Optimal Positioning Strategy for Multi-Camera Zooming Drones

Depósi o de In es igación de la Uni e sidad de Se illa h ps://idus.us.es/ This is an Accep ed Manusc ip o an a icle published by IEEE: M. Va gas, C. Vi as and T. Alamo, "Op imal Posi ioning S a egy o Mul i- Came a Zooming D ones," in IEEE/CAA Jou nal o Au oma ica Sinica, ol. 11, no. 8, pp. 1802-1818, Augus 2024, doi: 10.1109/JAS.2024.124455. “© 2024 IEEE. Pe sonal use o his ma e ial is pe mi ed. Pe mission om IEEE mus be ob ained o all o he uses, in any cu en o u u e media, including ep in ing/ epublishing his ma e ial o ad e ising o p omo ional pu poses, c ea ing new collec i e wo ks, o esale o edis ibu ion o se e s o lis s, o euse o any copy igh ed componen o his wo k in o he wo ks.” 1 Op imal posi ioning s a egy o mul i-came a, zooming d ones Manuel Va gas, Ca los Vi as and Teodo o Alamo, Membe , IEEE Abs ac —In he con ex o mul iple- a ge acking and su - eillance applica ions, his pape in es iga es he challenge o de e mining he op imal posi ioning o a single au onomous ae ial ehicle o agen equipped wi h mul iple independen ly-s ee able zooming came as o e ec i ely moni o a se o a ge s o in e es . Each came a is dedica ed o acking a speci ic a ge o clus e o a ge s. The key inno a ion o his s udy, in compa ison o exis ing app oaches, lies in inco po a ing he zooming ac o o he onboa d came as in o he op imiza ion p oblem. This enhancemen o e s g ea e lexibili y du ing mission execu ion by allowing he au onomous agen o adjus he ocal leng hs o he on-boa d came as, in exchange o a ying eal-wo ld dis- ances o he co esponding a ge s, he eby p o iding addi ional deg ees o eedom o he op imiza ion p oblem. The p oposed op imiza ion amewo k aims o s ike a balance among a ious ac o s, including dis ance o he a ge s, e icali y o iewpoin s, and he equi ed ocal leng h o each came a. The p ima y ocus o his pape is o es ablish he heo e ical g oundwo k o add essing he non-con ex na u e o he op imiza ion p oblem a ising om hese conside a ions. To his end, we de elop an o iginal con ex app oxima ion s a egy. The pape also includes simula ions o di e se scena ios, ea u ing a ying numbe s o onboa d acking came as and a ge mo ion p o iles, o alida e he e ec i eness o he p oposed app oach. Index Te ms—P ojec i e ans o ma ion, isual su eillance, isual objec acking, unmanned ae ial ehicle, con ex op imiza- ion. I. INTRODUCTION The combina ion o a emo ely pilo ed ai c a (RPAS) o unmanned ae ial ehicle (UAV) wi h an on-boa d s ee able came a, as a compound elec omechanical de ice ha can pu “eyes in he sky” [1], has been ex ensi ely s udied and success ully employed in a wide ange o applica ions. This ange om ea ly in elligence-ga he ing mili a y applica ions, o he mo e ecen widesp ead use in en e ainmen ac i i ies, co e ing in as uc u e inspec ion [2], [3], [4], sea ch and escue missions [5], a ic su eillance [6], [7], coope a i e o uncoope a i e a ge acking [8], [9], and a my iad o o he applica ions. The on-boa d came a is ypically moun ed on an ac ua ed gimbal, which se es a dual unc ion. Fi s , i ac s as a line-o - sigh s abilize , coun e ac ing he u ns and/o il s expe ienced by he ae ial pla o m. Second, i unc ions as a isual acking de ice, aimed a he desi ed a ge o in e es [10], [11], [12], [13], [14]. Fo a be e unde s anding o he imp essi e ea u es and pe o mance o con empo a y ae ial gimbaled zooming sys ems, such as he one moun ed on he DJI Zenmuse H20©, M. Va gas, C. Vi as and T. Alamo a e wi h he Depa men o Sys em Enginee ing and Au oma ion, Uni e si y o Se ille, Se ille, Spain. E-mail: {m a gas, i as, alamo}@us.es. please e e o he ollowing ideo: h ps://www.you ube.com/ embed/G6VRhckcMeo?s a =53&end=82. When he objec i e is o simul aneously keep ack o se e al a ge s ha migh be dis an om each o he , while obse ing each one o hem wi h su icien isual de ail, he e a e ypically wo app oaches. The i s app oach is s aigh o - wa d: deploying se e al ehicles wi h simila capabili ies, each one de o ed o a single a ge o a g oup o nea by a ge s [15], [16]. The second app oach, on he o he hand, ocuses on sha ed a en ion, whe e a single ehicle pe iodically changes i s came a’s aiming di ec ion among di e en a ge s o a oid losing ack o any o hem pe manen ly [17], [18], [19]. This second ca ego y, which has ecei ed less a en ion om he esea ch communi y, is ne e heless qui e a ac i e as i aims o accomplish he ask wi h minimal esou ce deploymen . This app oach is highly bene icial in e ms o implemen a ion cos and ene gy demands du ing missions. Addi ionally, i is less in usi e, a ea u e ha can be e y ele an in ce ain a eas o applica ion such as wildli e moni o ing o concealed su eillance ope a ions. In e ms o downsides, he sha ed-a en ion app oach aces inhe en cons ain s in he obse a ion pe iod o each a ge , in e sely co ela ed wi h he o al numbe o a ge s. Ano he challenge is he occu ence o ”blind lapses” du ing he a ge - swi ching ac ion, wi h no a ge wi hin he ield o iew o he in ol ed came a. This can become a comp omising ac o , especially in he p esence o a la ge numbe o sca e ed a ge s wi h a he unp edic able ajec o ies. Fu he mo e, while in e mi en obse a ion may su ice o es ima ing a ge ajec o ies, in applica ions whe e closely moni o ing and documen ing ac i i ies o beha io s is pa amoun , such gaps in obse a ion can pose a se ious handicap. In a ecen wo k [20], he concep o lying chameleons was in oduced as a mo e lexible