Do gla C nningham, Pe a Hof ed , Kla Mee , Ingo Schmi (H g.): INFORMATIK 15, Lec e No e in Info ma ic (LNI), Ge ell chaf fÈ Info ma ik, Bonn 15
Scale Space fo Cogni i e S em : A Po i ion Pape
Michael B e û1, Do gla W. C nningham and Ma in Welk
Ke o d :Scale pace , pa ial diffe en ial eq a ion , cogni i e em , pe cep ion
1 Scale Space and Cogni i e S em
Image , like he eal o ld, con ain info ma ion a man diffe en pa ial (and empo al) cale . In image p oce ing and comp e i ion, he cale pacei a cla ic app oach fo e abli hing an e plici ep e en a ion of he m l i- cale na e of image con en . In od - ced b Iijima in 19 [Ii ] and e en i el died hen in Japan [WII99], i ha ecei ed a en ion in e e n co n ie ince Wi kin o k [Wi8 ]. Scale- pace heo i an im- po an b ilding block fo model ofcogni i e em a i p f ndamen al concep of
he pe cep al acq i i ion of kno ledge in o a ma hema ical f ame o k [Ma8 , Li1 ].
The Idea Behind Scale Space .Rele an image fea e ( ch a he edge of a fo e , he nk of a ee, o he edge of a leaf) in an gi en image of a cene can be fo nd o e a ce ain ± a p io i nkno n ± ange of pa ial cale . Th , an image de c ip ion ha empha i e he hie a ch of cale ill be e ef l in nde anding image con- en . Ba icall , an image f i ep e en ed a an o de ed famil of inc ea ingl moo hed e ion of he o iginal image. Ini iall , de ail ob e able onl a a ®ne cale (e.g., di an lea e ) a e emo ed hile la ge- cale c e a e kep . A he amo n of moo hing i con in o l inc ea ed, mo e and mo e de ail ill be emo ed. Th , impo an image
c e a e o de ed b hei pa ial cale.
Le deno e no he famil of cale pace fo an inp image f b {T f : ≥ }, he e he pa ame e efe o he cale. Then he cale- pace eq i emen ha e¯ec he need of a hie a chical image ep e en a ion can be de c ibed ia h ee cla e of a iom [Al9 ]:
(i)a iom ela ed o he nde l ing c e of he cale pace ch a e.g. he emi-g o p p ope T+ f =T(T f)∀, ≥ (ii)a iom de c ibing impli®ca ion p ope ie ch a e.g. he ca ali p inciple hich a e ha c e fo nd a ome cale ⇤> can al a be aced back o ®ne cale < ⇤ (iii)a iom de c ibingin a iance ch a e.g. an la ion o o a ion in a iance.
The olde and be - nde ood cale pace [Ii ] i heGa ian cale pace. I i gene a- ed b he con ol ion of an inp image i h Ga ian ke nel Kσof inc ea ing a iance
1BTU Co b -Senf enbe g, In i e fo Ma hema ic , Pla de De chen Einhei 1, B ilding HG .51, 4 Co b , Ge man , b e @b- .de
BTU Co b -Senf enbe g, In i e fo Info ma ic , Kon ad-Wach mann-Alle 5, B ilding VG1c 1. , 4 Co b , Ge man , do gla .c nningham@b- .de
UMIT Hall T ol, In i e fo Biomedical Image Anal i , Ed a d-WallnÈofe -Zen m 1, Hall T ol, A ia, ma in. elk@ mi .a
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σ, i.e.T f:=K√ ⇤f. Eq i alen l , i can be a ed b a ol ion of hepa ial diffe en ial eq a ion(PDE) of linea diff ion =D he eDi he Laplace ope a o , i h he inp image a ini ial condi ion. He e, he cale pace pa ame e can be in e p e ed a e ol ion
ime. Thi de c ip ion immedia el gi e i e o a a ie of gene ali a ion ha a e ba ed on diffe en PDE and h capi ali e on he high ¯e ibili of PDE a a modelling ool.
E ample a e non-linea diff ion cale pace [We98] a ell a o-called mo phologi- cal cale pace [VDBS94]. Since PDE a i e al o a E le -Lag ange eq a ion , hi link
cale pace di ec l o a ia ional me hod in con in o - cale op imi a ion [SW ].
Rela ion o Vi al Pe cep ion and Cogni i e S em .A la ge bod of e ea ch i de- dica ed o e amining he ela ion hip be een ma hma ical a pec of cale pace and a pec of he h man i al em ( ee, e.g., [ HR , Li94]). Fo e ample, Koende ink ha ho n ha he ea l age of he i al co e eem o e a fo m of m l i cale dif- fe en ial geome o ep e en image [Ko9 ]. Like i e, he c e of ecep i e ®eld in he e ina and of he ea l i al co e can be modelled ing a Ga ian cale- pace.
