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# DIGITAL IMAGE PROCESSING ECE 468

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This 95 page Class Notes was uploaded by Marjorie Kulas on Monday October 19, 2015. The Class Notes belongs to ECE 468 at Oregon State University taught by S. Todorovic in Fall. Since its upload, it has received 25 views. For similar materials see /class/224421/ece-468-oregon-state-university in Engineering Electrical & Compu at Oregon State University.

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Date Created: 10/19/15

ECE 468 Digital Image Processing Lecture 9 Prof Sinisa Todorovic s39 isaeecsoregonstateedu 1 Oregnn State University Multiresolution Image Processing Informal motivation Images may show both very large and very small objects It may be useful to process the images at different resolutions Multiresolution Image Processing A more formal motivation An image is a 2D random process with locally varying statistics of pixel intensities Analysis of statistical properties of pixel neighborhoods of varying sizes may be useful Histogram of Small Pixel Neighborhoods Image Pyramids A representation of the image that allows its multiresolution analysis p l X 1JJ VCI Hapcx Level Mbase Dmvnsampler rows and columns cvcl j 39 approximation Upsumplcr rows and columns Interpolation til ler Prediction we Leicl in snl igc 3533 Example Image Pyramids a b FIGURE 73 Two image pyramids and their histograms a an approximation pyramid b a prediction residual pyramid illpul intng 1Given an image at levte 2 Filter the input and and downsample the filtered result by a factor of 2 This gives the image at level j1 3 Goto 1 4 Upsample and filter the image at level j1 this gives an approximation of the image at level j 5 Subtract this result from the image at level j this give the prediction residual at level j 6 Goto 1 Typical Filters For the multiresolution pyramid we use spatial filters Neighborhood averaging Lowpass Gaussian filter For the residual pyramid we use interpolation filters 0 bilinear bicubic Upsampling Downsampling Upsampling Inserting zeros f932ay2 3331 are even 0 ow Mm 7 Downsampling Discarding pixels f2iray Subband Image Coding a 1 FIGURE 76 mm a A twohand suhhand sodmg 21nd lccrxlingY syslcm and b its Spc cll39llm splitting pruperlim Wme me Iaaml High band x x 1 0 772 Tr Example Analysis Filter Bank l I1IIll II 1 11 13111 liiln I Lo I T H OJ l l l l l 1 l l ireleiuiJRJSnwu 7571711i1114i57 737271i11234ltn7 u n n 1 1 hhm 7H zll397 I 7m 114iiIllvlen 15IH 1quotIIHII T a r i ll l l 1 7371illl l l 3 J 5 n 7 7371711112 3 J 5 l 7 u 1 3l1ll 1 J 3 J 5117VH u a b C d e l FIGURE 75 Six functionally related filter impulse responses a reference response b sign reversal c and d order reversal differing by the delay introduced e modulation and 139 order reversal and modulation Subband Image Coding fipm I i for perfect reconstruction 1 lt2 lHuml lHile 900 1 h1n w 910 1 1hon Low baud High baud l l l l l 712 71 ho h1 go 91 crossmodulated filters HHWH Subband Image Coding for perfect reconstruction gem 1 h1n 910 1 1hon HIWW Low band High hand t t t 112 13 Subband Image Coding for perfect reconstruction 90W 1nh1n 910 1 1hon A i f2n h02n 2n murwn mam flpm 7 0 2n 1 Low band 1 High band A 2 1 h 2 Jr 1 7 2 1 i fhpnfn B1n n2 112 HHWH Subband Image Coding for perfect reconstruction gem 1 h1n 91W 1n1h0n Low band 1 1 1 1 1 112 A 1 f 2 ham 7 2n tHltwtt flpm i 0 7 2n 1 Hi I hand 2 A 1 f2n1h12n1 2n1 w W e 0 7 2n f f2nh02ng02n f2n 1h12n 1g12n 1 Huhu Subband Image Coding for perfect reconstruction 90W 1nh1n 910 1quot1hon Low band A i f2n lt ho 2n 2n H m flpm i 0 2n 1 Hi h band g A i f2n1h12n1 2n1 w fhpm 0 2n t t t 1 112 f f2nh02ng02n f2n 1h12n 1g12n 1 f f2n h02nh12n f2n 1 h12n1h02n 1 Subband Image Coding for perfect reconstruction gem 1 h1n 91W 1n1h0n fhpU f2nh02n 2n 0 2n1 aw HimU1 tHrtwtt Low band High hand 13M f2n1h12n1 2n1 0 2n 1 1 1 1 LI 112 7 f f2nh02ng02n f2n 1h12n 1g12n 1 f f2n h02nh12n f2n 1 h12n1 h02n 1 f h0n hl Subband Image Coding for perfect reconstruction 90W 1nh1n 910 1quot1hon rmer WW 1Huw1 1H1m1 Low band High band aw f2n1h12n1 2n1 0 2n 1 t t t to 112 17 f f2n lt h02n g02n f2n 1 lt h12n 1 g12n 1 f f2n h02nh12n f2n 1 h12n1h02n 1 f n them H1100 ffn Vector