CEE598 - Visual Sensing for Civil Infrastructure Eng. & Mgmt.
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1 CEE598 - Visual Sensing for Civil Infrastructure Eng. & Mgmt. Session 8- Linear Filters From Spatial Domain to Frequency Domain Mani Golparvar-Fard Department of Civil and Environmental Engineering 329D, Newmark Civil Engineering Lab mgolpar@illinois.edu Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign
2 Outline What we have learned so far: Review of lighting Reflection and absorption What is image filtering and how do we do it? Color models (if time allows) Today: Fourier transform and frequency domain Frequency view of filtering Sampling Reading: [FP] Chapters 7,8 Some slides in this lecture are courtesy to Prof. S. Savarese, Prof. D. Hoiem, prof F. Li, prof S. Lazebnik, and various other lecturers CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203 2
3 Review: image filtering g[, ] f [.,.] h[.,.] Credit: S. Seitz h[ m, n] g[ k, l] k, l f [ m k, n l] CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203 3
4 Review: image filtering g[, ] f [.,.] h[.,.] Credit: S. Seitz h[ m, n] g[ k, l] k, l f [ m k, n l] CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203 4
5 Review: image filtering g[, ] f [.,.] h[.,.] Credit: S. Seitz h[ m, n] g[ k, l] k, l f [ m k, n l] CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203 5
6 Review: image filtering g[, ] f [.,.] h[.,.] Credit: S. Seitz h[ m, n] g[ k, l] k, l f [ m k, n l] CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203 6
7 Review: image filtering g[, ] f [.,.] h[.,.] Credit: S. Seitz h[ m, n] g[ k, l] k, l f [ m k, n l] 7
8 Review: image filtering g[, ] f [.,.] h[.,.] Credit: S. Seitz h[ m, n] g[ k, l] k, l f [ m k, n l] 8
9 Review: image filtering in spatial domain Sobel CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
10 Difficulty in Spatial Domain Why does the Gaussian give a nice smooth image, but the square filter give edgy artifacts? Gaussian Box filter 0 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
11 Hybrid Images A. Oliva, A. Torralba, P.G. Schyns, Hybrid Images, SIGGRAPH 2006 Slides: Hoiem CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
12 Thinking in terms of frequency Why does a lower resolution image still make sense to us? What do we lose? Slides: Hoiem Image: CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203 2
13 Jean Baptiste Joseph Fourier ( ) had crazy idea (807): Any univariate function can be rewritten as a weighted sum of sines and cosines of different frequencies. Don t believe it? Neither did Lagrange, Laplace, Poisson and other big wigs Not translated into English until 878! But it s (mostly) true! called Fourier Series there are some subtle restrictions...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour. Laplace Lagrange Legendre Slides: Hoiem CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203 3
14 A sum of sines Our building block: Asin( x Add enough of them to get any signal f(x) you want! 4 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
15 Frequency Spectra example : g(t) = sin(2πf t) + (/3)sin(2π(3f) t) = + 5 Slides: Efros CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
16 Frequency Spectra 6 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
17 Frequency Spectra = + = 7 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
18 Frequency Spectra = + = 8 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
19 Frequency Spectra = + = 9 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
20 Frequency Spectra = + = 20 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
21 Frequency Spectra = + = 2 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
22 Frequency Spectra = A k sin(2 kt ) k 22 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
23 Example: Music We think of music in terms of frequencies at different magnitudes 23 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
24 Other signals We can also think of all kinds of other signals the same way Cats(?) xkcd.com FFT of a cat 24 Slides: Hoiem CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
25 Fourier analysis in images Intensity Image Fourier Image CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard,
26 Signals can be composed + = More: CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard,
27 Strong Vertical Frequency (Sharp Horizontal Edge) Diagonal Frequencies Strong Horz. Frequency (Sharp Vert. Edge) Log Magnitude Low Frequencies 27 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
28 Fourier Transform Fourier transform stores the magnitude and phase at each frequency Magnitude encodes how much signal there is at a particular frequency Phase encodes spatial information (indirectly) For mathematical convenience, this is often notated in terms of real and complex numbers Amplitude: A R I 2 2 ( ) ( ) Phase: tan I( ) R( ) Euler s formula: 28 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
29 Computing the Fourier Transform Continuous Discrete k = -N/2..N/2 Fast Fourier Transform (FFT): NlogN 29 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
30 The Convolution Theorem The Fourier transform of the convolution of two functions is the product of their Fourier transforms F[ g h] F[ g]f[ h] The inverse Fourier transform of the product of two Fourier transforms is the convolution of the two inverse Fourier transforms F [ gh] F [ g] [ h] Convolution in spatial domain is equivalent to multiplication in frequency domain! F 30
