Regularity Properties of p-distance. Transformations in Image Analysis

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1 Internatonal Journal o Contemorar Mathematcal Scences Vol 0, 05, no 3, HIKARI Ltd, wwwm-harcom htt://ddoor/0988/cms055 Reulart Proertes o -Dstance Transormatons n Imae Analss Aboubar Baoum*, Nashat Fared** and Rabab Mostaa*** *Deartment o Mathematcs, Al-Ahar Unverst, Caro, Et **Deartment o Mathematcs, An shams Unverst, Caro, Et ***Deartment o Mathematcs, An shams Unverst, Caro, Et Corht 05 Aboubar Baoum, Nashat Fared and Rabab Mostaa Ths s an oen access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch ermts unrestrcted use, dstrbuton, and reroducton n an medum, rovded the ornal wor s roerl cted Abstract In ths aer we etend the noton o nmal convoluton to a -nmal convoluton, and show that t s commutatve and assocatve oeraton on unctons We also use -nmal convoluton to et some - norms see [] that dene -dstances and to et some class o -conve nehborhoods whch are more arorate to handle wth and to et better erormance or mae rocessn 0 Our results enerale those n [8] when = Mathematcs Subect Classcaton: 46A03, 46A04, 46A6, 46A55, 5A5, 68M07 Kewords: -dstance, -conve uncton, -mdont conve uncton, -nmal convoluton, -reulart roertes, -semreulart roertes Introducton We ntroduced nvarant dstance results concernn the translaton nvarant dstance on an abelan rou X n secton, see [8] We stud n secton 3 the ostvel -homoeneous and -mdont convet 0 We have roved

2 44 Aboubar Baoum, Nashat Fared and Rabab Mostaa conve set and onl a [ a b a b] a, b X In addton, equvalent condtons to a uncton ben mdont conve on a -conve set are ven n rooston 33 In secton 4 we etend the noton o nmal convoluton to a -nmal convoluton, and show that ths -nmal convoluton s a commutatve and assocatve oeraton on unctons dened on an abelan rou X We have ntroduced n secton 5 the -reulart 0 We ve the condtons on a uncton on an abelan rou to be -sem reular Fnall, some equvalent concets on semreulart are ven n secton 6 see, Theorem 6 Basc dentons and Notatons Dstance transormatons o dtal maes are useul tools n mae analss A dstance transorm o a shae s the set o dstances rom a ven el to the shae, see [3, 4] The dstances can be measured n derent was, e, b aromatn the Eucldean dstance n the two-dmensonal mae, the Eucldean dstance between two els, and, ben - - Other dstances that have been used are the ctbloc dstance and the chess-board dstance ma, Denton Dstances and Metrcs Let X be an nonemt set, we shall measure dstance between onts n X, whch amounts to denn a real valued uncton on the Cartesan roduct X X Let us aree to call a uncton d : X X R, a dstance d s ostve dente, that s, d, 0 wth equalt recsel when =, and smmetrc, d, d,, X A dstance wll be called a metrc, n addton, t satses the tranle nequalt, d, d, d,,, X 3 I X s an abelan rou then the translaton-nvarant dstances are those whch sats, d a, a d, a,, X 4 Denton A -norm on a lnear sace E s a man rom E to R satsn: 0 and 0 and onl =0, 3, E s a scalar

3 Reulart roertes o -dstance transormatons n mae analss 45 On the Eucldean R n, a -norm 0<< s dened b n and so the -dstance between two onts,, n,,, n s ven b n d, 5 Denton 3A quas-norm on a lnear sace E s a man rom E to R satsn: 0 and 0 and onl =0 3 or all, E wth some constants For t s called a norm see, [,] It s well nown [9] that, or an >0,,, 0 wth Remar I I s a -norm wth 0, then 0 and 0 and onl =0 Usn 6 and notn that we et, 3 II In a smlar wa s a -norm wth >, then ma, 6 s a quas norm In act, s a norm Lemma [8]: There s a one to one corresondence between ostve dente even unctons and translaton nvarant dstances In act, an translaton nvarant dstance d on an abelan rou X denes a uncton d,0 on X whch s ostve dente, e 0, 0 when =0 7 And t s an even uncton e X 8 Conversel, a uncton whch satses 7, 8 denes a translaton nvarant dstance d, The ollown lemma wll be used n secton 3 Lemma [8]: Let d be a translaton-nvarant dstance on an abelan rou X and a uncton on X related to d as n lemma Then d s a metrc and onl s subaddtve e, X 9

