Maximal digital straight segments and convergence of discrete geometric estimators
Discrete geometric estimators approach geometric quantities on digitized shapes without any knowledge of the continuous shape. A classical yet difficult problem is to show that an estimator asymptotically converges toward the true geometric quantity …
Authors: F. De Vieilleville (LaBRI), Jacques-Olivier Lachaud (LaBRI), F. Feschet (LLAIC1)
Maximal digital straigh t segmen ts and on v ergene of disrete geometri estimators F rançois de Vieilleville, Jaques-Olivier La haud LaBRI, Univ. Bordeaux 1 351 ours de la Lib ération, 33405 T alene, F rane. {devieill,lahaud}labri.fr F abien F es het LLAIC1, Clermon t-F errand Campus des Cézeaux, 63172 Aubière Cedex, F rane feshetllai3.u-lermont1.fr No Institute Giv en Abstrat. Disrete geometri estimators approa h geometri quan tities on digitized shap es without an y kno wledge of the on tin uous shap e. A lassial y et diult problem is to sho w that an estimator asymptot- ially on v erges to w ard the true geometri quan tit y as the resolution inreases. W e study here the on v ergene of lo al estimators based on Digital Straigh t Segmen t (DSS) reognition. It is losely link ed to the asymptoti gro wth of maximal DSS, for whi h w e sho w b ounds b oth ab out their n um b er and sizes. These results not only giv e b etter insigh ts ab out digitized urv es but indiate that urv ature estimators based on lo al DSS reognition are not lik ely to on v erge. W e indeed in v alidate an h yp othesis whi h w as essen tial in the only kno wn on v ergene the- orem of a disrete urv ature estimator. The pro of in v olv es results from arithmeti prop erties of digital lines, digital on v exit y , om binatoris, on tin ued frations and random p olytop es. 1 In tro dution Estimating geometri features of shap es or urv es solely on their digitization is a lassial problem in image analysis and pattern reognition. Some of the geo- metri features are global: area, p erimeter, momen ts. Others are lo al: tangen ts, normals, urv ature. Algorithms that p erforms this task on digitized ob jets are alled disr ete ge ometri estimators . An in teresting prop ert y these estimators should ha v e is to on v erge to w ards the on tin uous geometri measure as the digitization resolution inreases. Ho w ev er, few estimators ha v e b een pro v ed to b e on v ergen t. In all w orks, shap es are generally supp osed to ha v e a smo oth b oundary (at least t wie dieren tiable) and either to b e on v ex or to ha v e a nite n um b er of inexion p oin ts. The shap e p erimeter estimation has for in- stane b een ta kled in [ 11 ℄. It pro v ed the on v ergene of a p erimeter estimator based on urv e segmen tation b y maximal DSS. The sp eed of on v ergene of sev eral length estimators has also b een studied in [ 4 ℄. Klette and uni¢ [10 ℄ surv ey results ab out the on v ergene (and the sp eed of on v ergene) of sev eral global geometri estimators. They sho w that disrete momen ts on v erge to w ard on tin uous momen ts. As far as w e kno w, there is only one w ork that deals with the on v ergene of lo al geometri estimators [3℄. The symmetri tangen t estimator app ears to b e on v ergen t sub jet to an h yp othesis on the gro wth of DSS as the resolution inreases (see Hyp othesis 41 ). The same h yp othesis en tails that a urv ature estimator is on v ergen t: it is based on DSS reognition and irumsrib ed irle omputation (see Denition 9 ). In this pap er, w e relate the n um b er and the lengths of DSS to the n um b er and lengths of edges of on v ex h ulls of digitized shap es. Using argumen ts related to digital on v ex p olygons and a theorem indued b y random p olytop es theory [1℄, w e estimate the asymptoti b eha viour of b oth quan tities. W e theoretially sho w that maximal DSS do not follo w the h yp othesis used in [3℄. Exp erimen ts onrm our result. The on v ergene theorem is th us not appliable to digital urv es. As a onsequene, the existene of on v ergen t digital urv ature estimators remains an op en problem. The pap er is organized as follo ws. First, w e reall some standard notions of digital geometry and om binatori represen tation of digital lines, i.e. patterns. The relations b et w een maximal segmen ts and edges of on v ex digital p olygons are then studied to get b ounds on maximal segmen ts lengths and n um b er. Finally , the asymptoti b eha viour of maximal segmen ts is dedued from the asymptoti b eha viour of on v ex digital p olygons. The gro wth of some DSS is th us pro v ed to b e to o slo w to ensure the on v ergene of urv ature estimation. This theoretial result is further onrmed b y exp erimen ts. 2 Maximal digital straigh t segmen ts W e restrit our study to the geometry of 4-onneted digital urv es. A digital ob jet is a set of pixels and its b oundary in R 2 is a olletion of v erties and edges. The b oundary forms a 4-onneted urv e in the sense used in the presen t pap er. Our w ork ma y easily b e adapted to 8-onneted urv es. In the pap er, all the reasoning are made in the rst o tan t, but extends naturally to the whole digital plane. The digital urv e is denoted b y C . Its p oin ts ( C k ) are assumed to b e indexed. A set of suessiv e p oin ts of C ordered inreasingly from index i to j will b e on v enien tly denoted b y [ C i C j ] when no am biguities are raised. 