Since contours of objects in igital images are istorte ue to igitiation noise an ue to segmentation errors, it is necessar to neglect the istortions w

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1 Continuit of the iscrete curve evolution L. J. Latecki a, R.-R. Ghaiall a, R. Lakamper a, an U. Eckhart a a Dept. of Applie Mathematics, Univ. of amburg Bunesstr. 55, 0146 amburg, German ABSTRACT Recentl Latecki an Lakamper (Computer Vision an Image Unerstaning 73:3, March 1999) reporte a novel process for a iscrete curve evolution. This process has various application possibilities, in particular, for noise removal an shape simplication of bounar curves in igital images. In this paper we prove thatthe process of the iscrete curve evolution is continuous: if polgon Q is close to polgon P, then the polgons obtaine b their evolution remain close. This result follows irectl from the fact that the evolution of Q correspons to the evolution of P if Q approimates P. This intuitivel means that rst all vertices of Q are elete that are not close to an verte of P, an then, whenever a verte of P is elete, then a verte of Q that is close to it is elete in the corresponing evolution step of Q. Kewors: iscrete curve evolution, polgon evolution, shape simplication, curve approimation 1. INTRODUCTION Latecki an R. Lakamper in Ref. 1 presente a shape similarit measure for object contours. An application of this measure to retrieval of similar objects in a atabase of object contours is emonstrate in Figure 1, where the user quer is given b a graphical sketch. Figure 1. Retrieval of similar objects base on a similarit measure of object contours. s: flatecki, ghaiall, lakaemper, eckhartg@math.uni-hamburg.e

2 Since contours of objects in igital images are istorte ue to igitiation noise an ue to segmentation errors, it is necessar to neglect the istortions while at the same time preserving the perceptual appearance at the level sucient for object recognition. To achieve this, the shape of objects is simplie b a novel iscrete curve evolution metho, before the similarit measure is applie. This approach allows us to reuce inuence of noise an to simplif the shape b removing irrelevant shape features without changing relevant shape features. The robustness of the iscrete curve evolution metho with respect to nois eformations has been verie b numerous eperiments (e.g., see Ref. or online emos on our web site 3 ). The continuit theorem (Theorem 1 in Section ) gives a formal justication of this fact: if a polgon Q is close to a polgon P,e.g.,Q is a istorte version of P, then the polgons obtaine b their evolution remain close. Thus, continuit guarantees us the stabilit of the iscrete curve evolution with respect to noise. Moreover, the igital curve evolution allows us to n line segments in nois images, ue to the relevance orer of the repeate process of lineariation (see e.g., Figure ). This fact follows from Theorems 1 an, which weprove in Section, since if polgon Q is close to polgon P, then rst all vertices of Q are elete that are not close to an verte of P (Theorem 1), an then, whenever a verte of P is elete, then a verte of Q that is close to it is elete in the corresponing evolution step of Q (Theorem ). Therefore, the linear parts of the original polgon will be recovere uring the iscrete curve evolution. Figure. (a)! (b): noise elimination. (b)! (c): etraction of relevant line segments. We assume that a close polgon P is given. In particular, an bounar curve in a igital image can be regare as a polgon without loss of information, with possibl a large number of vertices. We enote the set of vertices of P with V ertices(p ) an the set of eges with Eges(P ). The iscrete curve evolution prouces a sequence of polgons P = P 0 ::: P m such that jv ertices(p m )j 3, where j : j is the carinalit function. The process of the iscrete curve evolution is ver simple. The outline of the algorithm is the following. For ever evolution step i =0 ::: m; 1: 1. Each verte v in P i is assigne a relevance measure K(v P i ).. ApolgonP i+1 is obtaine after the vertices whose relevance measure is minimal are elete from P i. The relevance measure K(v P i ) is ene below. A few stages of our curve evolution are illustrate in Figure 3. The process of the iscrete curve evolution is guarantee to terminate, since in ever evolution step, the number of vertices ecreases b at least one. It is also obvious that this evolution converges to a conve polgon, since the evolution will reach astate where there are eactl three, two, one, or no vertices in P m. Onl when the set V ertices(p m ) is empt, we obtain a egenerate polgon equal to the empt set, which is triviall conve. Thus, we obtain for ever relevance measure K Proposition 1. The iscrete curve evolution converges to a conve polgon, i.e., there eists0 i m such that P i is conve, an if i<m,allpolgons P i+1 ::: P m are conve.

