse does not hold in general. For example, a fuzzy subset of R, given by its membership function µ (x) = +x, x R is quasiconvex, but it is not convex. Definition 0 (quasi) convex fuzzy hull, (qconv()) conv() of a fuzzy set is the smallest (with respect to relation ) (quasi) convex fuzzy set containing. If is a convex fuzzy set, then the convex fuzzy hull of is equal to. Definition digital fuzzy set is a fuzzy set which is defined on digital space D, i.e., on its point representation D = Z. Definition Let D (or Z ), be a digital fuzzy set. digital quasiconvex fuzzy hull QH() of is the smallest digital fuzzy set, given by the membership function µ, such that supp(qh()) = H DL (supp()), () µ(λp + ( λ)p ) min{µ(p ), µ(p )} () holds for every p, p H DL (supp()), and for all λ [0, ] such that λp + ( λ)p Z. Definition Let D (or Z ), be a digital fuzzy set. digital convex fuzzy hull H() of is the smallest digital fuzzy set, given by the membership function µ, such that supp(h()) = H DL (supp()), () µ(λp + ( λ)p ) λµ(p ) + ( λ)µ(p ) () holds for every p, p H DL (supp()), and for all λ [0, ] such that λp + ( λ)p Z. lgorithms Let be a digital fuzzy set, with membership function µ. Let us construct the DL-hull H DL (supp()), using the algorithm given in []. Let H DL (supp()) = = {(x, y ), (x, y ),...(x n, y n )},

3 n, be the digitalization of H DL (supp()). For (x, y ), let à (x, y, µ( )), where µ( ) = µ ( ) = µ. The sets and à = {Ã(x, y, µ( ),..., Ãn(x n, y n, µ( n )}, are the input sets to both our algorithms.. lgorithm for the Construction of a Digital Quasiconvex Fuzzy Hull QH() Let us present the algorithm for computing a digital quasiconvex fuzzy hull of a digital fuzzy set. s explained above, the input to the algorithm is the set. The idea of the algorithm is to process the input points iteratively, in the non-increasing order of their membership values, and to construct the set FH, by forcing α-cuts of FH to be convex sets. lso, we shall prove that by using this algorithm, the digital quasiconvex fuzzy hull QH() of can be obtained. lgorithm (for computing digital quasiconvex fuzzy hull): step 0: Initialize E = (E is the set of extremal points), H = ( H is the set of internal - nonextremal points). step : Order set by non increasing order of membership function µ and rename the points of as,,..., n, so that µ( ) µ( )... µ( n ). step : Let p = p(, ) be the line determined with points and. If µ( ) = µ( ), go to step. Otherwise, (if µ( ) > µ( )), go to step. step : Find all points of set p whose membership function is equal to µ( ) = µ( ). Find end points E and E of that set, remove them from, and store them in E. Remove from all points p, which belong to line segment E E, such that µ( p ) < µ(e ), and store them in H, with changed membership function µ( p ) = µ(e ) = µ(e ) (see Figure ). Go to step 5. step : Remove the point from and store it in E, with unchanged membership function. Set E E and µ(e ) = µ(e ) = µ( ). Rename points of set so that is the point with the largest membership function, next point is, etc. step : Find all points of set p which have equal membership function as point, together with points E and E (E i and E i ) (together with only one of those two points, if they are equal). Denote with E and E (E i and E i ) such points from all those points (if they exist), that any other point from p either coincides with one of them, or is between them. We also tae that the points of p are in the order E E E E (E i E i E i E i ), where the relation is one of relations: (congruence of points) or (be between two points). Store the points E and E (E i and E i ) in set E, if they are not already in E. If one of them is already an extreme point, assign to it the membership function of the other point, i.e. µ(e i ) = µ(e i ). For example, if E E E, we tae that µ(e ) = µ(e ) (see Figure ). Remove other points p from, and store them in H. Remove from all points q, which belong to the line segment E E (E i E i ), such that µ( q) < µ(e ) (µ( q ) < µ(e i )), and store them in H, with changed membership function µ( q ) = µ(e ) = µ(e ) (µ( q ) = µ(e i ) = µ(e i )). p l E E l - E E i p j q E E i p j Figure : First steps of lgorithm. current α-cut are all on a line. q l l Points of the step 5: Rename points of set so that is the point with the largest membership function, next point is, etc. If p, go to step, where input is again, and output is E i and E i. This iterative cycle is repeated as long as the previous condition is satisfied. Otherwise, denote with and the two end points from p E (E l i E l in Figure ). Introduce the set 0 = {, }. step 6: Join the point with points and from set 0 E. Remove from all points that are in the obtained triangle (together with point ), and store them in H, with changed membership function µ( ) = µ( ). Store in E the point with its membership function. This situation is illustrated in Figure. Set and µ( ) = µ( ) and store point in 0. In that way we get the set = 0 { }. Mar the derived triangle with M( ). step 7: In the next iteration (and those after if they are needed) we tae the first point (from the remain-