2.1 Standard line, digital straigh t segmen t, maximal segmen ts Denition 1. (R éveil lès [14 ℄) The set of p oints ( x, y ) of the digital plane veri- fying µ ≤ ax − by < µ + | a | + | b | , with a , b and µ inte ger numb ers, is al le d the standard line with slop e a/b and shift µ . The standar d lines are the 4-onneted disrete lines. The quan tit y ax − by is alled the r emainder of the line. The p oin ts whose remainder is µ (resp. µ + | a | + | b | − 1 ) are alled upp er (resp. lo w er) leaning p oin ts. The prinipal upp er and lo w er leaning p oin ts are dened as those with extremal x v alues. Finite onneted p ortions of digital lines dene digital str aight se gment . Sine w e w ork with restrited parts of C , w e alw a ys supp ose that indies are totally ordered on this part. Denition 2. A set of su essive p oints [ C i C j ] of C is a digital straigh t segmen t (DSS) i ther e exists a standar d line D ( a, b, µ ) ontaining them. The pr e di ate [ C i C j ] is a DSS is denote d by S ( i, j ) . The rst index j , i ≤ j , su h that S ( i, j ) and ¬ S ( i, j + 1) is alled the fr ont of i . The map asso iating an y i to its fron t is denoted b y F . Symmetrially , the rst index i su h that S ( i, j ) and ¬ S ( i − 1 , j ) is alled the b ak of j and the orresp onding mapping is denoted b y B . Maximal segmen ts form the longest p ossible DSS in the urv e. They are essen tial when analyzing digital urv es: they pro vide tangen t estimations [ 6,13 ℄, they are used for p olygonizing the urv e in to the minim um n um b er of segmen ts [7℄. Denition 3. A ny set of p oints [ C i C j ] is al le d a maximal segmen t i any of the fol lowing e quivalent har aterizations holds: (1) S ( i, j ) and ¬ S ( i, j + 1 ) and ¬ S ( i − 1 , j ) , (2) B ( j ) = i and F ( i ) = j , (3) ∃ k , i = B ( k ) and j = F ( B ( k )) , (4) ∃ k ′ , i = B ( F ( k ′ )) and j = F ( k ′ ) . F rom haraterizations (3) and (4) of Denition 3, an y DSS [ C i C j ] and hene an y p oin t b elongs to at least t w o maximal segmen ts (p ossibly iden tial) [ C B ( j ) C F ( B ( j )) ] and [ C B ( F ( i )) C F ( i ) ] . 2.2 P atterns and DSS W e here reall a few prop erties ab out p atterns omp osing DSS and their lose relations with on tin ued frations. They onstitute a p o w erful to ol to desrib e disrete lines with rational slop es [ 2,8℄. Sine w e are in the rst o tan t, the slop es are b et w een 0 and 1. Denition 4. Given a standar d line ( a, b, µ ) , we al l pattern of har ateristis ( a, b ) the su ession of F r e eman moves b etwe en any two onse utive upp er le aning p oints. The F r e eman moves dene d b etwe en any two onse utive lower le aning p oints is the pr evious wor d r e ad fr om b ak to fr ont and is al le d the rev ersed pattern . A pattern ( a, b ) em b edded an ywhere in the digital plane is ob viously a DSS ( a, b, µ ) for some µ . Sine a DSS on tains at least either t w o upp er or t w o lo w er leaning p oin ts, a DSS ( a, b, µ ) on tains at least one p attern or one r everse d p attern of harateristis ( a, b ) . Denition 5. W e al l simple on tin ued fration and we write: z = a/b = [0 , u 1 . . . , u i , . . . , u n ] with z = 0 + 1 u 1 + 1 . . . + 1 u n − 1 + 1 u n W e al l k -th on v ergen t the simple ontinue d fr ation forme d of the k + 1 rst p artial quotients: z k = p k q k = [0 , u 1 , . . . , u k ] . There exists a reursiv e transformation for omputing the pattern of a standard line from the simple ontinue d fr ation of its slop e [2℄. W e all E the mapping from the set of p ositiv e rationnal n um b er smaller than one on to F reeman-o de's w ords dened as follo ws. First terms are stated as E ( z 0 ) = 0 and E ( z 1 ) = 0 u 1 1 and others are expressed reursiv ely: E ( z 2 i +1 ) = E ( z 2 i ) u 2 i +1 E ( z 2 i − 1 ) (1) E ( z 2 i ) = E ( z 2 i − 2 ) E ( z 2 i − 1 ) u 2 i (2) In the follo wing, the omplexity of a pattern is the depth of its deomp osition in simple on tin ued fration. W e reall a few more relations: p k q k − 1 − p k − 1 q k = ( − 1) k +1 (3) ( p k , q k ) = u k ( p k − 1 , q k − 1 ) + ( p k − 2 , q k − 2 ) (4) W e no w fo us on omputing v etor relations b et w een leaning p oin ts (upp er and lo w er) inside a pattern. In the follo wing w e onsider a DSS ( a, b, 0) in the rst o tan t starting at the origin and ending at its seond lo w er leaning p oin t (whose o ordinate along the x -axis is p ositiv e). W e dene a/b = z n = [0 , u 1 , . . . , u n ] for some n . P oin ts will b e alled U 1 , L 1 , U 2 and L 2 as sho wn in Fig. 1. W e an state U 1 L 1 = U 2 L 2 and U 1 U 2 = L 1 L 2 = ( b, a ) . W e reall that the F reeman mo v es of [ U 1 L 1 ] are the same as those of [ U 2 L 2 ] . F urthermore F reeman mo v es b et w een U 1 and U 2 form the p attern ( a, b ) and those b et w een L 1 and L 2 form the r everse d p attern ( a, b ) . Prop osition 1. A p attern with an o dd omplexity (say n = 2 i + 1 ) is suh that U 1 L 1 = ( u 2 i +1 − 1)( q 2 i , p 2 i )+( q 2 i − 1 , p 2 i − 1 )+(1 , − 1) and L 1 U 2 = ( q 2 i − 1 , p 2 i +1) . Mor e over the DSS [ U 1 L 1 ] has E ( z 2 i ) u 2 i +1 − 1 as a left fator, and the DSS [ L 1 U 2 ] has E ( z 2 i − 1 ) u 2 i as a right fator. Pr o of. F rom Eq. (3) w e ha v e: p 2 i +1 q 2 i − p 2 i q 2 i +1 = ( − 1) 2 i +1+1 = 1 , whi h an b e rewritten as: aq 2 i − bp 2 i = 1 . ( q 2 i , p 2 i ) are learly the Bézout o eien ts of ( a, b ) . One an he k that p oin t ( b + 1 − q 2 i , a − 1 − p 2 i ) is L 1 : its remainder is a + b − 1 and its x -o ordinate while p ositiv e is smaller than b . W e immediately get U 1 L 1 = ( b + 1 − q 2 i , a − 1 − p 2 i ) . Using Eq. (4) yields: U 1 L 1 = (( u 2 i +1 − 1) q 2 i + q 2 i − 1 +1 , ( u 2 i +1 − 1) p 2 i + p 2 i − 1 − 1) . F rom L 1 U 2 = − U 1 L 1 + U 1 U 2 , w e further get Y X L 1 L 2 U 2 E ( z 2 i +1 ) E ( z 2 i ) E ( z 2 i ) E ( z 2 i ) E ( z 2 i − 1 ) p 2 i p 2 i − 1 q 2 i q 2 i − 1 U 1 O z 2 i +1 = [0 , 2 , 3 , 3] Fig. 1. A DSS ( a, b, 0) with an o dd omplexit y slop e, tak en b et w een origin and its seond lo w er leaning p oin t. that L 1 U 2 = ( q 2 i − 1 , p 2 i + 1) . F rom Eq. (1) E ( z 2 i ) u 2 i +1 − 1 is a left fator of [ U 1 U 2 ] but also of [ U 1 L 1 ] . W riting E ( z 2 i +1 ) as E ( z 2 i ) u 2 i +1 − 1 E ( z 2 i − 2 ) E ( z 2 i − 1 ) u 2 i +1 , and expanding L 1 U 2 as ( u 2 i q 2 i − 1 + q 2 i − 2 − 1 , u 2 i p 2 i − 1 + p 2 i − 2 + 1) with Eq. (4), w e see that E ( z 2 i − 1 ) u 2 i is a righ t fator of [ L 1 U 2 ] . ⊓ ⊔ Prop osition 2. A p attern with an even omplexity (say n = 2 i ) is suh that U 1 L 1 = ( q 2 i − 1 + 1 , p 2 i − 1 − 1) and L 1 U 2 = ( u 2 i − 1)( q 2 i − 1 , p 2 i − 1 )+ ( q 2 i − 2 , p 2 i − 2 )+ ( − 1 , 1) . Mor e over the DSS [ U 1 L 1 ] has E ( z 2 i − 2 ) u 2 i − 1 as a left fator, and the DSS [ L 1 U 2 ] has E ( z 2 i − 1 ) u 2 i − 1 as a right fator. 3 Prop erties of maximal segmen ts for on v ex urv es In this setion, w e study relations b et w een maximal segmen ts and digital edges of on v ex shap e digitization. The dilation of S b y a real fator r is denoted b y r · S . Let D m b e the digitization of step 1 /m , i.e. if S is a real shap e: D m ( S ) = ( m · S ) ∩ Z 2 . The length estimator based on the it y-blo k distane is written as L 1 . 3.1 Con v ex digital p olygon (CDP) Denition 6. A on v ex digital p olygon (CDP) Γ is a subset of the digital plane e qual to the digitization of its onvex hul l, i.e. Γ = D 1 (conv( Γ )) . Its v erties ( V i ) i =1 ..e form the minimal subset for whih Γ = D 1 (conv( V 1 , . . . , V e )) . The p oints on the b oundary of Γ form a 4- onne te d ontour. The numb er of verti es (or e dges) of Γ is denote d by n e ( Γ ) and its p erimeter by Per ( Γ ) . A C DP is also alled a lattie on v ex p olygon [16 ℄. An e dge is the Eulidean segmen t joining t w o onseutiv e v erties, and a digital e dge is the disrete seg- men t joining t w o onseutiv e v erties. It is lear that w e ha v e as man y e dges as digital e dges and as v erties. F rom haraterizations of disrete on v exit y [5℄, w e learly see that: Prop osition 3. Eah digital e dge of a CDP is either a p attern or a su ession of the same p attern whose slop e is the one of the e dge. In other wor ds, b oth verti es ar e upp er le aning p oints of the digital e dge. W e no w reall one theorem onerning the asymptoti n um b er of v erties of CDP that are digitization of on tin uous shap es. It omes from asymptoti prop erties of random p olytop es. Theorem 1. (A dapte d fr om Balo g, Bár ány [1℄) If S is a plane onvex b o dy with C 3 b oundary and p ositive urvatur e then D m ( S ) is a CDP and c 1 ( S ) m 2 3 ≤ n e ( D m ( S )) ≤ c 2 ( S ) m 2 3 wher e the onstants c 1 ( S ) and c 2 ( S ) dep end on extr emal b ounds of the urvatur es along S . Hen e for a dis c 1 and c 2 ar e absolute onstants. 3.2 Links b et w een maximal segmen ts and edges of CDP Maximal segmen ts are DSS: b et w een an y t w o upp er (resp. lo w er) leaning p oin ts la ys at least a lo w er (resp. upp er) leaning p oin t. The slop e of a maximal segmen t is then dened b y t w o onseutiv e upp er and/or lo w er leaning p oin ts. Digital edges are patterns and their v erties are upp er leaning p oin ts (from Prop. 3). Th us, v erties ma y b e upp er leaning p oin ts but nev er lo w er leaning p oin ts of maximal segmen ts. W e ha v e Lemma 1. A maximal se gment annot b e stritly ontaine d into a digital e dge. W e no w in tro due a sp eial lass of digital edge. Denition 7. W e al l supp orting edge , a digital e dge whose two verti es dene leftmost and rightmost upp er le aning p oints of a maximal se gment. Relations b et w een maximal DSS and digital edges are giv en b y the follo wing lemmas: Lemma 2. A supp orting e dge denes only one maximal se gment: it is the only one ontaining the e dge and it has the same slop e. If a maximal se gment ontains two or mor e upp er le aning p oints then ther e is a supp orting edge linking its leftmost and rightmost upp er le aning p oints with the same slop e. If a maximal se gment ontains thr e e or mor e lower le aning p oints then it has a supp orting edge . Lemma 3. If a maximal se gment is dene d by only two onse utive lower le an- ing p oints then it has one upp er le aning p oint whih is some vertex of the CDP by onvexity. Lengths of maximal segmen ts and digital edges are