Quality Improves with More Rays

Transcription

1 Recap

2 Quality Improves with More Rays Area Area 1 shadow ray 16 shadow rays CS348b Lecture 8 Pat Hanrahan / Matt Pharr, Spring 2018

3 pixelsamples = 1 jaggies

4 pixelsamples = 16 anti-aliased

5 Sampling and Reconstruction Basic signal processing Fourier transforms The convolution theorem The sampling theorem Aliasing and antialiasing Uniform supersampling Stochastic sampling CS348b Lecture 8 Pat Hanrahan / Matt Pharr, Spring 2018

6 Review: Basic Signal Processing

7 Sines and Cosines cos 2 x sin 2 x

8 Frequencies cos 2 fx f = 1 T f =1 cos 2 x f =2 cos 4 x

9 Constant Spatial Domain Frequency Domain

10 sin(2 /32)x 32 pixels per cycle Spatial Domain Frequency Domain

11 sin(2 /16)y 16 pixels per cycle? Spatial Domain Frequency Domain

12 sin(2 /16)y 16 pixels per cycle Spatial Domain Frequency Domain

13 sin(2 /32)x sin(2 /16)y Spatial Domain Frequency Domain

14 e r2 /16 2 Spatial Domain Frequency Domain

15 e r2 /32 2 Spatial Domain Frequency Domain

16 e x2 /32 2 e y2 /16 2 Spatial Domain Frequency Domain

17 Rotate 45 e x2 /32 2 e y2 /16 2 Spatial Domain Frequency Domain

18 Fourier Transforms The Fourier transform converts between the spatial and frequency domain Spatial Domain iω x F( ω) = f ( x) e dx 1 f ( x) = F( ω) e iω x dω 2π Frequency Domain Figures generated using fft2d.py CS348b Lecture 8 Pat Hanrahan / Matt Pharr, Spring 2018

19 Pat s Frequencies Spatial Domain Frequency Domain

20 Filtering: Low Pass Filter? Spatial Domain Frequency Domain

21 Filtering: Low Pass Filter Keep low frequencies Spatial Domain Frequency Domain

22 Filtering: Band Pass Filter Keep band of frequencies Spatial Domain Frequency Domain

23 Filtering: Band Pass Filter Keep band of frequencies Spatial Domain Frequency Domain

24 Filtering: High Pass Filter Keep high frequencies Spatial Domain Frequency Domain

25 Filtering by Convolution h( x) = f g = f ( x! ) g( x x! ) dx!

26 Convolution Theorem Convolution Theorem: Multiplication in the frequency domain is equivalent to convolution in the space domain. f g F G Symmetric Theorem: Multiplication in the space domain is equivalent to convolution in the frequency domain. f g F G CS348b Lecture 8 Pat Hanrahan / Matt Pharr, Spring 2018

27 Spatial and Frequency Domain Spatial Domain Frequency Domain

28 Math: Box and Sinc Functions T ( x) = $ 1! #! 0 " x x > T 2 T 2 sinc(x) sin x x

29 The Sampling Theorem

30 Simple Function f (x) F (ω)

31 Sampling: Multiply in Spatial Domain f (x) =

32 Some Magic The Fourier transform of a sequence of spikes is a sequence of spikes T 1/T III(t) = n=1 X n= 1 (t nt) III 1/T (!) = Comb or Shah function n=1 X n= 1 (! n/t)

33 Sampling: Convolve in Freq Domain F (ω) =

34 Reconstruction: Frequency Domain =

35 Recall T ( x) = $ 1! #! 0 " x x > T 2 T 2 sinc(x) sin x x

36 Reconstruction: Spatial Domain =

37 Recovering a Sampled Signal = =

38 Sampling Theorem This result is known as the Sampling Theorem, and Claude Shannon is credited with discovering it in 1949 A signal can be reconstructed from its samples without loss of information, if the original signal has no frequencies above 1/2 the sampling frequency For a given band-limited function, the rate it must be sampled is called the Nyquist Frequency CS348b Lecture 8 Pat Hanrahan / Matt Pharr, Spring 2018

39 Aliasing

40 Aliasing

41 Undersampling: Overlaps/Aliases =

42 Sampling a Zone Plate Zone plate: sin x y Left rings: signal Right rings: aliasing

43 Antialiasing Preventing Aliasing

44 Antialiasing by Prefiltering = = Frequency Space

45 Point vs. Area Sampling Point Exact Area Checkerboard sequence by Tom Duff

46 Uniform Supersampling Take n samples Average them together (filter = weighted average) Pixel = w Sample s s s Samples Pixel CS348b Lecture 8 Pat Hanrahan / Matt Pharr, Spring 2018

47 Point vs. Supersampled Point 4x4 Uniform Fewer aliases, but still present

48 Area vs. Supersampled Exact Area 4x4 Uniform Fewer aliases, but still present

49 Uniform Supersampling Increasing the number of samples moves each copy of the spectra further apart, thus there is less overlap This reduces, but does not eliminate, aliasing Pixel = w Sample s s s Samples Pixel CS348b Lecture 8 Pat Hanrahan / Matt Pharr, Spring 2018

50 Jittered Sampling Add uniform random jitter to each sample

51 Distribution of Extrafoveal Cones Monkey eye cone distribution Fourier transform Yellot

52 Jittered vs. Uniform Supersampling 4x4 Jittered Sampling 4x4 Uniform

53 Theory: Analysis of Jitter Non-uniform sampling n= s( x) = δ ( x x ) n n= x = nt + j n n jn Jittered sampling j( x) ~ j( x) " $ 1 x 1/ 2 = % $& 0 x > 1/ 2 J( ω) = sincω 1 2 2π n= 2 2 π n S( ω) = & 1 J( ω) ' + J( ω) δ ( ω ) T ( ) T 2 T 1 & 1 sinc 2 ω' δ ( ω ) = + T ( ) n=

54 Jittered Sampling

55 Poisson Disk Sampling

56 Integration Error when Sampling Integral Z I(f) = f(x) dx = F (0) Sampled integral Z I s (f) = f(x)s(x) dx = 1 N X f(xi ) f(x)s(x) =F (!) Z S(!) I s (f) = f(x)s(x) dx = F (!) S(!)!=0 Error = F (0) F (!) S(!)!=0 CS348b Lecture 8 Pat Hanrahan / Matt Pharr, Spring 2018

57 Non-uniform Sampling Uniform sampling The spectrum of uniformly spaced samples is also a set of uniformly spaced spikes Multiplying the signal by the sampling pattern corresponds to placing a copy of the spectrum at each spike (in freq. space) Aliases are coherent, and very noticable Non-uniform sampling Samples at non-uniform locations have a different spectrum; a single spike plus noise Sampling a signal in this way converts aliases into broadband noise Noise is incoherent, and much less objectionable May cause error in the integral Pat Hanrahan / Matt Pharr, Spring 2018 CS348b Lecture 8