3 Figure 3. A few stages of our curve evolution. The rst contour is a istorte version of the contour on www-site. 4 Now we give a precise enition of the iscrete curve evolution. Denition: Let an K min (P i )=minfk(u P i ):u V ertices(p i )g V min (P i )=fu V ertices(p i ):K(u P i )=K min (P i )g i.e., V min (P i )contains the vertices whose relevance measure is minimal in P i for i =0 ::: m; 1. For a given polgon P an a relevance measure K, we call iscrete curve evolution a process that prouces a sequence of polgons P = P 0 ::: P m,wherejvertices(p m )j3, such that Vertices(P i+1 )=V ertices(p i ) n V min (P i ): An algorithmic enition of the iscrete curve evolution is given in Ref. 5 (see also our www-site 3 ). The ke propert of this evolution is the orer of the substitution. The substitution is one accoring to a relevance measure K(v P i )=K(u v w), where u w are neighbor vertices of verte v in P i. The relevance measure K(v u w) isgiven b the formula K(v u w) = l 1l l 1 + l (1) where is the turn angle at verte v in P i, l 1 is the length of vu, an l is the length of vw. (Both lengths are normalie with respect to the total length of polgon P i.) The main propert of this relevance measure is the following The higher the value of K(v u w), the larger is the contribution to the shape of the polgon P i of arc vu[ vu. A motivation for this measure an its properties are escribe in Ref. 5. Observe that the relevance measure is not a local propert with respect to the polgon P, although its computation is local in P i for ever verte v. We prove in Theorem 1 that the iscrete curve evolution is continuous: if polgon Q is close to polgon P, then the polgons obtaine b their evolution are close. Moreover, we show that the evolution of Q will correspon to the evolution of P if Q approimates P (Theorem ). Before we state these theorems, we nee a few more enitions. Denition: Let p(p )=minf(e v) :E Eges(P ) an v Vertices(P ) an v 6 Eg, i.e., p(p ) is the minimal istance from vertices to eges the o not belong. For eample, in Figure 4 p = p(p ) is equal to the istance from verte B to ege CD. Denition: For ever E Eges(P )we call Strip (E) =B(E )=S E B( ) -strip of P aroun E an we enote the set of all strips of P b Strips (P )=fstrip (E) :E Eges(P )g. Figure 4 shows the set of all strips of P. Denition: Two strips Strip (E) anstrip (F )areajacent if sies E an F are ajacent, i.e., intersect in a verte of P.