Inner Product Given sequences Mn f2 71 i1 lt11 f2gt Z ffnf2n Subband Image Coding 1 p 1 w Subband Image Coding ho h1 go 91 are biorthogonal i1 lthz2n 1079706 5i j5n m 071 Example ltho2n M79106 0 Subband Image Coding 114quot 1 hMI 21 2w Lam ho h1 go 91 are orthonormal lthz2n 107970 5i j5n m 071 ltginagjn 2mgt 5i j5m m 071 Orthonormal Filter Bank Keven is the number of filter coefficients that must be even pl 1 1 1 Analysis fillei bank 1 Synthesis fillet bank 0 1 1 1 1 1 mn 91 1ngOKeven 1 n L fiwlnl lHMuull lHilmH Lm wm Highlmnd Z Keven 1 n7 Z 07 l l l l l 112 17 Orthonormal filter bank can be obtained from a single filter prototype Example Orthonormal Filter Bank 9101 1n90Keven 1 n Analysls mm bank giKeven 1 n 07 L filplnl lH11m11 1H1lwl 1 1 I hum 1111771 4111 llxli111nl 11 i o 11 To 1 D I T 1 ii i J 1 1 1 1 7 gt1 r 1 r 737171111 14 w 7 1717171111131w117 7 171411 I 114w nquot n n n 1 1 1 lumlhl77n 111711m1 11111l 11h11111 11 I T 11 0 u T I il l i l 1 1 1 1 1 iiililllll vlth1 l 34l17 712111114517 n Example Orthonormal Filter Bank 9101 1n90Keven 1 n giKeven 1 n 07 I II UII lm 4111 1mm Min 71 71 71 i3ilil l l I 3 J 5 h 7 3Zl U 1134 S I 7 3211 I 3 4 5 Ix u n u 1 1 l lululIIIu 15H 1Illn lintn1HI11I u lT ii 1 i 41 33W12145n 7 lm 1717101 5 4 5 e 7 71r 2 1n L I 4 lt r 7 18 Example Orthonormal Filter Bank phl39 9101 1ngOKeen 1 77 N giKeven 1 n 0 mm 1 blunimun 39 hAHnnl he T H W I 1 a V 1732 u2345 7 gtI171 Jl13451 7 1 H4 397 4 7m n H N Z n 1 II4IIIIII1Kliz 1 WIRHHHIHW 1 hhlmlwhlmlM al u T 0 0 I T1 1 J 4 l 7 311H 2 451 7 7 737171I11345h7 7 7271U2 45h n n u Example Orthonormal Filter Bank 9101 1n90Keven 1 n giKeven 1 n 07 I1NIH IH I lm IzH I U T o h y 1 i 3l1113515h7 3 5 quot 0101 om 1w Ilt I m 5 1ll 1 I JI11K 1 m 1 3 Z 17L2 4 5 7 1111 1 quot 77 JShT Example Orthonormal Filter Bank 9101 1n90Keven 1 n giKeven 1 n 07 Intmhim h 1 v x x v 1 I 131 4mm y l I 73gtlgt I DJ l Viilw7 01W h4un1K7 1 u u I1ngtlH Ill39 1 m ECE 468 Digital Image Processing Lecture 11 Prof Sinisa Todorovic sinisaeecsoregonstateedu 11 regnn State University Problem with Fourier Transform Available frequency content But not where that content is in the space domain of 2D signals or in the time domain for 1D signals ej27rw Problem with Fourier Transform Available frequency content But not where that content is in the space domain of 2D signals or in the time domain for 1D signals just scaling of the generating function We translation Scaling Functions Given g0 3 WkCE 2972942956 k Example Haar Scaling Functions wMv m mm w 7 n l l u i i 39 U 1 2 3 0 l 2 3 wu w 2 l W11 i 2 7 I l n J n l 2 3 U 1 2 u b c 1 e FIGURE 71 Some Hnm scaling functions Example Haar Scaling Functions Wm Fquot mm EU 1 l l u i 7 v X 0 l 2 3 U Z muvquot25l mm 3 2X 1 1 1 n v J l 2 3 0 I 2 05g017033 017133 025g017433 a b C d c 1 FIGURE 7 I Some Huar scaling functions Example Haar Scaling Functions 1211101 W 11111va 1m 7 11 1 1 I j J 1 1 1 11 7 7 1 1 1 139 u 1 2 3 0 1 2 111vw 2 21 10111111711017 1 1 I J 1 1 b 1 c d c f 1 1 1 1 FIGURE 7 l u 1 2 3quot 11 1 2 39 Some Hnm scaling functions 1mg V1 1 1 Function in V1 ll 1 5 P111 39 11 l 2 05g017033 017133 025g017433 Example Haar Scaling Functions mm m min m 7 11 1 1 J j l 1 1 m u i 1 i 1 1 V X o 1 2 3 0 1 2 3 2511111 12HZX 1mm 1 2X 71 1 1 1 1 a b 1 c 1 c 1 n 1 1 FIGURE 7 I 11 1 2 3quot U I 2 quot Some Ham scaling functions 11115 Vi 1111111116 1 1 1 l 2 11V in V1 11 1 51111 u 1 2 3 o 1 2 3 1 1 33 05801033 01133 025801433 Spo k wl z wl zmlw Nested Function Spaces Wcmcn FIGURE 712 39I lle neltletl lunction gtpucc spanned by A scaling l39uuclion Any Function If x 6 V 1 03 Zamkr k 93 ZBWHLWC k it ZVWHzMCC k Wavelet Function Spaces v2 m wl we we w VI Vu 3993 WU FIGURE 713 Thu rclaliuns hip hotwccn scaling and mum l39uncliun spucw Wavelet Function WW ltwjykw790jlgt 0 Vj1VjWJ Any Function If M 6 V1 Any Function If M 6 V1 75 Zaij71kz Z kwj71 ltgt k k W71 gtlt 3500 33 m s 3 H M86832 Miwig w w SL S 13 H M E TNViav M E rmiav M E rriav Na Na Na gtlt 3500 33 m s 13 H MQw TriRv MuwSTriRV SL S 13 H M Qw g xmviav M Rw n xmviav M Rw n xriav w 13 H Mgw u eisv M Mugging Ha Relations Between the Scaling and Wavelet Functions 2 aneijrlJL 7 n Z h n2gp2j1 n jk gt 90 Z h n e02 n Relations Between the Scaling and Wavelet Functions 2 aneijrlJL 7 