al e na i e in he con ex o ae ial mul i- a ge acking and su eillance. Unde his pa adigm, one single ae ial ehicle was equipped wi h se - e al independen ly s ee able came as o isually ack se e al a ge s in su icien de ail simul aneously. A con enien op i- miza ion p oblem was o mula ed, and se e al s a egies o i s esolu ion we e p oposed. Howe e , in his p e ious wo k, he p oblem was signi ican ly cu ailed by he assump ion o ixed ocal leng hs o he onboa d came as. As a esul , cap u ing a close iew o a speci ic a ge equi ed physically mo ing he ae ial ehicle close o i , po en ially comp omising he obse a ion o he o he a ge s. The p esen pape ex ends he p e ious wo ks by inco po- a ing he zooming e ec on he on-boa d came as, which can be au oma ically adjus ed in eal- ime acco ding o an op imal 2 c i e ion. This new ea u e in oduces a g ea deal o lexibili y o he concep ual sys em, signi ican ly enhancing i s capabi- li ies by adding new deg ees o eedom o he op imiza ion p oblem. Howe e , his enhancemen comes wi h a signi ican oll on he heo e ical de elopmen , as i b eaks away om he con ex na u e o he o iginal p oblem. To add ess his challenge, we in oduce an en i ely new o mula ion in his wo k ha no only ci cum en s his downside bu also ensu es he easibili y o he esul ing op imiza ion p oblem. The supe io mission capabili ies enabled by he in oduc- ion o he zooming e ec ha e been s udied in p e ious wo ks o he case o single-came a se ups [21], [22]. Howe e , a cau iona y no e mus be obse ed he e. The applica ion o high magni ica ion ac o s can make ine image s abiliza ion mo e challenging, ypically demanding he con luence o high- accu acy gimbal con ol and elec onic image s abiliza ion [23], [24]. Bea ing his la e conside a ion in o accoun , he p esen wo k is in ended o ake ad an age o hose enhanced capabili ies in he con ex o mul i-came a, mul i- a ge acking. The p oposed amewo k is well-sui ed o applica ions ac oss a ious domains. In high seas sea ch and escue missions, whe e people and small boa s may be ad i a e shipw ecks o disas e s, con inuous acking is c ucial o p io i izing escue e o s [25], [26], [27]. Simila ly, in su ei- llance missions in ol ing suspicious essels [28], such as hose used by smuggle s o illegal mig a ion g oups, simul aneous acking o mul iple essels migh be essen ial o ga he ing isual e idence and s eng hening bo de con ol [29], [30]. This app oach is also aluable o co e mili a y ope a ions [31], pe ime e de ense [32], [33], con oy p o ec ion [34], as well as o homeland secu i y missions and o he secu i y applica ions [35]. Mo eo e , in spo ing e en s [36], [37], cinema og aphy [38], [39], wildli e moni o ing [40], [41], and a ic su eillance [42], [43], he amewo k would allow o simul aneous acking and ilming o mul iple a ge s while minimizing in usion and maximizing co e age. The h ee main con ibu ions o he p esen pape can be summa ized as ollows: •A signi ican enhancemen o he exis ing concep o mul i-came a, mul i- a ge acking agen s, achie ed by in eg a ing he zooming capabili y in o he on-boa d acking came as. •In oduc ion o a no el op imiza ion app oach ailo ed o his enhanced amewo k, add essing he inhe en non-con exi y o he p oblem h ough con ex elaxa ion echniques while ensu ing i s easibili y. •Ex ension o p io esea ch o enable he al e na i e o acking clus e s a he han indi idual a ge s, b oade- ning he applicabili y and e icacy o he p oposed ame- wo k. I is impo an o emphasize ha his is a heo e ical wo k, ocused on highligh ing he po en ial applica ions o he p oposed concep in isual acking and su eillance o mul iple mo ing a ge s. The p oposed op imal guidance law is designed solely o p o ide he au onomous ehicle and he se o on-boa d came as wi h op imal se poin s o posi ion and ocal leng hs, based on he es ima ed loca ions o a ge s a each ins an . This wo k does no add ess any speci ic low-le el pa h planning s a egy in he 3D en i onmen [44], [45]. Such s a egies should accoun o he maneu e abili y and dynamic es ic ions o he ehicle i sel , which can a y signi ican ly depending on whe he dealing wi h holonomic ehicles like VTOL ( e ical ake-o and landing) mul i o o d ones o ehicles wi h s ong mo ion cons ain s such as ixed-wing ai c a s. These conside a ions a e beyond he scope o his pape and will be add essed in u u e wo ks, whe e a ge obse a ion and ehicle guidance can be in eg a ed in a uni ied manne . Table Iin oduces he no a ion used h oughou he pape . As a gene al ule, oman ype ace is ese ed o ec o s and ma ices. The es o he pape is o ganized as ollows. Sec ion II in oduces he wo king scena io, including he assump ions made on he di e en elemen s conside ed and he p inciples used o o mula e he op imiza ion p oblem. Sec ion III is dedica ed o p esen ing he o mula ion o he op imiza ion p oblem. Sec ion IV desc ibes he con ex elaxa ion o he s a ed p oblem and p o ides an in ui i e geome ic in e p e- a ion o his app oxima ion. I also ou lines he p oposed op imiza ion algo i hm. Sec ion V o e s some simula ion expe imen s demons a ing he pe o mance o he p oposed s a egy. Finally, Sec ion VI p esen s he conclusions o he wo k and ou lines u u e lines o esea ch. II. FRAMEWORK In his sec ion, we desc ibe he ope a ional amewo k, p oposing a single mul i-came a, zooming d one, e e ed o as he agen , as a sui able concep in he con ex o ae ial su eillance and mul i- a ge acking. Fi s , we ou line he speci ic ope a ional en i onmen