Transforms that are commonly used are separable

Transcription

1 Trasforms s Trasforms that are commoly used are separable Eamples: Two-dmesoal DFT DCT DST adamard We ca the use -D trasforms computg the D separable trasforms: Take -D trasform of the rows > rows ( ) Take -D trasform of colums of rows ( ) > ( ) EEE 58

2 The D Dscrete Fourer Trasform (D DFT) p ( ) The DFT cossts of samples of the DTFT at ω ω ( L ) L ( ) ω ω perodcty space (or tme) doma (ad frequecy doma) sequece ( ) eteded perodcally wth horzotal perod of ad vertcal perod of ( ) ( ) ( ) ( ) But we restrct ourselves to the ma perod to recover the orgal sgal EEE 58

3 D DFT. ( ) ( ) DFT ( ) ( ) j j e e ( ) ( ) j j e e EEE 58

4 D DFT Utary DFT (Symmetrc Form): y ( y ) ( ) ( ) j j e e ( ) ( ) j j e e ( ) e e j j Φ ( ) ( ) Φ Φ separable ( ) Φ e j where EEE 58

5 D DFT ote: ( ) ( ) c samples of perod betwee [ ) of DTFT DTFT perodc of perod D DFT of a mage: [) of DTFT ( ) 5 ( ) D DFT magtude EEE 58 ( )

6 D DFT ote: ( ) samples of perod betwee [) of DTFT c DTFT perodc of perod D DFT of a mage: ( ) 5 D DFT magtude EEE 58 ( )

7 D DFT For coveece use the shorthad otato W e j j a a W e D DFT Separable ca use oly D DFTs ( ) ( ) ( ) ( ) W ( ) ( ) W EEE 58

8 D DFT Matr Represetato: ˆ F where F are utary matrces ad F T F W But: ote: F symmetrc F utary ˆ T F F * F * F F F T F * T F F F * F F F W EEE 58

9 D DFT Wth the DFT lear shfts ad lear covolutos are replaced by crcular shfts ad crcular covolutos Crcularty s deoted usg modulo operator otato: () modulo If r () the r - ote: (-) (-) Crcular shft: ( ) ( ) ) m m m m W W ( ) Crcular covoluto same as lear covoluto but shftg correspods to crcular shft ad flppg correspods to crcular reverse (-) - ( ) h( ) ( ) ( ) EEE 58

10 D DFT Cojugate symmetry for real mages ( ) ( ) ( ) ( ) ( ) ( ) ( ) * * real ( ) ( ) ( ) ( ) * pck the > de lyg perod - Replace * EEE 58

11 D DFT Cojugate symmetry about - ( ) half of the coeffcets are redudat ad eed ot be stored - The DFT s a fast trasform DFT separable Trasformato ca be performed usg ( ) D DFTs for a mage Each D DFT of sze requres log comple multplcatos ad adds (CMADs) usg a FFT mplemetato DFT ca be performed log log log O( log ) operatos matr rows DFT matr colums DFT If O ( log )operatos EEE 58

12 The D Dscrete Cose Trasform (D DCT) e sc ete Cos e a s o ( C ) Advatages: Real trasform (for real mages) Real trasform (for real mages) Coeffcets early ucorrelated Ecellet Eergy Compacto There are dfferet types of DCTs but here we are terested There are dfferet types of DCTs but here we are terested a utary trasform type where the trasformato matrces are such that: C C Type IV ( ) ( ) ( ) ( ) ( ) ( ) DCT cos cos C C ( ) C where EEE 58

13 D DCT ( ) ( ) C ( ) C ( ) ( ) ( ) DCT cos cos Iverse ad Forward DCT are the same DCT separable ca use D trasforms D DCT of a mage ( ) D DCT magtude ( ) ote: A real trasform oly ygoes half way aroud the Ut Crcle EEE 58