n Z h n2gp2j1 n jk gt 90 Z h n e02 n Z hwn W2 n WW 1nh 1 n Haar Wavelet Functions My uWm tum mu 7 2 l l t I 7 l Haar Wavelet Functions awn L uMU tum Ht 7 2 l Haar Wavelet Functions WU JamiU DaliV IJJ39 2 i i l l i 0 1 7 i i 7 E1 i i 39 o 2 u 1 2 drum 4421 71 E 11 VU 619 m i 1 1 Function 0 0 i i In V1 71 E1 i i 1 2 1 1 2 x 1 1 WM JamiU DaliV IJJ39 2 l l i 0 1 7 i i 7 E1 i i 39 o 2 u 1 2 111111 EMZV 71 E 11 V1189 W i 1 1 Function 0 0 i i In V1 71 E1 i i 1 2 1 1 2 x MM e W M1 1 E W l 3V74MF1111 l 7 E 12 1 1 7 7 1 m E faw Mm 0 7 7 U U7I1lj 1 7 1 7 71 v lh 1511 71 x Z4 W11 0 2 0 1 2 1 Haar Wavelet Functions WU damn two w 7 2 i i l l i i i i i i 7 71 i i 39 0 l I U l 2 J W i 5 r2 m E V1 we W i l 1 Function 0 i i i In V1 71 71 i i i l 3 i l W x W E m 364 i 1 I was to 1 7 fmehmnm 0 7 7 U U7IIlj i 7 7 Expressed 71 mil5 m il 424 Wm Via V0 0 l 2 U l 2 Wavelet Series Expansion chok k jjo k 006 lt90jokxgt7fxgtgt ltWgtlt Wf 90jok Z Zdjk jk Example Haar Wavelet Series Expansion 1 yv1 V0 13900 05 05 1 07777777 07777777 705 1 705 0 02 5 075 1 0 035 05 075 l l 7 I W0 quot1 7f12 quotquot 14 m 14 012 0777 777 7777777 rm 7m 1 70x V W X 0 025 05 075 1 0 025 03 075 l I W 732 32 1 42320 316 110 110 7316 705 v 0 025 05 075 L 1 025 05 075 1 ECE 468 Digital Image Processing Lecture 2 Prof Sinisa Todorovic sinisaeecsoregonstateedu Oregnn Shh Universify Outline Image interpolation MATLAB tutorial Review of image elements Affine transforms of images Spatialdomain filtering Image Interpolation Bilinear N 1 Bicubic N 8 N N I I f7y Zzai y 10 j0 MATLAB Image Processing Toolbox Basic MATLAB Commands imread size whos imshow imwrite double uint8 im2uint8 im2bw plotimgsizeimg12 zerosmn onesmn randmn randnmn function outputs namefuncinputs return Basic MATLAB Commands ismember isempty intersect union for while meshgrid imadjust imhist Image Structure Image Elements Pixels 4adjacency 8adjacency madjacency Path directed undirected loop Region Connected set of pixels Region boundary inner and outer contour Foreground background Edge Connected pixels with high derivative values Interest points Tjunction Yjunction Highlights or specularities Lambertian surface isotropic reflectance Specular surface zero reflectance except at an angle Spatial Image Tranmormations Amne nansvmms Example I dZTKMO M 2D Translation displacement source S Savarese 13 2D Translation source S Savarese 13 2D Translation displacement y homogeneous coordinates a t 1 0 t P P t x I39 I39 yty 0 1 759 glJ 13 2D Translation P y R displacement t r y homogeneous coordinates x t 1 0 t P P t x I39 I39 yty 0 1 759 glJ x i tm 1 0 tm x gt Pl gt y ty 0 1 ty y 1 0 0 1 1 source S Savarese 13 2D Translation displacement y homogeneous coordinates y 1 x i tm 1 0 tr 33 gt P gt y i ty 0 1 ty y 1 0 0 1 1 RH translation matrix source S Savarese 13 2D Scaling source S Savarese 14 2D Scaling 8x 8x 0 0 w 8y y 0 8y 0 y 1 0 0 1 1 I 14 2D Scaling 85 w 8x 0 0 g 8y y 0 8y 0 y 1 0 0 1 1 HF scaling matrix sou roe S Savarese 14 2D Scaling scaling matrix source S Savarese 14 2D Scaling Translation Is the ordering important P SP P TSP P TP source S Savarese 15 2D Scaling Translation Is the ordering important P S P P T S P P T Pl 11l jf1lsgso 3W1 lll loo filo Slllll source S Savarese 15 2D Scaling Translation Is the ordering important P TSP O 1 O 1 ty 0 3y 0 y 1 O O 1 0 0 1 1 3x 0 tag scalin translation A O 89 t9 gmatrix source S Savarese 15 2D Rotation counterclockwise by angle 6 A P 008613 sin6y sin6930086y y quotquot i 1 E y c a gt cos 6 sin 6 O 33 I sin 6 cos 6 0 1 0 0 1 1 source S Savarese 16 2D Rotation counterclockwise by angle 6 A 0086U sin6y P sinQ U l COSQ y y 1 1 V LU LC 3086 sin6 O a sinQ 3086 O 1 O O 1 1 HF rotation matrix source S Savarese 16 2D Rotation counterclockwise by angle 6 3086 33 81116 3 1116 U l COSQ y 1 1 E y gt ICOSQ sin6 0 a i i 1999 950 1 0 0 1 1 rotation matrix source S Savarese 16 2D Rotation Scaling Translation i Icos Sin9 i 0 1 0 0 ifT9fi 95 0 0 11 0 y 1 O O 1 O O 1 O O 1 1 RH RH RH rotation translation scaling matrix matrix matrix gyl RS t y I 0 1 1 Estimating the Spatial Transform I I I I o o a mun a a a a a a a El I I I I inverse 000 a 39 estimation transform I I l error a a a a a Estimating the Spatial Transform W 7511 7512 7513 11 y 7521 7522 7523 3 1 