Rao Transforms: Application to the Restoration of Shift-Variant Blurred Images

Transcription

1 Rao rasors: Applcato to te Restorato o St-Varat Blurred Iages Dr. Muraldara SubbaRao ttp:// ural@ece.susb.edu ttp:// rao@tegralresearc.et Itroducto We descrbe cocrete oe-desoal D ad two-desoal D eaples o te practcal applcato o Rao rasors Rs []. e oe-desoal eaple s relevat to te restorato or deblurrg o st-varat oto blurred ages. We a potograp s captured b a ovg caera wt a te eposure perod sa. secod obects earer to te caera wll ave larger oto blur ta arter obects. s stuato ca arse we te caera s o a ovg plator suc as a car or a robot. A slar stuato.e. a st-varat deocus blur ca arse a laser barcode scaer or oe-desoal barcodes. We te plae o te barcode s slated stead o beg perpedcular to te drecto o vew te barcode age wll be blurred b a stvarat pot spread ucto SV-PSF. e two-desoal eaple s related to te restorato o ages or sgals degraded b a SV-PSF suc as blurred ages o a slated plae or a curved surace produced b a deocused caera sste. Oe Desoal case Let te orgal ocused age be ad te correspodg blurred age be g. e covetoal age blurrg odel ts case uses a st-varat pot spread ucto or kerel k. e blurred age g ad te kerel k are assued to be gve ad te proble s to solve or te ocused age. e covetoal blurrg odel s a tegral equato o te or: g k d It s a oe-desoal Fredol Itegral Equato o te Frst Kd. e above odel o blurrg as two probles. Frst t s dcult to d a closed-or verso orula tat s eplct ad uercall stable. For eaple te well-kow Sgular Value Decoposto SVD tecque s coputatoall epesve ad ustable. Secod te above odel does ot capture te pscal blurrg process a atural wa. It sees to pose ateatcal splct at te cost o drect atural odelg o te pscal blurrg process. We odel te blurred age g easured at as te su or tegral over all possble pot sources o te cotrbuto due to eac pot source located at -. e cotrbuto s gve b te product o te stregt o te sgal pot source - ad te value o a ew localzed st-varat pot spread ucto -. e ew odel o blurrg s: g d R M. SubbaRao 7 ural@ece.susb.edu

2 were k + ad RL 3 k IRL 4 e ew odel above s a tegral equato tat s eactl equvalet to te orgal tegral equato see [] or proo. Equato s reerred to as te Rao Itegral Equato RIE ad dees te Rao rasor R. Equato 3 dees te Rao Localzato rasor RL. Now te -t order partal dervatve o at wt respect to s deoted b. e -t dervatve o at wt respect to wll be deoted b d. 6 d e -t oet o te -t dervatve o s deed b d 7 Note tat te dervatve s wt respect to ad te oet s wt respect to. e orgal sgal wll be take to be soot or aaltc so tat t ca be epaded a alor seres. e alor seres epaso o - aroud te pot up to order N s N 5 a 8 were a. 9! e above equato s eact ad ree o a approato error we tsel s a poloal o degree less ta or equal to N. I ts case te dervatves o o order greater ta N are all zero. We as o-zero dervatves o order greater ta N te te above equato wll ave a approato error correspodg to te resdual ter o te alor seres epaso. s approato error usuall coverges rapdl to zero as N creases. I te lt as N teds to t te above seres epaso becoes eact ad coplete. Slarl te alor seres epaso o - aroud te pot up to order M s M a were a are as Eq. 9. Usg a trucated alor seres epaso as above gves ver accurate approatos a practcal applcatos suc as age deblurrg as te kerel ucto usuall cages sootl ad slowl wt respect to. M. SubbaRao 7 ural@ece.susb.edu

