Fourier Transforms. Convolutions. Capturing what s important. Last Time. Linear Image Transformation. Invertible Transforms.

Size: px

Start display at page:

Download "Fourier Transforms. Convolutions. Capturing what s important. Last Time. Linear Image Transformation. Invertible Transforms."

Agnes Stanley
5 years ago
Views:

orr Trasors Rq rad: Captr 7 92 &P Adso Soc ad ra (adot o) Opt rad: Hor 7 & 8 P 8 Last

coptatoay costy ad a copx oprato () () W wat to d a bttr xprsso A ar trasorato o t cto

o bass U Vctorzd a Trasord a PCA ICA orr Trasor or Wat Trasor or Strab Pyrad Trasor

1 orr Trasors Rq rad: Captr 7 92 &P Adso Soc ad ra (adot o) Opt rad: Hor 7 & 8 P 8 Last T Cooto trs: a/box tr Gassa tr t drc tr Lapaca o Gassa tr Ed Dtcto Cootos Cooto s coptatoay costy ad a copx oprato () () W wat to d a bttr xprsso A ar trasorato o t cto wos baor s spr (coptatoay capr) dr cooto Lar Ia Trasorato I aayz as t s ot s to a a ca o bass U Vctorzd a Trasord a PCA ICA orr Trasor or Wat Trasor or Strab Pyrad Trasor Irtb Trasors Captr wat s portat Sa bass ctos ar sd or t rs trasor U U Traspos ad copx cojat

A c st o bass Tass away ast s sow cas t a Ts ca o bass as a spca a Ja Baptst Josp orr (768-83) ad crazy da (87): Ay prodc cto ca

cad orr Srs A s o ss Or bd boc: As(ωx φ ) Add o o t to t ay sa (x) yo wat! How ay drs o rdo? Wat dos ac cotro?

orr Trasor W wat to drstad t rqcy ω o or sa So t s rparatrz t sa by ω stad o x: (x) (ω) orr Trasor Irs orr Trasor (ω) or ry ω ro

2 A c st o bass Tass away ast s sow cas t a Ts ca o bass as a spca a Ja Baptst Josp orr (768-83) ad crazy da (87): Ay prodc cto ca b rwrtt as a wtd s o ss ad coss o drt rqcs Do t b t? tr dd Lara Lapac Posso ad otr b ws ot trasatd to Es t 878! Bt t s tr! cad orr Srs A s o ss Or bd boc: As(ωx φ ) Add o o t to t ay sa (x) yo wat! How ay drs o rdo? Wat dos ac cotro? Wc o cods t coars s strctr o t sa? orr Trasor W wat to drstad t rqcy ω o or sa So t s rparatrz t sa by ω stad o x: (x) (ω) orr Trasor Irs orr Trasor (ω) or ry ω ro to (ω) ods t aptd A ad pas φ o t corrspod s As(ωx φ ) How ca od bot? Copx br trc! ( ω ) R( ω ) I ( ω ) 2 2 I( ω) A ± R( ω) I( ω) φ ta R( ω) W ca aways o bac: (x) T ad rqcy xap : (t) s(2 t) (/3)s(2 (3 t)) rqcy Spctra xap : (t) s(2 t) (/3)s(2 (3 ) t)

3 rqcy Spctra Usay rqcy s or trst ta t pas rqcy Spctra rqcy Spctra rqcy Spctra rqcy Spctra rqcy Spctra

4 rqcy Spctra rqcy Spctra A s(2 t ) T: Jst a ca o bass U (x) (ω) IT: Jst a ca o bass U - (ω) (x) Dtos Dtos Rca: ω x T xpota s cos( ωx) s( ωx) pas ca b codd by s/cos par As( ωx φ ) P cos( x) Q s( x) As( x φ ) 2 2 Α ± P Q So t s jst or sa (x) ts s at rqcy ω φ ta P Q

5 2D T trasor Ts s t atd trasor o t cta pc Ts s t pas trasor o t cta pc Ts s t atd trasor o t zbra pc

6 Ts s t pas trasor o t zbra pc Rcostrcto wt zbra pas cta atd Rcostrcto wt cta pas zbra atd a-ad Sc Ca ca spctr t rcostrct ost orato at ow rqcs!

