Sampling and Monte-Carlo Integration
Sampling and Monte-Carlo Integration Sampling and Monte-Carlo Integration MIT EECS 6.837 MIT EECS 6.837 Last Time Quiz solution: Homogeneous sum • Pixels are samples •(x1, y1, z1, 1) + (x2, y2, z2, 1) = (x +x , y +y , z +z , 2) • Sampling theorem 1 2 1 2 1 2 ≈ ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2) • Convolution & multiplication • This is the average of the two points • Aliasing: spectrum replication • General case: consider the homogeneous version of (x , y , z ) and (x , y , z ) with w coordinates w and w • Ideal filter 1 1 1 2 2 2 1 2 – And its problems •(x1 w1, y1 w1, z1 w1, w1) + (x2 w2, y2 w2, z2 w2, w2) = (x w +x w , y w +y w , z w +z w , w +w ) • Reconstruction 1 1 2 2 1 1 2 2 1 1 2 2 1 2 ≈ ((x1 w1+x2w2 )/(w1+w2) ,(y1 w1+y2w2 )/(w1+w2), • Texture prefiltering, mipmaps (z1 w1+z2w2 )/(w1+w2)) • This is the weighted average of the two geometric points MIT EECS 6.837 MIT EECS 6.837 Today’s lecture Ideal sampling/reconstruction • Antialiasing in graphics • Pre-filter with a perfect low-pass filter • Sampling patterns – Box in frequency • Monte-Carlo Integration – Sinc in time • Probabilities and variance • Sample at Nyquist limit • Analysis of Monte-Carlo Integration – Twice the frequency cutoff • Reconstruct with perfect filter – Box in frequency, sinc in time • And everything is great! MIT EECS 6.837 MIT EECS 6.837 1 Difficulties with perfect sampling At the end of the day • Hard to prefilter • Fourier analysis is great to understand aliasing • Perfect filter has infinite support • But practical problems kick in – Fourier analysis assumes
[Show full text]