tigh tly in tert wined, as sho wn b y the t w o next prop ositions. Prop osition 4. L et [ V k V k +1 ] b e a supp orting edge of slop e a b made of f p atterns ( a, b ) and let M S b e the maximal segmen t asso iate d with it (L emma 2). Their lengths ar e linke d by the ine qualities: L 1 ( V k V k +1 ) ≤ L 1 ( M S ) ≤ f + 2 f L 1 ( V k V k +1 ) − 2 and 1 3 L 1 ( M S ) ≤ L 1 ( V k V k +1 ) ≤ L 1 ( M S ) ≤ 3 L 1 ( V k V k +1 ) Pr o of. V erties V k and V k +1 are leftmost and rightmost upp er leaning p oin ts of M S . The p oin ts V k − ( b, a ) , V k +1 + ( b, a ) while learly upp er leaning p oin ts of the standard line going through [ V k V k +1 ] annot b elong to the CDP . Hene M S annot extend further of its supp orting edge of more than | a | + | b | − 1 p oin ts on b oth sides. Consequen tly L 1 ( M S ) ≤ L 1 ( V k V k +1 ) + 2( | a | + | b | − 1) . Using L 1 ( V k V k +1 ) = f ( | a | + | b | ) brings: L 1 ( V k V k +1 ) ≤ L 1 ( M S )) ≤ f +2 f L 1 ( V k V k +1 ) − 2 . W orst ases bring L 1 ( V k V k +1 ) ≤ L 1 ( M S ) ≤ 3 L 1 ( V k V k +1 ) ⊓ ⊔ Prop osition 5. L et M S k ′ b e a maximal se gment in the ongur ation of L emma 3 , and so let V k b e the vertex that is its upp er le aning p oint. The length of the max- imal se gment is upp er b ounde d by: L 1 ( M S k ′ ) ≤ 4 L 1 ( V k − 1 V k ) + L 1 ( V k V k +1 ) Pr o of. W e all L 1 , L 2 the leftmost and righ tmost lo w er leaning p oin ts and U 2 ≡ V k the upp er leaning p oin t (see Fig. 1). Supp ose that M S k ′ has a slop e with an o dd omplexit y (sa y 2 i + 1 ). Prop osition 1 implies L 1 ( L 1 U 2 ) = q 2 i + p 2 i . There is learly a righ t part of [ L 1 U 2 ] (i.e. [ L 1 V k ] ) that is on tained in [ V k − 1 V k ] and tou hes V k . The pattern E ( z 2 i − 1 ) u 2 i is a righ t fator of [ L 1 U 2 ] (Prop osi- tion 1 again). It is indeed a righ t fator of [ V k − 1 V k ] to o, sine it annot ex- tends further than V k − 1 to the left without dening a longer digital edge. W e get [ V k − 1 V k ] ⊇ E ( z 2 i − 1 ) u 2 i and immediately L 1 ( V k − 1 V k ) ≥ u 2 i L 1 ( E ( z 2 i − 1 )) = u 2 i ( q 2 i − 1 + p 2 i − 1 ) . F rom Eq. (4), w e ha v e: q 2 i + p 2 i = u 2 i ( q 2 i − 1 + p 2 i − 1 ) + q 2 i − 2 + p 2 i − 2 and q 2 i − 2 + p 2 i − 2 ≤ q 2 i − 1 + p 2 i − 1 . W e obtain immediately L 1 ( L 1 U 2 ) = q 2 i + p 2 i ≤ ( u 2 i + 1)( q 2 i − 1 + p 2 i − 1 ) . By omparing this length to the length of the digital edge [ V k − 1 V k ] , w e get L 1 ( L 1 U 2 ) ≤ u 2 i +1 u 2 i L 1 ( V k − 1 V k ) . Prop osition 1 and similar argumen ts on [ V k V k +1 ] brings L 1 ( U 2 L 2 ) ≤ u 2 i +1 u 2 i +1 − 1 L 1 ( V k − 1 V k ) . W orst ases are then L 1 ( L 1 U 2 ) ≤ 2 L 1 ( V k − 1 V k ) and L 1 ( U 2 L 2 ) ≤ 2 L 1 ( V k V k +1 ) . The ase where M S k ′ has a slop e with an ev en omplexit y (sa y 2 i ) uses Prop. 2 and is treated similarly . Sine M S has only one upp er leaning p oin t, it annot b e extended further than L 1 ( U 2 L 2 ) on the left and L 1 ( L 1 U 2 ) on the righ t (Lemma 2 ). W e th us get L 1 ( M S k ′ ) ≤ 4( L 1 ( V k − 1 V k ) + L 1 ( V k V k +1 )) . ⊓ ⊔ A pro of of the follo wing theorem based on pattern analysis is giv en in Ap- p endix B for limited spae reasons. A similar result related to linear in teger programming is in [15 ℄. It ma y also b e obtained b y viewing standard lines as in tersetion of t w o knapsa k p olytop es [ 9℄. Theorem 2. L et E b e a supp orting e dge whose slop e has a omplexity n , n ≥ 2 , then the maximal se gment ontaining E inludes at most n other e dges on e ah side of E . Corollary 1. The shortest p attern of a supp orting e dge for whih its maximal se gment may ontain 2 n + 1 digital e dge is z n = [0 , 2 , . . . , 2] . If the DCP is enlose d in a m × m grid, then the maximal numb er n of digital e dges inlude d in one maximal se gment is upp er b ounde d as: n ≤ log 4 m √ 2 / log (1 + √ 2) − 1 . Pr o of. The n um b er L = [0 , 2 , . . . , 2 , . . . ] is a quadrati n um b er equal to − 1 + √ 2 . Its reursiv e haraterization is U n = 2 U n − 1 + U n − 2 with U 0 = 0 and U 1 = 1 . Solving it leads to U n = √ 2 4 (1 + √ 2) n − (1 − √ 2) n . Hene asymptotially , U n ≈ √ 2 4 (1 + √ 2) n and lim n →∞ U n U n +1 = L . The shortest edge of slop e omplexit y n is learly an n -th on v ergen t of L . T o t in to an m × m grid, the omplexit y n is su h that U n +1 ≤ m . W e th us obtain that n ≤ log 4 m √ 2 / log (1 + √ 2) − 1 . ⊓ ⊔ Prop osition 6. Ther e exists at most two maximal se gments p er verti es in the ongur ation of L emma 3 with dier ent p arities of omplexity. Pr o of. W e rst pro v e that there is at most one maximal segmen t with only one upp er leaning p oin t on a v ertie of a DCP with an ev en omplexit y . Let us supp ose that M S 1 and M S 2 are t w o maximal segmen ts sharing a v ertie of the CDP (sa y U 2 ) with their slop es 3.3 Asymptoti n um b er and size of maximal segmen ts W e assume in this setion that the digital on v ex p olygon Γ is enlosed in a m × m grid. W e wish to ompute a lo w er b ound for the n um b er of edges related to at least one maximal segmen t. W e sho w in Theorem 3 that this n um b er is signian t and inreases at least as fast as the n um b er of edges of the DCP divided b y log