Midterm Summary Fall Yao Wang Polytechnic University, Brooklyn, NY 11201

Transcription

1 Midterm Summary Fa 3 Yao Wang Poytecnic University Brooyn NY

2 Topics Covered Image representation Coor representation Quantization Contrast enancement Spatia Fitering: noise remova sarpening edge detection Frequency domain representations FT DTFT DFT unitary transorms Impementation o inear itering using DTFT and DFT Yao Wang NYU-Poy EL53: Midterm Summary

3 Coor Image Representation Ligt is te visibe band o te EM wave Coor ue o a igt depends on te waveengt o te EM wave Iuminating igt vs. reecting igt Coor attributes: Brigtness uminance ue depends on te waveengt Saturation purity Any coor can be reproduced by mixing tree primary coors Iuminating and reecting igt sources oow dierent mixing rues Coor can be represented in many dierent coordinates or modes wit 3 components Yao Wang NYU-Poy EL53: Midterm Summary 3

4 Tri-cromatic Coor Mixing Tri-cromatic coor mixing teory Any coor can be obtained by mixing tree primary coors wit a rigt proportion Primary coors or iuminating sources: Red Green Bue RGB Coor monitor wors by exciting red green bue pospors using separate eectronic guns oows additive rue: R+G+BWite Primary coors or reecting sources aso nown as secondary coors: Cyan Magenta Yeow CMY Coor printer wors by using cyan magenta yeow and bac CMYK dyes oows subtractive rue: R+G+BBac Yao Wang NYU-Poy EL53: Midterm Summary 4

5 Coor Modes Speciy tree primary or secondary coors Red Green Bue. Cyan Magenta Yeow. Speciy te uminance and crominance SB or SI ue saturation and brigtness or intensity YIQ used in NTSC coor TV YCbCr used in digita coor TV Ampitude speciication: 8 bits per coor component or 4 bits per pixe Tota o 6 miion coors A x true RGB coor requires 3 MB memory Yao Wang NYU-Poy EL53: Midterm Summary 5

6 Coor Image Processing Appy contrast enancement or itering inear or non-inear to eac primary coor component independenty using te tecniques or monocrome images May cange te coor ue o te origina image Convert te tri-stimuus representation into a uminance / crominance representation and modiy te uminance component ony. For certain appications dierent operations may be appied to dierent coor components eiter in RGB domain or in YCC domain to obtain specia desired eects Yao Wang NYU-Poy EL53: Midterm Summary 6

7 Pseudo Coor Processing Given a gray-scae image we may dispay it in pseudo coors to better revea certain eatures Coor depends on image pixe vaue Oten used in medica image dispay weater maps vegetation maps Decompose an image into dierent requency bands and represent eac band wit dierent coors Yao Wang NYU-Poy EL53: Midterm Summary 7

8 Quantization Q Quantizer Q r 8 Decision Leves {t L+} Reconstruction Leves {r L} I t t [ + Ten Q r L eves need x R og bits returns te smaest integer tat is bigger tan or equa to x L r 7 r 6 r 5 r 4 t t t 3 t 4 t 5 t 7 t 8 r 3 r r t 6 Quantizer error r r r r 3 r 4 r 5 r 6 r 7 r 8 r t t t 3 t 4 t 5 t 6 t 7 t 8 Yao Wang NYU-Poy EL53: Midterm Summary 8

9 Uniorm Quantization Equa distances between adjacent decision eves and between adjacent reconstruction eves t t - r r - q Parameters o Uniorm Quantization L: eves L R B: dynamic range B max min q: quantization interva step size q B/L B -R Yao Wang NYU-Poy EL53: Midterm Summary 9

10 Uniorm Quantization: Functiona Representation stepsize q max - min /L Q r 7 max -q/ r 6 q Q + q min q + min r 5 r 4 r 3 x returns te biggest integer tat is smaer tan or equa to x r r r min +q/ t t t t 3 t 4 t 5 t 6 t 7 t 8 min max min I is caed te reconstruction eveindex wic indicates wic reconstruction eveis used or. q Yao Wang NYU-Poy EL53: Midterm Summary

11 Yao Wang NYU-Poy EL53: coor and quantization MSE o a Quantizer Mean square error MSE o a quantizer or a continuous vaued signa Were p is te probabiity density unction o MSE or a speciic image + L t t L t t q d p r d p Q Q E MSE } { σ N i M j q j i Q j i MN MSE σ

12 MSE o a Uniorm Quantizer or A Uniorm Source p / max σ min / B oterwise ma x η d min B min B max Uniorm quantization into L eves: q B / L B / L Error in eac bin iste same and is uniormy distributed in -q/ q/ σ q SNR q / σ R og og R og 6R db σ q B B R R q de L e σ q / q Every additiona bit increases te SNR by 6dB! Yao Wang NYU-Poy EL53: coor and quantization

13 Exampe: Nonuniorm Source Te pd o a signa is sown beow we want to quantize it to eves. Determine te partition and reconstruction eves tat minimizes te quantization error in terms o MSE. Aso compute te MSE and SNR. Go troug in cass. p / Exp Exp- Yao Wang NYU-Poy EL53: coor and quantization 3