and he unde lying assump- ions. Nex , we elabo a e on he ounda ional concep s behind he op imal e e ence-gene a ion s a egy o single agen s, which will be p oposed in he ollowing sec ion. A. Wo king Scena io As men ioned ea lie , he concep o in eg a ing mul i- came a se ups on o au onomous ae ial ehicles can be applied ac oss a wide ange o pla o ms, spanning om ligh mul i- o o sys ems o hea y ixed-wing UAVs. This se up ypically comp ises one cen al wide o ul a-wide angle came a, e- e ed o as C0, su ounded by n acking came as deno ed as Ci, whe e i=1, . . . , n. The cen al came a’s ole is o p o ide a pe ec ly zeni hal, low- esolu ion o e iew o he scene below, while he acking came as, equipped wi h mo o ized zoom lenses, a e able o acqui e images o dis an a ge s wi h he desi ed le el o isual de ail. Fig. 1illus a es his concep o a mul i o o ehicle wi h n=2 acking came as. Fig. 1also illus a es he coo dina e ames in ol ed in he se up. Fi s , {W} ep esen s he wo ld coo dina e ame de ined, o ins ance, ollowing he ENU (eas –no h–up) con igu a ion. I s o igin is ypically loca ed a a con enien s a- iona y poin on he g ound su ace. The nex e e ence ame, 3 TABLE I NOTATION Symbol Desc ip ion nNumbe o acking came as. mNumbe o nes ed in e als p esen in he op imiza ion index. {W}Wo ld coo dina e ame. {Ci}i- h came a ame (i={0,...,n},i= 0 e e s o cen al cam.). C jzi3D posi ion o i- h a ge wi h espec o {Cj}. zi3D posi ion o i- h a ge wi h espec o {W}. iEqui alen adius o i- h a ge in he scene. x3D posi ion o he ehicle wi h espec o {W}. x∗Op imal alue o x. x∗ cCon ex app oxima ion o he op imal alue o x. ˆx Value o xwhe e he con ex app oxima ion is made. ℓi(ˆx) Linea ay in R3, s a ing in ziand wi h di ec ion (ˆx−zi). pi, Vehicle’s ideal e e ence posi ion wi h espec o i- h a ge . x, y, h Componen s o x, ha is: x=[x, y, h] ⊤. XCon ex easibili y se cons aining he solu ion. hmin Minimum limi o he allowed al i ude ange. hmax Maximum limi o he allowed al i ude ange. diEuclidean dis ance o he i- h a ge : di:= ∥x−zi∥2. dScala dummy a iable. ηiRela i e o i- h a ge , uni a y ec o used in con ex elaxa- ion. iAc ual ocal leng h o he i- h came a. i, Re e ence alue o i. i,min Minimum allowed alue o i. i,max Maximum allowed alue o i. siEqui alen adius o i- h a ge ’s p ojec ion on he image plane. si, Re e ence alue o si. si,min Minimum allowed alue o si. si,max Maximum allowed alue o si. si,pix Equi alen o si, bu exp essed in pixels. si, ,pix Equi alen o si, , bu exp essed in pixels. si,min,pix Equi alen o si,min, bu exp essed in pixels. si,max,pix Equi alen o si,max, bu exp essed in pixels. Li,j j- h lowe limi imposed on some measu e o he i- h came a- a ge pai . Ui,j j- h uppe limi imposed on some measu e o he i- h came a- a ge pai . µi,νiWeigh ing coe icien s p esen in he op imiza ion index, e- la i e o he i- h came a- a ge pai . τi,j , σi,j Weigh ing coe icien s, ela i e o he i- h came a- a ge pai and he j- h nes ed in e al, p esen in he op imiza ion index. βiExponen in he p oposed op imiza ion index, linked o he i- h came a- a ge pai . αi,j Exponen in he p oposed op imiza ion index, linked o he i- h came a- a ge pai and o he j- h nes ed in e al. ρwiE ec i e pixel wid h o he i- h image senso , in me ic uni s pe pixel. ρhiE ec i e pixel heigh o he i- h image senso , in me ic uni s pe pixel. ρiRegula ized pixel leng h, o accoun o senso s wi h non- squa e pixels ρi:= √ρwiρhi. e1,e2,e3Vec o s de ining he canonical basis in R3. J(x) Ini ial op imiza ion index. J◦(x) Simpli ica ion o J(x), by emo ing he non-con ex e ms. Jc(x,ˆx) Con ex app oxima ion o J(x) a ound ˆx. Ψi,Λi,Γ iMain e ms composing he op imiza ion index o he i- h a ge . lDis ance be ween cen e s o a ge s o clus e s. Fig. 1. Illus a ion o he concep using a mul i o o pla o m, a cen al wide-angle came a on a 2-DOF gimbal, and wo acking came as wi h hei espec i e 3-DOF gimbals. Adap ed om [20]. deno ed as {B}, is igidly a ached o he ehicle’s body. Each came a has i s own coo dina e ame, deno ed as {Ci}. Each gimbal adjus s he ine ial o ien a ion angles necessa y o he co esponding came a by de ining he o ien a ion o {Ci}wi h espec o {W}. Fo a de ailed desc ip ion o he kinema ics in ol ed, please e e o [20]. We conside mul i- a ge acking as a wo-le el ask. A he bo om le el, whe e e he lying pla o m is posi ioned, he a ge aiming p oblem mus be sol ed. A he op le el, assuming ha he a ge s a e being co ec ly aimed o, a s ee ing s a egy mus be designed so ha he ehicle can ecei e an op imal posi ioning se -poin , as a unc ion o he cu en a ge dis ibu ion, acco ding o a gi en c i e ion. In his wo k, we s ic ly ocus on he second s age. B. Gene al assump ions and simpli ica ions He e, we ou line he a p io i condi ions ha a e assumed in he subsequen desc ip ion. Fi s , i is assumed ha an es ima e o he ehicle’s al i ude and global posi ion is a ailable a all imes, p o ided by he na iga ion senso y sys em. Addi ionally, in o ma ion abou he ine ial o ien a ion o each on-boa d came a is assumed o be a ailable. Fu he mo e, i will be assumed ha he na iga ion con ol sys em (in he case o he ae ial pla o m) o he co esponding o ien a ion con ol sys ems (in he case o each gimbal-came a se ) a e in p ope wo king o de , ul illing he p o ided a i ude, posi ion, o aiming se poin s as e icien ly as possible. In o de o simpli y subsequen geome ical desc ip ions, i is also pos ula ed ha he came a is a ached o he co e- sponding gimbal in such a way as o ensu e ha