4 A C B p G F γ D E Figure 4. Observe that if < p(p ), then the intersection of two non-ajacent -strips of P is empt. Denition: For a close polgon P an for ever v 0 V ertices(p ), there eist eactl two linear orers < P on V ertices(p )=fv 0 v 1 ::: v k+1 = v 0 g such thatv 0 < P v 1 < P ::: < P v k+1 = v 0 an v s v s+1 is an ege of P, where is the line segment joining point to. We enote the set of all such linear orers on Vertices(P )b O P. Clearl, jo P j =jv ertices(p )j. Clearl, the set V ertices(p ) together with one of the linear orers < P uniquel etermines Eges(P ). An linear orer < P O P can be etene to a linear orer on all points of P, i.e., on S Eges(P ). Denition: For an two points in a -strip S aroun an ege uv of P,wesa that < S i () < P (), where : S! uv is the metric projection. Denition: A polgon Q has the same orer as polgon P if there eists a function i : V ertices(q)! Strips(P ) an orer < P O P an an orer < Q O Q such that q i(q) forallq V ertices(q), q 1 < Q q i (i(q 1 ) < P i(q )) or (i(q 1 )=i(q ) an q 1 < i(q1) q in -strip i(q 1 )). For eample, see Figure 4. If <, where is the smallest istance between two vertices of Q, then no two vertices of Q can be projecte b to the same point. Denition: We sa that V ertices(p )isaneighbor (verte) of V ertices(p ) if line segment is an ege of P. We alsosa in this case that an are ajacent. Denition: A polgon Q is calle an ( )-approimation of polgon P if Q B(P ), Q has the same orer as polgon P, an the minimal length of eges of Q equal to. (see e.g., Figure 4).. CONTINUITY OF TE DISCRETE CURVE EVOLUTION Theorem 1 states that the iscrete curve evolution is continuous. Intuitivel, it sas that if polgon Q is sucientl close to a polgon P, then the polgons obtaine b theevolution of Q will remain close to P. This will be the case until a stage k of the evolution of Q is reache at which eactl one verte of Q k is containe in B(w ) forever verte w of P an all other vertices of Q are elete. Theorem 1. Continuit of the iscrete curve evolution. Let P be a simple polgon. For ever 0 <an ever 0 <<, there eist >0 an k Z + such that if polgon Q is ( )-approimation of P,then an 8i f0 1 ::: kg Q i B(P ) 9 bijection b : V ertices(q k )! V ertices(p ) 8v V ertices(q k ) 8w V ertices(p ) [v B(w ), w = b(v)]:

5 Proof: Let0<an 0 << be given. Let 0 be given b Theorem 3, below. Let be etermine b Proposition, below, for =minf 0 g. Let polgon Q be ( )-approimation of P. Then, b Proposition, Q is also a ( )-approimation of P for which aitionall hols that for ever verte v of P at least one verte of Q is containe in B(v ). an Since 0,we obtain b Theorem 3 for = that there eists an evolution stage k of Q such that 8i f0 1 ::: kg Q i B(P ) 9 bijection b : V ertices(q k )! V ertices(p ) 8v V ertices(q k ) 8w V ertices(p ) [v B(w ), w = b(v)]: Since, 8i f0 1 ::: kg Q i B(P ) an since at most one v V ertices(q k ) can be containe in B(w ) for w Vertices(P ), ue to <,we obtain which proves the theorem. 8v V ertices(q k ) 8w V ertices(p ) [v B(w ), w = b(v)] Theorem sas that if Q k is a sucientl close approimation of P an vertices of Q k correspon to vertices of P, then the iscrete curve evolution of Q k follows the evolution of P. Theorem. Let P = P 0 ::: P m be polgons obtaine from a polgon P in the course of iscrete curve evolution such that P m is the rst conve polgon an all minimal relevance measures are obtaine for onl one verte, i.e., jv min (P i )j =1for i =0 ::: m ; 1. There eists 0 < such that if polgon Q k be ( )-approimation of P an 9 bijection b : V