m . F rom this lo w er b ound, w e are able to nd an upp er b ound for the length of the smallest maximal segmen t of a DCP (Theorem 4). W e rst lab el ea h v ertex of the DCP as follo ws: (i) a 2-vertex is an upp er leaning p oin t of a supp orting edge, (ii) a 1-vertex is an upp er leaning p oin t of some maximal segmen t but is not a 2-v ertex, (iii) 0-verti es are all the remaining v erties. The n um b er of i -v erties is denoted b y n i . Giv en an orien tation on the digital on tour, the n um b er of edges going from an i -v ertex to a j -v ertex is denoted b y n ij . Theorem 3. The numb er of supp orting e dges and of 1-verti es of Γ ar e r elate d to its numb er of e dges with n e ( Γ ) Ω (log m ) ≤ n 1 + 2 n 22 . (5) A n imme diate or ol lary is that ther e ar e at le ast n e ( Γ ) /Ω (log m ) maximal se g- ments. Pr o of. F rom Theorem 2 and its Corollary 1 , w e kno w that a DSS hene a max- imal segmen t annot inlude more than Ω (log m ) edges. Hene there annot b e more than Ω (log m ) 0-v erties for one 1-v ertex or for one 2-v ertex. W e get n 00 ≤ ( n 1 + n 2 ) Ω (log m ) . W e dev elop the n um b er of edges with ea h p os- sible lab el: n e ( Γ ) = n 22 + n 02 + n 12 + n 20 + n 21 + n 00 + n 01 + n 10 + n 11 . Sine, n 02 + n 12 ≤ n 22 , n 20 + n 21 ≤ n 22 and n 01 + n 10 + n 11 ≤ 3 n 1 , w e get n e ( Γ ) ≤ 3 n 22 + n 00 + 3 n 1 . Noting that a 2-v ertex annot b e isolated b y def- inition of supp orting edges (Denition 7) giv es n 2 ≤ 2 n 22 . One inserted in n 00 ≤ ( n 1 + n 2 ) Ω (log m ) and ompared with n e ( Γ ) , w e get the exp eted result. ⊓ ⊔ W e no w relate the DCP p erimeter to the length of maximal segmen ts. Theorem 4. The length of the smal lest maximal se gment of the DCP Γ is upp er b ounde d: min l L 1 ( M S l ) ≤ Ω (log m ) Per( Γ ) n e ( Γ ) . (6) Pr o of. W e ha v e P er ( Γ ) = P n e L 1 ( E i ) . W e no w ma y expand the sum on sup- p orting edges (22-edges), on edges tou hing a 1-v ertex, and on others. Edges tou hing 1-v erties ma y b e oun ted t wie, therefore w e divide b y 2 their on tri- bution to the total length. X n e L 1 ( E i ) ≥ X n 22 L 1 ( E 22 j ) + 1 2 X n 1 L 1 ( E ?1 k − 1 ) + L 1 ( E 1? k ) (7) F or the rst term, ea h supp orting edge indexed b y j (a 22 -edge) has an asso- iated maximal segmen t, sa y indexed b y j ′ . F rom Prop osition 4 , w e kno w that L 1 ( E 22 j ) ≥ 1 3 L 1 ( M S j ′ ) . F or the seond term, ea h 1-v ertex indexed b y k is an upp er leaning p oin t of some maximal segmen t indexed b y k ′ . Prop osition 5 holds and L 1 ( E ?1 k − 1 ) + L 1 ( E 1? k ) ≥ 1 4 L 1 ( M S k ′ ) . Putting ev erything together in Eq. (7), w e get: X n e L 1 ( E i ) ≥ 1 3 X n 22 L 1 ( M S j ′ )+ 1 8 X n 1 L 1 ( M S k ′ ) ≥ ( 1 3 n 22 + 1 8 n 1 ) min l L 1 ( M S l ) ≥ 1 8 ( n 1 +2 n 22 ) min l L 1 ( M S l ) Inserting the lo w er b ound of Theorem 3 in to the last inequalit y onludes. ⊓ ⊔ 4 Asymptoti prop erties of shap es digitized at inreasing resolutions W e ma y no w turn to the main in terest of the pap er: studying the asymptoti prop erties of disrete geometri estimators on digitized shap es. W e therefore onsider a plane on v ex b o dy S whi h is on tained the square [0 , 1 ] × [0 , 1 ] (w.l.o.g.). F urthermore, w e assume that its b oundary γ = ∂ S is C 3 with ev ery- where stritly p ositiv e urv ature. This assumption is not v ery restritiv e sine p eople are mostly in terested in regular shap es. F urthermore, the results of this setion remains v alid if the shap e an b e divided in to a nite n um b er of on v ex and ona v e parts; ea h one is then treated separately . The digitization of S with step 1 /m denes a digital on v ex p olygon Γ ( m ) insrib ed in a m × m grid. W e rst examine the asymptoti b eha vior of the maximal segmen ts of Γ ( m ) , b oth theoretially and exp erimen tally . W e then study the asymptoti onver gen e of a disrete urv ature estimator. 4.1 Asymptoti b eha vior of maximal segmen ts The next theorem summarizes the asymptoti size of the smallest maximal seg- men t wrt the grid size m . Theorem 5. The length of the smal lest maximal se gment of Γ ( m ) has the fol- lowing asymptoti upp er b ound: min i L 1 ( M S i ( Γ ( m ))) ≤ Ω ( m 1 / 3 log m ) (8) Pr o of. Theorem 4 giv es for the DCP Γ ( m ) the inequalit y min i L 1 ( M S i ( Γ ( m ))) ≤ Ω (log m ) Per( Γ ( m )) n e ( Γ ( m )) . Sine Γ ( m ) is on v ex inluded in the subset m × m of the dig- ital plane, its p erimeter Per ( Γ ( m )) is upp er b ounded b y 4 m . On the other hand, Theorem 1 indiates that its n um b er of edges n e ( Γ ( m )) is lo w er b ounded b y c 1 ( S ) m 2 / 3 . Putting ev erything together giv es min i L 1 ( M S i ( Γ ( m ))) ≤ Ω (log m ) 4 m c 1 ( S ) m 2 / 3 whi h is one redued what w e w an ted to sho w. ⊓ ⊔ Although there are p oin ts on a shap e b oundary around whi h maximal seg- men ts gro w as fast as O ( m 1 / 2 ) (the ritial p oin ts in [12 ℄), some of them do not gro w as fast. A loser lo ok at the pro ofs of Theorem 4 sho ws that a sig- nian t part of the maximal segmen ts (at least Ω (1 / (log m )) ) has an a v erage length that gro ws no faster than Ω ( m 1 / 3 log m ) . This fat is onrmed with ex- p erimen ts. Fig. 2 , left, plots the size of maximal segmen ts for a disk digitized with inreasing resolution. The a v erage size is loser to m 1 / 3 than to √ m . 