0 0 1 1 What is the minimum number of point pairs to find matrix T AXB if detA 7A 0 gt X A lB or if detATA 7A 0 e X ATA 1ATB Re writing the Equation of Transformation a 2511 2512 2513 3 2521 2522 7523 1 0 0 1 1 Re writing the Equation of Transformation ac 1511 2512 2513 3 2521 2522 7523 1 0 0 1 1 33i397511 yi t12 14513 04521 04522 04523 73 04511 04512 04513 M4521 yi39tzz 1t23y Re writing the Equation of Transformation 95 1511 1512 1513 95 y i 1521 1522 1523 y 1 0 0 1 1 3614511 yi t12 11513 07521 0391522 04233624 0391511 01512 01513 3614521 yi t22 139t23y i yi 1 0 0 0 1513 0 0 0 95 yi 1 1521 e1 e1 Cd Summary of Affine Transforms TABLE 22 Af ne lramformmiuns based on Eqv 21x43 Transmnmmm mine Maul T CoIJI39tlrnace Example Name qumllous ILIcnlily 1 u u 1 I l I 139 w n n l Scullng c u n 39 L U U u u y mu u u l Rul dliun cm H 1quot U 1mm H 7 w Sin U 14quot CUM H 7CngtH wxinu Tmmlulion I U u v 1 H n r w 1V 1 l Shcdrh39crlimlb l H 0 Y L W 11 1 0 V 39w H H ShU df llmrimnl dl I M U v 0 I 0 v z w n n I SpatialDomain Filtering of Images SpatialDomain Filtering of Images g y Tf7 y transformed input input Basic Operations on Images Addition Multiplication 24 Example Averaging Noisy Measurements 901311 rm Way 1 K may Ezgxxiy a b c d e f FIGURE 216 a Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise be Results oi averaging 5 10 20 50 and 10 noisy images respectively Original image courtesy ofNASA 25 Example Shading Correction Way friyhriy abc FIGURE 229 Shading certaction a Shaded SEM image of a tungsten filament and suppol39l magnified approximately 130 times bl The snarling pattern tc Product or a l by tlrc reciprocal ol39b Original image courtesy of Mr Michael Shaffer Department of Geological Sciences University of Oregon Eugene Example Masking Way friyhriy 39 M abc FIGURE 230 a Digilnl denlal Xray imageb ROI mask 0quotb39nlalil1gl66lll willl finillgh while mrlcsponds lo I and black corresponds to U c Pl39oducl nfzl and h Example Rescaling Intensity Values fx7 y minfx7 34 maxl x 11 minl x 11 906711 K output 28 Example Local Averaging Smoothing mayb 2 mm m y ENy Il x m a a 1 V x y The Value of his pixel quot the average Value of the pixels in SA 29 Output intensity level 3 Example Image Negatives 9x7yL1f7y Example Gamma Correction 3L4 L2 1 4 567 7 0 L4 L2 3L4 L 71 Input intensity level r Example Gamma Correaion Nem Class Humewmk we gt Many am emahzamn 24th 3 a m gt Many am spatmcamn 24th 3 a 2 Spatial cummunch anchHEWanDn 2mm 3 42 r amume andshawpemngspanaquotHtaws exttmk aa Humewmk 2 ECE 468 Digital Image Processing Lecture 4 Prof Sinisa Todorovic sinisaeecsoregonstateedu 11 Oregnn Slat University Disclaimer The following slides are just excerpts from the textbook You should learn all material presented in chapter 4 in the textbook CFT of Rectangular Pulse Wm M T r4T a b FIGURE 413 a A 2D function and b a section of its spectrum not to scale The block is longer along the Iaxis so the spectrum is more contracted along the naxis Compare with Fig 44 sin7rMT sin7r1T F ATZ M7 1 WMT 7r1T Sampling Theorem NATAZU 1 Sampling Theorem Footprint of an ideal Iowpass box filler 4 nulzx anx u 7 00 m n Fuz TZ EGOWEOOFUL le Aliasing FIGURE 416 Aliasing in images In a and b1he lengths of the sides of the squares are 16 and 6 pixels respectively and aliasing is Visually negligible In c and d the sides of the squaies aJe 09174 and 114798 iixels aspeciively and the lesuhs SllOW signi canl aliasing Nole 111m Ll masquerades as a quotnormalquot image Aliasing Due to Subsampling abc FIGURE 417 Illustration of aliasing on resampled images a A digital image with negligible visual aliasing b Result of resizing the image to 50 of its original size by pixel deletion Aliasing is clearly visible c Result of blurring the image in a with a 3 X 3 averaging filter prior to resizing The image is slightly more blurred than b but aliasing is not longer objectionable Original image courtesy of the Signal Compression Laboratory University of California Santa Barbara Fm Centering DFT a b ng backtoback c d periods meet here i 39 FIGURE 423 H u Centering the ip M2 0 VI2 1 J kMZ Mi XM Fourier transform a A 1D DFT showing an infinite T b k b k number of periods W0 ac tO ZIC periods meet here quot 51111333 o tame y multiplying fx F