4 46 Aboubar Baoum, Nashat Fared and Rabab Mostaa 3 Postvel -homoeneous and -mdont convet Denton 3We call a dstance d on an abelan rou X a ostvel - homoenous t satses, d m, m m d,, X, m N, 0 3 O course, s the even uncton d,0 related to a translaton-nvarant dstance d, so ben ostvel -homoeneous, e m d m,0 m X, m N mles that, m m X, m Z 3 We call a uncton o one varable a ostvel -homoeneous, m m and, o two varables s called a ostvel - homoeneous m, m m, We etend the concet o conve and mdont conve uncton to -conve and -mdont conve uncton or 0<< Denton 3 let E be a vector sace over the eld R o real numbers A subset A o E s sad to be -conve the arc sement [, ] { ; 0} s contaned n A or ever choce o, A For = we et the denton o conve set [5, 0] Denton 33A uncton on an abelan rou X wth values n the etended real number sstem, s called: Conve on a conve set t s satses 0 Conve on a -conve set, 0, t s satses 0, 3 -Conve on a conve set, 0, t s satses 0, 4 -Conve on a -conve set, 0, t s satses 0 Denton 34 A uncton on an abelan rou X wth values n the etended real number sstem, s called: Mdont conve on a conve set t s satses [ ], X Mdont conve on a -conve set t s satses

5 Reulart roertes o -dstance transormatons n mae analss 47 [ ], X 3 -Mdont conve on a conve set t s satses [ ], X 4 -Mdont conve on a -conve set t s satses [ ], X Eamle3 In R the set E { X, Y : } s /-conve,,, E we show that For,,, E In act Eamle3 A smlar arument shows that the set B {, : } s a -conve set The ollown theorems 3, 3, 33, and 34 ve some roertes concernn the derent tes o -mdont convet unctons Theorem3 A homoenous uncton s a -mdont conve on a -conve set and onl a [ a b a b] a, b X Proo Snce s a -mdont conve uncton on a -conve set then [ ], X Puttn a and b then a b, a b 33 Hence, a [ a b a b] [ a b a b] Conversel, let a [ a b a b] and usn 33 we et,

6 48 Aboubar Baoum, Nashat Fared and Rabab Mostaa [ ] [ ] [ ] Then s a -mdont uncton on a -conve set Theorem 3A -homoeneous uncton s mdont conve on a -conve set and onl a [ a b a b] a, b X Proo Snce s mdont conve on a -conve set then [ ] Usn 33 we et, a [ a b a b] [ a b a b] [ a b a b ] Conversel, let a [ a b a b] and usn 33 we et, [ ] [ ] [ ] Then s mdont conve on a -conve set Theorem 33 I s homoeneous and -mdont conve on -conve set then s mdont conve on a -conve set I s -homoeneous and mdont conve on a -conve set then mdont conve on a -conve set Proo: Snce s - mdont conve uncton on a -conve set t ollows [ ] Usn 33 we et, a [ a b a b] [ a b a b] [ a b a b] Rsn to the th ower we et: a [ a b a b] Snce s a mdont conve uncton on a -conve set then a [ a b a b] s

