Monte Carlo Methods: Anti-aliasing in computer graphics

Prof: Michael MascagniPresented by: Aneesh Sharma

Date: June 20, 2011

Introduction Problem description – Aliasing Anti-aliasing Sampling techniques Results Discussion and analysis Conclusion References

Everything in computer is discrete.

The process of mapping a continuous

function to a discrete one is called

sampling.

To represent or render an image in

computer graphics we need sampling.

Due to the fact that the world is sampled and then rendered, possibilities for errors exist.

There are basically two types of aliasing:◦ Temporal aliasing

Because of the limitation of number of frames per second.

E.g. car’s wheel appears to rotate too slowly or even backwards

◦ Spatial aliasing Because of the fact that only a finite number of rays

can be sampled from the scene.  E.g. representing a high-resolution image at a lower

resolution. (downsampling)

One of the causes of aliasing in computer graphics is the fact that we looses information while down-sampling the data.

Technique of minimizing the distortion when representing continuous objects in discrete systems.

Left: aliased image, right: antialiased image.

focus: different sampling techniques.

Because pixels are discrete, computer graphics(CG) is inherently a sampling process.

Ray tracing, and other forms of point sampling are important techniques in CG.

The purpose of ray tracing is to recreate photo-realistic 3D images on a 2D computer screen. 

Because of regularly spaced samples, these techniques suffer from Aliasing.

Discrepancy analysis measures sample point equi-distribution.

Gives an idea of how sampling pattern captures details in a picture.

Two low discrepancy sequences:◦ Hammersley◦ Halton

Each non-negative integer k, is represented using prime base p.

So, a k-th d-dimensional Hammersely point is

Where n = total number of Hammersely points.

Problem with Hammersley pointset: non-heirarchial.

Means, different values of n results in different sets of points.

Halton pointset: heirarchial

φ function is heirarchial, and so are the halton point sets.

The work considers 2-D plane, and uses different prime base for the experiment.

Total number of sampled points considered = 500.

Comparison between random, Hammersley, and Halton point sets.

Solution: Scrambling – method used RR2 (reverse radix sampling)

Hammersley points with p=2 gives the most uniform distributed patterns.

Halton point set gives patterns with varying uniformity and regularity.

Halton point set suffers from holes when higher dimensions (3-D or more) is considered.

As p1 increases hammersley points tends to align and reduce their usefulness.

Halton point doesn’t suffer from this problem and it allows incremental sampling.

Further, author applied these methods to ray tracing applications with significant improvement in pixel error.

The complexity of both Hammersley and Halton points generation algorithms is O( NlogpN ).

These sampling techniques are applied to path planning paradigm.

3 link (equal length) manipulator (robotic arm).

End location: (19.501032, 9.984035)

Distance to goal: 0.499223 cm

Path length: 56.182854 cm Path steps: 53 Iterations: 3024 Used samples: 1565/13516

(11.5789 %) Computation time: 0.067694

s

End location: (20.028568, 9.795012)

Distance to goal: 0.206970 cm

Path length: 63.798946 cm

Path steps: 65 Iterations: 3185 Used samples:

1642/14230 (11.539 %) Computation time:

0.013087 s

Halton sampling Random Sampling

Computer graphics Monte Carlo volume rendering

◦ Medical x-ray images◦ The quality of images is strongly influenced by

the transfer function, which maps a data value onto a sampling probability.

Quasi-Monte Carlo integration

1. Robert A. Cross. Sampling patterns optimized for uniform distribution of edges. In Graphics Gems V, pages 359–363. AP Professional, 1995.

2. D. P. Dobkin and D. Eppstein. Computing the discrepancy. In Proceedings of the 9th ACM Symposium on Computational Geometry, pages 47–52, 1993.

3. D. P. Dobkin and D. P. Mitchell. Random-edge discrepancy of supersampling patterns, In Graphics Interface, pages 62–69, 1993.

4. StefanHeinrich andAlexanderKeller. Quasi-monte carlomethods in computer graphics, part i: The qmc buffer. Technical report, University of Kaiserslautern, 1994.

5. R. L. Cook, Stochastic Sampling in Computer Graphics. ACM Transactions on Graphics, Vol. 5, No. 1, pages 51-72, 1986.

6. T. T. Wong, W. S. Luk, and P. A. Heng, Sampling with Hammersley and Halton Points, Graphics tools: The jgt editors' choice, 2005.

7. KennethChiu, Peter Shirley, and ChangyawWang. Multi-jittered sampling. In GraphicsGems IV, pages 370–374. AP Professional, 1994.