4 Figure : does not belong to the line p. Figure : Triangle M( ) is extended with point into triangle. ing points, if they exist) from set. Because of the renaming of points of the set, that point is. If =, then go to step 8. Store in E the point with unchanged membership function, and remove from. If on the derived line segments or in the interior of the derived polygon in current iteration there are left some of the extreme points with the same membership function µ( ), remove them from set E and store them in set H with unchanged membership function. From points of set = {,, } ( r = {,,,..., i, i, i+,..., j, j, j+,..., m }) and point we mae another set ( r ), which contains vertices of the convex hull of set { } ( r { }). In Figure, = {,,, }, in Figures and 5, = {,, }. In derived set ( r ) we also rename indexes and denote point with in the case lie in Figure, or with in the cases as in Figures and 5 (with i for some index i in the general case). Figure 5: Triangle M( ) is extended with point into triangle. If point is colinear with some two neighboring points from ( r ), lie in Figure 5 and the middle point of that three points (in Figure 5, it is ) has membership function equal to µ( ), remove it from E and store it in H. i + i i - i + j- j j+ m Figure 6: does not belong to the convex polygon M( r ). new convex polygon is constructed. Figure : Triangle M( ) is extended with point into quadrangle. Denote with M( ) (M( r )) polygon determined with points from ( r ). Remove from all points from (M( ) \ M( )) ( (M( r ) \ M( r ))), and store them in H, with improved membership function, from µ( ) to µ( ), if that value is less than µ( ) (it is possible that µ( ) < µ( ) or µ( ) = µ( )). step 8: F H = E H. The end. Let us denote by δ ( F H ) the digital fuzzy set which corresponds to a fuzzy set F H, i.e., the digital fuzzy set whose support consists of a set of pixels P = δ (p), where p supp( F H ), with the same membership function (µ(p ) = µ(p)). Theorem The digital fuzzy set δ ( F H ), derived by the previous algorithm, is the digital quasiconvex

5 fuzzy hull QH() of set. Proof. ny α-cut of the set FH is by construction either a point or a line segment (this is the case when this line segment belongs to the line p = p(, ) from the algorithm), or a polygon M( r ) with its interior. In both cases, this α-cut is a convex set, and so by Theorem the set FH is a convex set. By construction, FH is the smallest fuzzy set such that () holds. supp( FH ) is equal to the support of, which is by construction (in the preprocessing step) the digitalization of the DL-hull of the support supp() of the fuzzy set. Figure 7: Illustration of the lgorithm.. lgorithm for the Construction of a Digital Fuzzy Convex Hull H() Let us present the algorithm for computing a digital convex fuzzy hull H() of a digital fuzzy set. s already explained, the input to the algorithm is the set. Let us denote by 0 the set embedded in R, i.e., 0 = { (x, y, 0),..., n (x n, y n, 0)}, and let à = {Ã(x, y, µ ),..., Ãn(x n, y n, µ n )}, with µ = µ( ), =,..., n. The idea of the algorithm is to adapt the gift-wrapping method for the computation of the (discrete) convex hull H(S) of a set S of points in R. When we apply the adapted algorithm on the set S = à 0, we obtain its convex hull H(S), and we simultaneously construct the fuzzy set FH. The adaptation is based on the fact that we do not need the bottom and the side facets of the convex hull H(S) of S, and only the upper part of H(S) needs to be constructed. Because of the special geometric position of the points in S, at each step of the gift-wrapping algorithm we need to tae into account only the subset of input points, and also we may eliminate some points of S from any further consideration at each step of the algorithm. lgorithm (for computing digital convex fuzzy hull): step : Initialize E = (E is the set of extremal points), L = (L is the list of unprocessed facets, and the unprocessed edges of the facets), H = ( H is the set of internal - nonextremal points). step : Let à à be the point with maximum z coordinate, and let à à be such that the angle between a vector parallel to the line p(ã, Ã), and vector = (0, 0, ), is minimal. Next, let σ0 = σ 0 (Ã, Ã, Ã) be the plane determined by non-colinear points Ã, Ã, and à from Ã, such that the angle between the normal vector to σ 0, and the vector is minimal. (We rename the points, if necessary.) Note that all points of à are either in the plane σ 0, or below it. Let us find all