tr orr doa Wat s a ood rprstato or a aayss?

ot wr Px doa rprstato ts yo wr (px ocato) bt

dscrpto o a ts wat s app wr Appcato to Ia

arat o dscrt orr trasor Ra brs ast ptato Us

tsty T top-t cos rprst ow rqcs t botto rt

7 tr orr doa Wat s a ood rprstato or a aayss? orr trasor doa ts yo wat (txtra proprts) bt ot wr Px doa rprstato ts yo wr (px ocato) bt ot wat Wat a a rprstato tat s yo a oca dscrpto o a ts wat s app wr Appcato to Ia coprsso Lossy Ia Coprsso (JPEG) Coprsso s abot dd drcs ro t tr a wr yo ca t s t Boc-basd Dscrt Cos Trasor (DCT) Us DCT JPEG A arat o dscrt orr trasor Ra brs ast ptato Us DCT JPEG T rst coct B() s t DC copot t ara tsty T top-t cos rprst ow rqcs t botto rt rqcs Boc sz sa boc astr corrato xsts btw bor pxs ar boc bttr coprsso soot ros

Ia coprsso s DCT DCT abs a coprsso by coctrat ost a orato t ow rqcs Loos portat a o ( rqcs) by ctt B() at botto rt T dcodr copts t rs DCT IDCT Qatzato Tab 3 5 7 9 3 5 7 5 7 9 3 5 7 9 7 9 3 5 7 9 2 9

8 Ia coprsso s DCT DCT abs a coprsso by coctrat ost a orato t ow rqcs Loos portat a o ( rqcs) by ctt B() at botto rt T dcodr copts t rs DCT IDCT Qatzato Tab JPEG coprsso coparso 89 2 Wy s t orr doa partcary s? It ts s t ct o ar cootos Tr s a ast aort or pror t DT aow or ct sa tr T orr doa ors a atrat doa or drstad ad apat t a T Cooto Tor T orr trasor o t cooto o two ctos s t prodct o tr orr trasors T rs orr trasor o t prodct o two orr trasors s t cooto o t two rs orr trasors Cooto spata doa s qat to tpcato rqcy doa! orr trasor o cooto Cosdr a (crcar) cooto o ad orr trasor o cooto Ta DT o bot sds DT ( )

9 ( ) DT Wrt t DT ad cooto xpcty orr trasor o cooto ( ) DT o t xpot orr trasor o cooto ( ) DT Ca arabs t s ( ) ( ) orr trasor o cooto ( ) DT Pror t DT (crcar bodary codtos) ( ) ( ) G orr trasor o cooto ( ) DT orr trasor o cooto Pror t otr DT (crcar bodary codtos) ( ) ( ) G H G Cooto rss T -d T: O(o) coptato t wr s br o saps 2-d T: 2(o) wr s br o pxs o a sd Cooto: K 2 wr K s br o saps r Say 2 K 2-d T: w cooto s 2 2

10 B otato or orr aayss S was ar ctors o t cooto oprator otato or orr aayss: Sap T sap rd s a prodc strctr orr s prtty ood at ad tat W saw tat a s wa as sros probs wt sap Sap s a ar procss Sap Dsty I w r cy sap dsty s o Sap Dsty I w scty sap t sa t ay b sta or sot spr dr rcostrcto (tat's aas!) Ipt Rcostrctd Sap Tor W sap a sa at dscrt tras t sap rqcy st b ratr ta twc t st rqcy o t pt sa ordr to b ab to rcostrct t ora prcty ro t sapd rso (Sao yqst Wttar Koto) Rcap: otato or s was Brr s was s sp Yo t t sa s wa jst scad dow T s ctos ar t ctors o t cooto oprator Sap s was s trst Gt aotr s wa ot cssary t sa o! (aas) I w rprst ctos (or as) wt a s o s was cooto ad sap ar asy to stdy

Priority Search Trees - Part I

Priority Search Trees - Part I .S. 252 Pro. Rorto Taassa oputatoal otry S., 1992 1993 Ltur 9 at: ar 8, 1993 Sr: a Q ol aro Prorty Sar Trs - Part 1 trouto t last ltur, w loo at trval trs. or trval pot losur prols, ty us lar spa a optal