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1 Preparaton for the examnaton 5LIN0 Vdeo processng Gerard de Haan Avalable materal: Lectures x h durng 7 weeks Book (Dgtal Vdeo Post Processng) Verson Dec. 04 Except Chapter 6 Questons n every chapter, to exercse for exam Avalable from Marja de Mol, Flux 4.3 (Eu.50) Demo software (VdProc) Downloadable from bookssoftware (password ) Sldes: Your notes (hardly necessary when you learn from the book and do the exercses) You may brng the book to the exam! VdProc 3 Schedule lectures 5LIN0 4 Week Week Week 3 Week 4 Bascs (Ch, 3) Vdeo dsplays (Ch 9) Flterng (Ch 4) Pcture-Rate Converson (Ch 7) Week 5 Week 6 Week 7 Week 8 De-nterlacng (Ch 8) Moton Estmaton (Ch 0) Object Detecton (Ch ) X Shftng and addng mages 5 What f we average an mage wth a shfted verson? 6 What f we subtract an mage wth a shfted verson? = Orgnal mage pel shfted mage average mage A frequency s suppressed f we shft over 80 degree. Therefore, the frequency has a complete cycle of 4 pxels In other words, suppresson occurs at a quarter of the samplng frequency pel + = Averagng wth a Orgnal mage pel dagonally shfted mage Subtracton suppresses DC (coarse spatal detal A frequency wth a complete cycle of 4 pxels doubles n ampltude In other words, enhancement occurs at a quarter of the samplng frequency Subtracton wth a pel dagonally shfted mage pel - =

2 7 How do we shft an mage? (scannng assumed) 8 f v (c/ph) Delay f t (c/pp) Image scanned from top-left to bottom rght pxel horzontal shft lne vertcal shft pcture same poston, temporal shft Lnear flterng of dscrete mages f h (c/pw) 9 What s a lnear flter? A lnear flter s a system wth an nput and an output that: Contans s, multplers and adders = weghted sum of ed copes of nput/output sgnal 0 Example lnear flter x[k-3] x[k-] x[k-] x[k] w 3 w w w ww3 ww4 0 y [ k] w x[ k n] n sum n y[k] Meanng of : Smple notaton From horzontal to vertcal flterng Defnton: (,, ) flter weghted sum of horzontally neghbourng pxels Defnton: flter weghted sum of vertcally neghbourng pxels Vertcally shftng requres lne-s Orders of magntude more expensve than a pxel- Temporal flterng requres pcture-s Agan orders of magntude more expensve Change pxel-s lne-s Lne- can be mplemented wth a lne-memory ¼ 0 Lne Lne Lne ½ sum ¼

3 out Vdeo course: Bascs: Flterng 3 3 What f you flter an mage both n H- and V-dmenson? Flter support w w w3 w4 w5 w6 4 What f we re-use output samples? Recursve flterng ww 3 ww ww3 ww4 0 w7 w8 w9 n sum wr4 wr3 wr w 3r w r w r wr w 0r mage mage y [ k] w. x[ k n] w. y[ k m] n n m mr 5 6 Some mpulse responses (,,) Cascadng flters (,) Sngle whte pxel nput (,0,) 0 7 The cascadng of two (, ) flters A 8 Why a trangle? (convoluton!) tme A ½ ½ A ½ ¼ ¼ tme tme Example cascade: Cascade of (,) and (, ):

4 out Vdeo course: Bascs: Flterng 4 9 Smple calculaton of mpulse response of cascade 0 Smple calculaton of mpulse response of cascade Cascade of and 0 Cascade of (,) and (, ): Cascade of and : ¼ f s Cascade of and (cascade of and and and ): ¼ f s ½ f s ½ f s Smple calculaton of mpulse response of cascade Cascade of (horzontal) and a(vertcal) Cascade of (horzontal) and a (vertcal) 4 Qualtatve analyss of lnear flters 3 What s mage flterng? (D convoluton) 4 Effect of the box flter on a natural mage Flter support w w w3 w4 w5 w6 n Assgn weghted sum to output mage Orgnal (nput) Box-flter output w7 w8 w9 mage mage Move the flter-support (=coeffcent-matrx,.e. mpulse response) over each pxel n the mage, multply the entres by the pxels, sum together, and assgn result to central pxel of output mage E.g. 3x3 box -flter 9 Effect s averagng

