Digital Image Processing

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1 Digital Image Processing Part 3: Fourier Transform and Filtering in the Frequency Domain AASS Learning Systems Lab, Dep. Teknik Room T109 (Fr, 11-1 o'clock) Course Book Chapter

2 Fourier transform properties, revisited Conjugate Symmetry y Spectrum symmetric about the origin F( u, v) F but not * ( u, v) F( u, v) F * ( u, v)

3 Contents Nyquist-Shannon Sampling Theorem

4 5 Nyquist-Shannon Sampling Theorem Aliasing / Undersampling, Moiré Pattern decrease resolution

5 5 Nyquist-Shannon Sampling Theorem How to avoid Aliasing Problems? y y sampling continuous function (irradiance in the camera) discrete grid number of samples relative to the function seems important a signal sampled too slowly is misrepresented by the samples from "Computer Vision A Modern Approach", Forsyth and Ponce, Prentice Hall, 00

6 5 Nyquist-Shannon Sampling Theorem Sampling a Signal in 1D f() Reconstruction of the Original Continuous Signal y which sample rate? y how to derive the continuous signal from the samples? y how to model the sampling process?

7 5 Nyquist-Shannon Sampling Theorem Sampling a Signal in 1D f() How to Model the Sampling Process? y continuous model of a sampled signal needed y sum of delta functions (D: "bed-of-nails function") sampling process = multiplication with a sampling function f III ()

8 5 Nyquist-Shannon Sampling Theorem Considering Band-Limited Signals f() F F(u) u -w w y signal band-width: band/range of non-ero frequencies y a band-limited signal is constrained in terms of how fast it can change

9 5 Nyquist-Shannon Sampling Theorem Fourier Transform of a Sampled Signal f()f III () y sampling = multiplication with the sampling function in the spatial domain

10 5 Nyquist-Shannon Sampling Theorem Fourier Transform of a Sampled Signal f()f III () F F(u)F III (u) Fourier transform of a "Dirac comb" is again a Dirac comb ( Poisson summation) y sampling = multiplication with the sampling function in the spatial domain y equals convolution in the frequency domain

11 5 Nyquist-Shannon Sampling Theorem Fourier Transform of a Sampled Signal f()f III () F F(u)F III (u) w y sampling = multiplication with the sampling function in the spatial domain y equals convolution in the frequency domain u

12 5 Nyquist-Shannon Sampling Theorem Fourier Transform of a Sampled Signal f()f III () F F(u)F III (u) w y non-overlapping support of the "shifted Fourier Transforms" we can reconstruct the signal from the sampled versions u

13 5 Nyquist-Shannon Sampling Theorem Reconstruction of the Signal from "Computer Vision A Modern Approach", Forsyth and Ponce, Prentice Hall, 00

14 5 Nyquist-Shannon Sampling Theorem Reconstruction of the Signal from "Computer Vision A Modern Approach", Forsyth and Ponce, Prentice Hall, 00

15 5 Nyquist-Shannon Sampling Theorem Reconstruction of the Signal Inverse Fourier Transform Cut out by multiplication with bo filter from "Computer Vision A Modern Approach", Forsyth and Ponce, Prentice Hall, 00

16 5 Nyquist-Shannon Sampling Theorem Reconstruction of the Signal Inverse Fourier Transform Cut out by multiplication with bo filter from "Computer Vision A Modern Approach", Forsyth and Ponce, Prentice Hall, 00

17 5 Nyquist-Shannon Interpolation Formula sinc Eact Recovering From Samples y convolution of the samples with sinc sin( ) 1 f scaled so that the ero-crossings occur at the sampling instants f n ( ) n sinc, n : 0 : 0 sinc() F 1 w H(u) u w

18 5 Nyquist-Shannon Sampling Theorem Reconstruction of the Signal y but if support regions do overlap? we can't reconstruct the signal Fourier transform in the regions that overlap can't be determined

19 5 Nyquist-Shannon Sampling Theorem Fourier Transform of a "Dirac comb" f III () F F III (u) Δ w 1/Δ u y reciprocal behaviour of Δ and Δu

