Kevin Steele

CS 6650

Programming Assignment #1: Monte Carlo Integration, Density Estimation, Metropolis

 

 

Part 0: Analytic Integration of function

 

All images were computed using 32-bit floats with no gamma correction. I originally generated them using doubles, and those images were minutely better. However, the computation speed gain using floats more than made up for the slightly poorer image quality.

 

 

Animations

 

These animations show how the function behaves under a phase shift. Each frame shows the function phase shifted by 2pi/90 from the previous frame (90 frames in each animation). The stratified MCI solution exhibits animated aliasing, and the pseudo-bspline MCI solution substitutes the aliasing for animated noise.

 

The animations are Quicktime format, and are best viewed as looped movies.

 

MCI Stratified 100 Samples per Pixel

MCI BSpline 100 Samples per Pixel

MCI BSpline 2000 Samples per Pixel

 

 

Part 1: MC Integration, 16 random samples

 

 

 

Difference Image from Analytic Solution (intensity scale factor: 2.2)

 

Difference Image from Analytic Solution (intensity scale factor: 10)

 

Part 2: MC Integration, 16 stratified samples

 

 

 

Difference Image (intensity scale factor: 6.71)

Difference Image (intensity scale factor: 10)

Part 3: MC Integration, 100 random samples

 

 

 

Difference Image (intensity scale factor: 5.1)

Difference Image (intensity scale factor: 10)

Part 4: MC Integration, 100 stratified samples

 

 

 

Difference Image (intensity scale factor: 31.88)

Difference Image (intensity scale factor: 10)

Part 5: MC Inegration, 100 random samples, bspline filter

 

 

 

 

 

 

 

Part 6: Density Estimation, 100 random samples per pixel on average

 

The scaling for image intensity (power) was done using the analytic integral of the function, rather than an MCI approximation. This improves the accuracy of this particular image, but is not practical in the general case.

 

 

Difference Image (intensity scale factor: 8.5)

Difference Image (intensity scale factor: 10)

Part 7: Density Estimation, 100 stratified samples per pixel on average

 

 

 

Difference Image (intensity scale factor: 42.5)

 

 

 

Difference Image (intensity scale factor: 10)

 

 

 

Part 8: Metropolis, 100 random samples per pixel on average

 

The area of mutation for each sample in this image was the entire image. Each sample point had an equal probability to transport to any position in the image. This allowed the samples to converge much faster, though such a large proportional area may not be practical or possible in the general case.

 

Difference Image (intensity scale factor: 9.11)

 

 

 

Difference Image (intensity scale factor: 10)