any gimbal- induced o a ion occu s a ound he came a nodal poin , hus elimina ing no iceable pa allax e ec s. In p ac ice, came as a e ypically moun ed o minimize g a i a ional ine ia, making ha pa icula se up unlikely. Howe e , due o he expec ed signi ican p oximi y be ween he heo e ical nodal poin and he cen e o g a i y ela i e o he dis ances o he objec s 4 o in e es , his app oxima ion emains alid. Mo eo e , i is easonable o assume ha he eal- ime ope a ion o he isual- eedback gimbal con ol loop will add ess any misalignmen issues. On he o he hand, he e ec o lens dis o ion will be igno ed in ou desc ip ion. This will no en ail any loss o gene ali y in he p oposal, p o ided ha , in he p esence o signi ican lens dis o ion, images will unde go a ec i ica ion p ocess be o e u he p ocessing. Wi h ega d o a ge posi ion es ima ion and aiming, we a e only in e es ed in he posi ion and appa en size on he image o each a ge o clus e o a ge s, ega dless o hei o ien a ion. In he case whe e a came a is acking a single a ge , he e e ed posi ion will be de ined by he appa en cen e o i s silhoue e in he image, while in he case o acking clus e s, a sui able aiming cen e will be de ined as a poin o in e es . On he o he hand, as men ioned abo e, he desc ip ion o any low-le el a i ude con ol o he ae ial pla o m o any low- le el gimbal con ol is ou o he scope o he pape , as hese can be conside ed issues la gely sol ed om a heo e ical and echnical poin o iew [46], [47], [7], [48]. Fo he sake o simplici y in he desc ip ion, i is assumed ha he dis ances be ween he posi ion o any came a ame {Ci}and he own ehicle’s body ame {B}a e negligible, compa ed o he dis ances om he ae ial pla o m o any one o he a ge s. In ac , o he in ended p ac ical applica ions, i is expec ed ha his di e ence comes in se e al o de s o magni ude. I will be assumed ha i a pa icula a ge is wi hin he ield o iew o one o mo e onboa d came as, i s absolu e posi ion wi h espec o {W}can be es ima ed a any ime. This es ima ion may be achie ed by adhe ing o he la - ea h assump ion, as demons a ed in [20], which elimina es he need o s e eo, ele a ion maps, o any o he ange- ga he ing mechanism. This simpli ica ion is app oxima ely alid in ce ain con ex s and has been p e iously exploi ed by nume ous o he au ho s, as e idenced in [17], [49], [12], [50], [28], [13], [19]. No ably, o sho e sea ch and escue (SAR) ope a ions [25], [26], [27] ep esen a clea example whe e he la -ea h assump ion can be e ec i ely le e aged. Fo ou pu poses, i his assump ion holds, and wi h knowledge o he absolu e posi ion o he ae ial ehicle and he ine ial o ien a ion o he obse ing came a(s), he absolu e posi ion o he a ge can be de i ed (please e e , o ins ance, o he desc ip ion p o ided in [20]). Simila ly, an es ima ion o he a ge ’s size in he eal wo ld could be geome ically de i ed. In he desc ip ion ha ollows, howe e , we will no necessa ily s ick o he la -ea h assump ion, as any o he dep h es ima ion mechanism would be equally alid o ou pu poses. Du ing ini ializa ion, i is an icipa ed ha he downwa d- acing, wide-angle came a (C0) will ha e all he a ge s o in e es wi hin i s ield o iew, enabling he es ima ion o he posi ion o each a ge wi h espec o his came a, deno ed as C0zi, whe e i= 1, . . . , n. Using his ec o and he known kinema ics o he came a a angemen , i becomes easible o es ima e he posi ion o he a ge wi h espec o he acking came a ha will be assigned o i 1,Ci. Subsequen ly, he co esponding came a is o ien ed owa ds he a ge in a manne ha posi ions i app oxima ely a he cen e o i s ield o iew (please e e again o [20] o he de ailed compu a ional desc ip ion). Finally, he isual se oing module o ha gimbal-came a se could be ini ia ed o add ess any misalignmen s in he ini ial aiming owa ds he a ge . When employing a clus e ing s a egy, i can be di ec ly implemen ed wi hin he wo-dimensional space o he cen al came a’s iew. Mo eo e , he posi ions co esponding o he aiming cen e o each clus e should be con e ed om image coo dina es o 3D posi ions. As i can be unde s ood om he desc ip ion in he p e ious pa ag aph, when clus e ing is u ilized, i is jus a ma e o eplacing he poin o in e es o a single a ge by he aiming cen e o he app op ia e clus e . C. Gene al guidelines o op imal posi ioning This sec ion delinea es he guidelines o con igu ing he s a egy o he op imal se poin p o ision o he mul i-came a, mul i- a ge acking sys em unde in es iga ion. As we place special emphasis on a ge obse a ion, he objec i e is no me ely o keep he ehicle as close as possible o he se o a ge s, bu o do so while p omo ing a ce ain deg ee o e icali y o he iewpoin wi h espec o each one o hem. Simul aneously, we aim o ensu e an op imal balance be ween he dis ance o each a ge and he ocal leng h o he espec i e came a. Essen ially, he op imiza ion p oblem in ol es quan i a i ely ha monizing hese h ee aspec s. As s a ed in [20], en o cing e icali y may be o pa icula in e es o se e al easons: •The absence o any cons ain o conce n ega ding e icali y would g an un es ic ed lexibili y o ob ain images o he a ge om ully zeni hal o ully ho izon al iewpoin s. The subsequen a iabili y in he a ge ’s ap- pea ance could pose la ge challenges o he image-based acking algo i hms, po en ially making he acking ask mo e complex. •In a scena io whe e mul iple a ge s o unspeci ied heigh a e in mo ion, enabling acking o such a ge s wi hou imposing minimum e icali y and al i ude cons ain s can po en ially esul in an inc eased likelihood o occlu- sions. These occlusions may be due o unin e es ing objec s obs uc ing he a ge s, in e e ence be ween he a ge s hemsel es, o e en he isk o one onboa d came a obs uc ing he line o sigh o ano he . •In cases whe e he goal is o su ey a ge s wi h minimal in usion, e en wi h he in en ion o going unno iced by he a ge s hemsel es, i is ad an ageous o apply a minimum al i ude h eshold while op imizing e ical alignmen ela i e o he a ge se . •When wo king wi h clus e s ins ead o isola ed a ge s, we will obse e ha ou iso opic me ic o assessing he obse a ion quali y o a gi en clus e becomes mo e accu a e as he pe spec i e o he iewpoin s becomes less p ominen . 