7 Reulart roertes o -dstance transormatons n mae analss 49 Usn 33 we et, [ ] Rsn to ower / we et, [ ] [ ] Theorem 34For a non neatve uncton we et: I set s mdont conve on conve set then s mdont conve on -conve I s -mdont conve on -conve set then s mdont conve on conve set Proo To clam, assume that s mdont conve on conve set then, [ ] Rsn to th ower, and notcn that 0 we et rom 6: [ ] [ ] To clam, note that as s -mdont conve on -conve set t ollows that [ ] Rsn to ower we et: [ ] th [ ] In what ollows we ve the relaton between a ostvel -homoeneous uncton and a -mdont conve uncton or 0 For classcal case o =, see [8] Lemma 33 A ostvel -homoeneous uncton s a mdont conve on a - conve set t s subaddtve Proo Let be a subaddtve and a ostvel - homoeneous uncton then P Hence, [ ] The ollown result etends the classcal one o [8] Prooston 33 Let be a subaddtve uncton on an abelan rou X satsn 0 where 0<< The ollown three condtons are equvalent: s mdont conve on -conve set and 0=0

8 50 Aboubar Baoum, Nashat Fared and Rabab Mostaa X s ostvel -homoeneous Proo Let be mdont conve on -conve set then, b lemma 3 s mdont conve on -conve set, e [ ] Tan = and notn that 0=0 we et, [ ] [ 0] So, On the other hand, Hence, Snce Hence, s subaddtve then rom we et [ ] Also, so, 0 0 Snce s ostvel -homoeneous uncton then b tan m= we et, From subaddtvt o m m m N We show that, m m we see that, m m or certan m then t s true that: In act, m m m m The rst and last elements o ths nequalt are equal so we have, m m B nducton holds or all, N In the ollown secton we etend the denton o nmal convoluton to - nmal convoluton 4 Metrcs dened b -nmal convoluton Denton4 Let, be two unctons dened on an abelan rou X wth values n the etended real lne [, ] For 0 we dene the -Inmal convoluton o and as ollows: n [ ], X 4 X

9 Reulart roertes o -dstance transormatons n mae analss 5 For = see [8] and [0] Remar 4 One can dene a -nmal convoluton or as ollows, Lemma 4 A uncton n[ ] on an abelan rou s subaddtve and onl t satses the nequalt In case 0 0, then Proo We clam: I s sub addtve s sub addtve then, Hence, n[ ] On the other hand, I In artcular, then, n so, [n ] e Fnall, we alwas have that tae e elds s subaddtve 0, so 0 0 t ollows A ood eamle s obtaned b aln lemma 4 or a - norm I we, then n Theorem 4-nmal convoluton s a commutatve and assocatve oraton on unctons ProoI Commutatvt: Lettn =- we et, n II Assocatvt: n = h n[ h ] = n[ n h ] = n n[ h ] =

10 5 Aboubar Baoum, Nashat Fared and Rabab Mostaa = h ] n[ n h ] n n[ = h ] n n[ Puttn ~, = h ~ ] ~ n n[ ~ = n[ ] h h A -old convoluton can be dened b n[ ], 3 where the nmum s over all choces o elements X such that Theorem 4 Let ] [0, : X be a uncton on an abelan rou X satsn 0=0 Dene sequence o unctons b uttn,, Then the sequence s decreasn and ts lmt 0 lm s subaddtve Moreover, dom =Ndom e, s nte recsel n the sem rou enerated b dom Proo We wll rove that the sequence s decreasn we tae =0 n the denton o n Net we shall rove that Let X, Y be ven wth, and a ostve number Then there est numbers, such that and We have, n So, P Snce s arbtrar, the nequalt ollows