14 MMSE Quantizer For any pd te quantizer tat minimizes MSE is nown as Minima MSE MMSE quantizer. Given simpe pd soud now ow to determine te decision and reconstruction eves to minimize MSE Given a quantizer and signa pd soud now ow to cacuate MSE Yao Wang NYU-Poy EL53: coor and quantization 4

15 Contrast Enancement An requenty used important operation ow to te te contrast o an image rom its istogram? Given a istogram can setc a transormation tat wi enance te contrast istogram equaization istogram speciication Yao Wang NYU-Poy EL53: Midterm Summary 5

16 istogram vs. Contrast p p p a Too dar b Too brigt c We baanced Yao Wang NYU-Poy EL53: Midterm Summary 6

17 Enancement o Too-Dar Images g g max New istogram max g Origina istogram max Transormation unction: og unction: gc og + Or Power aw: g c ^r <r< g max g Yao Wang NYU-Poy EL53: Midterm Summary 7

18 Enancement o Too-Brigt Images g g max New istogram max g Origina istogram max Transormation unction: Power aw: gc ^r r> g max g Yao Wang NYU-Poy EL53: Midterm Summary 8

19 Enancement o Images Centered near te Midde Range g g max New istogram max g Origina istogram max Transormation unction g max g Yao Wang NYU-Poy EL53: Midterm Summary 9

20 istogram Equaization Transorms an image wit an arbitrary istogram to one wit a at istogram Suppose as PDF p F Transorm unction continuous version g pf t dt g is uniormy distributed in istogram Equaization Yao Wang NYU-Poy EL53: Midterm Summary

21 istogram Speciication Wat i te desired istogram is not at? p F istogram Equaization g s istogram Equaization z p Z z p G g g g P S s s s z Speciied istogram p F d z p Z z dz z s g s g Yao Wang NYU-Poy EL53: Midterm Summary

22 Image Fitering Appications: Noise remova image smooting ow-pass itering Edge enancement deburring ig-empasis Edge detection ig-pass itering Contrast enancement is accompised by point operation i.e. eac pixe vaue is canged based on its origina vaue not its neigboring pixes but te transormation unction depends on te overa istogram o te image Image itering reers to canging te coor vaue o one pixe based on te coor vaues o tis pixe and its neigbors Yao Wang NYU-Poy EL53: Midterm Summary

23 Tree Ways o Impementing Linear Fitering Spatia domain Weigted average o adjacent pixes inear convoution Weigts or te iter depends on te desired itering eect Frequency domain FT DTFT Design spatia iter based on desired requency response Low pass ig pass ig empasis Convoution teorem: * ó F Frequency domain DFT Perorm itering in DFT domain Convoution teorem: circuation ó F Reation between circuation convoution and inear convoution Fiter mass in te DFT domain must be designed propery so tat te corresponding iter in te spatia domain is rea Te iter mas soud enjoy te symmetry property o rea signas Yao Wang NYU-Poy EL53: Midterm Summary 3

24 Yao Wang NYU-Poy EL53: Midterm Summary 4 Linear Convoution o Continuous Signas D convoution Equaities D convoution α α α α α α d x d x x x x x x x x x x δ δ β α β α β α β α β α β α d d y x d d y x y x y x

25 Yao Wang NYU-Poy EL53: Midterm Summary 5 Linear Convoution o Discrete Signas D convoution D convoution Separabe itering Row irst ten coumns m m m n m m m n n n * n m n m n m n m *

26 Interpretation as weigted average o neigboring sampes ow soud we design te weigts? Smooting: sum coe > ig empasis: sum some coe < Edge detection ig pass: sum Yao Wang NYU-Poy EL53: Midterm Summary 6

27 Ex: Smooting by Averaging Repace eac pixe by te average o pixes in a square window surrounding tis pixe g m n m n + m n + m + n + m n + m n + m n + m + n + m n + m + n + Trade-o between noise remova and detai preserving: Larger window -> can remove noise more eectivey but aso bur te detais/edges Yao Wang NYU-Poy EL53: Midterm Summary 7

28 Yao Wang NYU-Poy EL53: Midterm Summary 8 Directiona Edge Detector ig pass in one direction and ow pass in te ortogona direction Prewitt edge detector Sobe edge detector [ ] [ ] 3 3 ; 3 3 y x [ ] [ ] 4 4 ; 4 4 y x

29 Specia Considerations in Impementation Boundary treatment I is MxN is KxL convoved image g is M+K-xN +L- Instead o expanding te image size we modiy itering operations at te boundary Symmetric expansion Leave te boundary pixes uncanged Renormaization Fitered vaues may not be integer and may ave negative vaues may ave a smaer or arger dynamic range tan origina To save resuting image in an 8-bit unsigned car ormat we normaize a vaues to -55 g g-g_min/g_max-g_min * 55 Yao Wang NYU-Poy EL53: Midterm Summary 9