Mo eo e , he p eci e la o of ecep i e ®eld i e on he e ina i e ac l ha i needed o c ea e a cale- pace ep e en a ion. Mo e in ig ingl , he all igno ed fac ha he e a e man mo e connec ion feeding f om highe i al a ea do n in o lo e i al a ea
han feed p a d f om lo e o highe a ea i con i en i h he need of cale- pace con c ion [ HR ], and ha made e e al p edic ion (e.g., he empo al modi®ca ion of
ecep i e ®eld c e) ha ha e ince been e i®ed [AB ].
Di ec ion fo F e Re ea ch.Recen in e e in cale- pace e ea ch goe in n me- o di ec ion , ee e.g. he confe ence ol me [JFA15]. One ®eld of in e iga ion i he in eg a ion of diffe en colo model ela ed o h man colo pe cep ion. Ano he end i
he e en ion of cale- pace model b non-local concep ha allo o e ploi imila i- ie be een c e occ ing a diffe en loca ion in an image. O he o k aim a inco po a ing ep e en a ion of e e info ma ion in o cale- pace f ame o k . On he applica ion ide, he e a e ne a emp o he anal i of a dio ignal b cale- pace me hod [LF15].
We belie e ha he po en ial of he cale pace ie poin ha ill no been e ha ed. I can be con ide ed a pi o al concep fo he heo e ical nde anding of diffe en app oa- che o ignal and image p oce ing. I abili o p o ide ma hema ical ingen model fo p oce e of h man pe cep ion oge he i h he e a ili of he nde l ing ma hema-
ical concep le e pec ha i can be ed fo e en mo e de ailed modelling of cogni i e p oce e . On hi ba i , i can g ide he de elopmen of h man-comp e in e ac ion p o- ce e .
Li e a
[AB ] Angel cci, A. B e loff, P. C.: Con ib ion of feedfo a d, la e al and feedback connec ion o he cla ical ecep i e ®eld cen e and e a-cla ical ecep i e ®eld -
o nd of p ima e V1 ne on . P og e in B ain Re ea ch, 154:9 ±1 , . [Al9 ] Al a e , L i G icha d, F ÂedÂe ic Lion , Pie e-Lo i Mo el, Jean-Michel: A iom
and f ndamen al eq a ion of image p oce ing. A chi e fo a ional mechanic and anal i , 1 ( ):199± 5 , 199 .
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Do gla C nningham, Pe a Hof ed , Kla Mee , Ingo Schmi (H g.): INFORMATIK 15, Lec e No e in Info ma ic (LNI), Ge ell chaf fÈ Info ma ik, Bonn 15 [Ii ] Iijima, Tai o: Ba ic heo on no mali a ion of a pa e n (in ca e of pical one- dimen ional pa e n). B lle in of he Elec o echnical Labo a o , : 8± 88, 19 . [JFA15] J.-F.A jol, M.Nikolo a, N.Papadaki , H g. Scale Space and Va ia ional Me hod in
Comp e Vi ion. Sp inge , Cham, 15.
[Ko9 ] Koende ink, Jan J.: The b ain a geome engine. P chological Re ea ch, 5 ( ± ):1 ± 1 , 199 .
[LF15] Lindebe g, Ton F ibe g, Ande : Scale- pace heo fo a di o ignal . In: Scale Space and Va ia ional Me hod in Comp e Vi ion, S. ±15. Sp inge , 15.
[Li94] Lindebe g, Ton : Scale- pace heo in comp e i ion. Kl e , Bo on, 1994.
[Li1 ] Lindebe g, Ton : A comp a ional heo of i al ecep i e ®eld . Biological c be ne- ic , 1 ( ):589± 5, 1 .
[Ma8 ] Ma , D.: Vi ion: A Comp a ional In e iga ion in o he H man Rep e en a ion and P oce ing of Vi al Info ma ion. MIT P e , 198 .
[SW ] Sche e , O ma Weicke , Joachim: Rela ion be een eg la i a ion and diff ion ®l- e ing. Jo nal of Ma hema ical Imaging and Vi ion, 1 (1):4 ± , .
[ HR ] e Haa Romen , B.M.: F on end i ion and m l i- cale image anal i . Kl e , Do - de ech , .
[VDBS94] Van Den Boomgaa d, Rein Sme lde , A nold: The mo phological c e of image : The diffe en ial eq a ion of mo phological cale- pace. Pa e n Anal i and Machine In elligence, IEEE T an ac ion on, 1 (11):11 1±111 , 1994.
[We98] Weicke , Joachim: Ani o opic diff ion in image p oce ing, Jgg. 1. Te bne S ga , 1998.
[Wi8 ] Wi kin, And e P.: Scale-Space Fil e ing. In: P oceeding of he 8 h In e na ional Join Confe ence on A i®cial In elligence. S. 1 19±1 , 198 .
[WII99] Weicke , Joachim I hika a, Seiji Imi a, A hi: Linea cale- pace ha ® been p opo ed in Japan. Jo nal of Ma hema ical Imaging and Vi ion, 1 ( ): ± 5 , 1999.
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