ertices(q k )! V ertices(p ) 8v Vertices(Q k ) 8w Vertices(P ) [v B(w ), w = b(v)] then for the evolution stages i f0 1 ::: m; 1g of P we have: v V ertices(q k+i ) n V ertices(q k+i+1 ), b(v) V ertices(p i ) n V ertices(p i+1 ): Proof: Let an let C i =minfk(u P i ) ; K min (P i ):u Vertices(P i ) an K(u P i ) 6= K min (P i )g C min < minfc i : i =0 ::: m; 1g: If u w are neighbors of verte v in P i, then the relevance measure is enote b K(v P i )=K(u v w). Since K(u v w) is a continuous function of l 1 = jvuj l = jv wj, an the turn angle at v, it is also continuous with respect to the position of vertices: 8C >0 9 1 > 0[u 0 B(u 1 ) v 0 B(v 1 ) w 0 B(w 1 )] ) jk(u v w) ; K(u 0 v 0 w 0 )j <C: We take to be the 1 > 0 obtaine for Cmin an for all triples of vertices v u w in P. Thus, we have [u 0 B(u ) v 0 B(v ) w 0 B(w )] ) jk(u v w) ; K(u 0 v 0 w 0 )j < C min an 8v V ertices(q k ) 8w V ertices(p ) [v B(w ), w = b(v)]:

6 Thus, for ever there vertices v u w in Q k wehave [u B(b(u) ) v B(b(v) ) w B(b(w) )] ) jk(u v w) ; K(b(u) b(v) b(w))j < C min : ence for u 1 v 1 w 1 u v w vertices in Q k K(b(u 1 ) b(v 1 ) b(w 1 )) ; K(b(u ) b(v ) b(w )) >C min ) K(u 1 v 1 w 1 ) ; K(u v w ) > 0: Consequentl, jv min (Q k+i )j = 1 for i =0 ::: m; 1. Let u V min (P i ) an V ertices(p i ) n V min (P i ). Then K( P i ) ; K(u P i ) > C min, an therefore, K(b ;1 () Q k+i ) ; K(b ;1 (u) Q k+i ) > 0. Therefore, u V min (P i ) implies b ;1 (u) V min (Q k+i ), an consequentl, for ever i f0 1 ::: m; 1g: v V ertices(p i ) n V ertices(p i+1 ) i b ;1 (v) V ertices(q k+i ) n V ertices(q k+i+1 ): Theorem 3. For ever polgon P an ever 0 <,there eist 0 < 0 < p 5 such that for ever < 0 an for ever ( )-approimation Q of P with the propert that for ever verte v of P at least one verte of Q is containe in B(v ), there eists k such that: 8i f0 1 ::: kg Q i is ( ) ; approimation of P an 9 bijection b : V ertices(q k )! V ertices(p ) 8v V ertices(q k ) 8w V ertices(p ) [v B(w ), w = b(v)]: Proof: LetP be a polgon with the minimal turn angle an with p = p(p ) (see Figure 4). Let Q be ( )-approimation of P for an < p 5. We ivievertices of Q into two tpes: We call ever verte q of Q that oes not belong to B(v ) for an verte v of P Tpe 1 verte. We call ever verte q of Q that is containe in B(v ) for some verte v of P Tpe verte. We will show that there eists 0 < p 5 ( 0 epens on ) an an evolution stage k of Q such that for ever < 0 all Tpe vertices remain an all Tpe1vertices are elete from Q k. Observe that at most one verte q of Q can be containe in B(v ), since <, where is the minimal length of eges of Q. ence, for ( )-approimation Q of P eactl one q of Q is containe in B(v ) for ever verte v of P. Step 1 We rstshow in Lemma 4 (below) that if is a Tpe 1 verte, then an its irect neighbors V ertices(q) are in the same -strip S of P an is between with respect to > Q, i.e., either > Q > Q or > Q > Q. This implies that is between with respect to the orer > S of points in -strip S. Then we show in Lemma 5 (below) that if three points are in the same -strip S of P an is between with respect to > S, then the relevance measure K( ) of [ is boune b a function m 4 of. This implies that K( ) m 4 ( ) forever Tpe 1 verte of Q. Lemma 1. Let V ertices(q) an i() =Strip (uv), where u v V ertices(p ) an v > P u (see Figure 5). If b B(v ) \ V ertices(q) an an b are neighbors in Q, thanb> Q. Proof: LetS = i() anuv be the sie of P containe in S. Let S v be the line segment of length perpenicular to uv such thats v is containe in S an v S v (see Figure 5).

