Image Sampling. Moire patterns. - Source: F. Durand

Transcription

1 Image Sampling Moire patterns Source: F. Durand -

2 Any questions on project 1? For extra credits, attach before/after images how your extra feature improved results! 2

3 Image Scaling This image is too big to fit on the screen. How can we reduce it? How to generate a halfsized version?

4 Image sub-sampling 1/8 1/4 Throw away every other row and column to create a 1/2 size image - called image sub-sampling

5 Image sub-sampling 1/2 1/4 (2x zoom) 1/8 (4x zoom) Why does this look so crufty?

6 Even worse for synthetic images

7 From Time to Frequency Domain example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t) = + [Slide from Srinivasa Narasimhan]

8 Frequency Spectra? = [Slide from Srinivasa Narasimhan]

9 Frequency Spectra? = + = [Slide from Srinivasa Narasimhan]

10 Frequency Spectra = + = [Slide from Srinivasa Narasimhan]

11 Frequency Spectra = + = [Slide from Srinivasa Narasimhan]

12 Frequency Spectra = + = [Slide from Srinivasa Narasimhan]

13 Frequency Spectra = + = [Slide from Srinivasa Narasimhan]

14 Frequency Spectra = [Slide from Srinivasa Narasimhan]

15 Fourier Transform Fourier Transform (FT) Inverse Fourier Transform (IFT)

16 Questions? 16

17 Fourier Transform and Convolution Spatial Domain (x) Frequency Domain (u) [Slide from Srinivasa Narasimhan]

18 Convolution filter -1 1 Image Correlation output

19 Correlation (review) filter Fourier -1Transform! 1 Image Fourier Transform! Correlation output

20 Correlation (review) H H11 F F11 20

21 Correlation (review) H H11 G = H x F G11 F F11 Multiply per cell Inverse Fourier Transform! G11 = H11 x F11 21

22 H11 Correlation (review) H g F F11 Multiply per cell

23 Fourier Transform and Convolution Spatial Domain (x) Frequency Domain (u) [Slide from Srinivasa Narasimhan]

24 Fourier Transform and Convolution Spatial Domain (x) Frequency Domain (u) [Slide from Srinivasa Narasimhan]

25 Fourier Transform and Convolution Spatial Domain (x) Frequency Domain (u) So, we can find g(x) by Fourier transform IFT FT FT [Slide from Srinivasa Narasimhan]

26 Fourier Transform Pairs (I) [Slide from Srinivasa Narasimhan]

27 Fourier Transform Pairs (II) Comb function [Slide from Srinivasa Narasimhan]

28 Questions? 28

29 Sampling Theorem Continuous signal: (Real world signal) Shah function (Impulse train): Sampled function: (What the image measures) [Slide from Srinivasa Narasimhan]

30 [Slide from Srinivasa Narasimhan] Sampling Theorem Sampled function:

31 Fourier Transform Pairs (II) [Slide from Srinivasa Narasimhan]

32 [Slide from Srinivasa Narasimhan] Sampling Theorem Sampled function:

33 [Slide from Srinivasa Narasimhan] Sampling Theorem Sampled function: *

34 [Slide from Srinivasa Narasimhan] Sampling Theorem *

35 [Slide from Srinivasa Narasimhan] Sampling Theorem We can recover from

36 Nyquist Frequency [Slide from Srinivasa Narasimhan] Aliasing We can recover F(u) when no interference that is when ( ) > 2 (Nyquist Frequency)

37 Nyquist Frequency [Slide from Srinivasa Narasimhan] ( ) > 2 (Nyquist Frequency) Condition met if you sample fine enough (X0 is small) or if image does not have high frequency signal

38 Sampling and the Nyquist rate (summary) Aliasing occurs when sampling rate is not high enough To avoid aliasing: sampling frequency must be 2 * max frequency in the image

39 Questions? 39

40 2D example Good sampling Bad sampling

41 2D example Don t wear high-frequency texture for a photo-shoot... Good Sampling (Proper image viewer) Bad Sampling (Crappy image viewer - preview) 41

42 How to fix this? 1/8 1/4 1/2

43 Subsampling with Gaussian pre-filtering G 1/4 G 1/8 Gaussian 1/2 Solution: filter the image, then subsample

44 Subsampling with Gaussian pre-filtering Gaussian 1/2 Solution: filter the image, then subsample G 1/4 G 1/8

45 Real Comparison 45

46 Sometimes we want many resolutions Known as a Gaussian Pyramid [Burt and Adelson, 1983] In computer graphics, a mip map [Williams, 1983] Gaussian Pyramids have all sorts of applications in computer vision We ll talk about these later in the course

47 The Gaussian Pyramid Low resolution blur blur down-sample down-sample blur down-sample down-sample blur High resolution [Slide from Srinivasa Narasimhan]

48 How big is the combined pyramid? [Slide from Srinivasa Narasimhan]

49 How big is the combined pyramid? [Slide from Srinivasa Narasimhan]

50 Questions 50

51 Image Resampling So far, we considered only power-of-two subsampling What about arbitrary scale reduction? How can we increase the size of the image?

52 Image resampling So far, we considered only power-of-two subsampling What about arbitrary scale reduction? How can we increase the size of the image?

53 Image resampling So far, we considered only power-of-two subsampling What about arbitrary scale reduction? How can we increase the size of the image?

54 Image resampling So far, we considered only power-of-two subsampling What about arbitrary scale reduction? How can we increase the size of the image?

55 Image resampling Piecewise linear approximation 1 d = 1 in this example

56 Image resampling Piecewise linear approximation 1 d = 1 in this example

57 Piece-wise Linear Approximation [ Wikipedia ]

58 Piece-wise Linear Approximation = [ Wikipedia ]

59 Piece-wise Linear Approximation in 2D Bilinear interpolation = linear combination of [ Nishino ]

60 Piece-wise Linear Approximation in 2D Bilinear interpolation [ Nishino ] = linear combination of Weights are rectangle areas

61 Bicubic interpolation Bilinear uses nearby 2x2 = 4 samples Bicubic uses nearby 4x4 = 16 samples 61

62 Comparisons Let s interpolate! 62

63 Comparisons Nearest-neighbor Bilinear interpolation Bicubic interpolation 63

64 For project 1, to handle scale/rotation Pick an appropriate level in the pyramid Interpolate colors to handle rotation 64

65 Temporal Sampling Effects 65

66 Content aware image resizing 66