20 5 Nyquist-Shannon Sampling Theorem Sampling Theorem y there should be no overlap between the repetitions of the FT of the signal F III (u) y the sampling interval should be more than double of the highest frequency (1/w) present in the signal w 1/Δ u w 1 1 w

21 5 Nyquist-Shannon Sampling Theorem f()f III () F F(u)F III (u) w u f III () F F III (u) Δ 1/Δ u

22 5 Nyquist-Shannon Sampling Theorem Aliasing / Undersampling f() F(u) u f III () F(u)F III (u) u Δ aliasing

23 5 Nyquist-Shannon Sampling Theorem Aliasing / Undersampling, Moiré Pattern

24 5 Nyquist-Shannon Sampling Theorem Avoiding Moiré Patterns? y anti-alias filter: use a low-pass filter before down-sampling

25 5 Nyquist-Shannon Sampling Theorem Avoiding Moiré Patterns y anti-alias filter: use a low-pass filter before down-sampling eample: no low-pass filter used Fourier Transform (scaled + 'tile'd) from "Computer Vision A Modern Approach", Forsyth and Ponce, Prentice Hall, 00

5 Nyquist-Shannon Sampling Theorem Avoiding Moiré Patterns y anti-alias filter: use a low-pass filter before down-sampling eample: mild Gaussian

26 5 Nyquist-Shannon Sampling Theorem Avoiding Moiré Patterns y anti-alias filter: use a low-pass filter before down-sampling eample: mild Gaussian low-pass filter used ( = 1 piel) Fourier Transform (scaled + 'tile'd) from "Computer Vision A Modern Approach", Forsyth and Ponce, Prentice Hall, 00

27 5 Nyquist-Shannon Sampling Theorem Avoiding Moiré Patterns y anti-alias filter: use a low-pass filter before down-sampling eample: more aggressive Gaussian low-pass filter used ( = piels) Fourier Transform (scaled + 'tile'd) from "Computer Vision A Modern Approach", Forsyth and Ponce, Prentice Hall, 00

28 Contents Fast Fourier Transform

29 6 Discrete Fourier Transform (DFT) Direct Calculation of the DFT F( u) 1 M M 1 0 f ( ) e ju/ M y for each Fourier coefficient M comple multiplications and M 1 comple additions y altogether: M multiplications and (M 1) additions comple mult. = 4 real multiplications + real additions comple addition = real additions 4 M real multiplications and 4 M M + real additions ~ M

30 6 Fast Fourier Transform (FFT) 1D-DFT requires ~M operations F( u) 1 M M 1 0 f ( ) e ju/ M FFT requires only ~ M log M operations y divide-and-conquer algorithm y avoid calculation of the same products multiple times

31 6 Fast Fourier Transform (FFT) FFT requires only ~ M log M operations y the FT can be divided into successive two-point transforms, which reduces the number of operations to M log M

32 6 Fast Fourier Transform (FFT) FFT requires only ~ M log M operations y the FT can be divided into successive two-point transforms, which reduces the number of operations to M log M y eample M = 51 DFT: ~M = 6144 operations FFT: ~ M log M = 4608 operations computational advantage: ~M / M log M = 57

33 Contents 1. Fourier Transform, Discrete Fourier Transform Comple Numbers. Filtering in the Frequency Domain y Convolution, Convolution Theorem y Relation Between Spatial and Frequency Filters 3. Recovering Intrinsic Images, Homomorphic Filtering 4. Properties of the Fourier Transform y Correlation 5. Nyquist-Shannon Sampling Theorem 6. Fast Fourier Transform

34 Digital Image Processing Part 4: Image Restoration AASS Learning Systems Lab, Dep. Teknik Room T109 Course Book Chapter

35 Contents 1. Image Degradation and Restoration. Noise Models 3. Noise Reduction 4. Image Restoration Linear, Position Invariant (LPI) Degradation Processes Inverse Filtering, Deconvolution Wiener Filtering

36 Contents Image Degradation and Restoration

37 1 Image Degradation and Restoration Restore a Degraded Image y restoration using knowledge of the degradation process y modelling the degradation process ( inverse methods) noise, bad focusing, non-linear sensors, motion, y obtain an estimate of the original image Comparison with Image Enhancement y image enhancement: make the image "better" in some way (heuristic) y image restoration: often involves formulation of a performance measure more objective methods