7 u S v / b v S Figure 5. b v Figure 6. a u Since b B(v ), (b S v). The iagonal of rectangle with sie of length an is equal to p 5. Since < p 5, this iagonal is shorter than. Therefore, if b is containe in such a rectangle no other verte of Q can be containe in it. Since i() =Strip (uv), we obtain b> Q. Lemma. Let V ertices(q) an i() = Strip (uv), where u v V ertices(p ) an v >u(see Figure 6). If a B(v ) \ V ertices(q) an b B(u ) \ Vertices(Q), thanb Q Q a. Proof: Ifi(a) 6= i(), then i() > P i(a) implies > Q a. If i(b) 6= i(), then i(b) > P i() implies b> Q. It remains to consier the case i(a) =i() =i(b). We showthatb> Q b showing that the inverse orer > Q b leas to inconsistenc. The proof of > Q a is analog. We assume that > Q b. Then an b cannot be neighbor vertices of Q b Lemma 1. Thus, there eists V ertices(q) such that is neighbor of b an > Q > Q b (see Figure 6). Yet, if i() > P i(), then b> Q. If i() =i(), then b> Q b Lemma 1. Lemma 3. Let be a Tpe 1 verte of Q. If V ertices(q) is a neighbor of (i.e., is an ege of Q), then i() =i() or B(v ) i() for some verte v of P (see Figure 7(a)). i() i() (a) v a (b) v u i() Figure 7. Proof: We assume that i() 6= i(). Then i() > P i() ori() > P i(), sa i() > P i(), which implies > Q. First we show that the intersection i() \ i() is not empt, i.e., i() an i() are neighbor strips. If this were not the case (see Figure 7(b)), then there eists such thati() 6= i(), i() 6= i(), i() \ i() 6=, an \ i() is nonempt. Since is a Tpe 1 verte of Q, there eists u i() \ i() (this intersection is equal to B(v ) for some v P ). If i(u) =i(), then > Q u> Q. If i(u) =i(), then > Q u> Q,b Lemma. Since > Q u> Q contraicts the fact that is an ege of Q, weobtainthati() ani() are neighbor strips.

8 Therefore, there eists v i() \ i() averte of P an a B(v )=i() \ i() averte of Q (see Figure 7(a)). B Lemma, a> Q. Since <p, B(v ) oes not intersect an strip of P other than i() an i(). ence, i(a) =i() ori(a) =i(). If i(a) =i(), then > Q a,sincei() > P i(). If i(a) =i(), then = a or > Q a,b Lemma. If = a, then = a B(v ). If >a,then>a>, which isinconstant with the fact that is a irect neighbor of. Lemma 4. Let be atpe 1 verte of Q. If Vertices(Q) are ierent neighbors of then i() an is between with respect to < Q. Proof: Since V ertices(q) are ierent neighbors of, clearl, is between with respect to < Q. If i() =i() =i(), then i() is triviall true. If i() 6= i(), then B(v ) i() for some verte v of P,b Lemma 3. Consequentl i(). Similarl, i() ifi() 6= i(). Lemma 5. Let points be in the same -strip S of P an be between with respect to > S. Then the relevance measure K( ) of [ is boune b a function ; m 4 of : K( ) m 4 ( ) = + arcsin( ) : + Proof: LetS be the maimal line segment containe in S that contains points an (). Clearl, the length of S is (see Figure 8(a)). S is similarl ene. Since is between with respect to > S, lies between S an S. ence, the situation in Figure 8(b) is not possible. Let L be the istance between S an S an let R S be the maimal rectangle containe in S with sies S an S. We call L the length of R an the height ofr. S S (a) L (b) Figure 8. Let 0 0 be the enpoints of the sie of R which has length L an which is further awa form than the other sie of length L (Figure 9(a)). Since j 0 jjj, j 0 jjj, an turn angle 0 0 is not smaller than turn angle, we obtain K( 0 0 ) K( ), where jabj is the length of the line segment ab. ence it is sucient to compute the relevance measure for = 0 an = 0 (Figure 9(b)). Let M be the sie of length L of R that is ierent from sie 0 0 an let 0 be the metric projection of on M (Figure 9(b)). Since j 0 0 j j 0 j, j 0 j j 0 j an the angle is not smaller than 0 0, we obtain K( ) K( 0 0 ). If 0 is moving on sie M, thenk( ) K( ) for R. Since K( ) is onl a function of 0,we will enote it b m( 0 )=K( ). Since 0 is moving on sie M, we can assume that 0 [0 L]. Thus, we have K( ) m( 0 ), where R an 0 [0 L]. The function m() with (0 L)isgiven b theformula (see Figure 10(a)) q +(L ; ) p + arctan( L; ) + arctan( ) m() = q +(L ; ) + p () +