3 Eaple: We coclude our oe-desoal dscusso wt a specc eaple were we let N M ad let te orgal kerel k be a Gaussa tat s k ep πσ σ were ep e. e kerel above s a global kerel. It s localzed usg te Rao Localzato rasor RL to dee a ew local kerel as Eq. 3.e. k + wc becoes ep πσ σ For otatoal coveece we deote ρ. 3 σ ereore ρ ρ ep 4 π e alor seres epaso o aroud te pot up to order M s +. 5 It ca be sow tat we s as deed Eq. 4 ρ a ρ ρ were ρ s te dervatve o ρ wt respect to. Note tat te above ucto s a eve ucto o as t volves ol. s ucto s setrc wt respect to.e. -. ereore all odd oets o wt respect to wll be zero ad wt M ad N te R becoes g + d. 7 Splg we get g Sce all odd oets o ad are zero we set s 6 3. sples te proble. Furter we ave or ts case ad bot rst ad all ger dervatves o secod dervatves wt respect to o dervatve o ad are alwas zero. Also te rst ad ger ad are all zero. Ol te rst a ot be zero. It wll be deoted b 3. Sgcat splcato o a ateatcal proble suc as ere s lkel a practcal applcatos. Wt te above splcatos we get M. SubbaRao 7 3 ural@ece.susb.edu

4 g akg dervatves o te above equato oce ad twce we get g + + ad g. We treat te above equatos 9 to as tree algebrac equatos te tree ukows ad. e ca be easl solved troug successve elato ad back substtuto. I ts partcular eaple wc correspods to a tpcal practcal applcato te process o solvg becoes trval. We solve or to obta g g g 3 e soluto above ca be urter spled b otg tat 4 σ ad ρ ρ σ ρ 3 σ 3 ρ ρ 3 us we ave obtaed Eq. te Iverse Rao rasor IR or a case tat s useul practcal applcatos. It s a closed-or soluto up to secod order ters. Soluto up to a order N ca be obtaed slarl. A soluto or s gve ters o te dervatves o g at ad oets o dervatves o te localzed kerel. I all our searc o relevat researc lterature we ave ever see suc a closed-ro soluto or te Fredol Itegral Equato o te Frst Kd. s s a local soluto ad coverges rapdl or ts partcular eaple. A ew oter ore coplcated eaples are preseted te book [] but te sple eaple ere llustrates te potetal power o Rs. I atr otato te orward ad verse R or ts case ca be wrtte as g / g 3 / g / 3 g 3 / g g I te put ucto s a poloal ad te value o N s cose to be te sae as te degree o te poloal te ro te teor oe sees tat te recostructo sould be perect. s as bee vered sulato eperets. M. SubbaRao 7 4 ural@ece.susb.edu

5 wo-desoal case A st-varat deocused age wll be deoted b g. e st-varat pot spread ucto SV-PSF wll be deoted b were ad are st-varace varables ad ad are spread ucto varables. e orgal ucorrupted ocused put age wll be deoted b. e Rao rasor ts case s 48 g dd. e ollowg otato wll be used to represet partal dervatves o g ad te oets o : g g k k k d d 5 or. Usg te above otato te alor seres epaso o aroud up to order N ad aroud te pot up to order M are gve b were C k p a C N 53 M 54 a C C ad k! p! k p! C deotes te boal coecets deed b ad a ad a are costats as deed Eq. 9. Substtutg te above epressos to te Rao rasor o Eq. 48 ad splg we get N M + + g a C a C 55 e above equato ca be rewrtte as M. SubbaRao 7 5 ural@ece.susb.edu

