technische universität münchen fakultät für informatik computer graphics ss 2014 sampling...

Technische Universität München

Fakultät für Informatik

Computer GraphicsSS 2014

Sampling

Rüdiger WestermannLehrstuhl für Computer Graphik und Visualisierung


Computer Graphics

2

Overview

• So far:– Image synthesis

• Ray tracing; models, transformations, shading & lighting, textures, acceleration

• Today– Aliasing and antialiasing techniques

• Prefiltering • Supersampling • Postfiltering• Stochastic and adaptive sampling


Computer Graphics

3

Sampling

• Mathematically, point sampling using regularly spaced sample points is the multiplication of the function with a comb function

0 xT 2T


Computer Graphics

4

Sampling

• Recall: image synthesis means point sampling of a continuous signal– Image contains samples of a continuous signal at a

discrete set of positions– Pixel spacing determines the frequencies (the size of

details) that can be reconstructed– Undersampling the image (taking too less samples to

allow for the reconstruction of the signal from the samples) causes aliasing artifacts (alias = ghost)


Computer Graphics

5

Aliasing

• Original scene and luminosity (brightness) distribution along a scan line

5


Computer Graphics

6

Aliasing

• Point sampling the scene at pixel centers

6


Computer Graphics

7

Aliasing

• The rendered image

7


Computer Graphics

8

Aliasing

• Jagged profiles

8


Computer Graphics

9

Aliasing

• Loss of details

9


Computer Graphics

10

Aliasing

• Note: the sampling frequency decreases with increasing distance to the viewpoint

d0 d1 d2 d3viewpoint


Computer Graphics

11

Aliasing

• With increasing distance to the viewer and slope of the surface, an ever larger surface area falls in-between adjacent rays


Computer Graphics

12

Aliasing

• Cause of aliasing– Sampling frequency is not high enough to cover all

details– It is below the Niquist limit

Shannons Sampling Theorem: The signal has to be sampled at a frequency that is equal to or higher than two times the highest frequency in the signal


Computer Graphics

13

Aliasing

• Aliasing artefacts– Spatial aliasing

– Temporal aliasing


Computer Graphics

14

Antialiasing

• How to avoid aliasing caused by an undersampling of the signal, i.e. the sampling frequency is not high enough to cover all details– Supersampling - increase sampling frequency– Prefiltering - decrease the highest frequency in the

signal, i.e. filter the signal before sampling– Postfiltering – filtering after sampling, but just blurres

the image


Computer Graphics

15

Antialiasing

• Supersampling - increase sampling frequency– Use more rays per pixel, i.e. virtually increase the

resolution of the pixel raster– e.g. use 4x4 rays per pixel and compute average of all

16 colors as final pixel color– Sharp edges are washed out– OK, but doesn´t eliminate aliasing because sharp edges

contain infinitely high frequencies


Computer Graphics

16

Antialiasing

• Supersampling– Regular supersampling


Computer Graphics

17

Antialiasing

• Filtering: average weighted samples


Computer Graphics

18

Antialiasing

• Filtering example


Computer Graphics

19

Antialiasing

• Filtering - jittered instead of regular sampling


Computer Graphics

20

Antialiasing

• Regular sampling– Visibility of aliases also caused by the regular sampling

grid– Human visual system is sensitive against regular

structures, but rather insensitive against high frequency noise

• Stochastic supersampling– Place samples randomly within pixel– Alias frequencies are converted to noise– But can result in clusters of sample


Computer Graphics

21

Antialiasing

• Poisson-disk sampling– Random generation of samples with

limit for the minimum distance between samples

• Jittered sampling– Random jittering from regular grid points

• Stratified random sampling– Regular partitioning of pixel region– One random sample per partition


Computer Graphics

22

Antialiasing

• Comparison– Regular, 1x1

Regular 3x3

Regular, 7x7

Jittered, 3x3

Jittered, 7x7

22


Computer Graphics

23

Antialiasing

• Example:

23


Computer Graphics

24

Antialiasing

• Example:

24


Computer Graphics

25

Antialiasing

• Example:

25


Computer Graphics

26

Antialiasing

• Example:

26


Computer Graphics

27

Antialiasing

• Prefiltering – Antialiasing before sampling (mainly used in texture

mapping)• Filtering (smoothing) a signal to remove details below the

frequency which is used to sample the signal


Computer Graphics

28

Antialiasing

• Prefiltering combines color contributions into a pixel


Computer Graphics

Antialiasing in texture mapping

• Many texels fall onto one pixels

29


Computer Graphics


• Mip-Mapping:prefiltered levels of detail (LOD) in a pyramid

• At every level:average 2x2texels from thefiner levelinto one texel30


Computer Graphics


31


Computer Graphics


32


Computer Graphics


• Mip-Mapping: how does it work?– When a fragment is texture mapped, the mip-map level

at which the texel size is equal to the pixel size is computed

– From this level the texture is then sampled

– It remains to be answered how the level is computed

33


Computer Graphics


• We want to know what the size of one texel wrt the size of one pixel is – this allows estimating how many texels fall into one pixel

34


Computer Graphics


• Computing the mip-map level:Check screen pixel “size” in texture coordinates

35less than one texel per pixelwe call this magnification

more than one texel per pixelwe call this minification

u,v: fragments texture coordinatesx,y: pixel coordinates


Computer Graphics


36


Computer Graphics


37 without MipMapping with MipMapping


Computer Graphics


38Isotropic filtering

(mipmapping)Anisotropic filtering


Computer Graphics


• Mip-Mapping– The mip-map only stores a discrete set of levels – If pixel size matches texel size at a level in between, it

is interpolated between the two adjacent levels

39

Sample 0

Sample 1

interpolateassignSelect

resolution

Bilinear in texture + linear between levels

= trilinear


Computer Graphics

40

Summary

• Quality of rendering strongly depends on antialiasing algorithms used– Typically, supersampling in combination with

prefiltering is used• Supersampling and mipmapping

• We are now ready to implement a high quality and efficient ray tracing algorithm

• What comes next is an alternative image synthesis approach based on the projection of geometry onto the image plane

technische universität münchen fakultät für informatik computer graphics ss 2014 sampling...

Documents