4.2 Asymptoti on v ergene of disrete geometri estimators A useful prop ert y that a disrete geometri estimator ma y ha v e is to on v erge to w ard the geometri quan tit y of the on tin uous shap e b oundary when the dig- itization grid gets ner [3 ,4,10 ℄. It ma y b e expressed as follo ws, Denition 8. L et F b e any ge ometri desriptor on the shap e S with b oundary γ and digitizations Γ ( m ) . The disr ete ge ometri estimator E asymptotially on v erges towar d the desriptor F for γ i |E ( Γ ( m )) − F ( γ ) | ≤ ǫ ( m ) with lim m → + ∞ ǫ ( m ) = 0 . (9) 1 10 100 1000 10 100 1000 max min mean 4x^(1/3) 4x^(1/2) 0.15 0.2 0.25 0.3 0.35 0.4 0 100 200 300 400 500 600 700 Mean absolute error Deviation of mean absolute error Fig. 2. F or b oth urv es, the digitized shap e is a disk of radius 1 and the absissa is the digitization resolution. Left: plot in log-spae of the L 1 -size of maximal segmen ts. Righ t: plot of the mean and standard deviation of the absolute error of urv ature estimation, | ˆ κ − 1 | (exp eted urv ature is 1). W e no w reall the denition of a disrete urv ature estimator based on DSS reognition [3℄. Denition 9. L et P b e any p oint on a disr ete ontour, Q = B ( P ) and R = F ( P ) ar e the extr emities of the longest DSS starting fr om P ( al le d half-tangen ts ). Then the urv ature estimator b y irumirle ˆ κ ( P ) is the inverse of the r adius of the ir le ir umsrib e d to P , Q and R , r es ale d by the r esolution m . Exp erimen ts sho w that this estimator rather orretly estimates the urv ature of disrete irles on aver age ( ≈ 10 % error). It is indeed b etter than an y other urv ature estimators prop osed in the litterature. Theorem B.4 of [3 ℄ demon- strates the asymptoti onver gen e of this urv ature estimator, sub jet to the h yp othesis: Hyp othesis 41 Half-tangents on digitize d b oundaries gr ow at a r ate of Ω ( √ m ) with the r esolution m . Ho w ev er, with our study of maximal segmen ts, w e an state that Claim. Hyp othesis 41 is not v eried for digitizations of C 3 -urv es with stritly p ositiv e urv ature. W e annot onlude on the asymptoti on v ergene of the urv ature estimator b y irumirle. Pr o of. It is enough to note that half-tangen ts, b eing DSS, are inluded in max- imal segmen ts and ma y not b e longer. F urthermore, sine maximal segmen ts o v er the whole digital on tour, some half-tangen ts will b e inluded in the small- est maximal segmen ts. Sine the smallest maximal segmen ts are no longer than Ω ( m 1 / 3 log m ) (Theorem 5 ), the length of some half-tangen ts has the same up- p er b ound, whi h is smaller than Ω ( √ m ) . ⊓ ⊔ The asymptoti on v ergene of a urv ature estimator is th us still an op en problem. F urthermore, preise exp erimen tal ev aluation of this estimator indi- ates that it is most ertainly not asymptotially on v ergen t, although it is a- tually on a v erage one of the most stable disrete urv ature estimator (see Fig. 2, righ t). F ormer exp erimen tal ev aluations of this estimator w ere a v eraging the urv ature estimates on all on tour p oin ts. The on v ergene of the a v erage of all urv atures do es not indue the on v ergene of the urv ature at one p oin t. 5 Conlusion W e sho w in this pap er the relations b et w een edges of on v ex h ulls and maximal segmen ts in terms of n um b er and sizes. W e pro vide an asymptotial analysis of the w orst ases of b oth measures. A onsequene of the study is the refutation of an h yp othesis related to the asymptoti gro wth of maximal segmen ts and whi h w as essen tial in pro ving the on v ergene of a urv ature estimator based on DSS and irumirles [3 ℄. Our w ork also applied to digital tangen ts sine their on v ergene relies on the same h yp othesis. The existene of a on v ergen t disrete estimator of urv ature based on DSS is th us still a hallenging problem and w e are urren tly in v estigating it. Referenes 1. An tal Balog and Imre Bárán y . On the on v ex h ull of the in teger p oin ts in a dis. In SCG '91: Pr o e e dings of the seventh annual symp osium on Computational ge ometry , pages 162165. A CM Press, 1991. 2. J. Berstel and A. De Lua. Sturmian w ords, lyndon w ords and trees. The or et. Comput. Si. , 178(1-2):171203, 1997. 3. D. Co eurjolly . A lgorithmique et gé ométrie p our la ar atérisation des ourb es et des surfa es . PhD thesis, Univ ersité Ly on 2, Déem bre 2002. 4. D. Co eurjolly and R. Klette. A omparativ e ev aluation of length estimators of digital urv es. IEEE T r ans. on Pattern A nal. and Mahine Intel l. , 26(2):252257, 2004. 5. Ch ul E.Kim. Digital on v exit y , straigh tness, and on v ex p olygons. IEEE T r ans. on Pattern A nal. and Mahine Intel l. , 6(6):618626, 1982. 6. F. F es het and L. T ougne. Optimal time omputation of the tangen t of a disrete urv e: appliation to the urv ature. In Disr ete Ge ometry and Computer Imagery (DGCI) , v olume 1568 of LNCS , pages 3140. Springer V erlag, 1999. 7. F. F es het. and L. T ougne. On the Min DSS Problem of Closed Disrete Curv es. In A. Del Lungo, V. Di Gesù, and A. Kuba, editors, IW CIA , v olume 12 of Ele toni Notes in Disr ete Math. Elsevier, 2003. 