points à from Ã, which belong to σ 0. Let us denote by S 0 the convex polygon, which is the convex hull of the set of such points, and by Ẽ, Ẽ,..., Ẽr, the set of vertices of S 0. (This can be done by finding the projection of such points on the Oxy plane, and using some two dimensional convex hull algorithm.) Let us remove the points E, E,..., E r, which correspond to vertices Ẽ, Ẽ,..., Ẽr of S 0, from into the set E of extremal points. Each time we remove some points from, we remove simultaneously the corresponding points à from the set Ã. convex polygon S 0 is a facet of the convex hull H(S). We put S 0 in the list L of facets of H(S), together with the list (ẼẼ, ẼẼ,, Ẽr Ẽr, ẼrẼ) of edges of S 0, ordered counter-clocwise. We remove from the points, which correspond to the remaining (non-extremal) points à σ 0, and put them in the set H.We find points l pr(s 0 ), for which Ãl(x l, y l, µ l ) σ 0. We remove the corresponding points l from and we put them in H with new membership value µ( l ) so that (x l, y l, µ( l )) σ 0. step : Let Ẽ Ẽ be one edge of the polygon S 0. Let us find a vertical plane through edge ẼẼ, and let πẽ Ẽ be the halfspace determined by that plane, which does not contain S 0. Let Ã, be the point from the set Ã, such that à belongs to πẽ Ẽ, and that the angle between the normal vector to the plane σ = r(ẽ, Ẽ, Ã) and vector = (0, 0, ) is minimal. Let us find the convex hull S of the set of points of B = à π Ẽ Ẽ σ. We remove the projections

6 of the vertexes of S from, and we put them in E. We remove from the points, which correspond to the remaining (non-extremal) points à of S, and put them in the set H.We find points l pr(s ), for which Ãl(x l, y l, µ l ) σ. We remove the corresponding points l from and we put them in H with new membership value µ( l ) so that (x l, y l, µ( l )) σ. We put the convex polygon S in the list L, together with the list of its unprocessed edges ordered counterclocwise. We remove the edge ẼẼ from L. We repeat step with each unprocessed edge of S 0. We iteratively process all facets which we retrieve from the list L, by iteratively processing the unprocessed edges of such facets. The iteration process stops when =, or L =. The end. F H = E H. Let us denote by δ ( F H ) the digital fuzzy set which corresponds to a fuzzy set F H, i.e., the digital fuzzy set whose support consists of a set of pixels P = δ (p), where p supp( F H ), with the same membership function (µ(p ) = µ(p)). Theorem The digital fuzzy set δ ( F H ), derived by the previous algorithm, is the digital convex fuzzy hull H() of set. Proof. By the preprocessing step of the lgorithm, H DL(supp()) = = {(x, y ), (x, y ),...(x n, y n )}, so the equation of Definition is satisfied. The convex hull H( 0 Ã) = H(0 ÃF H) (where à F H = {(x, y, µ ) (x, y ) FH }) obtained by our modification of the gift-wrapping method is convex, so the fuzzy hypograph of FH is convex. By Theorem, fuzzy set FH (with membership function µ determined by the algorithm) is also convex, i.e., condition of Definition is satisfied. By construction, δ ( F H ) is the smallest fuzzy set satisfying the conditions of Definition. Thus, QH() = δ ( F H ). Conclusion and Future Wor We introduced the notions of a digital quasiconvex fuzzy hull QH(), and of a digital convex fuzzy hull H(), of a given digital fuzzy set. QH() and H() are both defined on a set of pixels, and they both have the same support H DL (). QH() is a quasiconvex Figure 8: Illustration of the gift-wrapping algorithm. set, while H() is a convex set. We proposed an algorithm for the construction of both digital fuzzy hulls. We intend to extend this wor by considering other notions of convexity of digital fuzzy sets, such as α- convexity. References [] N. E. mmar, Some Properties of Convex Fuzzy Sets and Convex Fuzzy Cones, Fuzzy Sets and Systems 06, 8-86, 999. [] B. B. Chaudhuri,. Rosenfeld, On the Computation of the Digital Convex Hull and Circular Hull of a Digital Region, Pattern Recognition, Vol., No., pp , 998. [] L. Čomić, N. M. Ralević, On Convex Hulls of Fuzzy Sets, 6 th International Symposium Interdisciplinary Regional Research, Hungary- Romania-Yugoslavia, Novi Sad -. otobar 00. [] N. M. Ralević, V. Ćurić, M. Janev, lgorithm for computing the digital convex fuzzy hull, th Serbian-Hungarian Joint Symposium on Intelligent Systems, September 9-0, 006, Subotica, Serbia and Montenegro (006), [5] J. G. lir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and pplications, Prentice Hall PTR Upper Saddle River, New Jersey, 995. [6] F. P. Preparata, M. I. Shamos, Computational Geometry: n Introduction, Springer Verlag, New Yor, 985. [7] N. M. Ralević, Digitals objects and convex fuzzy sets, cta Mechanica Slovaca (accepted). [8] X. Yang, Note on Convex Fuzzy Sets, Fuzzy Sets and Systems 5, 7-8, 99. 5