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ECE 484 Digital Image Processing Lec 06 - Fourier Transform & Image Sampling

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ECE 484 Digital Image Processing Lec 06 - Fourier Transform & Image Sampling Zhu Li Dept of CSEE, UMKC Email: [email protected], Ph: x 2346. http://l.web.umkc.edu/lizhu Office Hour: Tu/Th 2:30-4pm@FH560E slides created with WPS Office Linux and EqualX equation editor Z. Li, ECE 484 Digital Image Processing, 2019. p.1 Outline Recap of Lec 05 Fourier Transform - Freq Domain Representation of Images Image Sampling . Nyquist Sampling Rate and Aliasing . Super-Resolution and Interpolation Summary Z. Li, ECE 484 Digital Image Processing, 2019. p.2 Re-Cap: Linear Filters Convolution 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 20 30 30 30 20 10 0 0 0 90 90 90 90 90 0 0 0 20 40 60 60 60 40 20 0 0 0 90 90 90 90 90 0 0 0 30 60 90 90 90 60 30 1/9 11/9 11/9 11/9 11/9 1/9 0 0 0 90 90 90 90 90 0 0 0 30 50 80 80 90 60 30 0 0 0 90 0 90 90 90 0 0 0 30 50 80 80 90 60 30 11/9 11/9 11/9 = 0 0 0 90 90 90 90 90 0 0 0 20 30 50 50 60 40 20 * 0 0 0 0 0 0 0 0 0 0 10 20 30 30 30 30 20 10 0 0 90 0 0 0 0 0 0 0 10 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Padding k xk Complexity: . M*N*K2 Z. Li, ECE 484 Digital Image Processing, 2019. p.3 Properties of Linear Filtering Convolution properties . Shift-Invariant: f(m-k, n-j)*h = g(m-k, n-j), if f*h=g . Associative: f*h1*h2 = f*(h1*h2) this can save a lot of complexity . Distributive: f*h1 + f*h2 = f*(h1+h2) useful in SIFT’s DoG filtering. Applications . Scale space filtering with successive Gaussian . DoG filtering with difference of Gaussian blurred images * = Z. Li, ECE 484 Digital Image Processing, 2019. p.4 Image Smoothing/De-noising Average Smoothing (Box Filter - also an acceleration scheme) Gaussian Smoothing Z. Li, ECE 484 Digital Image Processing, 2019. p.5 Image Sharpening Sharpen an image: blurred image unit impulse (identity) scaled Gaus Laplacian of impulse sian Gaussian Let’s add it back: + α = original detail sharpened Z. Li, ECE 484 Digital Image Processing, 2019. p.6 Edge Detection Difference of Gaussian (DoG) Laplacian of Gaussian (loG) Sobel Z. Li, ECE 484 Digital Image Processing, 2019. p.7 Linear Filter Acceleration Separable Filtering 2D convolution (center location only) The filter factors into a product of 1D filters: Perform convolution = along rows: * Followed by convolution = along the remaining column: * Z. Li, ECE 484 Digital Image Processing, 2019. p.8 Linear Filter Acceleration Approx Separable Filter via SVD kernel h U S V sig. significant t n a e c noise i noise s f i = i o n n g i s DoG approximation (HW-2) Z. Li, ECE 484 Digital Image Processing, 2019. p.9 Outline Recap of Lec 05 Fourier Transform - Freq Domain Representation of Images Image Sampling . Nyquist Sampling Rate and Aliasing . Super-Resolution and Interpolation Summary Z. Li, ECE 484 Digital Image Processing, 2019. p.10 Fourier Series Any periodic contiunous signals can be expressed as linear combinations of sinusoid While coefficients are: Joseph Fourier 1768-1830 Z. Li, ECE 484 Digital Image Processing, 2019. p.11 Even and Odd Functions An arbitrary function can be expressed in term of even and odd functions E(-x) = E(x) O(-x) = -O(x) Z. Li, ECE 484 Digital Image Processing, 2019. p.12 Fourier Cosine Series Because cos(mt) is an even function, we can write an even function, f(t), as: where series Fm is computed as Here we suppose f(t) is over the interval (–π,π). Z. Li, ECE 484 Digital Image Processing, 2019. 13 Fourier Sine Series Because sin(mt) is an odd function, we can write any odd function, f(t), as: where the series F’m is computed as Z. Li, ECE 484 Digital Image Processing, 2019. 14 Fourier Series So if f(t) is a general function, neither even nor odd, it can be written: Even component Odd component where the Fourier series is Z. Li, ECE 484 Digital Image Processing, 2019. 15 The Fourier Transform Let F(m) incorporates both cosine and sine series coefficients, with the sine series distinguished by making it the imaginary component: Let’s now allow f(t) range from – to , we rewrite: F(u) is called the Fourier Transform of f(t). We say that f(t) lives in the “time domain,” and F(u) lives in the “frequency domain.” u is called the frequency variable. Z. Li, ECE 484 Digital Image Processing, 2019. 