2D Signal Processing
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1 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Introdction to.. D Signal Processing lectre Andrzej Leśniak Proessor at aclt o Geolog Geophsics and Enironmental Protection AG Uniersit o Science and Technolog D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Signal processing in reqenc domain Pair o continos orier transorms - D case i e π d iπ e d I[ ] Pair o continos orier transorms - D case i π + e dd i π + e dd jφ jφ e e [ R + I ] R + I I I arctg φ arctg R R P φ P [ ]
2 D Signal Processing D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Smmar o some important properties o the D orier transorm. Proos o most o them is elemantar. [ ] β α β α + + [ ] β α α β β α [ ] e i β α π β α + [ ] [ ] cos α ω α ω α + + i n n n [ ] n n n i dpdt t p dd dd dd G dd g [ ] G g nr Propert Epression orier transorm is linear Scaling propert 3 Translation propert 4 Modlation theorem 5 Dierentiation propert 6 Spectrm dierentiation propert 7 Integration propert 8 Parceal theorem 9 Raleigh theorem Conoltion theorem D Signal Processing D Signal Processing Lectres7 TU reiberg Andrzej Leśniak i i e e π π i K K + + sin cos sin cos i i i e π π θ θ θ + D discrete orier transorm Eample: transormation o the rectangle nction
3 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak M M iπ M + e D discrete orier transorm M iπ M + e I nction is real the common case or digital images - the ollowing ormlas are tre: KM KM K K Amplitde spectrm is smmetrical and phase specrtm asmmetrical. Comments : dwo dierent was to present amplitde and phase spectrm: D Signal Processing Lectres7 TU reiberg Andrzej Leśniak D rectangle nction Similaril to D case the relation between sampling rates in time and in reqenc domain can be written as: M Direct reslt o the magnitde spectrm calclation - point is located in p-let corner not in the middle o the image To increase the resoltion in the reqenc domain we can add zeros to the ends o the rows and colmns o image. On the let the reslt o epansion o the table rom size 3 3 to size 56 X 56
![be proed: + [ ] M Center o the orier tran](/docs-images/87/96370130/images/4-2.jpg "sorm e.a.")
4 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak I It is common practice to mltipl the inpt image b the nction - +. It is not diiclt to show sing the propert o translation in reqenc domain that the ollowing eppression can be proed: + [ ] M Center o the orier transorm e.a. point is now in the point with coordinates M/+ i /+ Aerage ale o the image can be ealated as a center ale o the spectrm M M iπ M + e M M D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Is the igre shape related to the image in magnitde spectrm log ω ω
5 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak There is no simple relations onl or igres with high smmetr we can ind somw simmilarities D Signal Processing Lectres7 TU reiberg Andrzej Leśniak In more complicated images there is diiclt to ind some reglarities. On some images the edges can be seen. Scanning electron microscop SEM o damadged integrated circit. In orier amplitde spectrm the pictre o the images with orientation o 45 are seen.
6 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak iltering in reqenc domain Basic steps :. Mltipl image b coeicients - + to center the spectrm. Compte discret two-dimensional orier transorm 3. Mltipl spectrm b ilter nction transer nction 4. Compte inerse discret two-dimensional orier transorm - 5. Etract real part o the reslt 6. Mltipl the reslt b coeicients - + to obtain the inal reslt D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Selected ilters and their properties otch ilter or others M Aerage ale o the reslt o iltering is eqal to. The image on the let is normalized in the rage o [55].