6 were N 56 g S M + + S a C a C 57 We ca ow wrte epressos or te varous partal dervatves o g as or ad g S. N p q pq p q 58 p+ q N. Note tat p q M p+ q S S a C a C p q p q + p + q p q + p+ q. p q 6 p q e above equato or g or pq N ad p+ q N p q costtute N + N + / equatos as a ukows equatos or p q g. e sste o ca be epressed atr or wt a sutable R coecet atr o sze N + N + / rows ad colus. ese equatos ca be solved p q eter uercall or algebracall to obta ad partcular. e soluto or were S ca be epressed as N S g 6 are te verse R coecets or te -desoal case. Eaple: We preset a soluto or te case o N ad M or te case o a -D Gaussa SV-PSF gve b + ep We wll dee a ew paraeter ρ as ρ σ. πσ σ 6 63 M. SubbaRao 7 6 ural@ece.susb.edu

7 ereore te SV-PSF ca be wrtte as ρ ρ + ep π For ts case as te -D case a oet paraeters ad ter dervatves becoe zero. Speccall Slarl ρ ρ +. ρ ρ ρ +. ρ We see tat ad are bot rotatoall setrc wt respect to ad. ereore all odd oets are zero.e. Also s odd or s odd. 67 dd 68 or all ad tereore all dervatves o wt respect to ad are zero. Also sce M all dervatves o o order ore ta wt respect to ad are zero. I suar 3 3. ereore we get R to be g e above equato gves a etod o coputg te output sgal g gve te put sgal. It ca be wrtte a or slar to Eq. 56 to obta te R coecets S. We ca derve te verse R or ts case usg Equato 69. As te -desoal case we cosder te varous dervatves o g Eq. 69 ad solve or te dervatves o as ukows. I ts partcular eaple we rst solve or ters o oter ters usg Eq. 69. e we take te dervatve o te epresso or wt respect to M. SubbaRao 7 7 ural@ece.susb.edu

8 ad solve or. Net we take te dervatve o. e we take te dervatve wt respect to o wt respect to ad solve or ad ad solve or ad respectvel. Slarl we take dervatves wt respect to o ad ad solve or ad respectvel. Fall we back substtute tese results ad elate ad to get te ollowg eplct soluto or ters o te dervatves o g ad oets o te dervatves o as below : 3 g g g g g Furter splcato o te above equato s possble due to rotatoal setr e.g. ad. e above equato gves a eplct closed-or orula or restorg a age blurred b a st-varat Gaussa pot spread ucto. e above equato ca be wrtte a or slar to Equato 6 or verse R to obta te verse R coecets. It s clear ro te above dscusso tat te etod or pleetg te orward ad verse R or te two-desoal case s slar to te oe-desoal case eplaed earler. Eperets: Several sulato eperets were doe to ver te teor above. e eperets cossted o bot D ad D cases. Frst te ukow ucto was cose to be a poloal o a certa order e.g or a se ucto o a certa perod e.g. S te te kerel was cose to be oe o Gaussa or rect ad a Cldrcal te D case wt a alor seres epaso up to order M or. e order N o te poloal was vared 3 to 8 ad te perod o te se ucto was vared s to s deret eperets. e spread paraeter sga o te SV- PSF te deret cases was vared learl e.g. s.5+.. e aaltc epressos or te blurred age ad te restored age were plotted a terval e.g. _- to _a wt sapled pots. As epected we te ukow ucto was a poloal te soluto or was eact. However te case o se uctos due to trucato o te seres epaso as epected te soluto ad sall errors. s error creased we te rato o te paraeter sga to te perod o te se wave creased. e error was sall up to a rato o.. wo eaples o D put uctos are as below N 3 M. s.5 + cos.5 N 4 M 5 ad 5. a a M. SubbaRao 7 8 ural@ece.susb.edu