38 1 Image Degradation and Restoration Model of the Image Degradation Process f(,y) H deg (,y) + g(,y) n(,y) y f(,y): y H deg (,y): y n(,y): y g(,y): "ideal" input image (original image) degradation function (imaging system, ) additive noise degraded image: g(,y) = H deg [f(,y)] + n(,y)

39 1 Image Degradation and Restoration Model of the Image Restoration Process ^ f(,y) restoration filter g(,y) y g(,y): ^ y f(,y): degraded image: g(,y) = H deg [f(,y)] + n(,y) estimate of the original image (restored image)! H and H deg are not clearly distinguished in GW

40 Contents Noise Models

41 Noise Models Model of the Image Degradation Process f(,y) H deg (,y) + g(,y) n(,y)

42 Noise Models Model of the Image Degradation Process H deg = I f(,y) + g(,y) n(,y) Noise Models y noise is assumed to be uncorrelated with the image process independent of spatial coordinates y and independent of piel brightness no correlation between piel values and noise component

43 Noise Models Gaussian Noise (Normal Noise) p 1 ( ) ( ) e p() mean and variance

44 Noise Models ) ( 1 ) ( e p Gaussian Noise (Normal Noise) = 0, = 1 ) ( 1 ) ( e p

45 Noise Models = 0, = 1 Gaussian Noise (Normal Noise) p 1 ( ) ( ) e y parameters: (average value) and (standard deviation) sufficient to describe the distribution y central limit theorem many small effects add independently into an observed variable y used to model sensor noise (low light levels)

46 Gaussian Noise Noise Models f = imread('bubbles.tif'); imshow(f,[]);[m N] = sie(f); ) ( 1 ) ( e p ) ( 1 ) ( e p

47 Gaussian Noise Noise Models f = imread('bubbles.tif'); imshow(f,[]);[m N] = sie(f); ) ( 1 ) ( e p ) ( 1 ) ( e p

48 Noise Models Gaussian Noise p 1 ( ) ( ) ( ) e f = imread('bubbles.tif'); imshow(f,[]);[m N] = sie(f); fgn = imnoise(f, 'gaussian', 0, 0.01); figure; imshow(fgn,[]);

49 Noise Models Uniform p( ) b 1 0 a : a b : otherwise 1 b a p() mean and variance a b ( b 1 a ) a b

50 Noise Models a = 0, b = 1 Uniform p( ) b 1 0 a : a b : otherwise

51 Noise Models a = 0, b = 1 Uniform p( ) b 1 0 a : a b : otherwise y parameters: a, b y generated by a "standard" random number generator

52 Noise Models Uniform p( ) b 1 0 a : a b : otherwise y parameters: a, b y generated by a "standard" random number generator

53 Noise Models Salt and Pepper (Impulse) y bipolar impulse noise Pa : a p( ) Pb : b 0 : otherwise p() P b P a a b

54 Noise Models P a = 0.5, a = 0; P b = 0.5, b = 1 Salt and Pepper (Impulse) y bipolar impulse noise Pa : a p( ) Pb : b 0 : otherwise

55 Noise Models P a = 0.5, a = 0; P b = 0.5, b = 1 Salt and Pepper (Impulse) y bipolar impulse noise Pa : a p( ) Pb : b 0 : otherwise y parameters: a, b y modelling of large impulse corruptions (often a, b are saturated values: 0 and 55)

56 Noise Models Salt and Pepper (Impulse) y bipolar impulse noise Pa : a p( ) Pb : b 0 : otherwise f = imread('bubbles.tif'); imshow(f,[]);[m N] = sie(f);

57 Noise Models Salt and Pepper (Impulse) y bipolar impulse noise Pa : a p( ) Pb : b 0 : otherwise f = imread('bubbles.tif'); imshow(f,[]);[m N] = sie(f);