16 The Inverse Fourier Transform We go from f(t) to F(u) by Fourier Transform Given F(u), f(t) can be obtained by the inverse Fourier transform Inverse Fourier Transform Z. Li, ECE 484 Digital Image Processing, 2019. 17 Fourier Transform Example f(x) with limited time domain support Z. Li, ECE 484 Digital Image Processing, 2019. p.18 2D Fourier Transform A straight extension of 1D FT: . F(u,v) is complex in general, can be written as, . |F(u,v)| is the magnitude spectrum . tan-1(FI(u,v)/FR(u,v)) is the phase spectrum Z. Li, ECE 484 Digital Image Processing, 2019. p.19 2D FT Example Spatial domain block: u . f(x,y) = 1 if inside [-X, X] and [-Y, Y] v Z. Li, ECE 484 Digital Image Processing, 2019. p.20 FT example - Gaussian Gaussian Kernel has Gaussian spectrum Z. Li, ECE 484 Digital Image Processing, 2019. p.21 FT example - Circular Function Z. Li, ECE 484 Digital Image Processing, 2019. p.22 FT Example - Impulses in spatial domain Question: what would be F(u,v) of : Z. Li, ECE 484 Digital Image Processing, 2019. p.23 Freq Interpretation The central part of FT, i.e. the low frequency components are responsible for the general gray-level appearance of an image. The high frequency components of FT are responsible for the detail information of an image. Z. Li, ECE 484 Digital Image Processing, 2019. 24 FT properties 2D FT properties Z. Li, ECE 484 Digital Image Processing, 2019. p.25 Fourier Analysis of Pictures Repeated patterns . Lunar Orbital Image (1966) . Repeated patterns from rolling shutter . remove artifacts in freq domain Z. Li, ECE 484 Digital Image Processing, 2019. p.26 Magnitude vs Phase Magnitude seems to carry more information, also easy to explain in frequency Phase is also carrying important information Z. Li, ECE 484 Digital Image Processing, 2019. p.27 Discrete Fourier Transform DFT real(A) imag(A) Forward transform u=0 Inverse transform basis u=7 n n Z. Li, ECE 484 Digital Image Processing, 2019. p.28 DFT vs. DCT 1D-DCT 1D-DFT real a imag a a ( ) ( ) u=0 u=0 u=7 u=7 Z. Li, ECE 484 Digital Image Processing, 2019. n=7 p.29 DFT spectrum DFT domain representation and processing real-valued input Z. Li, ECE 484 Digital Image Processing, 2019. p.30 2-D Fourier basis Fourier basis: low freq is in the middle real imag real( ) imag( ) Z. Li, ECE 484 Digital Image Processing, 2019. p.31 LP, HP and BP in Freq Domain Filtering in frequency domain BP filter to remove repetitive noises LP filter and smoothing Z. Li, ECE 484 Digital Image Processing, 2019. p.32 Outline Recap of Lec 08 Image Sampling . subsampling . interpolation Summary Z. Li, ECE 484 Digital Image Processing, 2019. p.33 Image Scaling Need to resample images This image is too big to fit on the screen. How can we reduce it? How to generate a half- sized version? Z. Li, ECE 484 Digital Image Processing, 2019. p.34 Image sub-sampling sub sample without filtering, what is wrong ? 1/8 1/4 Throw away every other row and column to create a 1/2 size image - called image sub-sampling Z. Li, ECE 484 Digital Image Processing, 2019. p.35 Image sub-sampling Aliasing... 1/2 (2x zoom) 1/4 1/8 (4x zoom) Why does this look so crufty? Z. Li, ECE 484 Digital Image Processing, 2019. p.36 Even worse for synthetic images Aliasing effect Z. Li, ECE 484 Digital Image Processing, 2019. p.37 Sampling and the Nyquist rate Aliasing can arise when you sample a continuous signal or image . occurs when your sampling rate is not high enough to capture the amount of detail in your image . Can give you the wrong signal/image—an alias . formally, the image contains structure at different scales . called “frequencies” in the Fourier domain . the sampling rate must be high enough to capture the highest frequency in the image To avoid aliasing: . sampling rate > 2 * max frequency in the image . This minimum sampling rate is called the Nyquist rate Z. Li, ECE 484 Digital Image Processing, 2019. p.38 Sampling in Time Domain Computer needs a discrete representation of signals • Many signals originate as continuous-time signals, e.g. conventional music or voice • By sampling a continuous-time signal at isolated, equally-spaced points in time, we obtain a sequence of numbers Ts n {…, -2, -1, 0, 1, 2,…} t T Ts is the sampling period. s s(t) Sampled analog impulse waveform train Z.
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