11 Reulart roertes o -dstance transormatons n mae analss 53 Theorem43 [8]: There s a translaton nvarant metrc d on X such that d,0 or all X and as n theorem 3 Then the lmt o the sequence also satses ths nequalt, d,0, so that t s ostve dente I s smmetrc, s also smmetrc and denes a metrc d, d, on the sub rou ZA = NA Corollar 4[8]: Let W be a nte set n an abelan rou X contann the orn and let be a uncton on X wth 0=0, tan the value outsde W and nte ostve values at all onts n W/{0}Then lm s a ostve dente subaddtve uncton I W s smmetrc and -=, then denes a metrc on the subrou ZW=NW o X enerated b A Eamle 4Let W { Z, }, and dene the rme dstances as,0 0, a 0,, b 0 Then b awe et,0 a But b a, then,0 b a, so that,0,0 a In act, b the denton o -nmal convoluton, P P P,0,0,,,, b For an, K,0 b, so that actuall,0 b Ths s because we tae non ero stes to o rom the orn to,0, the dstance assned to the ath s at least P b Eamle4 We alwas have, and t ma haen that or some el W Let or nstance,0 a,, c, and etend b relecton and ermutaton o the coordnates Then,0 3,0,,, 3 c, So 3c a we et,0 3 c a,0 ths s undesrable, because we eect the rme dstance ornall dened between the orn and,0 W to survve and to be equal to the dstance dened b the mnmum over all aths It s thereore natural to requre that = everwhere n W For = see, [6] and also, [3], [4] Prooston4 [8]: let be as n corollar 4 Then the sequence s ontwse statonar, e, or ever or all 5 -Reulart Proertes X there s an nde such that We sa that an arbtrar uncton s -semreular, m mles m 5

12 54 Aboubar Baoum, Nashat Fared and Rabab Mostaa s sad to be -reular or an ont m wth W and mn and an reresentaton m wth A but not all equal to or 0 we have a strct nequalt m Thus -reulart means that the arc sement s the unque mnmal ath rom 0 to m, whereas -semreulart means that the arc sement rom 0 to m s mnmal, but not necessarl the onl mnmal ath For = we et the concet reulart due to [8] In what ollows we ve a sucent condton to have a -sem reular uncton Prooston 5 Let : X [, ] be a uncton on an abelan rou X Assume that there ests a uncton such that, s a ostvel -homoeneous s a -mdont conve 3 arees wth wherever s less than Then s -sem reular Proo Let m We shall rove that m when dom I one o the, ths nequalt certanl holds; on the other hand, dom or all, we now that and, so that the nequalt ollows rom the subaddtvt o : m m m 6 -Semreulart o Dstances n Two Dmensons In ths secton we let X be the mae lane Z, we can then embed t nto the Eucldean lane R and use also unctons dened there We dene a rme dstance on all ras Rw as ollows: sw s w or sr and w an element o W Ths maes sense two derent ras Rw and Rw ntersect onl at the orn, so we assume ths to be true Here we denote b R the set o all nonneatve real numbers, so that Rw s the ra rom the orn throuh w: R w { tw; t R, t 0} We then etend to all o R so that t becomes lnear n each sector dened b two nehborn vectors n W To mae ths recse we let the nonero elements o W be w,, w enumerated n the R counterclocwse drecton, so that w and w dene a sector ree rom elements o W, and so on, untl the sector dened b w and w In the sector dened b w and w we dene sw tw Here, o course, s w w t w, shall be understood as s, t R w = 6

13 Reulart roertes o -dstance transormatons n mae analss 55 The uncton wll then actuall be ecewse lnear on R, and denes a dstance there as well as on Z The ollown lemma wll be used to rove theorem 6 Lemma 6Let be a real valued uncton dened on an abelan rou X such that 0 wth equalt onl or 0 ; 6 And, X 63 Dene a dstance on X b d, Then d s a metrc and onl s -mdont conve on a conve set Proo The roertes and o a metrc ben obvousl ullled, the onl queston can be whether d satses the tranle nequalt 3 I s - mdont conve we et d, d, d, So the tranle nequalt s true Conversel, suose now that the tranle nequalt holds, then we clam that s -mdont conve We note that, d,0 d, d,0 The ollown result ves some equvalent concets on semreulart Theorem 6: Let a nte smmetrc set W n Z be ven, 0 {,,, 0 W w w w }, where w 0 Assume that two ras Rw Rw ntersect onl at the orn and that W contans two lnearl ndeendent vectors Let a smmetrc uncton be ven wth nte ostve values n W /{0}, 0 0and the value outsde W Then dene on R to be equal to on W, to be a ostvel -homoeneous and ecewse lnear n each sector whch does not contan an ont rom W n ts nteror Elctl, ths means that we dene b 6 above The ollown ve condtons are equvalent: A The rme dstance uncton s -semreular B The uncton s -mdont conve -conve n R ; C The restrcton o to Z s -mdont conve; D The dstance d, s a metrc on R ; E The dstance d, s a metrc on Z and