1Wi hou loss o gene ali y, each a ge (o clus e ) is allo ed a numbe ha ma ches he numbe o he acking came a o which i is assigned. 5 On he o he hand, ega ding he op imal ade-o be ween dis ance o a gi en a ge and he ocal leng h ea u ed by he co esponding came a, one migh simply belie e ha ying o exploi , as much as possible and wi hou any penal y, he a ailable zooming ange o he came a is, om an economic o ene ge ic poin o iew, he bes app oach. Howe e , his migh esul in excessi e p essu e on he pa o he aiming de ice in e ms o i s accu acy o s abili y speci ica ions. Al e na i ely, ying o ind a con enien balance be ween a oidance o unnecessa y ehicle displacemen s and pushing he zooming mechanism o consis en ly wo k on i s op limi , could be a wise policy. III. PROBLEM FORMULATION This sec ion in oduces he p oposed s a egy o p o iding e e ences o he mul i-came a agen , comp ising a sui able combina ion o he ehicle’s posi ion ela i e o he se o a ge s and he speci ic alues o he ocal leng hs. The main idea is ha he p oposed op imiza ion index will consis o a e m o each a ge / acking came a pai . Each o hese e ms will include se e al sub e ms, each imposing penal ies o iola ing speci ic condi ions, which will be des- c ibed in de ail. We deno e by x∈R3 he main decision a iable, which ep esen s he absolu e posi ion o he ae ial ehicle wi h espec o {W}. Acco ding o one o he assump ions made in Sec ion II-B, i can also be used as an app oxima ion o he posi ion o any onboa d came a. We assume he p esence o n a ge s, wi h posi ions zi,i= 1, . . . , n, also e e ed o {W}. Each acking came a is in cha ge o acking one pa icula a ge and each a ge is moni o ed by a single acking came a. The scala i ep esen s he equi alen adius o he i- h a ge , and is used as a con enien measu e o he size o each a ge in he eal wo ld. On he o he hand, dideno es he Euclidean dis ance o he i- h a ge , di=∥x−zi∥2. The ocal leng h o he i- h came a is deno ed by i. I will be assumed ha a p e e ed e e ence alue o his pa ame e has been a p io i chosen wi hin he wo king ange i, ∈[ i,min, i,max]. S a ing om he pe spec i e p ojec ion p inciple, we can exp ess he ollowing equi alence o a ios: di i = i si , i = 1, . . . , n. (1) whe e si ep esen s he equi alen adius o he p ojec ed image o he objec in me ic uni s. This dis ance can be mo e con enien ly ob ained om i s equi alen exp essed in pixels, si,pix:si≈ρisi,pix, whe e ρi ep esen s he egula ized pixel leng h o he co esponding came a. This app oxima ion becomes exac o image senso s wi h squa e pixels. Suppose he ocal leng h is se o he e e ence alue i, . The dis ance di, , a which he adius o he a ge in he image equals a p ede ined desi ed alue, si, , is ob ained om he ollowing exp ession: di, i = i, si, , i = 1, . . . , n. We conclude ha di, = i i, si, , i = 1, . . . , n ep esen s he mos con enien dis ance o each a ge , as i ensu es he desi ed size o he a ge in he image when using he e e ence ocal leng h. Acco ding o ou discussion in Subsec ion II-C, e ical iews a e o en p e e ed, so he ideal posi ion pi, o he came a wi h ega d o he i- h a ge would be pi, = zi+  0 0 1  di, = zi+ e3di, , i = 1, . . . , n. As a s a ing poin , we could ob ain he posi ion xby sol ing he op imiza ion p oblem min n X i=1 µi∥x−pi, ∥2 2,(2) The weigh ing ac o s µi∈R≥0se e o po en ially p io i ize some a ge s o e o he s. This uncons ained, s ic ly con ex op imiza ion p oblem has an explici solu ion. By di e en ia ing wi h espec o x, we ob ain ha he solu ion x∗sa is ies he ollowing equa ion: n X i=1 2µi(x∗−pi, )=0. Tha is, x∗= n P i=1 µipi, n P i=1 µi .(3) The p oblem s a emen can be imp o ed in di e en ways. Fo example, cons ain s could be included in he ehicle’s posi ion wi h espec o {W}, i.e., min x∈X n X i=1 µi∥x−pi, ∥2 2. Fo ins ance, a es ic ed al i ude ange could be speci ied: X=x∈R3:hmin ≤x⊤e3≤hmax.(4) Ano he imp o emen has o do wi h he e icali y o he iewpoin s. I should be no ed ha he al i ude p o ided by (3) does no accoun o he dispe sion o he a ge s. To p io i ize al i ude in cases whe e he a ge s a e widely sca e ed, a new e m could be in oduced, such as he ollowing: n X i=1 νi(∥x−zi∥2−(x −zi)⊤e3)2.