u 1L3 H x V L a A assigns c A 2D DFT showing an infinite number of periods The solid area is 001N21N71a the M X N data array Fu 2 Am 1 obtained with Eq 45 15This array Hquot 7 consists of four quarter periods d A Shifted DFT Foul backtoback Obtained by Maw eriods meet here 1 Periods ofthe DFT P Emltllilygljyg f 11 V D M gtlt Ndata array Fm 7 bison computing M N Fu 39U The data fx gt U nowcontainsone 7 2 7 2 complete centered period as in b Example i i i i Translation in Space gt No change in Spectrum Rotation in Space gt Rotation in Spectrum Example a h c d e f FIGURE 427 it Woman h Phase angle c Woman reconstructed using only the phase angle l Womnn reconstructed usingY only the spectrum 6 Reconstruction using the phase angle corresponding to the woman and the spectrum corresponding t the rectangle in Fig1243 1 Reconstruction using the phase of the rectangle tllLi the spectrum of the woman Circular Convolution g ftm m C h Li 3 0 FIGURE 428 Lult m culttmn 0 200 400 0 200 cmn39ululion nt Mm Wquot two discrctc t39uncliom ttlalnincd using the Z upptonch ml i u 200 400 0 z 400 5 n 3 mim M7quot result in cl is cumct Right column Convolution ut the sumo l l l 430 tunctiuns hut huim tulungtntn axe t t t t how tJuta train 0 200 adjacent periods fx gx pm ttC wmpurou nd cnm 1200 yielding an 60 incur c convolution X y r 0 200 400 600 300 0 200 400 multTnuhttnn the cmruct result function pudding nnts t lac uwtl ECE 468 Digital Image Processing Lecture 7 Prof Sinisa Todorovic sinisaeecsoregonstateedu Oregon State University Model of Image Degradation Restoration FIGURE 5 gtnyi Amodeloflhe chrudulinn R I image 11 V function krialllfllfllgltrn f V V legradntion H L lestorutiun Noise mew RESTORA TION process DECRADA TION 901379 hm y x y 7701379 Guv HuvFuv Nuv Image Restoration by Inverse Filtering FIGURE 5 A model 01 he chmdmim EMAquot inume fix y igt unclmn Rugtlur1xll0n cl k nlionl H restoration proces L Now nu v RESTORA TIDN DEGRADA TION Guv Hu vFu v Nuv Image Restoration by Inverse Filtering FIGURE 5 I A model of the chrudulinn R V n image I39I1VI Igt l39uncliun kli lllllltglclt39quot legrmintion H lealorulion process DECRADA TION Guv Hu vFu v Nuv Image Restoration by Inverse Filtering FIGURE 5 A model 01 he chmdmim 31Lquot image fix quotI Igt I39unclmn 7 cle 1d iunr H I39eslornlion process If Nolm mu m RESTORA TION DEGRADA TION Guv Hu vFu v Nuv Fuv Fuv m Inverse Filtering Fuv Fuv W Bad news Even when Huv is known there is always unknown noise Often Huv has values close to zero Example Inverse Filtering Atmospheric turbulence effect Huv exp k u M22 v N2256 Example Inverse Filtering 0 FIGURE 527 Restoriu Fig 5251 with Eq 5 71 a Result of using t e full lter 12 Result with H cut off outside al39adius of Wiener Filtering Mean Square Error Filtering Incorporates both Degradation function 0 Statistical characteristics of noise Assumption noise and the image are uncorrelated Optimizes the filter so that MSE is minimized e2 E mm m N Wiener Filtering Guv HuvFuv Nuv degraded output input Fu7 21 Hwu7 2Gu7 21 restored 1 Hu7v2 Hm 1Vu7v2 Fu7vl2 lmum IHWWW Signal to Noise Ratio 1 Hu7v2 HWU7U 2 HWW Hu7v2 Signal to Noise Ratio 1 Hu7v2 HWU7U 2 HWW Hu7v2 inverse filter Signal to Noise Ratio u v 1 Hu7vgy H 7 Hu7vHu WP Nu7v 7 Fu7v2 inverse filter Signal to Noise Ratio 1 H Mg JVum I U72 Hu7U l2 N39U 739U 7 Fu7vl2 inverse filter unknown M 1N 1 Fu7vl2 i u0 110 SNR 7 M71 N4 Nu7vl2 Approximation Flu1411 1 WWW 2 z 1 lHvallz 1 HM HWW lgiuLZZlL HM HWNZ m Example Wiener Filtering ahc FIGURE 528 Comparison of inverse and VVlaner ltering a Result of full inverse ltering of Fig 525b b Radially limited invelse lter result 2 Wiener lter result Exaple Wiener Filteing g h i u FIGURE 529 a 8 bit image corrupted by motion blur and additive noise b Result of inverse filtering c Result of Wiener filtering d f Same sequence but with noise variance one order of magnitude less g i Same sequence but noise variance reduced by five orders of magnitude from a Note in ii how the deblurred image is quite visible through a curtain of noise Midterm Exam Review m Average 668 Median705 STD 142 Proiected Total Average 855 Median 871 STD 61 ECE 468 Digital Image Processing Lecture 8 Prof Sinisa Todorovic sinisaeecsoregonstateedu Oregon State University