8. G.H. Hardy and E.M. W righ t. A n intr o dution to the the ory of numb ers . Oxford Univ ersit y Press, fourth edition, 1960. 9. A. S. Ha y es and D. C. Larman. The v erties of the knapsa k p olytop e. Disr ete Applie d Mathematis , 6:135138, 1983. 10. R. Klette and J. uni¢. Multigrid on v ergene of alulated features in image analysis. Journal of Mathemati al Imaging and Vision , 13:173191, 2000. 11. V. K o v alevsky and S. F u hs. Theoretial and exp erimen tal analysis of the auray of p erimeter estimates. In Förster and Ru wiedel, editors, Pr o . R obust Computer Vision , pages 218242, 1992. 12. J.-O. La haud. On the on v ergene of some lo al geometri estimators on digitized urv es. Resear h Rep ort 1347-05, LaBRI, Univ ersit y Bordeaux 1, T alene, F rane, 2005. 13. J.-O. La haud, A. Vialard, and F. de Vieilleville. Analysis and omparativ e ev alu- ation of disrete tangen t estimators. In E. Andrès, G. Damiand, and P . Lienhardt, editors, Pr o . Int. Conf. Disr ete Ge ometry for Computer Imagery (DGCI'2005), Poitiers, F r an e , LNCS. Springer, 2005. T o app ear. 14. J.-P . Rév eillès. Géométrie disrète, alul en nombr es entiers et algorithmique . Thèse d'etat, Univ ersité Louis P asteur, Strasb ourg, 1991. 15. V. N. Shev henk o. On the n um b er of extreme p oin ts in linear programming. Kib er- netika , 2:133134, 1981. In russian. 16. K. V oss. Disr ete Images, Obje ts, and Funtions in Z n . Springer-V erlag, 1993. A Pro of of Prop osition 2 Prop osition 2 : A p attern with an even omplexity (say n = 2 i ) is suh that U 1 L 1 = ( q 2 i − 1 + 1 , p 2 i − 1 − 1) and L 1 U 2 = ( u 2 i − 1)( q 2 i − 1 , p 2 i − 1 )+ ( q 2 i − 2 , p 2 i − 2 )+ ( − 1 , 1) . Mor e over the DSS [ U 1 , L 1 ] has E ( z 2 i − 2 ) u 2 i − 1 as a left fator, and the DSS [ L 1 , U 2 ] has E ( z 2 i − 1 ) u 2 i − 1 as a right fator. Pr o of. F rom Eq. (3 ) w e ha v e: p 2 i q 2 i − 1 − p 2 i − 1 q 2 i = ( − 1 ) 2 i +1 = − 1 , whi h an b e rewritten as: a ( − q 2 i − 1 ) − b ( − p 2 i − 1 ) = 1 and ev en tually a ( q 2 i − q 2 i − 1 ) − b ( p 2 i − p 2 i − 1 ) = 1 . W e learly obtain the Bézout o eien ts. F rom its remainder w e get the relativ es o ordinates of L 1 , as: U 1 L 1 = ( q 2 i − 1 + 1 , p 2 i − 1 − 1) . F rom L 1 U 2 = − U 1 L 1 + U 1 U 2 w e get : L 1 U 2 = (( u 2 i − 1) q 2 i − 1 + q 2 i − 2 − 1 , ( u 2 i − 1) p 2 i − 1 + p 2 i − 2 + 1) . Using E ( z 2 i ) = E ( z 2 i − 2 ) u 2 i − 1 +1 E ( z 2 i − 3 ) E ( z 2 i − 1 ) u 2 i − 1 and U 1 L 1 = ( u 2 i − 1 q 2 i − 2 + q 2 i − 3 + 1 , u 2 i − 1 p 2 i − 2 + p 2 i − 3 − 1 ) , it is lear that E ( z 2 i − 2 ) u 2 i − 1 is a left fator of the DSS [ U 1 L 1 ] . F rom Eq. (2) and L 1 U 2 w e learly see that E ( z 2 i − 1 ) u 2 i − 1 is a righ t fator of the DSS [ L 1 U 2 ] . ⊓ ⊔ B Pro of of Theorem 2 Lemma 4. W e al l P n a p attern of omplexity n whose F r e eman o de is E ( z n ) . One an build strit right and left fators ( al le d r esp e tively R and L ) of P n suh that: (i) [ RP n ] , [ P n L ] and [ RP n L ] ar e DSS of slop e z n , (ii) R and L ar e p atterns (or su essions of the same p attern) , (iii) RP n , P n L and RP n L ar e not p atterns, (iv) the slop e of R is gr e ater than that of P n and the slop e of P n is gr e ater than that of L , (v) maximal omplexity of slop e of R and L dep ends on p arity of n : Complexity of P n maximal omplexity of R maximal omplexity of L 2 i + 1 2 i + 1 2 i 2 i 2 i − 1 2 i (vi) Complexity of fators obtaine d by substr ating R or L fr om P n dep ends on p arity of n : Complexity of P n omplexity of P n r R omplexity of P n r L 2 i + 1 2 i 2 i + 1 2 i 2 i 2 i − 1 Pr o of. Sine R and L are strit fators of P n , their F reeman mo v es are ompat- ible with those of E ( z n ) , giving same slop e when R , P n and L are put together. Th us [ RP n L ] is a DSS of slop e z n . F rom digital straigh tness w e learly ha v e digi- tal on v exit y (see [5 ℄). Upp er leaning p oin ts of this DSS are lo ated at extremities of P n . W e simply ho ose among strit fators R and L those that are patterns so that they t desriptions giv en in Eq. (1) and Eq. (2). W e ma y no w desrib e them giv en the parit y of n . Consider the ase where n is o dd (sa y n = 2 i + 1 ), from Eq. (1) w e get: R = E ( z 2 i ) u 2 i +1 − r E ( z 2 i − 1 ) and L = E ( z 2 i ) u 2 i +1 − l with r > 0 and l > 0 . If R and L are longer patterns, they are not an ymore strit fators of P 2 i +1 . W e see that R is a pattern of omplexit y 2 i + 1 and that L is a suession of the pattern E ( z 2 i ) , with a omplexit y of 2 i . The slop e of R equals z ′ 2 i +1 = [0 , u 1 , . . . , u 2 i , u 2 i +1 − r ] = p ′ 2 i +1 q ′ 2 i +1 . F rom Eq. (4) w e get that p 2 i +1 q 2 i +1 = p ′ 2 i +1 + r p 2 i q ′ 2 i +1 + r q 2 i . The sign of z ′ 2 i +1 − z 2 i +1 is that of p ′ 2 i +1 q 2 i − q ′ 2 i +1 p 2 i , and is p ositiv e (see Eq. ( 3)). Th us the slop e of R is greater than that of P 2 i +1 . Same reasoning applied to z 2 i +1 − z 2 i brings that the slop e of P 2 i +1 is greater than that of L . F ator obtained b y substrating R from P 2 i +1 equals E ( z 2 i ) r and substrating L from P 2 i +1 giv es E ( z 2 i ) l E ( z 2 i − 1 ) . Consider no w that n is ev en (sa y n = 2 i ), from