5 5 5 Box Flter Smoothng due to averagng neghborng pxels Frequency doman: leaves LF unaltered and attenuates HF So, box flter s a low-pass flter A 0 Spatal: Box pos 9 Frequency doman: snc-functon H 0 f 6 Not only low-pass flters the frequency doman Transfer of flter Spectrum nput mage Spectrum of output mage A H A = Low-pass 0 f 0 f 0 f A H A = Hgh-pass 0 f 0 f 0 f A H A = Band-pass 0 f 0 f 0 f 7 Hgh-pass flters 8 Effect of the hgh-pass flter on a natural mage Turn a LP-flter nto a HP-flter Subtract smoothed mage from orgnal = remove LF Result: only HF Orgnal (nput) Hgh-pass flter output +8 Impulse-response found wth matrx subtracton: Example: LPF = 3x3 Box: Edge Enhancement 30 Effect of the edge-enhance, or peakng, flter HP-flters: Hgh values at edges, Low values n constant regons Addng hgh frequences back to the mage enhances edges Possble approach: Image = Image + [Image smooth(image)] Low-pass Hgh-pass [- - ]- - - x/

6 6 3 Edge enhancement, peakng, or unsharp maskng 3 Negatve flter coeffcents may lead to problems! Negatve coeffcents can lead to negatve mage values Most mage formats don t support ths No negatve lght. Also, boostng HF may gve values above peak-whte Solutons: Clppng: Chop off values below mn or above max Offset: Add a constant to move the mn value to 0 Re-scale: Rescale the mage values to fll the range (0,max) Non-lnearty Spols black Spols contrast 33 What about the mage boundares? At mage poston (0,0), you need e.g. pxel (-,-), whch doesn t exst Opton : Make output mage smaller Opton : Replcate edge pxels Opton 3: Reflect mage about edge 5LIN0 Vdeo processng Gerard de Haan Purpose of flters The purpose of flters Removal of spectral components E.g. for alas preventon, or remove nterference Enhancement of spectral components Edge/feature enhancement Removal of nose Balance between suppresson of nose and relevant mage components Interpolaton Image scalng, geometrcal deformatons (warpng)

7 7 37 Nose removal 38 Alas preventon H-LPF V-LPF The smple way. Nearest pxel Flterng and mage re-szng (D case) Orgnal pxels: Requred lower densty grd: Requred hgher densty grd: Repeated pxels! Dropped pxels! 4 Spatal scalng 4 Bt better: tap lnear nterpolaton Scaled usng nearest pxel nterpolaton Trangular shape of mpulse response (weght lnearly deceases wth dstance) Length mpulse response s 3 (always pxels n support), Suppose we need a sample at x=4.75: Result s: 0.5F(4) F(5) Scaled wth proper flters x