Image Reconstruction from Projections Xray computed tomography Xraying an object from different directions gt 3D object representation FIGURE 535 Four generations of CT scanneisThe dotted arrow lines indicate incremental linear motion The clotted arrow Subject i39otatioriThe Detector the subject s head indicates linear motion perpendicular to the plane oftie paperTh double arrows in a and 1 indicate that the sourcedetector unit is translated anc then brought back into its original position Example Backprojecting a 1D signal Absorption profile a b 7 C d e FIGURE 532 a Flat region a showing a simple Beam 4 object an input parallel beam and a detector stripl b Result of back projecting the sensed strip data ie the 1D absorp tion pro le c The weam and detectors rotated by 90 d Backprojection e The sum of b and dThe inten sity Where the back projections intersect is twice the intensity of the individual backprojections As We Increase the Number of Backprojections 39A 7 halo effect ahc def FIGURE 533 1 Same as Fig i 3 hHe Reconstruction 39 Ul 73and4 39 eclions45quot 21 a f Recunalluctiun with 32 haekpmjucv lions 5625quot apart note the blurring E ample Backprojecting a 1D signal a b c d e f FIGURE 534 a A region with two objects bHd Reconstruction using L 2 and 4 backprqiections 45 apart 5 Reconstruction with 32 backproject39ions 5625D apart f Reconstruction with 64 backprojections 2 8125D apart Projection r azcb 330050ysin0 p FIGURE 536 Normal representation of a straight line Radon Transform y39 A 10ml gm at in i llie projection omplcle pl OJEClIOn gip Bk for a fixed angle A point in the projection Pja k is the raysum along 1005611c ysinek p Radon Transform oo oo gp fzy6zcos9 ysin 7pdzdy 700 700 continuous space coordinates M71 N71 90 9 Z fI7 WW 0089 ysin9 7 p 0 10 y discrete space coordinates key tool for reconstruction from projections fx7y 0 Example Radon Transform A 7 2y2 r2 ow 391 FIGURE 538 A disk and a plot of its Radon transform derived analytically Here we were able to plot the transform because it depends only on one variable When 2 depends on both p and H the Radon transform becomes an image whose axes are p and 0 and the intensity of a pixel is proportional to the value ofg at the location of that pixel Sinogram Image of Radon Transform lSU 135 H 90 H l FIGURE 539 Two images and their sinog rams Radon transforms Each row of a sinogram is a proiection along the corresponding angle on the vertical axis Image c is called the SheppLagmi plinnlonr In its original form the contrast of the phantom is quite low It is shown enhanced here to facilitate viewing Properties of Objects from Sinogram Sinogram symmetric Object symmetric Sinogram symmetric about image center Object symmetric and parallel to x and y axes Sinogram smooth Object has uniform intensity Computed Tomography CT Key objective Obtain a 8D representation of a volume from its projections How Backproject all projections and sum them all in one image By stacking all images we obtain the 8D volume Backprojection from the Radon Transform A poiulgipr H in action gm HA7 APT V m pr0cc0n Cninplcle pl oi rm 2 rim angle 0 Given point gpj 0k Backprojection copy the value of gpj 6k on the entire line Vp 2 fgkxy gxcos 6k ysin6k k W fewwd Backprojection from the Radon Transform V A poinl gtp 24k in q the pmiuclion C omplclc prowcllulL gm Hk rm 2 fixed unglc 0 Given point gpj 0k Backprojection copy the value of gpj 6k on the entire line Vp 2 f0klt y 930 cos 6k ysin6k k gt Z Laminogram Obtained from Sinogram Backprojection for a specific angle fek 337 y 933 COS 9k ySin6k7 9k Summation over all theta f7y Zf67y 60 Laminogram Obtained from Sinogram Backprojection for a specific angle f6k7y 933 COS 9k ySin6k7 9k Summation over all theta f7y gfgtay 60 Example Laminograms Significant improvements can be obtained by reformulating backprojections Relating 1D Fourier Transform of the projection with 2D Fourier Transform of the image from which the projection was obtained 1D Fourier Transform of the Projection 0a 0 9p 0e j2 quotdp 1D Fourier Transform of the Projection 0a 0 0 9p 0e j2 quotdp by definition Cw fx7y6x cos ysin 7 pe j2quot pdx dy dp 1D Fourier Transform of the Projection 0a 0 9p 0e j2 quotdp by definition Cw fx7y6x cos ysin 7 