(5) As we can see, o any pa icula a ge , his e m penalizes he dispa i y be ween he dis ance o he ehicle, gi en by he no m o (x−zi), and he p ojec ion o his same ec o on o he e ical di ec ion. 6 Fo con enience, he new op imiza ion p oblem can be ew i en by g ouping bo h p e ious se o e ms oge he as ollows: min x∈X n X i=1 Γi(x,zi)(6) wi h Γi(x,zi) = µi∥x−pi, ∥2 S+νi(∥x−zi∥2−(x −zi)⊤e3)βi. To p o ide mo e gene ali y o he app oach, a weigh ed Eu- clidean no m has been in oduced o penalize he disc epancy (x−pi, ). We hus assume ha S∈R3×3is posi i e de ini e and ha ∥x−pi, ∥2 S= (x −pi, )⊤S (x −pi, ). Addi ionally, he squa e o m in (5) has been gene alized by allowing any βi≥1as he exponen . P ope y 1: Suppose ha µi≥0, νi≥0, βi≥1,i= 1, . . . , n and Sis a posi i e de ini e ma ix. Then he unc ion n X i=1 Γi(x,zi)is con ex. P oo : Fi s , we no ice ha µi∥x−pi, ∥2 Sis, by cons uc- ion, a s ic ly con ex e m. Rega ding he second e m in Γi, since he Euclidean no m is con ex and we a e sub ac ing om i a linea o m o he a iable, his e y di e ence is also con ex. Addi ionally, we obse e ha his di e ence is always non-nega i e. Acco ding o a p ope y o con ex unc ions (see, o ins ance, Example 3.13 o [51]), i g(x) is a con ex and nonnega i e unc ion o xand βi≥1, hen g(x)βi is con ex. The e o e, n X i=1 Γi(x,zi)is con ex as i esul s om he non-nega i e weigh ed sum o con ex unc ions. Ano he se o ele an cons ain s a ises when conside ing bounds on he ocal dis ances. Suppose ha one desi es o impose ha each ocal leng h ilies wi hin i s ope a ional ange [ i,min, i,max]: i,min ≤ i≤ i,max, i = 1, . . . , n. (7) As discussed ea lie , in acco dance wi h he desi ed adius o he a ge p ojec ion on he image, si, , he e exis s an op imal a io be ween he dis ance o he a ge and he ocal leng h: di i = i si, , i = 1, . . . , n. (8) We in e ha , in o de o gua an ee he op imal a io di i , while p ese ing he cons ain s (7), i is ad isable o impose he ollowing condi ions: i si, i,min ≤di≤ i si, i,max, i = 1, . . . , n. By g adually elaxing he equali y cons ain imposed by (8), one could also conside imposing limi s on he adius siin he image, ha is si,min ≤si≤si,max. Again, om he pin-hole came a model (1), and he cons ain s si∈[si,min, si,max], i∈[ i,min, i,max]we ob ain i si,max i,min ≤di≤ i si,min i,max, i = 1, . . . , n. Viola ing he p e ious inequali ies indica es ha he cons ain si∈[si,min, si,max]canno be sa is ied e en i he ocal leng h is adjus ed o i s ope a ional limi s (ei he he a ge is oo close o oo dis an ). To simpli y he exp essions in he pape , we deno e by Li,j and Ui,j a ious lowe and uppe bounds on he dis ance di= ∥x−zi∥2 o he i- h a ge . Le us deno e Li,1= i si, i,min, Ui,1= i si, i,max , i=1, . . . , n. Li,2= i si,max i,min, Ui,2= i si,min i,max , i=1, . . . , n. We no ice ha , by cons uc ion, di, ∈[Li,1, Ui,1]⊆[Li,2, Ui,2], i = 1, . . . , n. We now enume a e a ious si ua ions conce ning each a ge , anging om op imal posi ioning o non-accep able posi ioning: •x = pi, : This ep esen s op imal e ical alignmen wi h espec o a speci ic a ge , as well as he op imal ocal leng h and p ojec ed image size. •di=di, : Op imal ocal leng h and image size. •di∈[Li,1, Ui,1]: Focal leng h wi hin he ope a ional ange and op imal image size. •di∈[Li,2, Ui,2]: Bo h ocal leng h and image size a e wi hin hei ope a ional anges. •di∈ [Li,2, Ui,2]: The size in he image will no mee he cons ain s, e en when se ing he ocal leng h o he limi s o he ope a ional ange. A. Cos unc ions Ψiand Λi When he numbe o a ge s is g ea e han 3, i is gene ally no possible o choose xin such a way ha he ehicle is placed a he op imal dis ance wi h espec o e e y a ge . We now de ine a gene al class o unc ions ha could be used o penalize he ehicle no being placed a he op imal dis ance wi h espec o he i- h a ge . Conside , as in he p e ious subsec ion, ha o each a ge , we ha e a nes ed se o in e als ha include he desi ed dis ance di, . Tha is, di, ∈[Li,0, Ui,0]⊆[Li,1, Ui,1]⊆[Li,2, Ui,2]⊆. . .⊆[Li,m, Ui,m]. Acco ding o ou p e ious desc ip ion, he i s in e al could be de ined as: Li,0=Ui,0=di, . This cons i u es a me e gambi o allow a mo e gene al and well- ounded desc ip ion o he nes ed-in e als s a egy. We p opose penalizing unc ions Ψi(·)and Λi(·) ha assign a posi i e cos when di=∥x−zi∥2is no included in he nes ed se o in e als. Fo any scala a iable, d, hese unc ions will be de ined as: Ψi(d) = m X j=0 τi,j (max{0, d −Ui,j})αi,j (9) Λi(d) = m X j=0 σi,j (max{0, Li,j −d})αi,j ,(10) whe e 7 •The scala s τi,j ≥0and σi,j ≥0se e o po en ially assign di e en weigh s o each a ge (i.e. p io i izing some a ge s). They also allow us o penalize some iola ions mo e han o he s. •The posi i e in ege s αi,j ≥1a e exponen s, which will ypically be chosen equal o 1 o 2. Example: Suppose ha m= 2,τi,0=σi,0= 1,τi,1=σi,1= 2,τi,2=σi,2= 10. Suppose also ha one makes Li,0=di, , Ui,0=di, , Li,1= i si, i,min , Ui,1= i si, i,max , Li,2= i si,max i,min , Ui,2= i si,min i,max. Then Ψi(di)+Λi(di)=|di −di, |+ 2 X j=1 τi,jmax{di −Ui,j , Li,j −di,0}αi,j . The i s e m penalizes he de ia ion be ween di=∥x−zi∥2 and di, , he second e m penalizes de ia ions om he op imal choice o ocal leng h o di, and he hi d e m penalizes dis ances whe e he a ge ’s speci ica ions o image size canno be me wi hin he ope a ional ange o he ocal leng h. We no ice ha Λi(∥x−zi∥2)is a non-inc easing unc ion o ∥x−zi∥2, since i penalizes being oo close o he i- h a ge . The le el se s o he unc ion a e non-con ex, because he inequali y Λi(∥x−zi∥2)≤ρ ansla es in o a cons ain o he o m ∥x−zi∥2≥ ρ, which is non-con ex. The opposi e happens wi h unc ions Ψi(∥x−zi∥2), which, as s a ed in he ollowing p ope y, a e con ex wi h espec o x. This con exi y will be ele an when conside ing nume ical op imiza ion me hods o ob ain he op imal alue o x. P ope y 2: Suppose ha τi,j ≥0,αi,j ≥1,i= 1, . . . , n, j= 1, . . . , m. De ine unc ion Ψ : R≥0→R≥0as Ψi(d) = m X j=0 τi,j (max{0, d −Ui,j})αi,j , i = 1, . . . , n. Then Ψi(∥x−zi∥2)is con ex wi h espec o x. P oo : By de ini ion, Ψi(∥x−zi∥2) = m X j=0 τi,j (max{0,∥x−zi∥2−Ui,j})αi,j . We no ice ha Ψi(∥x−zi∥2)consis s o he non-nega i e weigh ed sum o e ms max{0,ΥUi,j (x)}αi,j , whe e ΥUi,j (x) = ∥x−zi∥2−Ui,j , i = 1, . . . , n, j = 1, . . . , m. (11) Since he Euclidean no m is con ex wi h espec o i s a gumen (see, e.g., subsec ion 3.1.5 o [51]), we can in e ha he unc ions ΥUi,j (x)a e con ex. Fu he mo e, due o he con exi y o ΥUi,j (x)and he inequali y αi,j ≥1, we can conclude ha max 0,ΥUi,j (x)αi,j is also con ex ( he p oo o his claim is p esen ed in Appendix A). The e o e, Ψi(|x−zi|2)is con ex since i esul s om he non-nega i e weigh ed sum o con ex unc ions. B. P oposed op imiza ion p oblem De ine he cos unc ion J(x) = n X i=1 Ψi(∥x−zi∥2)+Λi(∥x−zi∥2)+Γi(x,zi).