58 Noise Models Salt and Pepper (Impulse) y bipolar impulse noise Pa : a p( ) Pb : b 0 : otherwise f = imread('bubbles.tif'); imshow(f,[]);[m N] = sie(f); fsap = imnoise(f, 'salt & pepper', 0.05); figure; imshow(fsap,[]);

59 Rayleigh mean and variance Noise Models a a e a b p b a : 0 : ) ( ) ( p() a b a 4 a b / 4 ) (4 b

60 Noise Models a = 0, b = 1 Rayleigh p( ) b a 0 e ( a) b : : a a

61 Noise Models a = 0, b = 1 Rayleigh p( ) b a 0 e ( a) b : : a a y parameters: a, b y distribution of the length of a D vector with normally distributed components y characterising noise in range data y approimating skewed histograms also: spectrum of random comple numbers

62 Noise Models Rayleigh p( ) b a 0 e ( a) b : : a a y parameters: a, b y distribution of the length of a D vector distributed components y characterising noise in range data y approimating skewed histograms

63 Noise Models Eponential p( ) a e 0 a : : 0 0 p() a mean and variance 1 1 a a

64 Noise Models a = 1 Eponential p( ) a e 0 a : : 0 0

65 Noise Models a = 1 Eponential p( ) a e 0 a : : 0 0 y parameter: a y special case of Erlang (Gamma) noise (b = 1) y often used to model time intervals (>0!) between randomly occurring events (radioactive decay)

66 Noise Models Eponential p( ) a e 0 a : : 0 0 y parameter: a y special case of Erlang (Gamma) noise y often used to model time intervals (>0!) between randomly occurring events

67 Erlang (Gamma) Noise mean and variance Noise Models p() K a b 1 1) ( 1 1)! ( 1) ( b b e b b a K 0 : 0 0 : 1)! ( ) ( 1 e b a p a b b a b a b

68 Noise Models a =, b = 5 Erlang (Gamma) Noise p( ) b a ( b b1 e 1)! 0 a : : 0 0

69 Noise Models a =, b = 5 Erlang (Gamma) Noise p( ) b a ( b b1 e 1)! 0 a : : 0 0 y sum of b eponentially distributed variables y eponential distribution with parameter a

70 Contents Noise Reduction

71 3 Noise Reduction Additive Noise Additive Noise y assumed to be uncorrelated with the image y assumed to be the only degradation present in an image Estimation of Noise Shape and Noise Parameters y from images of a "flat" gray surface y from patches in the image where the gray value is approimately constant (as featureless as possible)

72 3 Noise Reduction Additive Noise Estimation of Noise Shape and Noise Parameters y from images of a "flat" gray surface f = imread('bubbles.tif'); selpol = roipoly(f);

73 3 Noise Reduction Additive Noise Estimation of Noise Shape and Noise Parameters y from images of a "flat" gray surface f = imread('bubbles.tif'); selpol = roipoly(f); figure; imhist(f(selpol));

74 3 Noise Reduction Additive Noise Estimation of Noise Shape and Noise Parameters y from images of a "flat" gray surface f = imread('bubbles.tif'); selpol = roipoly(f); figure; imhist(f(selpol));

75 3 Noise Reduction Additive Noise Restoration in Case of Additive Noise Only y spatial filtering: mean arithmetic, geometric, harmonic (Pythagorean means) cannot compensate for salt & pepper noise contra-harmonic (Lehmer) mean can compensate for salt or pepper noise order-statistics well suited for salt & pepper noise median, median with repeated passes,... midpoint alpha-trimmed filter

76 Digital Image Processing Part 4: Image Restoration AASS Learning Systems Lab, Dep. Teknik Room T109 Course Book Chapter

77 Contents 1. Image Degradation and Restoration. Noise Models 3. Noise Reduction 4. Image Restoration Linear, Position Invariant (LPI) Degradation Processes Inverse Filtering, Deconvolution Wiener Filtering

78 3 Noise Reduction Adaptive Mean Filter

79 3 Noise Reduction Adaptive Mean Filter fˆ, y g(, y) n g(, y) m L L

80 3 Noise Reduction Adaptive Mean Filter fˆ, y g(, y) n g(, y) m L L local mean

81 3 Noise Reduction Adaptive Mean Filter fˆ, y g(, y) n g(, y) m L L local variance local mean