31 Properties of Fourier Transforms Linearity Fourier transform of a real signal is symmetric about the origin The energy of the signal is the same as the energy of its Fourier transform See Szeliski Book (3.4) CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203 3
32 Filtering in spatial domain * = 32 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
33 Filtering in frequency domain FFT FFT = Inverse FFT 33 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
34 Fourier Matlab demo 34 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
35 FFT in Matlab Filtering with fft im =... % im should be a gray-scale floating point image [imh, imw] = size(im); fftsize = 024; % should be order of 2 (for speed) and include padding im_fft = fft2(im, fftsize, fftsize); % ) fft im with padding hs = 50; % filter half-size fil = fspecial('gaussian', hs*2+, 0); fil_fft = fft2(fil, fftsize, fftsize); % 2) fft fil, pad to same size as image im_fil_fft = im_fft.* fil_fft; % 3) multiply fft images im_fil = ifft2(im_fil_fft); % 4) inverse fft2 im_fil = im_fil(+hs:size(im,)+hs, +hs:size(im, 2)+hs); % 5) remove padding Displaying with fft figure(), imagesc(log(abs(fftshift(im_fft)))), axis image, colormap jet CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard,
36 Filtering Why does the Gaussian give a nice smooth image, but the square filter give edgy artifacts? Gaussian Box filter 36 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
37 Gaussian 37 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
38 Box Filter 38 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
39 Sampling Why does a lower resolution image still make sense to us? What do we lose? Image: CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard,
40 Subsampling by a factor of 2 Throw away every other row and column to create a /2 size image 40 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
41 Aliasing problem D example (sinewave): Source: S. Marschner CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203 4
42 Aliasing problem D example (sinewave): Source: S. Marschner CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard,
43 Aliasing problem Sub-sampling may be dangerous. Characteristic errors may appear: Wagon wheels rolling the wrong way in movies Checkerboards disintegrate in ray tracing Striped shirts look funny on color television Source: D. Forsyth CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard,
44 Aliasing in video 44 Slide by Steve Seitz CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
45 Aliasing in graphics Source: A. Efros 45
46 Sampling and aliasing CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
47 Nyquist-Shannon Sampling Theorem When sampling a signal at discrete intervals, the sampling frequency must be 2 f max f max = max frequency of the input signal This will allows to reconstruct the original perfectly from the sampled version v v v good bad 47 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
48 Anti-aliasing Solutions: Sample more often Get rid of all frequencies that are greater than half the new sampling frequency Will lose information But it s better than aliasing Apply a smoothing filter 48 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
49 Algorithm for downsampling by factor of 2. Start with image(h, w) 2. Apply low-pass filter im_blur = imfilter(image, fspecial( gaussian, 7, )) 3. Sample every other pixel im_small = im_blur(:2:end, :2:end); 49
50 Anti-aliasing Forsyth and Ponce 2002 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard,
51 Subsampling without pre-filtering /2 /4 (2x zoom) /8 (4x zoom) Slide by Steve Seitz 5
52 Subsampling with Gaussian pre-filtering 52 Gaussian /2 G /4 G /8 Slide by Steve Seitz CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard,
53 Why does a lower resolution image still make sense to us? What do we lose? Image: 53 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
54 Clues from Human Perception Early processing in humans filters for various orientations and scales of frequency Perceptual cues in the mid-high frequencies dominate perception When we see an image from far away, we are effectively subsampling it Early Visual Processing: Multi-scale edge and blob filters 54 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
55 Hybrid Image in FFT Hybrid Image Low-passed Image High-passed Image 55 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
56 Things to Remember Sometimes it makes sense to think of images and filtering in the frequency domain Fourier analysis Can be faster to filter using FFT for large images (N logn vs. N 2 for autocorrelation) Images are mostly smooth Basis for compression Remember to low-pass before sampling 56 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
57 Practice question. Match the spatial domain image to the Fourier magnitude image B A C D E 57 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
58 Next class Template matching Image Pyramids Filter banks and texture Denoising, Compression 58 CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. Mani Golparvar-Fard, 203
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