9 M M L L (a) (b) Figure 9. where L an are constant. It is eas to obtain b analing function m that m is smmetric with respect to = L, m obtains its minimum at point = L,anmis monotone ecreasing for (0 L ] an monotone increasing for [ L L). Let 0 be the point [0 L] for which the istance to is equal to (see Figure 10(b)). Since <, 0 belongs to interval (0 L). Since m is smmetric, m is ene for [ 0 L; 0 ]. Therefore, m reaches its maimum for = 0, an consequentl, K( ) m( 0 ). arctan(/) 0 arctan(/(l-) (a) L (b) L Figure 10. Since the position of point 0 is a function of an, we will treat m( 0 ) as a function of L, which wewill enote b m. Consequentl, we obtain K( ) m ( L ) =m( 0 ) for [0 L] [0 ]. It remains to n an upper boun of the function m ( L ) for L (0 1). To achieve this, we n a more han formula for m ( L ) =m( 0 ), i.e., we nm 3 ( t) =m ( L ), where t [0 1) ant + is the istance from 0 to (see Figure 11). The function m 3 is given b ( + t) m 3 ( t) = arcsin( ) + arcsin( +t ) + + t : (3) 0 t+ Figure 11. We willshow that ; m 3 ( t) m 4 ( ) = + arcsin( ) (4) + where m 4 ( ) is equal m 3 with t =0. When we multipl m 4 ( ) ; m 3 ( t) 0b( + )( + + t) an simplif it, we get ( + + t) ; t arcsin( ) ; ( + ) ( + t) arcsin( ) 0: (5) + t

10 Since arcsin, we obtain (5) ( + )+t ; t ; ( + ) =0 (6) It follows from Lemmas 4 an 5 that K( ) m 4 ( ) forever Tpe 1 verte of Q. Moreover, it hols lim!0 m 4 ( ) =0forever. Step We show that the relevance measure of a Tpe verte is boune from below. It follows from Lemma 6 that if is a Tpe verte, then either the irect neighbors Vertices(Q) of are in ierent strips or is not between with respect to > Q. Lemma 6. Let b be Tpe verte, i.e., b V ertices(q) an b B(v ) for some verte v of P. Let V ertices(q) be the irect neighbors of b. Then either i() 6= i() or b is not between with respect to > Q. Proof: We assume that i() =i(). Then, b Lemma 1, either both b> Q an b> Q or both b< Q an b< Q. Thus, b is not between with respect to > Q. Let be Tpeverte. Since i() i() [ i(), if i(), then i() [ i(). Since minimum on a larger set can onl be smaller, it is sucient to consier the case in which i() 6= i() [ i(). Therefore, we assume that i() 6= i(). Since we seek an lower boun of K, ank is monotone increasing with respect to the length of line segments an, it is sucient to consier the case in which = an =. For ever position of (a Tpeverte), the turn angle is the smallest if lie on the outer bounar of i() [ i(), see Figure 1(a). Since K is monotone increasing with respect to the turn angle, we assume that lie on the outer bounar of i() [ i(). β (a) (b) Figure 1. Now we show that the smallest value of K( ) is obtaine if lies at the insie corner of i()[i(), see Figure 1(b). Since = an =, wehave K( ) =K(), where is the turn angle, see Figure 1(b). Since K() is a monotone ecreasing function of the istance between an, itreaches the minimum when this istance is maimal, see Figure 13, which is the case if lies in the insie corner of i() [ i(). β Figure 13.