8 8 43 Longer flter? 44 What are the effects of the flter length? Trangular shape of mpulse response (weght lnearly decreases wth dstance) Now flter support sze s 4 (max 3 pxels n support) Suppose we need a sample at x=4.75: Length = Assume the flter sze s 3, and we need a sample at x=4.75 We need the mage samples, F(x), from x=4, x=5 and x=6 x We need the flter value, H(s), at s=-0.75, s=0.5 and s=.5 We compute: F(4)*H(-0.75)+F(5)*H(0.5)+F(6)*H(.5) Of course: The samplng theorem These were example attempts Does theory provde any requrements for the nterpolaton flter? 0 f s f s We have a contnuous sgnal We sample t to obtan a dscrete representaton Samplng theorem: we can reconstruct the contnuous sgnal from ts dscrete representaton, provded t contaned no frequences above half the samplng frequency We can re-sample the contnuous sgnal at any rate! T s tme 47 The mpulse response of such flter looks lke ths 48 Re-samplng It s a sync-functon The flter has to be approxmated as t has an nfnte length Makng an mage larger s lke re-samplng the orgnal sgnal at a hgher densty Approxmaton means removng smaller coeffcents Example for BW = f s / [ ] [ ] [ ]... Orgnal More samples Orgnal spacng = bgger Reducng an mage n sze s lke re-samplng at lower densty In theory, steps: Reconstructon of the contnuous sgnal Followed by re-samplng wth the new samplng frequency In practce: sample rate converson n the dscrete doman

9 9 49 The general prncple of sample rate converson 50 Spectra n samplng rate converson (down-samplng) Intermedate nterpolator decmator grd grd grd Samplng-rate converter As a consequence of the tme-dscrete nature of the processng, the nput and output samplng frequency have a ratonal relaton. results from an nteger up-samplng and an nteger sub-samplng 0 frequency f s f s 0 frequency f s Alas! 0 frequency f s Repeat on output rate wll overlap wth base-band unless the nterpolatng LPF suppresses part of spectrum 5 Sgnals n sample rate converson (down-samplng 3/4) 5 Spectra n sample rate converson (up-samplng) nput 0 frequency f s f s 3f s Interp. Intermed. Repeats at nput rate must be suppressed wthout suppressng base-band 0 frequency f s output to prevent alas or blur at output rate 0 frequency f s3 53 Sgnals n sample rate converson (up-samplng, 3/) 54 Requrements for the nterpolaton (scalng) flter nput F(nT) sgnal sgnal Interpolaton flter Interp. output 3 4 nt nt Up-samplng: LPF passes base-band sgnal and suppresses the nput repeat spectra where possble at ntermedate rate Down-samplng: LPF suppresses part of the base-band and repeat spectra to avod foldng nto the new base-band output spectrum

10 0 55 Poly-phase flterng: the flter s desgned at the ntermedate frequency, at the output frequency a varyng set of coeffcents s used 56 Poly-phase flterng In practce, we do not use an ntermedate samplng rate. Orgnal pxels: C-6 C-5 C-4 C-3 C- C- C0 C C C3 C4 C5 C6 [ ] a X b X c X d X e X f X C-6 C-5 C-4 C-3 C- C- C0 C C C3 C4 C5 C6 Requred pxel on other densty grd: 4 n 5 The flter coeffcents, a b c d e f.., depend on the poston of the requred pxel n 57 Applcaton: Aspect rato converson 58 4:3 mage on a wde-screen (6:9) pcture tube Lnear scalng optons: Accept sde-panels Accept geometrcal dstorton 59 4:3 mage on a wde-screen (6:9) pcture tube 60 Wde-screen mage on a 4:3 pcture tube Non-lnear scalng optons: Lnear scalng optons: Accept geometrcal dstorton Panorama vew Accept letter-box Geometrcal dstorton Partal mage loss

11 magnfcaton Vdeo course: Bascs: Flterng 6 Wde-screen mage on a 4:3 pcture tube 6 Optons n wde-screen converson Non-lnear scalng optons: Dstortons concentrated evenly dstrbuted n sde panels Usually magnfcaton between 0.5 and.0.33 Constant horzontal stretch Panorama.0 Amaronap vew 0.75 Constant horzontal compress Amaronap Horzontal poston 64 5LIN0 Vdeo processng Recaptulaton Gerard de Haan 65 output, adders, multplers and s w w w3 w4 sum wr4 wr3 wr wr y [ k] w. x[ k n] w. y[ k m] n n m mr 66 Purpose of flters Removal of spectral components E.g. for alas preventon, or removal of nterference sgnal Enhancement of spectral components Edge/feature enhancement Removal of nose Balance between nose suppresson and suppresson of relevant mage components Interpolaton Image up- and down-scalng, geometrcal deformatons

