CS4495/6495 Introduction to Computer Vision. 2C-L3 Aliasing

Transcription

1 CS4495/6495 Introduction to Computer Vision 2C-L3 Aliasing

2 Recall: Fourier Pairs (from Szeliski)

3 Fourier Transform Sampling Pairs FT of an impulse train is an impulse train

4 Sampling and Aliasing

5 Sampling and Reconstruction

6 Sampled representations How to store and compute with continuous functions? Common scheme for representation: samples S. Marschner

7 Reconstruction Making samples back into a continuous function for output (need realizable method) for analysis or processing (need mathematical method) Amounts to guessing what the function did in between S. Marschner

8 1D Example: Audio low frequencies high

9 Sampling in digital audio Recording: sound to analog to samples to disc Playback: disc to samples to analog to sound again S. Marschner

10 Sampling and Reconstruction Simple example: a sign wave S. Marschner

11 Undersampling What if we missed things between the samples? S. Marschner

12 Undersampling Simple example: undersampling a sine wave unsurprising result: information is lost S. Marschner

13 Undersampling Simple example: undersampling a sine wave unsurprising result: information is lost surprising result: indistinguishable from lower frequency S. Marschner

14 Undersampling Simple example: undersampling a sine wave Low frequency also was always indistinguishable from higher frequencies S. Marschner

15 Undersampling Aliasing: signals traveling in disguise as other frequencies S. Marschner

16 Aliasing in video S. Seitz

17 Aliasing in images

18 What s happening? Input signal: Plot as image: x = 0:.05:5; imagesc(sin((2.^x).*x)) Alias! Not enough samples

19 Antialiasing Sample more often Join the Mega-Pixel craze of the photo industry But this can t go on forever Make the signal less wiggly Get rid of some high frequencies Will loose information But it s better than aliasing

20 Preventing aliasing Introduce lowpass filters: remove high frequencies leaving only safe, low frequencies to be reconstructed S. Marschner

21 (Anti)Aliasing in the Frequency Domain

22 Impulse Train Define a comb function (impulse train) in 1D as follows where M is an integer c o m b [ x ] [ x k M ] M 1 k c o m b [ x ] 2 x B.K. Gunturk

23 FT of Impulse Train in 1D 1 c o m b ( x ) 1 2 c o m b ( u ) 2 Remember: x Scaling f a x u F a a u B.K. Gunturk

24 Impulse Train in 2D (bed of nails) c o m b ( x, y ) x k M, y ln M, N k l

25 FT of Impulse Train in 2D (bed of nails) Fourier Transform of an impulse train is also an impulse train: x k M, y ln k l k 1 k l u, v M N M N l c o m b ( x, y ) M, N As the comb samples get further apart, the spectrum samples get closer together! c o m b 1 1, M N ( u, v ) B.K. Gunturk

26 FT Impulse Train in 1D 1 c o m b ( x ) 1 2 c o m b ( u ) 2 Remember: x Scaling f a x u F a a u B.K. Gunturk

27 Sampling low frequency signal

28 B.K. Gunturk f(x) F(u) comb M (x) M Multiply (sample): f x comb M (x) x x x comb 1 (u) M 1 M Convolve: F u comb 1 (u) M u u u

29 B.K. Gunturk f(x) F(u) x f x comb M (x) u F u comb 1 (u) M x M 1 M u No problem if the maximum frequency of the signal is small enough

30 B.K. Gunturk Sampling low frequency signal f x comb M (x) F u comb 1 (u) M M x W 1 M 1 2 M u If there is no overlap, W < 1 2M, the original signal can be recovered from its samples by low-pass filtering.

31 Sampling high frequency signal f(x) F(u) x W W u < f x comb M x > F u comb 1 (u) M Overlap: The high frequency energy is folded over into low frequency. It is aliasing as lower frequency energy. And you cannot fix it once it has happened. 1 M u

32 Sampling high frequency signal f(x) F(u) f ( x ) * h ( x ) x W W u u Antialiasing filter Anti-aliasing filter [ f ( x ) * h ( x )] c o m b M ( x ) Apply low pass u 1 M B.K. Gunturk

33 Sampling high frequency signal Without anti-aliasing filter: f ( x ) c o m b ( x ) M W u With anti-aliasing filter: 1 M [ f ( x ) * h ( x )] c o m b M ( x ) u 1 M B.K. Gunturk

34 Aliasing in Images

35 Image half-sizing Suppose this image is too big to fit on the screen. How can we reduce it e.g. generate a half-sized version? S. Seitz

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

37 Image sub-sampling 1/2 1/4 (2x zoom) 1/8 (4x zoom) Aliasing! What do we do? S. Seitz

38 Gaussian (lowpass) pre-filtering Solution: filter the image, then subsample G 1/4 G 1/8 Gaussian 1/2 S. Seitz

39 Subsampling with Gaussian pre-filtering Gaussian 1/2 G 1/4 G 1/8 S. Seitz

40 Compare with... Original G 1/8 (4x zoom) Subsample 1/8 (4x zoom) S. Seitz

41 Campbell-Robson contrast sensitivity curve The higher the frequency the less sensitive human visual system is

42 Lossy Image Compression (JPEG) Block-based Discrete Cosine Transform (DCT) on 8x8

43 Using DCT in JPEG The first coefficient B(0,0) is the DC component, the average intensity The top-left coeffs represent low frequencies, the bottom right high frequencies

44 Image compression using DCT DCT enables image compression by concentrating most image information in the low frequencies Quantization Table

45 Image compression using DCT Lose unimportant image info (high frequencies) by cutting B(u,v) at bottom right The decoder computes the inverse DCT IDCT Quantization Table

46 JPEG compression comparison 89k 12k