9 Cocluso Equato 7 gves a closed-or soluto or te restorato o a st-varat deocused age. It as bee vered eperetall troug sulatos. s localzed soluto to st-varat age restorato ca be easl eteded to ger order local poloal approatos ad a deret odels o SV-PSF Gaussa cldrcal rect etc.. e resultg coputatoal approac s to 3 orders o agtude aster ta te classcal SVD approac [34]. e etod ere as bee pleeted o actual ages wt sulated blur ad vered. s etod olds uc prose a applcatos. Apped I ts secto we preset closed-or eplct epressos or te oets o te dervatves o te SV-PSF or deret cases. D Gaussa e PSF as te ollowg or : ep πσ σ t ereore te oet s epressed as te ollowg tegral A ep d πσ σ A Sce te lts o tegrato do ot deped o te varable we ca tercage te tegrato wt respect to ad deretato wt respect to. ereore to get te dervatves o te oets tat s we copute te tegral A ad te deretate te. e ollowg steps lead us to a geeral orula or. ep d πσ σ + ep d πσ σ ep s a oddeve ucto s oddeve. We ake te ollowg σ σdt substtuto : t tereore σ t ad d. Wt ts substtuto te σ t above equato becoes: + + σ t ep t dt πσ M. SubbaRao 7 9 ural@ece.susb.edu

10 + σ + Γ A3 π I te eperets we ave cose σ to var learl as σ.5 +. ad N. ereore ad σ tereore +.5. σ σ D Rectagular PSF e PSF or te rectagular case as te ollowg or or A4 Oterwse t ereore te oet ca be epressed as: d A5 As see ro te orula A5 te lts o tegrato ow do deped o te varable So we caot take dervatves o A5 to get s. Below we derve a geeral orula or uder te assupto tat s learl varg tat s secod ad ger order dervatves o are all zero. d d A6 d Now uder te assupto tat s lear we ave or te dervatve: d! + d Substtutg A7 to A6 we get! + d A7 +! + + A8 D Gaussa e PSF as te ollowg or : M. SubbaRao 7 ural@ece.susb.edu

11 M. SubbaRao 7 ural@ece.susb.edu + ep σ πσ A9 ereore te t oet s epressed as te ollowg tegral σ πσ d d + ep A We splt te double tegral as a product o two sgle tegrals σ σ πσ d d ep ep We tegrate eac as we dd or te oe desoal case. e result s as ollows: [ ] π σ Γ + Γ + A We deretate A to get te dervatves. D Rectagular PSF e PSF or te rectagular case as te ollowg or or A Oterwse ereore te t oet ca be epressed as: d d A3 As te Oe-Desoal case te lts o tegrato deped o. ereore to get te dervatves o te oets we eed to deretate uder te tegral. e orula or te dervatves o te oets s: + d d A4 Uder te assupto tat vares learl wt ad +! A5 Here s te rst partal dervatve o wt respect to ad s te rst partal dervatve o wt respect to. ereore puttg A5 to A4 ad tegratg we get

12 ! [ + + ] A6 D Cldrcal PSF: e dervato s ver slar to te Rectagular case. So we wll skp ost o te steps ere. e PSF or te cldrcal case as te ollowg or B R A7 πr Oterwse e oets ad ter dervatves ca be obtaed b evaluatg te ollowg tegral + da A8 B R πr We ake te assupto tat R vares learl wt ad. ereore te partal dervatves o are as ollows: + + +! R + R A9 + + πr πr o copute A8 we swtc to polar co-ordates tat s we replace b r cosθ ad b r sθ ad te area eleet da b rdrdθ. ereore te tegral A8 becoes! R R π R r cos θ s πr θdrdθ were + +! R π π + + R R R + rg A + rg cos θ s θdθ ca be coputed separatel. Reereces. "Rao rasors: eor ad Applcatos" b M. Subbarao Rao U.S. Coprgt Regstrato No. X Jue 5. Purcase at ttp:// M. Subbarao Passve ragg ad rapd autoocusg U.S. patet No Sept W. K. Pratt Dgtal Iage Processg Secod Edto Secto.3 pages 376 to 38 Jo Wle ad Sos 99 ISBN A. K. Ja Fudaetals o Dgtal Iage Processg Secto 8.9 pages 99 to 3 Pretce-Hall Ic. 989 ISBN M. SubbaRao 7 ural@ece.susb.edu