12 67 The general prncple of sample rate converson 68 Poly-phase flterng In practce, we do not use an ntermedate samplng rate. grd nterpolator Intermedate grd decmator Samplng-rate converter grd Orgnal pxels: a X b X c X d X e X f X As a consequence of the tme-dscrete nature of the processng, the nput and output samplng frequency have a ratonal relaton. results from an nteger up-samplng and an nteger sub-samplng Requred pxel on other densty grd: 4 n 5 The flter coeffcents, a b c d e f.., depend on the poston of the requred pxel n Dfferent flter types Non-lnear flters Lnear flters Rank-order flters Hybrd flters Morphologcal flterng Adaptve flters 7 Lnear flters not very effectve aganst shot nose Lnear flters fne n case small nose values are more frequent than hgh nose values E.g. Gaussan nose Shot nose (salt and pepper nose,..) s characterzed by relatvely few extreme nose values and (almost) no small nose values E.g. mpulses from gnton of combuston engnes, or flm defects, defectve pxels To know f a pxel s extreme we have to rank adjacent pxel values 7 Lnear and rank-order flters The dfference Ths s what a lnear flter outputs: y w x w x... w n x n Let the flter support (vector) defne the nput pxels used: T S( x) ( x, x,..., xn) and the coeffcent vector defnes the weghtng: W w, w,..., w ) ( n Now we can brefly defne the fltered output pxel as: y( x) W. S( x)

13 3 73 Lnear and rank-order flters The dfference 74 Lnear and rank-order flters The dfference So, the lnear flter s defned as: y( x) W. S( x) If we now defne the ordered (ranked) support: T Sr ( x) ( x( ), x(),..., x( n) ) wth: x x x ( ) ()... ( n) Then the rank-order flter s defned by: y( x) W r. S ( x) r The pxel-weghts n a lnear flter are determned by the spatotemporal poston of the pxel relatve to the output poston The pxel-weghts n a rank-order flter are determned by the rank number of the pxel after orderng all values n the support. Examples: mnmum, maxmum, mdpont = (max + mn)/ medan, α-trmmed mean 75 Effectveness aganst shot nose 76 Less extreme dstrbutons: The α-trmmed-mean flter Orgnal 3x3 box 3x3 medan The general rank-order flter s defned by: y( x) W r. S ( x) r The medan flter, especally effectve for shot nose: W med (0,...,...,0) The α-trmmed-mean flter, long-tal dstrbuted nose, but less extreme: (0,...0,,,,,,0...,0) W Medan flter, especally effectve for shot nose: y( x) W med. Sr ( x) W med (0,...,...,0) α-central pxels are averaged, extremes gnored In an alternatve termnology, α dentfes the percentage of pxels that s trmmed-off 77 Shot nose + Gaussan nose reducton 78 The max and mn flter (morphologcal flterng) Orgnal sgnal 5x5 mn flter (dlaton) 5x5 mn- followed by 5x5 max-flter (closng) Orgnal 3x3 medan Orgnal sgnal 5x5 max flter (eroson) 5x5 max- followed by 5x5 mn-flter (openng) 3x3 box 3x3 -trmmed mean