11 For these positions of, wehave K( ) =( ; arcsin ) where is the turn angle of i() [ i(), see Figure 14. Since, weobtain K( ) k( ) =( ; arcsin ) (7) for an Tpe verte an its neighbors. Moreover, it hols lim!0 k( ) = for ever an. Η α Η Figure 14. Since lim!0 m 4 ( ) = 0, there eists 0 < 0 < p 5 such that, for ever 0 << 0, Therefore, we have proven that ; + arcsin( ) = m 4 ( ) <k( ) =( ; arcsin + ) : (8) K( ) m 4 ( ) <k( ) K( ) (9) for ever 0 << 0,ever Tpe 1 verte 1, an ever Tpe verte. If the process of evolution is continue until the relevance measure m 4 ( ) is obtaine, then all Tpe 1 vertices are elete an all Tpe vertices remain, which proves the theorem. In the course of the proof of Theorem 3, we have proven the following two inequalities: Corollar 1. For ever Tpe 1 verte 1 an its neighbors ; 1 1 in polgon Q: K( ) + arcsin( ) : (10) + For ever Tpe verte an its neighbors in polgon Q: ( ; arcsin ) K( ): (11) Proposition. Let be a turn angle of polgon P for which epression q 1 sin ( ) + 1 4cos ( ) = q 1 sin ( + 1 ) 4cos ( ) is maimal. If an polgon Q is a ( )-approimation of P, then Q is a ( )-approimation of P for which aitionall hols that for ever verte v of P at least one verte of Q is containe inb(v ). Proof: Let polgon Q be a ( )-approimation of P. We want to n the smallest such that for ever verte v of P at least one verte of Q is containe in B(v ).

12 Let v be verte of P with turn angle. We n such a at verte v. Let e be the inner corner of the two -strips S 1 an S that contain v, see Figure 15. Then the line ve ivies the inner angle of S 1 [ S in two halves. Let ab be the line segment perpenicular to line ve such thata an b lie on the outsie bounar of S 1 [ S. Let be the length of line segment vb. Since Q B(P ), there must be a verte of Q in S 1 [ S above the line ab. Therefore, there must be a verte of Q in B(v ). It remains to compute the length of. Let point c be the perpenicular projection of point b on the inner sie of S in S 1 [ S, see Figure 15. Let u be neighbor verte of v in P such thatuv S 1. Let point f be the perpenicular projection of point e on uv. As can be seen in Figure 15, = jvej + jebj. Since cos( ) in the triangle vef, weobtain s 1 = jebj =sin( ) in the triangle ebc an jvej = sin(90o ; )= sin ( ) + 1 4cos ( ) u 180 γ v γ η f a e b γ/ δ S δ 1 c S 180 γ 180 γ Figure 15. REFERENCES 1. L. J. Latecki an R. Lakamper, \Contour-base shape similarit," in Proc. of Int. Conf. on Visual Information Sstems, D. P. uijsmans an A. W. M. Smeulers, es., Springer-Verlag, Lecture Notes in Computer Science 1614, pp. 617{64, Amsteram, June L. J. Latecki an R. Lakamper, \Polgon evolution b verte eletion," in Proc. of Int. Conf. on Scale-Space Theories in Computer Vision, Corfu, Greece, September L. J. Latecki, R. Lakamper, an U. Eckhart, \ 4. www site, \ 5. L. J. Latecki an R. Lakamper, \Conveit rule for shape ecomposition base on iscrete contour evolution," Computer Vision an Image Unerstaning 73, pp. 441{454, 1999.