![reqenc and space domains M M δ e M M s s M de s δ M s δ de iπ M + M M h δ m n h m n h m n h I[ δ ] h mask Two ersions o the conoltion theorem](/docs-images/87/96370130/images/7-2.jpg "Deinition o the D conoltion Using the properties o the implse nction and and conoltion theorem we can established that the ilters in spatial and")
7 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Lowpass ilter nction and the reslt o lowpass iltering. The clear image aeraging eect is isible. ighpass ilter nction and the reslt o highpass iltering. Usall so the aerage ale o the reslt is eqal zero. The image is darker than original image and sharp. The image shold be normalized or some ales shold be added to it to increase the energ o the image. D Signal Processing Lectres7 TU reiberg Andrzej Leśniak M M h m n h m n m n h h Deinition o the implse nction δ δ δ Relations between iltering in reqenc and space domains M M δ e M M s s M de s δ M s δ de iπ M + M M h δ m n h m n h m n h I[ δ ] h mask Two ersions o the conoltion theorem Deinition o the D conoltion Using the properties o the implse nction and and conoltion theorem we can established that the ilters in spatial and reqenc domains constitte the a orier transorm pair
8 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Model o the iltering in reqenc domain G Ideal lowpass ilter perspectie plot image and radial cross section the ilter cannot be realized with electronic components it can be onl implemented in compter. dla dla D D D > D Becase the centre o the reqenc rectangle was moed rom point to the point M/ / the distance rom point to the centre is eqal to: D M + D distance rom point to the centre o the reqenc rectangle D ct-o reqenc D Signal Processing Lectres7 TU reiberg Andrzej Leśniak The image o size 5 5 piksels and its orier spectrm. Circles hae radis eqal to : piksels and enclose and 99.5 % o the energ o the image. M P T PT P α M P T image amplitde spectrm % P P T
9 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Reslts o the ideal lowpass iltering with ct o reqencies set at radis ales o i 3 piels. or larger ct o reqencies the aeraging eect is smaller. In igres 345 the ringing eect is clearl isible. D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Real lowpass Btterworth ilter. + D D n D M +
10 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Original image and reslts o the iltering with Btterworth low pass ilter o order with ct o reqencies at radis o i 3 piels. There is no ringing eect. D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Spatial representations o Btterworth low pass ilter o order 5 and with ct o reqencies at radis 5 piels and corresponding gre leels proiles throgh the center o the ilters. Ringing increases as a nction o ilter order.
11 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Real lowpass Gass ilter. D e D D σ D M + or DD transer nction reach the ales.67 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Original image and reslts o the iltering with Gassian low pass ilter with ct o reqencies at radis o i 3 piels. There is no ringing eect.
12 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Eample o application o lowpass iltering to the machine perception. Image on the let is eample o the tet o poor resoltion a transmission photocop historical material We can see the distorted shapes and broken characters. Application o the low pass ilters ills the gaps. Althogh the characters ater iltering are blrred the can be easier recognized atomaticall. Gassian ilter D 8 image size D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Eample o lowpass iltering in printing and pblishing indstr cosmetic processing Image o size 8 73 piels Gassian iltering D 3 Gassian iltering D 8 The aim o the processing is to obtain smoother and soterlooking reslts e. Smooth sharp o ine skin lines and small blemishes.
13 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak ighpass sharpening ilters P LP Eamples o the transer nctions o highpass sharpening ilters: Ideal ilter Btterworth ilter 3 Gass ilter perspectie plot image radial cross section D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Spatial representations o the basic highpass ilters and corresponding greleels proiles: Ideal ilter Btterworth ilter 3 Gassian ilter
14 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Reslts o ideal highpass iltering o testing image with ct-o reqenc D eqal to 5 3 and 8 piels. dla dla D D D > D D ct-o reqenc Ringing eect is clearl isible in the irst and the second image. D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Reslts o Btterworth highpass iltering o testing image with ct-o reqenc D eqal to 5 3 and 8 piels and order n. + D D n D M + Compare this ormla to Btterworth lowpass ilter In that case there is no ringing eect in the images.
15 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Reslts o Gassian highpass iltering o testing image with ct-o reqenc D eqal to 5 3 and 8 piels. D D e D M + In that case there is also no ringing eect in the images. D Signal Processing Lectres7 TU reiberg Andrzej Leśniak Bandpass and Bandstop ilters D W < D D > D D D W D + W + W Ideal ilter W width o the band n D W + D D ep D D D W Btterworth ilter Gassian ilter
![or D D D other [ ] [ ] M / + / M / + + / D + + D](/docs-images/87/96370130/images/16-4.jpg "D n D D ep D D")
16 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak a Image corrpted b the sinsoidal noiseb power spectrm o the image; isolated points ronded the centar are spectrm o sinsoidal noise c transer nction o the Btterworth ilter d Image ater iltration oise rejected rom the image D Signal Processing Lectres7 TU reiberg Andrzej Leśniak D notch ilters D radis center o the ilter ; otice the necessar smmetr condition D D or D D D other [ ] [ ] M / + / M / + + / D + + D D n D D ep D D
17 D Signal Processing Lectres7 TU reiberg Andrzej Leśniak a Satelite image with horizontal scannere lines b Amplitde spectrm o image a c otch ilter that pass onl noise component connected with horizontal lines d Inerse orier transorm showing noise component in spatial domain e Reslt o the notch ilter rejection iltering dierence between image a and d
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