14 D DCT ( ) ( ) C ( ) C ( ) ( ) ( ) DCT cos cos Iverse ad Forward DCT are the same DCT separable ca use D trasforms D DCT of a mage ( ) D DCT magtude ( ) ote: A real trasform oly ygoes half way aroud the Ut Crcle EEE 58

15 D DCT Dsadvatage: The DCT does ot have ce covoluto ad modulato propertes. DCT ca be related to DFT ad computed va FFTs DCT related to DFT of symmetrc eteso of a sequece O ( ) operatos eeded for a mage. log Fast trasform. EEE 58

16 D DCT Matr Represetato C { ( )} C where C ( ) cos ( ) C utary ad real C orthogoal C real ad orthogoal C - C T ad C C * ˆ C T C C C ˆ ˆ * * T T CC EEE 58

17 D DCT ote: f C C C orthogoal ˆ C T C CC ˆ T Iterpretato of DCT DFT perodc eteso of sequeces DCT symmetrc ad perodc eteso of sequeces (ma perod symmetrc) EEE 58

18 D DCT Cosder a mage ( ) - g h I d e f a b c - Symmetrcally eted mage about - ad - aes - a b c d e f g h g h d e f a b c c b a f e d h g h g f e d c b a EEE 58 - y( ) ( ) ( ) ( -- ) ( -- )

19 D DCT ow take DFT of y( ) ote that ths s a pot DFT DCT ca be epressed terms of DFT Ca use FFT to make a fast DCT mplemetato wth O( ) operatos for a mage log Compare DFT to DCT ( ) 5 ( ) ( ) ( ) EEE 58

20 D DST (Dscrete Se Trasform) S ( sc ete S e a s o ) Defto: ( ) ( ) ( )( ) ( )( ) DST s s ˆ Propertes smlar to DCT (separable real fast trasform) but also symmetrc faster trasform ( ) ( )( ) ( ) T S S Φ Φ s Very good eergy compacto but less tha DCT ( ) S S Φ EEE 58

21 D DST Iverse ad forward trasforms are the same ( ) ˆ ( ) s ( )( ) s ( )( ) DST Matr represetato ˆ S T S S S S symmetrc where S ˆ S subde correspods to drectos S subde correspods to drectos sce utary S ad S real ad symmetrc ( ) ( ) { S ( )} s EEE 58

22 D Dscrete adamard a d Trasform Based o a matr descrpto (Typcally defed terms of a matr) Very effcet: matr elemets (ad thus elemets of bass vectors) take oly the bary values ± well-suted for DSP ca be mplemeted wthout multplers Symmetrc trasform Defto: For a mage (( ) wth (power of ) T ˆ log *T s also real ad orthogoal - T ˆ where are matrces where ad usually 3 work o mage blocks of sze respectvely EEE 58

23 D Dscrete adamard Trasform sc ete ada a d a s o geerated usg the roecker product ad the core matr by the followg recurso Eample: EEE 58

24 D Dscrete adamard a d Trasform Frequecy terpretato Lets look at the 4 4 matr: low frequecy bass vector hgh h frequecy bass vector md frequecy bass vector md frequecy bass vector Sequecy # of sg chages Recall that for susodal sgals frequecy ca be defed terms of zero crossg or sg chages The bass vectors of a adamard matr ca be obtaed by samplg a class of fuctos called the Walsh fuctos whch ca take oly the bary values ± ad whch form a complete orthoormal bass L adamard trasform also called Walsh- adamard trasform 3 EEE 58

25 D Dscrete adamard a d Trasform Sequecy umber of zero crossgs of a Walsh fucto trastos (sg chages) a bass vector of Geerally we order the bass vectors (row or colums sce T ) terms of creasg sequecy Ths s called Walsh-ordered adamard d Trasform (f bass vectors ordered) d) EEE 58

26 D Dscrete adamard a d Trasform Trasform terpretato Orgal trasformed o ce covoluto ad modulato propertes but very fast trasform. adamard epressed terms of roecker products ad also as sparse matrces t ca be mplemeted O( log ) addtos ad subtractos. EEE 58