14 4 79 Relevance: moton detecton sgnals 80 Combnaton of lnear and rank-order: The hybrd flter Orgnal sgnal 5x5 max flter (dlaton) 5x5 max- cascaded wth 5x5 mn-flter (closng) If we concatenate the lnear and ranked supports: S T h( x) ( x, x,..., xn, x(), x(),..., x( n) ) and also the coeffcent vectors defnes the weghtng: Wh ( w, w,..., wn, wr, wr,..., wrn) Orgnal sgnal 5x5 mn flter (eroson) 5x5 mn- cascaded wth 5x5 max-flter (openng) Then the hybrd flter s defned by: y( x) W. S h ( x) h 8 Combnaton of lnear and rank-order: The blateral flter 8 Blateral flter example: edge-preservng nose flter In the lnear flter, weghts depend on poston relatve to centre: w k k f( c k) In a rank-order flter, they may depend on the smlarty wth current pxel: w f ( x x ) k c In the blateral flter the weght s defned by: w N f ( c k). f ( x k. k c x ) Where N s selected such that the sum of the coeffcents s Functons f and f may be e.g. Gaussan or trangular functons Optmzaton s often a problem Flter optmzaton Brefly back to lnear flters to prepare for adaptve flterng Store vdeo Program enhancement algorthm Play enhanced vdeo Subjectve assessment Subjectve assessment s tme consumng Huge desgn space requres automatc optmzaton

15 5 85 It seems straghtforward 86 Bottleneck elmnated? Desred output Processng output Store vdeo Play enhanced vdeo (a-b) /N.S MSE Program enhancement algorthm MSE optmzaton Mnmze a sum of squared pxel dfferences by varyng all parameters 87 MSE-optmal (non-recursve) lnear flterng Cross-correlaton matrx The lnear flter s defned by: Auto-correlaton matrx y w x w x w3 x3... w n x n Compare the output wth orgnal mage to know error: e y o y For a mnmal MSE, the frst dervatve should be zero: e e e xe 0 w w X X... X n w Y If we now defne: X X... X n w Y X j x x j Y x y o We then get: X n X n... X nnwn Yn 88 MSE-optmal (non-recursve) lnear flterng We have to solve the followng equaton: X X... X n X X... X We thereto wrte: n X n w Y X n w Y X nnwn Yn X.W Y And conclude upon the followng optmal weghts: W X X.W X Y 89 We can use ths for nose reducton Orgnal nosy mage 3-taps flter support 90 We may also use t for de-blurrng nosy mages Orgnal blurred nosy mage Concluson: LMSE-flter enhances nose less Therefore, also reduces the blur less MSE-best compromse 3-taps LMSE-flter 3-taps averagng flter 3x3LMSE-flter 3x3 peakng-flter

16 6 9 How does ths compare wth nverse flterng? 9 Possble to calculate an nverse flter for perfect deblurrng. Typcally a recursve flter Infntely hgh gan results, f blurrng flter has a zero response for a gven frequency Consequence s that nose wll be amplfed unlmted Orgnal blurred nosy mage De-blurrng wth nverse flter Problem wth LMSE-flters 93 Images wth dentcal MSE Nosy Blurred Nosy Compressed Blurred Sharpened 5LIN0 Vdeo processng Compressed Sharpened Gerard de Haan Adaptve flterng Traned Flters (optmzng adaptve flters)

17 7 97 Images wth dentcal MSE Nosy Nosy Blurred Compressed Blurred Sharpened 98 What s wrong wth MSE? The mstake s n the averagng! Some pcture parts get a lot better wth a method that s poor on the average! characterze mage parts that can be treated dentcally Compressed Sharpened and ndvdually optmze (LMSE!) parts wth the same character! 99 Classfcaton based flterng 00 Support wth nput pxels x x x 3 x 4 x 5 x 6 x 7 x 8 x 9 Classfcaton LUT Lnear flterng y Classfcaton examples y w x w x... w c c c x nc n 0 Local edge drecton classfcaton 0 Local contrast classfcaton Image Edge drectons (colour) Image Local contrast (colour) 0 o 50 o 90 o 30 o 70 o Low Hgh

18 8 03 Local sharpness classfcaton Image Local sharpness (colour) 04 Codng the local structure to classfy support data Classfcaton code s concatenaton of pxels reduced to sngle bt:,( x xav ) ADRC( x) 0,( x xav ) wth x av n n x Low Hgh Archtecture of the hybrd flter Not only lnear flterng Classfcaton LUT Non-lnear flterng Concept can be extended to traned rank-order and blateral flters Traned Mxng of processng optons Not only flterng metrcs Classfcaton Processng opton Processng opton LUT Mxer Processng opton