14 56 Aboubar Baoum, Nashat Fared and Rabab Mostaa Proo In our case the uncton s contnuous, so -mdont convet s equvalent to -convet We also note that on Nw Nw s the uncton constructed rom as n Corollar 4 Indeed, w mw w m w m w mw w mw It ma haen that or certan els In act B mles C b tan the restrcton rom R to Z I C holds, t ollows rom the -homoenet that or all vectors wth ratonal comonents, and then or all vectors b contnut The roo that D and E are equvalent s o course smlar Net we shall rove the equvalence o B and D and o C and E; the roo s the same Havn establshed the equvalence o B, C, D and E, we shall now see that B mles A Indeed I s -conve, then s -sem reular b Prooston 4 n ths case we have everwhere n R Fnall we shall rove that A mles B Thus, assume that the rme dstance uncton s semreular We have to rove that s -conve, but snce t s ecewse lnear n the sectors dened b the vectors n W, t s enouh to rove that the lnear nterolaton, o between the ras Rw and Rw les above on the ra, R The uncton, s ven b sw tw s w t w s t R,,, Now the value o, at a ont w and w wth s, t sw tw n the sector dened b N s recsel the lenth assned to the ath rom 0 to sw ollowed b the sement rom that ont to sw tw Suose now that the latter ont s on the ra R w ; thus sw tw rw or some r R B semreulart, the value o that lenth s not smaller than the value o at rw, assumn s, t and r to be nteers Ths means that, at the ont rw In eneral, s, t N, r wll be a ratonal number, Thereore r wll be an nteer we choose s and t as multles o some nteer In vew o the ostve homoenet o and, we must then have, on the whole ra R w, whch, as we remared, means that s -conve Acnowledments The authors would le to than roessor COKselman or hs nd nterest and useul dscussons

15 Reulart roertes o -dstance transormatons n mae analss 57 Reerences [] A Baoum, Boundn subsets o some metrc vector sace Arve or mathematcs, Volume 8, Number -, htt://ddoor/0007/b [] A Baoum, Foundatons o comle analss n non locall conve saces, Functon theor wthout convet condton, Mathematcal studes 93, North Holand, Elsever 00 [3] unla Boroeors, Dstance transormaton n arbtrar dmensons, Comuter Vson, rahcs, and Imae Processn 7, htt://ddoor/006/ [4] unla Boroeors, Dstance transormaton n dtal maes, Comuter Vson, rahcs, and Imae Processn 34, htt://ddoor/006/s [5] C O Kselman, Conve unctons on dscrete sets, lette and unc, 004 [6] C O Kselman, Dtal eometr and mathematcal morholo, lecture notes, Usala Unverst, deartment o mathematcs, 004 [7] C O Kselman, Dtal Jordan curves theorems Dscrete eometr or comuter maer, 9 th nternatonal conerence, Usala, Sweden, htt://ddoor/0007/ _5 [8] C O Kselman, Reulart roertes o dstance transormatons n mae analss, comuter vson and mae understandn 64:3996, htt://ddoor/0006/cvu [9] A Petch, Nuclear Locall conve saces, Erebnsse der Mat68 97 [0] R Rocaellar, Conve analss, Prnceton, 97 Receved: March, 05; Publshed: Arl 3, 05