pe j2quot pdx dy dp foryeij27rwmcoseysin0dx dy 1D Fourier Transform of the Projection 0a 0 9p 0e j2 quotdp by definition Cw fx7y6x cos ysin 7 pe j2quot pdx dy dp foryeij27rwmcoseysin0dx dy Fw cos 0 no 51110 1D Fourier Transform of the Projection Cw oo gpt9e j2m pdp by definition 00 9 fz7 y6z cos 9 y sin 7 26277 de dy dp oo oo fI7 yeij21rwa cos9y sin 9dr dy 700 700 Fw cos 07 w sin 0 Fourier Slice Theorem Fourier Slice Theorem 1 2D Fourier l1 IliSlDl lii F IL v x 11 1D Fourier 1 zusorm 1D FT a slice of 2D FT Reconstruction Using Filtered Backprojections by definition 3379 Fuvej2 w ydu dv Reconstruction Using Filtered Backprojections by definition lmy Fuve quotltwvygtdu dv u wcos v wsin dudv wdwd 27r oo 33 Fw cos0wsin Hej2 050ysmew dw d0 0 0 Reconstruction Using Filtered Backprojections by definition 3379 Fuvej2 w ydu dv u wcos v wsin dudv wdwd 27r oo 33 Fw cos0wsin Hej2 050ysmew dw d0 0 0 by Fourier Slice Theorem 27r 0 Way Gw0e w 05 9ysm 9w dw d0 0 O Reconstruction Using Filtered Backprojections Gw7 0 1800 G w7 0 Reconstruction Using Filtered Backprojections Gw7 0 1800 G w7 0 f7y wGw76ej27rwvcos6ysin6 dw d0 0 0 Reconstruction Using Filtered Backprojections Gw7 0 1800 G w7 0 f7y wGw76ej27rwvcos6ysin6 dw d0 0 0 my 7r mllew70ej2W dw d6 300 cos 6y sin 6 Reconstruction Using Filtered Backprojections Cw 0 1800 G w7 0 fa7y wGw70ej27rwvcos6ysin6 dw d0 0 0 may llew70ej2de d0 0 0 300 cos 6y sin 6 1D filtering Box Ramp Filter a IN C d 6 FIGURE 542 a Frequency domain plot of the filler iwi after band limiling il wilh a bux lillci b Spatial Immin W H I 39 l c lJl39CScnlailOlL Fl th39Li lL bWWI cl Hammng domain Lil mi39m windowngv funcl ion 1 Windnwed lamp liltei formed as he producl of a and c e Spalial l39c lJl39CSclllailOll of he pi39oducl note he decrease in ringing v VSpulinl damn in Frequuncy dam a i n Frcqucncy Llnmui n Algorithm for Filtered Backprojection 1Given projections gpe obtained at each fixed angle a 2 Compute Gwe 1D Fourier Transform of each projection gpe 3 Multiply Gwe by the filter function le modified by Hamming window 4 Compute the inverse of the results from 3 5 Integrate sum over a all results from 4 22 Examples HE J naive backprojection zoom ramp filter windowed ramp filter windowed ramp filter ramp filter 23 ECE 468 Digital Image Processing Lecture 5 Prof Sinisa Todorovic sinisaeecsoregonstateedu 111 Oregnn 5m University Disclaimer The following slides are just excerpts from the textbook You should learn all material presented in chapter 4 in the textbook Filtering in the Frequency Domain Fuv ab FIGURE 429 a SEM image of a damaged integrated circuit h Fourier spectrum of a Original image courtesy of Dr J M Hudak Brockhouse Institute for Materials Research McMaster University Hamilton Ontario Canada 0 u 01 0 Hu v i 1 otherwise f 1Hu vFu7 12 Filtering in the Frequency Domain H Hu v HUI 139 f A l I39t I39 t t t r Wotan WW t 39 750 I 0 0 o c tt Va will e lb 3 abc def FIGURE 43 Top row trequeney domain filters Bottom row corresponding ltered images obtained LLlting ELI 4i7l We used a 035 in c to obtain t the height of the lter itsell is 1 Compare 1 with Fig 42921 Steps in Frequency Domain Filtering a h c d e f g h FIGURE 436 UAn M x N inmgef b Padded image fpol size P X Q C Result of multiplying fp by l l d Spectrum of F c Centered Gaussian lnwpzm lter H of size x Q l Spectrum 01 the product HFp 51 gl the product of 1Hy and the real purl ol39 the IDFT of HFP 11 Final result g obtained by cropping the first M rows and N columnsuf gp Steps in Frequency Domain Filtering 1 Input txy of size MXN 2 Compute padding tpxy of size PxQ where P 2M Q 2N 3 Multiply tpxy1XY to center its DFT 4 Compute DFT of tpxy1XY gt Fuv 5 Use tilter Huv of size PxQ with center at coordinates P2Q2 6 Multiply elementwise Guv HuvFuv 7 Compute the real part of IDFT gpxy 2 real DFTGuv 1XY 8Crop the top left MXN region to get gxy Ideal Lowpass Filter D Hill 2 1 DUI v a b c FIGURE 440 at Perspective plot of an ideal lowpass filter transfer function h Filter displayed as an image c Filter radial cross seetior Du711 g D0 Du7 1 gt Do Example Lowpass Filtering aaaaaaaa u b FIGURE 441 ziTesl paillern of size 688 X 188 pixels and b its Fourier spectrumThe spectrum is dnuble the image size due to pudding but is shown