82 3 Noise Reduction Adaptive Mean Filter needs to be estimated fˆ, y g(, y) n g(, y) m L L local variance local mean

83 Noise Reduction Adaptive Mean Filter y behaviour based on the local variance (~ average contrast) L L n m y g y g y f ), ( ), (, ˆ 3

84 Noise Reduction Adaptive Mean Filter y behaviour based on the local variance (~ average contrast) y if the noise has ero variance ( n = 0) then f(,y)=g(,y) L L n m y g y g y f ), ( ), (, ˆ ^ 3

85 3 Noise Reduction Adaptive Mean Filter fˆ, y g(, y) n g(, y) m y behaviour based on the local variance (~ average contrast) ^ y if the noise has ero variance ( n = 0) then f(,y)=g(,y) y if the local variance L is large compared to the noise variance n then ^ f(,y)g(,y) preserve edges L L

86 3 Noise Reduction Adaptive Mean Filter fˆ, y g(, y) n g(, y) m y behaviour based on the local variance (~ average contrast) ^ y if the noise has ero variance ( n = 0) then f(,y)=g(,y) y if the local variance L is large compared to the noise variance n then ^ f(,y)g(,y) preserve edges y if L n then f(,y)m L (mean over the neighbourhood) average out noise L L

87 3 Noise Reduction Adaptive Median Filter y subimage of varying sie y tries to recognie S&P noise piels (as min/ma values) Step A y epand subimage until median is not S&P noise median min AND median ma, continue with step B y or subimage has maimal sie piel = median Step B y piel = y if y min AND y ma y piel = median otherwise

88 3 Noise Reduction Periodic Noise Periodic Degradation Function H(,y) y additive, periodic (approimately sinusoidal) noise peaks in the Fourier transform F

89 3 Noise Reduction Periodic Noise Periodic Degradation Function H(,y) y additive, periodic (approimately sinusoidal) noise peaks in the Fourier transform F

90 3 Noise Reduction Periodic Noise Periodic Degradation Function H(,y) y applying a band-reject filter (sets the peaks to ero) y removes peaks in the Fourier transform in pairs Butterworth band-reject filter

91 3 Noise Reduction Periodic Noise Notch Filtering y periodic noise applying a notch filter y must appear in symmetric pairs about the origin ideal notch reject filter Gaussian notch reject filter not shown: Butterworth notch reject filter

92 Noise Reduction Periodic Noise Optimal Notch Filtering y estimate principal interference frequencies "by eye" using a notch pass filter H np (u,v) y subtract a weighted portion of the interference pattern y optimal value of the weight w? ), ( ), ( ), ( ˆ v G u v u H v u N np ), ) ˆ(, ( ), ( ), ( ˆ y n y w y g y f )}, ( ˆ { ), ( ˆ 1 v u N y n F 3

93 Noise Reduction Periodic Noise Optimal Notch Filtering Optimality Criterion y minimise variance of the estimate f(,y) over a (a+1)(b+1) neighbourhood y assume that w(,y) is approimately constant over the (a+1)(b+1) neighbourhood w( + s, y + t) w(,y) y w y w y, 0,,! ^ ), ˆ( ), ( ˆ ), ˆ( ), ( ), ) ˆ(, (, y n y n y n y g y n y g y w 3

94 Noise Reduction Periodic Noise Optimal Notch Filtering y tries to distinguish periodic and random noise y periodic large weight if degraded image is highly correlated with the noise estimate: cov[g(,y),n(,y)] is large y random small weight if the variance of the noise estimate over the neighbourhood is large (irregular noise) ), ˆ( ), ( ˆ ), ˆ( ), ( ), ) ˆ(, (, y n y n y n y g y n y g y w ), ) ˆ(, ( ), ( ), ( ˆ y n y w y g y f 3

Computer Vision & Digital Image Processing

Computer Vision & Digital Image Processing Image Restoration and Reconstruction I Dr. D. J. Jackson Lecture 11-1 Image restoration Restoration is an objective process that attempts to recover an image