19 Sharpenng and de-nosng Applcaton examples Nose flter wthout classfcaton Traned Flter output PxelPlus, PerfectPxel, Dgtal Realty Creaton, Up-scalng combned wth sharpenng Interlace artfact removal Standard up-scalng Traned Flter output Lne-average nput Traned Flter 3 Enhancement of dgtal vdeo Traned Flter output 4 Mcroscopy resoluton enhancement : RED : RED GREEN BLUE

20 0 5 Up-scalng of MRI-mages 6 Conclusons MSE (%) 00 Classfcaton LUT 80 In Flter Out MSE Smlar archtecture for many functons 0 0 HD BLn Fourer TF Desgn methodology replaces heurstcs 7 The value of traned flterng 8 Automatc optmzaton Desgn methodology replaces heurstcs for tunng No thnkng faster Desgner creatvty stll for fndng the relevant classes Neghbourhood Selecton 9 Neghbourhood selecton 0 Adaptve flterng: Neghbourhood Selecton So far, we assumed that ALL pxels n the support are combned under ALL crcumstances We may also propose to adapt the set of pxels that are combned to expectatons about local correlaton, usng a socalled neghbourhood selecton technque A neghbourhood shall be defned as a sub-set of the support Varous optons exst to exclude pxels from the support K-nearest, sgma nearest, symmetrcal nearest, Pxels n the neghbourhood are then combned Weghted averagng, rank-order, etc. support Select neghbourhood from support E.g. only pxels smlar to current pxel neghbourhood Combne pxels n neghbourhood Lnear flter Weghted average Rank-order flter Mn, max, medan

21 Support and neghborhood examples Case Case Non-adaptve: Standard neghborhood Case Case Example 3x3 support contans all 9 pxels Center pxel s the current pxel For dfferent technques the neghborhood pxels wll be shown wth a red border In a standard neghborhood all pxels from the support end up n the neghborhood 3 Effect of the standard neghbourhood selecton (5x5) 4 Neghbourhood selecton: K () nearest neghbours Case Case In a K () nearest neghborhood, pxels plus the central pxel, end up n the neghborhood 5 Effect of the k () nearest neghbourhood selecton (5x5) 6 Neghbourhood selecton: Sgma nearest neghbours Case Case In a Sgma nearest neghborhood, the number of [pxels n the neghborhood depends on how smlar the pxels are

22 Grey-level Grey-level Grey-level Vdeo course: Bascs: Flterng 7 Effect of the sgma nearest selecton (5x5) 8 Neghbourhood selecton: Symmetrc nearest neghbours Case Case In a Symmetrc nearest neghborhood, half of the pxels plus the central pxel, end up n the neghborhood 9 Neghbourhood selecton: Symmetrc nearest neghbours 30 Neghbourhood selecton: Symmetrc nearest neghbours 3 Pxel,, and 3 n neghborhood E.g. averagng leads to an output value hgher than the central pxel Pxel 4, 5, and 6 n neghborhood E.g. averagng leads to an output value lower than the central pxel 4 Poston Poston 3 Neghbourhood selecton: Symmetrc nearest neghbours 3 The effect of symmetrc nearest neghbour flterng (5x5) Result s a steeperedge In ths case symmetrc neghborhood selecton, wth pxel averagng leads to a sharper edge! Poston

