Matrix Formulation of Image Restoration Problem

Transcription

1 atri Formlatio o Imae estoratio Problem -D Case: We will cosider te -D ersio irst or simplicity: m m* m + m We will assme tat te arrays ad ae bee zero-padded to be o size were let + let. eceort we will ot eplicitly metio te zero-paddi. e deradatio eqatio: m k k m k + m ca be writte i matri-ector orm as ollows: + were

2 oweer sice te arrays ad are zero-padded we ca eqialetly set: 3 3 otice tat te secod matri is circlat; i.e. eac row o is a circlar sit o te preios row. ample: A let o array 3 B let o array A + B 4 say 4.

3 otice tat. Ideed

4 eceort we will se so tat we ca apply properties o circlat matrices to. -D Case: Sppose are arrays ater zero-paddi. e deradatio eqatio ca be writte i matri-ector ormat as ollows: + were

5 3 ote tat is a block-circlat matri wit blocks. ac block is itsel a circlat matri. Ideed te matri is a circlat matri ormed rom te -t row o array m: 3 Gie te deradatio eqatio: +

6 or obectie is to recoer rom obseratio. We will assme tat te array m sally reerred to as te blrri ctio ad statistics o te oise m are kow. e problem becomes ery complicated i array m is kow ad tis case is sally reerred to as blid restoratio or blid decooltio. otice tat ee we tere is o oise; i.e. m or te ales o m were eactly kow ad matri is iertible compti directly wold ot be practical. ˆ ample: Sppose 56. ereore ad wold be a by matri to be ierted! atrally direct iersio o wold ot be easible. Bt as seeral sel properties; i particlar: is block circlat. is sally sparse as ery ew o-zero etries. We will eploit tese properties to obtai ˆ more eicietly. I particlar we will derie te teoretical soltios to te restoratio problem si matri alebra. oweer we it comes to implemeti te soltio we ca resort to te Forier domai taks to te properties o circlat matrices.

7 Costraied least sqares ilteri restoratio ecall tat te kowlede o blr ctio m is essetial to obtai a meail soltio to te restoratio problem. Ote kowlede o m is ot perect ad sbect to errors. Oe way to alleiate sesitiity o te reslt to errors i m is to base optimality o restoratio o a measre o smootess sc as te secod deriatie o te imae. We will approimate te secod deriatie Laplacia by a matri Q. Ideed we will irst ormlate te costraied restoratio problem ad obtai its soltio i terms o a eeral matri Q. Later dieret coices o matri Q will be cosidered eac ii rise to a dieret restoratio ilter. Sppose Q is ay matri o appropriate dimesio. I costraied imae restoratio we coose ˆ to miimize Q ˆ sbect to te costrait ˆ + ˆ. ˆ. ecall te deradatio eqatio Itrodctio o matri Q allows cosiderable leibility i te desi o appropriate restoratio ilters we will discss speciic coices o Q later. So or problem is ormlated as ollows: mi Qˆ sbect to ˆ or ˆ

8 A brie reiew o matri dieretiatio Sppose ad is a ctio o two ariables. e I or some matri A ad some ector b te were sperscript deotes matri traspose. b A b A b A b A A

9 ecall rom calcls tat sc a costraied miimizatio problem ca be soled by meas o Larae mltipliers. We eed to miimize te ameted obectie ctio J ˆ : J ˆ Qˆ + α ˆ were α is a Larae mltiplier. We set te deriatie o J ˆ wit respect to ˆ to zero. J ereore ˆ ˆ Q Qˆ α ˆ Q Q + α ˆ α Q Q + α α α Q Q + γq Q + were γ is cose to satisy te costrait α ˆ. We will ow se te aboe ormlatio to derie a mber o restoratio ilters.

10 Psedo-ierse Filteri e psedo-ierse ilter tries to aoid te pitalls o applyi a ierse ilter i te presece o oise. Cosider te costraied restoratio soltio wit Q ˆ I. is ies Q γ Q + ˆ γi +. It ca be implemeted i te Forier domai by te ollowi eqatio: F ˆ G were *. + γ + γ e parameter γ is a costat to be cose. ote tat γ ies s back te ierse ilter. For γ > te deomiator o is strictly positie ad te psedo-ierse ilter is well deied.

11 Psedo-Ierse Filteri eample Zero-mea Gassia oise wit ariace σ. 3 m m S. ˆ m γ.75 S.76 γ. S. 338 γ.5 S.447 γ I. Filter S. 7

12 iimm ea Sqare rror Wieer Filter is is a restoratio teciqe based o te statistics mea ad correlatio o te imae ad oise. We cosider eac elemet o ad as radom ariables. Deie te correlatio matrices e matrices ad are real ad symmetric wit all eieales bei o-eatie. e D-DF o te correlatios ad are called te power spectra ad are deoted by S ad S respectiely. ecall te costraied restoratio soltio ie by Coose matri Q sc tat { } { } ad Q Q + γ ˆ Q Q

13 I a sese we are tryi to miimize te oise-to-sial ratio. e costraied restoratio is te ie by is ca be implemeted si DF as ere [ ] F S is te power spectral desity o te imae m ad [ ] S is te power spectral desity o te oise m. e restoratio ilter is called te parametric Wieer ilter wit parameter γ Special cases: γ : Wieer Filter γ : Ierse Filter γ : Parametric Wieer Filter + γ ˆ [ ] [ ] [ ] were ˆ * * S S S S G G S S F + γ + γ + γ

14 Accordi to te costraied restoratio ilter deried earlier parameter γ sold be cose to satisy ˆ. oweer coice o γ yields a optimal ilter i te sese o miimizi te error ctio e { [ m ˆ m ] }. I oter words setti γ yields a statistically optimal restoratio. Implemetatio o te parametric Wieer ilter reqires kowlede o te imae ad oise power spectra S ad. I particlar we eed te so called sial-to-oise ratio S ρ S S. is is ot always aailable ad a simple approimatio is to replace ρ by a costat ρ. I tis case te Wieer ilter is ie by * + γ ρ S ote tat as ilter. ρ o oise te Wieer ilter teds to te ierse

15 Wieer ilter eample + + r σ σ ρ or lo σ σ m db ρ 5.9dB ρ 5.9dB ρ 5.9dB m ˆ m γ ˆ m γ.

16 Parametric Wieer Filter eample eect o parameter γ m ˆ m γ. ˆ m γ. ˆ m γ ˆ m γ 5 ˆ m γ 5 Small ales o γ reslt i better blr remoal ad poor oise ilteri. Lare ales o γ reslt i poor blr remoal ad better oise ilteri.