in half size so that it fits in the pageThe superimposed circles have radii equal to 103060 W and 461 with respect to the fullsize spectrum image These radii enclose 871 93 957 97539 and 092 of the padded image power respectively Example Lowpass Filtering Vii I l I l l aaaaaaaa will quotOll a a IIIIIIIII nobII annual 39a39f39 4395quot aaaaaaa aaaaaaaa Results at filtering Ming lLl Fs with cum 1mm d 460 as Shown m Fig 44L b Th 7 393 nsunr 442 in Onginal image nil m u i 22 and mm of the mint iespeclhely frcqnencies Set at mdii Ines 1 power rammed by these mm L Butterworth Lowpass Filter HUI v H1U A DUI v a l 0 FIGURE 444 a Perspective plot of a Butterworth lowpass filter transfer function h Filter displayed as an image c Filter radial cross sections of orders I through 4 Butterworth Lowpass Filter ab FIGURE 443 a Representation in the spatia domain of an ILPF of radius 5 and size 1000 X 1000 Ringing for large n ho zontalline passing through the center of the image Butterworth Lowpass Filter Ringing for large n II it i i l i a b c d FIGURE 446 atd Spatial representation of BLPFS of order 12 5 and 20 and corresponding intensin pro les through the center of the iiiers the size in all cases it will Y HM and the Cutoff frequency is 5 Observe how ringing increases as a function of filter ui dei Example Butterworth Lowpass Filter n I I I u a 39 I ll aaaaaaaa quotOII III a a IIIIll l nnllc IJ aaaaaaa fo 39a39f39 ll aaaaaaaa aaaaaaaa 2 b c d e r new 445 a Oliginai image Him Resule of filming using BLPFs of order 2 with cmofffmquencies m me radii shown in g 441 Compm 8 mm Fig 442 Lowpass Gaussian Filter HUI v H1 l39 11 1n 11 L607 D0 30 7 D1 JIJ lt f DU 100 My 7 W V 39 n 4 ll DUI L a be FIGURE 447 u Perspective plot of 21 GLPF transfer function h Filler displayed an an image 4 Filler radial cross sections for various values 0139 Du Hm eXP 203 Example Lowpass Gaussian Filter l aaaaaaaa 0 J a H llllll I I IJ Ll a aaaaaa a mm aaaaaaaa ab rd m a aaaaaaaa e f FIGURE 448 a Olignml mugs In m Resulzs 0139 Menu 5mg CLI Fs wnh cuto flequencics ax lhe mdii shown in Fig 441 Compare win Figs 442 and 445 Example Lowpass Gaussian Filter Historicaiiy certain computer programs were w tten using onw two digits rather than fOU39 to de ne the appncame year Accommglv the companyis saftware way recognize a date using quot00 as 1900 rathEr than the year Ea Histormaw mite 391 computw pv39ogrmns yawr xursttew using rmay Lwrquot 119m rather Man four EC de ne the applniab x veal irfllr vgiw the Compar y r softwazzy may recoqnzze a fate usmq 3900quot as 1900 rather ban the 20m as t Example Lowpass Gaussian Filter After this goes sharpening for Hollywood a b c FIGURE 450 3 Original image 784 X 732 pixels 1 Result of filtering using a GLPF with Do 100 1 Result of filtering using a GLPF with Do 80 Note the reduction in fine skin lines in the magni ed sections in b and c 17 Highpass Filtering in the Frequency Domain HHpuv1 HLpuv Highpass Filters 11mm Hu17 a 1 ltl i Hquot u 4 mu 1 ll mu 0 u11 i v 10 D 1 DUI v n quot120 llu v i 1 10 1 v a b 6 Din 1 d e f g h i quot FIGURE 452 Top row Perspective plot image representation and cross section of a typical ideal highpnss filter Middle and bottom rowsThe same sequence for typical Butterworlli and Gaussian ltiglipass filters Ideal Butterworth Gaussian Example Highpass Filtering 20 Laplacian Filter Homework 4 Huv 47r2u2 112 21va FIGURE 457 1 Thumb print b Result of highpass filtering a 6 Result of thresholding h Original image courtesy of the US National Institute of Standards and Technology 21 Image Sharpening in the Frequency Domain 9x7y aw i cVz w Guv Fuv HuvFuv c 1 needs scaling gxy F11 47ru2 l 112Fu 11 Image Sharpening in the Frequency Domain Unsharp Masking 9367 y f y cw y 17 w my y 71HLpu7 vFu7 22 g7y 7 11 01 HLPU7UFU7U gm Fl l cHHPW vFU7 11 high frequency emphasis filter Unsharp Masking in the Frequency Domain ab Cd FIGURE 459 a A chest Xray image b Result of highpass ltering with a Gaussian lter c Result of highfrequencyemphasis filtering using the same filter Ll Result of performing histogram equalization on c Original image courtesy of Dr Thomas R Gest Division of Anatomical Sciences University of Michigan Medical School

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