(12) Suppose also ha he decision a iable x∈R3is cons- ained o belong o a gi en non-emp y con ex se X, e.g. (4). The subsequen op imiza ion p oblem co esponds o he cons ained minimiza ion o he p oposed index P(x) : x∗= a g min x∈X J(x).(13) One o he i ues o he s a ed op imiza ion p oblem is ha i is always easible, which is o special ele ance in his kind o applica ions. The cos unc ion J(x) consis s o h ee di e en ypes o e ms, which a e analyzed in he ollowing: (i) Ψ(∥x−zi∥2): This e m, in iew o P ope y 2, is con ex wi h espec o xand non-dec easing wi h ∥x−zi∥2. (ii) Λi(∥x−zi∥2): As men ioned in he p e ious subsec ion, his e m is non-con ex and non-inc easing wi h ∥x− zi∥2. (iii) Γi(x,zi): Acco ding o P ope y 1, his e m is also con ex on x. Since Xis assumed o be con ex, he only sou ce o non- con exi y in he p oposed op imiza ion p oblem a e he e ms Λi(∥x−zi∥2),i= 1, . . . , n. We de ail in he ollowing sec ion how o build con ex uppe app oxima ions o Λi(∥x−zi∥2). These a e hen used o o mula e a con ex app oxima ion o he en i e o iginal p oblem. IV. CONVEX RELAXATION As commen ed be o e, he unc ions Λi(∥x−zi∥2) = m X j=0 σi,j (max{0, Li,j − ∥x−zi∥2})αi,j , o i= 1, . . . , n, a e no con ex wi h espec o x. We a e, ne e heless, in e es ed in a aining a con enien con ex app oxima ion o such unc ions. In gene al e ms, con exi ica ion o e s an a ac i e al e - na i e om a p ac ical poin o iew, as i will allow a mo e e icien implemen a ion o u u e p ac ical applica ions. By app oxima ing non-con ex op imiza ion p oblems wi h con ex p oblems, con exi ica ion educes compu a ional complexi y and acili a es he de elopmen o scalable and obus solu- ions. Speci ically, his sec ion will demons a e how o de i e uppe con ex bounds o each unc ion Λi(|x−zi|2). The esul ing app oxima e con ex op imiza ion algo i hm will be e alua ed unde a ious scena ios in Sec ion V. By cons uc ion, Λi:R≥0→R≥0is a non-inc easing unc ion on i s a gumen . Tha is, da≤dbimplies Λi(da)≥Λi(db), i = 1, . . . , n. (14) Suppose ha ηi∈R3, sa is ies ∥ηi∥2≤1. Then, η⊤ i(x −zi)≤ ∥ηi∥2∥x−zi∥2≤ ∥x−zi∥2. 8 F om his and he non-inc easing na u e o Λi:R≥0→R≥0 wi h espec o i s a gumen , we ob ain Λi(η⊤(x −zi)) ≥Λi(∥x−zi∥2),(15) ∀η∈ { η∈R3:∥η∥2≤1},∀x∈R3. The ollowing p ope y shows how o ob ain η1,η2,...,ηnin such a way ha he uppe bound Λi(η⊤ i(x −zi)) coincides wi h he o iginal unc ion Λi(∥x−zi∥2)a a gi en poin ˆx. P ope y 3: De ine he unc ions ηi:R3→R3,i= 1, . . . , n, as ηi(ˆx) =    ˆx −zi ∥ˆx −zi∥2 i ˆx = zi 0o he wise. Then, Λi((x −zi)⊤ηi(ˆx)) ≥Λi(∥x−zi∥2),∀x∈R3, i = 1, . . . , n. Mo eo e , he equali y holds along he ay ℓi(ˆx) = {zi+ ληi(ˆx) : λ∈R≥0}. Tha is, gi en i∈ {1, . . . , n}, Λi((x −zi)⊤ηi(ˆx)) = Λi(∥x−zi∥2),∀x∈ℓi(ˆx). P oo : We i s p o e he i s claim. Gi en i∈ {1, . . . , n}, we conside wo cases, ˆx = zi, and ˆx =zi: (i) ˆx = zi: In his case ηi(ˆx) = 0 ∈R3. F om ηi(ˆx) = 0 and he non-inc easing na u e o Λi:R≥0→R≥0, see (14), we ob ain Λi((x −zi)⊤ηi(ˆx)) = Λi(0) ≥Λi(∥x−zi∥2),∀x∈R3. (ii) ˆx = zi: We ha e ha ˆx = ziimplies ∥ηi(ˆx)∥2= 1. Thus, i ollows om inequali y (15) ha Λi((x −zi)⊤ηi(ˆx)) ≥Λi(∥x−zi∥2),∀x∈R3. We now p o e he second claim. As be o e, we conside wo cases, ˆx = zi, and ˆx =zi: (i) ˆx = zi: In his case ηi(ˆx) = 0 and he ay ℓi(ˆx) collapses o a single poin , ℓi(ˆx)={ˆx}. Thus, he equali y Λi(∥x− zi∥2) = Λi((x −zi)⊤ηi(ˆx)) = Λi(0) holds i ially o all x∈ℓi(ˆx). (ii) ˆx = zi: Fo e e y x = zi+ληi(ˆx) ∈ℓi(ˆx), we ha e ∥x−zi∥2=∥zi+ληi(ˆx) −zi∥2 =|λ| ∥ηi(ˆx)∥2=|λ|. (x −zi)⊤ηi(ˆx) = (zi+ληi(ˆx) −zi)⊤ηi(ˆx) =ληi(ˆx)⊤ηi(ˆx) = λ=|λ|, whe e he las equali y holds because, acco ding o he de ini ion o he ay ℓi(ˆx),λ≥0. Hence, o e e y x∈ ℓi(ˆx), we ha e (x −zi)⊤ηi(ˆx) = ∥x−zi∥2. and om his, we inally conclude Λi((ˆx −zi)⊤ηi(ˆx)) = Λi(∥ˆx −zi∥2),∀x∈ℓi(ˆx). The p e ious p ope y s a es ha an uppe bound o index J(x) in oduced in (12) can be ob ained by eplacing Λi(∥x− zi∥2)wi h Λi((x −zi)⊤ηi(ˆx)). As i is claimed in he nex p ope y, he ob ained uppe bound is con ex on x. P ope y 4: Gi en ˆx ∈R3, de ine he unc ions ηi:R3→ R3,i= 1, . . . , n, as ηi(ˆx) :=    ˆx −zi ∥ˆx −zi∥2 i ˆx = zi 0o he wise. Then, Jc(x,ˆx) := n X i=1 Ψi(∥x−zi∥2)+Λi((x−zi)⊤ηi(ˆx))+Γ(x,zi), (16) sa is ies (i) Jc(ˆx,ˆx) = J(ˆx),∀ˆx ∈R3. (ii) Jc(x,ˆx) ≥J(x),∀x∈R3,∀ˆx ∈R3. (iii) Jc(x,ˆx) is a s ic ly con ex unc ion wi h espec o x, o all ˆx ∈R3. P oo : By cons uc ion, we ha e (ˆx −zi)⊤ηi(ˆx) = ∥ˆx − zi∥2,i= 1, . . . , n. Thus, n X i=1 Λi((ˆx −zi)⊤ηi(ˆx)) = n X i=1 Λi(∥ˆx −zi∥2),∀ˆx ∈R3. F om his equali y, we di ec ly in e he i s claim o he p ope y. Addi ionally, in iew o P ope y 3, we ha e Λi((x −zi)⊤ηi(ˆx)) ≥Λi(∥x−zi∥2),∀x∈R3, i = 1, . . . , n. which p o es he e aci y o he second claim. In o de o inish he p oo , we i s no ice ha , as