30 Fourier Transorm For Discrete Time Sequence DTFT D Forward Transorm Inverse Transorm F u n / n e jπun jπun n F u e du / D Forward Transorm F u v m n m n e jπ mu+ nv Inverse Transorm / / jπ mu+ nv m n F u v e dudv / / Separabe impementation Transorm eac row irst wit D DTFT ten eac coumn Yao Wang NYU-Poy EL53: Midterm Summary 3

31 Design Fiters Based on Desired Frequency Response Convoution teorem x y* x y F u v u v Fiter xy designed based on desired requency response uv Sarp transitions in requency domain ó very ong iters in spatia domain Appy a window unction to smoot te transition band ó sorter iters Giver a spatia iter we can use DTFT to better understand its itering eect requency response Separabe iters Design orizonta and vertica iters separatey For images we usuay use very sort iters Yao Wang NYU-Poy EL53: Midterm Summary 3

32 Discrete Fourier Transorm DFT: I DTFT or Finite Duration Signas te signa is ony deined or n Fourier transorm becomes: F' N n nexp... N jπn : Samping F' at Forward transorm DFT : N F F' N N n Inversetransorm IDFT: n N N /N...N-and rescaing yieds : F exp jπ N nexp n jπ N n n... N... N Yao Wang NYU-Poy EL53: Midterm Summary 3

33 Circuar Convoution Teorem Circuar convoution n n N n N n- N n n 3 4 n 3 4 n n n- Convoution teorem n n F n+ n+4 n+ n+3 Yao Wang NYU-Poy EL53: Midterm Summary 33

34 D Discrete Fourier Transorm Deinition Assuming m n m M- n N- is a inite engt D sequence F m n MN MN M N m n M N m n e F e Comparing to DTFT F u v m n m n e m n jπ + M N m n jπ + M N jπ mu+ nv... M m... M u M n v / / jπ mu+ nv m n F u v e dudv / / N... N... N ;. Yao Wang NYU-Poy EL53: Midterm Summary 34

35 Periodicity Property o D DFT Periodicity F F < or M < or N. -M -M/ M/ M -N A C A C B D B D M A C A C N B D N- M- B D M-N- -N/ N/ N Low Frequency ig Frequency Yao Wang NYU-Poy EL53: Midterm Summary 35

36 Cacuate Linear Convoution Using DFT D case n is engt N n is engt N gn n* is engt N N +N -. To use DFT need to extend n and n to engt N by zero padding. n * n Convoution gn DFT DFT DFT F x Mutipication G Yao Wang NYU-Poy EL53: Midterm Summary 36

37 Wat is a Linear Transorm Represent an image or an image boc as te inear combination o some basis images and speciy te inear coeicients. + t t t 3 t 4 Yao Wang NYU-Poy EL53:Midtemr Review 37

38 One Dimensiona Linear Transorm Let C N represent te N dimensiona compex space. Let N- represent N ineary independent vectors in C N. For any vector є C N N were t B [ Bt... N ] t t t. t N t B A and t orm a transorm pair Yao Wang NYU-Poy EL53: Midterm Review 38

39 Yao Wang NYU-Poy EL53: Midterm Review 39 Ortonorma Basis Vectors OBV { N-} are OBV i > < δ N N t t t > < > >< < A B t N. I BB B B B B or B is unitary

40 Yao Wang NYU-Poy EL53: Midterm Review 4 Deinition o Unitary Transorm Basis vectors are ortonorma Forward transorm Inverse transorm A B t > < N N n n n t * [ ] t A Bt t N N N t n t n

41 Two Dimensiona Unitary Transorm { } is an ortonorma set o basis images Forward transorm Inverse transorm Yao Wang NYU-Poy EL53: Midterm Review 4 > < * M m N n n m F m n T F M N M N T or n m T n m F F

42 Yao Wang NYU-Poy 4 Exampe o D Unitary Transorm / / / / / / / / / / / / / / / / T T T T F EL53: Midterm Review

43 Separabe Unitary Transorm Let M- represent a set o ortonorma basis vectors in C M Let g N- represent anoter set o ortonorma basis vectors in C N Let g T or mn mg n. Ten wi orm an ortonorma basis set in C MxN. Transorm can be perormed separatey irst row wise ten coumn wise Yao Wang NYU-Poy EL53: Midterm Review 43

44 Yao Wang NYU-Poy EL53: Midterm Review 44 Exampe o Separabe Unitary Transorm Exampe D DFT / / / / / / / / / / / / / / / /. / / / / T T T T N n j M m j N n M m j e N n g e M m e MN m n π π π +

45 Wy Using Transorm? Wen te transorm basis is cosen propery Many coeicients ave sma vaues and can be quantized to w/o causing noticeabe artiacts Te coeicients are uncorreated and ence can be coded independenty w/o osing eiciency. Yao Wang NYU-Poy EL53: Midterm Review 45

46 Midterm Exam Logistics Scedued time: /8/3 :-:5 R75 Cosed-boo seet o notes aowed doube sided OK Specia Oice our Wed /3 4-6 PM MTC 9.7 Tur /4 4-6PM MTC 9.7 Yao Wang NYU-Poy EL53: Midterm Summary 46