23 3 33 The effect of symmetrc nearest neghbour flterng (5x5) 34 Adaptve flterng: Neghbourhood Selecton Select neghbourhood Combne pxels n from support neghbourhood support E.g. only pxels close to current pxel neghbourhood Lnear flter Weghted average Rank-order flter Mn, max, medan 35 Prepare yourself for the exam Before: Chapter, 3, and 9 Today: Chapter 4 I recommend you read the text Book avalable at Flux 4:3 Book may be used durng exam And try the exercses n the book: Chapter, 3 and 9 Chapter 4 You have to download VdProc (w3.es.ele.tue.nl/~dehaan/ ) Send me e-mal for password (G.d.Haan@tue.nl) 36 Quanttatve analyss of lnear flters 37 Frequency response of a lnear flter 38 Frequency response of a lnear flter H(0) H() H(N) sum y [ k] h( ) x[ k ] Gven a system wth mpulse response h() Gven an nput sgnal = complex snusod transfer jk x( k) e The output sgnal s: y( k) h( ). x( k ) And the frequency response:, k h( ). e j H ( e ) h( ). e j( k) jw e jk h( ). e j

24 4 39 Example: the flter 40 For all flters wth a symmetrc mpulse response Followng the defnton: j jw jw 0 jw H( e ) h( ). e e e e e jw cos t jsnt Substtutng: We get: H( ) cos( t ) jsn( t ) cos t jsnt Whch smplfes to: 4 H( ) cos t Symmetrc: h()=h(-), =..L L The frequency response s L j j. H( e ) h( ). e... a.cos(. ) L 0 Real-valued (=zero-phase) transfer functon Ths translates nto a constant for all frequences L ½.f s f 4 Example: the ( ) - flter 4 Example: the flter Followng the defnton: j jw 0 jw H( e ) h( ). e e e Substtutng: e jw cos t jsnt We get a complex frequency transfer (non-zero phase): H( ) cos t jsnt We can calculate the magntude of the transfer: H( ) R I cos t cos t sn t So, we have: Ths smplfes to: And we remember from modulaton: So, we conclude: H( ) cos t cos t sn t H( ) cos t cos cos cos( ) cos( ) 4cos ( t / ) cost H( ) cos( t / ) ½.f s f 43 What f we re-use output samples? Recursve flterng h(0) h() h(n) sum h r (M) h r () y ( k) h( ). x( k ) h ( m). y( k m) m r 44 What about a recursve flter? Now the output sgnal s: y( k) or: M N 0 h( ). x( k ) M h ( m). y( k m) r m ( h r ( m)). y( k) h( )). x( k ) m or: h( n) n y( k). x( k) hr ( m) m jn h( n). e n and fnally: H ( j) j h ( m). e m r m x( k) e jk

25 5 45 Inverse flterng Assume a sgnal has been fltered wth a (½ ½) - flter Can we nvert the operaton (de-blur)? j jw 0 jw H( ) ( e ) h( ). e 0.5e 0. 5e We are lookng for H - (e jω ),.e.: j H () e ) jw 0 jw h() ( ). e e e ( e Ths s a recursve flter!: ( jw ) 46 The nverse of a (½ ½) - flter H(0)= H() H(N) sum h r (M) h r ()=- 47 So ths s equvalent to a copper wre: 48 Consequences of nfnte gan nverse flter Result of cascade: Same nose, but nverse of ½ 0 ½ ½ ½ sum - + ½.f s f But f there was some nose: Worse than the nosy blurred: f ½.f s Snce the (½ ½) flter has a zero transfer for the Nyqust frequency, the nverse flter has an nfnte gan there! 49 Solutons for de-blurrng nosy mages 50 Prepare yourself for the exam The value of the nverse flter s lmted because of nose amplfcaton The nose easly domnates the percepton f blurrng process suppresses parts of the spectrum sgnfcantly Norbert Wener ( ) proposed the MSE-optmal soluton by addng a nose-dependng term n the denomnator The result s commonly referred to as the Wener Flter MSE-best soluton modellng blur AND nose Last week: Chapter and Chapter 3 Today: Chapter 4 I recommend you read the text Book avalable at Pt9:4 And try the exercses n the book: Chapter -3 Chapter 4 You have to download VdProc (w3.cs.ele.tue.nl/~dehaan/ ) Send me e-mal for password (G.d.Haan@tue.nl)