Eq. (2) w e get: R = E ( z 2 i − 1 ) u 2 i − r and L = E ( z 2 i − 2 ) E ( z 2 i − 1 ) u 2 i − l . If R and L are longer patterns, they are not an ymore strit fators of P 2 i . Clearly , R has a omplexit y of 2 i − 1 and that of L equals 2 i . The slop e of L equals z ′ 2 i = [0 , u 1 , . . . , u 2 i − 1 , u 2 i − l ] = p ′ 2 i q ′ 2 i . F rom Eq. (4) w e get that p 2 i q 2 i = p ′ 2 i + lp 2 i − 1 q ′ 2 i + lq 2 i − 1 . The sign of z 2 i − z ′ 2 i is that of q ′ 2 i p 2 i − 1 − p ′ 2 i q 2 i − 1 , and is p ositiv e (see Eq. (3)). Th us the slop e of P n is greater than that of L . Same reasoning applied to z 2 i − 1 − z 2 i brings that the slop e of R is greater than that of P n . F ator obtained b y substrating R from P 2 i equals E ( z 2 i − 2 ) E ( z 2 i − 1 ) r and substrating L from P 2 i giv es E ( z 2 i − 1 ) l . It is no w lear that slop es are stritly dereasing from R to P n and from P n to L whatev er the parit y of n . ⊓ ⊔ Theorem 6. L et E b e a supp orting e dge whose slop e has a omplexity n , n ≥ 2 , then the maximal se gment ontaining E inludes at most n other e dges on e ah side of E . Pr o of. W e onstrut 2 n digital edges around E : ( R i ) 1 ≤ i ≤ n at left of E , ( L i ) 1 ≤ i ≤ n at righ t of E . These edges are su h that [ R n . . . R i . . . R 1 E L 1 . . . L j . . . L n ] is a DSS of slop e z n = a/ b and has no other upp er leaning p oin ts but those lo ated on E . E ma y on tain sev eral times the pattern E ( z n ) . It is lear that R n . . . R i . . . R 1 (resp. L 1 . . . L j . . . L n ) has to b e a righ t (resp. left) strit fator of E ( z n ) . Moreo v er R i is a righ t strit fator of E ( z n ) r R 1 . . . R i − 1 and L i is a left strit fator of E ( z n ) r L 1 . . . L i − 1 . F rom Prop osition 3 an y of the digital edges ( R i ) 1 ≤ i ≤ n and ( L i ) 1 ≤ i ≤ n is a pattern or a suession of the same pattern. F rom Eq. ( 1) and Eq. (2) t w o suessiv e digital edges with same omplexit y (sa y n ) annot form a righ t or left strit fator of a pattern with same omplexit y . Th us omplexities of ( R i ) 1 ≤ i ≤ n and ( L i ) 1 ≤ i ≤ n are dereasing when i inreases. Moreo v er to fulll on v exit y prop erties, slop es of edges are dereasing from R n to L n . W e no w build ( R i ) 1 ≤ i ≤ n when n is o dd (sa y n = 2 i + 1 ). F rom Lemma 4 , R 1 has a omplexit y that equals 2 i + 1 and R 2 is a righ t strit fator of a pattern whose omplexit y equals 2 i . Applying Lemma 4 brings R 2 with a omplexit y of 2 i − 1 . Applying the same reasoning reursiv ely brings other edges as sho wn on T able 1 . Lemma 4 also implies dereasing slop es and giv e upp er b ounds in omplexit y of fators. Construtions for the three other ases are giv en in T ables 1 and 2 and follo w the same reasoning. T o satisfy full deomp osition ea h ( u k ) 1 ≤ n has to b e equal or greater than 2 . If this ondition is not meet for some k , than steps asso iated with it (e.g. an y fators on taining u k − r j or u k − l j as p o w ers of some pattern) are skipp ed. This onludes the pro of. ⊓ ⊔ T able 1. Construtions of ( R i ) 1 ≤ i ≤ n and ( L i ) 1 ≤ i ≤ n giv en n o dd . Construtions of ( R i ) 1 ≤ i ≤ n when n = 2 i + 1 F ator F reeman mo v es Complexit y R 1 E ( z 2 i ) u 2 i +1 − r 1 E ( z 2 i − 1 ) 2 i + 1 R 2 E ( z 2 i − 1 ) u 2 i − r 2 2 i − 1 R 3 E ( z 2 i − 2 ) u 2 i − 1 − r 3 E ( z 2 i − 3 ) 2 i − 1 R 4 E ( z 2 i − 3 ) u 2 i − 2 − r 4 2 i − 3 . . . . . . . . . R 2 j E ( z 2 i +1 − 2 j ) u 2 i +2 − 2 j − r 2 j 2 i + 1 − 2 j R 2 j +1 E ( z 2 i − 2 j ) u 2 i +1 − 2 j − r 2 j +1 E ( z 2 i − 1 − 2 j ) 2 i + 1 − 2 j . . . . . . . . . R 2 i +1 0 u 1 − r 2 i +1 1 1 Construtions of ( L i ) 1 ≤ i ≤ n when n = 2 i + 1 F ator F reeman mo v es Complexit y L 1 E ( z 2 i ) u 2 i +1 − l 1 2 i L 2 E ( z 2 i − 2 ) E ( z 2 i − 1 ) u 2 i − l 2 2 i L 3 E ( z 2 i − 2 ) u 2 i − 1 − l 3 2 i − 2 L 4 E ( z 2 i − 4 ) E ( z 2 i − 3 ) u 2 i − 2 − l 4 2 i − 2 . . . . . . . . . L 2 j E ( z 2 i − 2 j ) E ( z 2 i +1 − 2 j ) u 2 i +2 − 2 j − l 2 j 2 i + 2 − 2 j L 2 j +1 E ( z 2 i − 2 j ) u 2 i +1 − 2 j − l 2 j +1 2 i − 2 j . . . . . . . . . L 2 i +1 0 u 1 − l 2 i +1 0 T able 2. Construtions of ( R i ) 1 ≤ i ≤ n and ( L i ) 1 ≤ i ≤ n giv en n ev en. Construtions of ( R i ) 1 ≤ i ≤ n when n = 2 i F ator F reeman mo v es Complexit y R 1 E ( z 2 i − 1 ) u 2 i − r 1 2 i − 1 R 2 E ( z 2 i − 2 ) u 2 i − 1 − r 2 E ( z 2 i − 3 ) 2 i − 1 R 3 E ( z 2 i − 3 ) u 2 i − 2 − r 3 2 i − 3 R 4 E ( z 2 i − 4 ) u 2 i − 3 − r 4 E ( z 2 i − 5 ) 2 i − 3 . . . . . . . . . R 2 j E ( z 2 i − 2 j ) u 2 i +1 − 2 j − r 2 j E ( z 2 i − 1 − 2 j ) 2 i + 1 − 2 j R 2 j +1 E ( z 2 i − 1 − 2 j ) u 2 i − 2 j − r 2 j +1 2 i − 1 − 2 j . . . . . . . . . R 2 i 0 u 1 − r 2 i 1 1 Construtions of ( L i ) 1 ≤ i ≤ n when n = 2 i F ator F reeman mo v es Complexit y L 1 E ( z 2 i − 2 ) E ( z 2 i − 1 ) u 2 i − l 1 2 i L 2 E ( z 2 i − 2 ) u 2 i − 1 − l 2 2 i − 2 L 3 E ( z 2 i − 4 ) E ( z 2 i − 3 ) u 2 i − 2 − l 3 2 i − 2 L 4 E ( z 2 i − 4 ) u 2 i − 3 − l 4 2 i − 4 . . . . . . . . . L 2 j E ( z 2 i − 2 j ) u 2 i +1 − 2 j − l 2 j 2 i − 2 j L 2 j +1 E ( z 2 i − 2 − 2 j ) E ( z 2 i − 1 − 2 j ) u 2 i − 2 j − l 2 j +1 2 i − 2 j . . . . . . . . . L 2 i 0 u 1 − l 2 i 0
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