s a ed in P ope y 2, he e ms Ψi(∥x−zi∥2)a e con ex on x. We now p oceed simila ly o he p oo o P ope y 2 o show ha he e ms Λi((x −zi)⊤ηi(ˆx)) a e also con ex on x. No e ha Λi((x−zi)⊤ηi(ˆx)) consis s o he non-nega i e weigh ed sum o e ms o he o m max{0,ΥLi,j (x,ˆx)}αi,j , whe e ΥLi,j (x,ˆx) = Li,j −(x −zi)⊤ηi(ˆx) (17) i= 1, . . . , n, j = 1, . . . , m. a e con ex unc ions since hey a e linea unc ions on x. The con exi y o max{0,ΥLi,j (x,ˆx)}αi,j is p o en in Appendix A. F om he con ex na u e o hese unc ions, we conclude ha Λi((x −zi)⊤ηi(ˆx)) is con ex on xsince i esul s om he non-nega i e weigh ed sum o con ex unc ions. This, along wi h he con exi y o he e ms Γi(x,zi)and Ψi(·), es ablished by P ope ies 1and 2 espec i ely, p o es he hi d claim. A. G aphical in e p e a ion I should be clea by now ha ou p oblem un olds in he Euclidean 3D space. Howe e , in o de o p o ide an in ui i e g aphical in e p e a ion o he app oxima ion p o ided by he con ex elaxa ion, in his subsec ion, we will make a 2-do educ ion o he p oblem, assuming ha all a ge s keep he same cons an alue o he second coo dina e (le us assume ha o all a ge s his componen emains equal o ze o) while hey mo e a ound. By d opping his second componen , we will be able o plo a 3D su ace ep esen ing he o iginal op imiza ion index J(x) desc ibed by (12), as well as he 15 he ac ual image co e age, while he g een ci cum e ences in he bo om- igh subplo s e lec he e e ence alues si, . The ile named es ComplexScena io.mp4, included as supplemen a y ma e ial, p o ides he comple e simula ion o his expe imen in ideo o ma . VI. CONCLUSIONS AND FUTURE WORK A no el app oach o he p oblem o mul i- a ge acking and su eillance by means o a single au onomous ae ial agen is p esen ed. Based on he p emise ha such an agen is equipped wi h se e al independen ly-s ee able zooming came- as, he objec i e is o in e he bes e e ence posi ion o he agen a each ins an , ensu ing op imal moni o ing o a se o designa ed a ge s. By exploi ing he la -ea h assump ion and he knowledge o he agen ’s al i ude measu emen , he posi ion and equi- alen adius o he a ge s o clus e s can be ob ained as a no malized plana oo p in on he scene below. Ideal moni- o ing o a a ge o clus e is hen o mula ed as he i o his equi alen oo p in p ojec ion o a e e ence ci cle cen e ed on he image acqui ed by he acking came a alloca ed o ha a ge o clus e . Simul aneously, i is essen ial o ensu e ha he solu ion espec s he allowed al i ude band o he ehicle and ha he esul ing ocal leng h o e e y came a emains wi hin i s ope a ional ange, as long as i is easible. The esul ing cons ained op imiza ion p oblem is ound o be in insically non-con ex and non-di e en iable. An o iginal app oxima ion o he p oblem is de eloped h ough con ex elaxa ion o cope wi h his non-con ex na u e while ensu ing he easibili y o he p oblem. Simula ed expe imen s demons- a e he g ea e lexibili y o he p oposed scheme o adap o a ied a ge sca e ing condi ions and signi ican di e ences in a ge o clus e sizes. As pa o u u e wo ks, immedia e in e es is ocused on in eg a ing an op imal con ol law o e ec i ely guiding he ehicle, conside ing i s dynamic cha ac e is ics alongside he equi emen s o ideal a ge moni o ing desc ibed in his pape . Ano he immedia e u u e ask is ex ending he p oblem o a collabo a i e mul i-agen amewo k. Gene alizing he solu ion o coo dina e a eam o mul i-came a agen s in scena ios wi h many widely sca e ed a ge s will enable isual su eillance mos e ec i ely. Addi ionally, he e is in e es in e o mula ing he p oblem o u ilize pu e isual in o ma ion di ec ly, wi hou elying on any 3D econs uc ion me hods, which can be seen as he main limi a ion in he p oposed app oach. APPENDIX A PROOF OF THE CONVEXITY OF (max{0,Υ(x)})α Th oughou he desc ip ion gi en in his appendix, (x), h(x) and Υ(x) ep esen gene ic scala unc ions o ec o a iable, while g(·) ep esen s a gene ic scala unc ion o scala a iable. The scala αwill be used as a gene ic exponen , while µis used as weigh ing ac o . P ope y 6: Suppose ha Υ : Rnx→Ris con ex. Then (x) = (max{0,Υ(x)})αis con ex in Rnx o e e y α≥1. P oo : The p oo elies on s anda d esul s om he composi ion o con ex unc ions (see, e.g. subsec ion 3.2.4. o [51]), and i is included he e o comple eness. Deno e h(x) = max{0,Υ(x)}. Since h(x) is he poin wise maximum o wo con ex unc ions, i is also con ex (see, e.g. subsec ion 3.2.3. o [51]). F om he con exi y o h(x) we ha e h(µxa+ (1 −µ)xb)≤µh(xa) + (1 −µ)h(xb), ∀xa,∀xb,∀µ∈[0,1]. De ining g:R→Ras g(y) = yαwe ob ain (x) = (max{0,Υ(x)})α=hα(x) = g(h(x)), whe e, by cons uc ion, h(x) ∈R≥0,∀x. We analyze he i s and second de i a i es o g(y) = yα o non-nega i e alues o yand α≥1: g′(y) = αyα−1≥0,∀y≥0 g′′(y) = α(α−1)yα−2≥0,∀y≥0. F om he e, we in e ha g(y) = yαis a nondec easing and con ex unc ion in R≥0 o e e y α≥1. Suppose now ha xaand xba e wo a bi a y ec o s in Rnx,µ∈[0,1], and α≥1. Taking in o accoun ha h: Rnx→R≥0is con ex in Rnx, and ha , gi en α≥1,g(y) = yαis non dec easing and con ex in R≥0, we ob ain, (µxa+ (1 −µ)xb)=(h(µxa+ (1 −µ)xb))α =g(h(µxa+ (1 −µ)xb)) ≤g(µh(xa) + (1 −µ)h(xb)) ≤µg(h(xa)) + (1 −µ)g(h(xb)) =µ (xa) + (1 −µ) (xb). We no ice ha he i s inequali y is due o he con exi y o h(·)and he non-dec easing na u e o g(·)in R≥0. The second inequali y is due o he con exi y o g(·)in R≥0. Thus, we ha e p o ed ha (µxa+ (1 −µ)xb)) ≤µ (xa) + (1 −µ) (xb),∀xa, ∀xb,∀µ∈[0,1]. 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