Cs 445 645 introduction to computer graphics
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CS 445 / 645 Introduction to Computer Graphics. Lecture 20 Antialiasing. Environment Mapping. Used to model a object that reflects surrounding textures to the eye Polished sphere reflects walls and ceiling textures Cyborg in Terminator 2 reflects flaming destruction

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Cs 445 645 introduction to computer graphics

CS 445 / 645Introduction to Computer Graphics

Lecture 20


Environment mapping

Environment Mapping

  • Used to model a object that reflects surrounding textures to the eye

    • Polished sphere reflects walls and ceiling textures

    • Cyborg in Terminator 2 reflects flaming destruction

  • Texture is distorted fish-eye view of environment

  • Spherical texture mapping creates texture coordinates that correctly index into this texture map

Materials from nvidia

Materials from NVidia

  • http://developer.nvidia.com/object/Cube_Mapping_Paper.html

Sphere mapping

Sphere Mapping

Blinn newell lattitude mapping

Blinn/Newell Lattitude Mapping

Cube mapping

Cube Mapping



  • Pipelining of multiple texture applications to one polygon

  • The results of each texture unit application is passed to the next texture unit, which adds its effects

  • More bookkeeping is required to pull this off



What is a pixel

What is a pixel?

  • A pixel is not…

    • A box

    • A disk

    • A teeny tiny little light

  • A pixel is a point

    • It has no dimension

    • It occupies no area

    • It cannot be seen

    • It can have a coordinate

A pixel is more than a point, it is a sample



  • Most things in the real world are continuous

  • Everything in a computer is discrete

  • The process of mapping a continuous function to a discrete one is called sampling

  • The process of mapping a continuous variable to a discrete one is called quantization

  • Rendering an image requires sampling and quantization





Line segments

Line Segments

  • We tried to sample a line segment so it would map to a 2D raster display

  • We quantized the pixel values to 0 or 1

  • We saw stair steps, or jaggies

Line segments1

Line Segments

  • Instead, quantize to many shades

  • But what sampling algorithm is used?

Area sampling

Area Sampling

  • Shade pixels according to the area covered by thickened line

  • This is unweighted area sampling

  • A rough approximation formulated by dividing each pixel into a finer grid of pixels

Unweighted area sampling

Unweighted Area Sampling

  • Primitive cannot affect intensity of pixel if it does not intersect the pixel

  • Equal areas cause equal intensity, regardless of distance from pixel center to area

Weighted area sampling

Weighted Area Sampling

  • Unweighted sampling colors two pixels identically when the primitive cuts the same area through the two pixels

  • Intuitively, pixel cut through the center should be more heavily weighted than one cut along corner

Weighted area sampling1




Weighted Area Sampling

  • Weighting function, W(x,y)

    • specifies the contribution of primitive passing through the point (x, y) from pixel center



  • An image is a 2D function I(x, y) that specifies intensity for each point (x, y)

Sampling and image

Sampling and Image

  • Our goal is to convert the continuous image to a discrete set of samples

  • The graphics system’s display hardware will attempt to reconvert the samples into a continuous image: reconstruction

Point sampling an image

Point Sampling an Image

  • Simplest sampling is on a grid

  • Sample dependssolely on valueat grid points

Point sampling

Point Sampling

  • Multiply sample grid by image intensity to obtain a discrete set of points, or samples.

Sampling Geometry

Sampling errors

Sampling Errors

  • Some objects missed entirely, others poorly sampled

Fixing sampling errors

Fixing Sampling Errors

  • Supersampling

    • Take more than one sample for each pixel and combine them

      • How many samples is enough?

      • How do we know no features are lost?

150x15 to 100x10

200x20 to 100x10

300x30 to 100x10

400x40 to 100x10

Unweighted area sampling1

Unweighted Area Sampling

  • Average supersampled points

  • All points are weighted equally

Weighted area sampling2

Weighted Area Sampling

  • Points in pixel are weighted differently

    • Flickering occurs as object movesacross display

  • Overlapping regions eliminates flicker

Signal theory

Signal Theory

  • Convert spatial signal to frequency domain


Pixel position across scanline

Example from Foley, van Dam, Feiner, and Hughes

Signal theory1

Signal Theory

  • Represent spatial signal as sum of sine waves (varying frequency and phase shift)

  • Very commonlyused to representsound “spectrum”

Fourier analysis

Fourier Analysis

  • Convert spatial domain to frequency domain

    • Let f(x) indicate the intensity at a location in space, x (pixel value)

    • u is a complex number representing frequency and phase shift

      • i = sqrt (-1) … frequently not plotted

    • F(u) is the amplitude of a particular frequency in a signal

      • In this case the signal is f(x)

Fourier transform

Fourier Transform

  • Examples of spatial and frequency domains

Nyquist sampling theorem

Nyquist Sampling Theorem

  • The ideal samples of a continuous function contain all the information in the original function if and only if the continuous function is sampled at a frequency greater than twice the highest frequency in the function

Nyquist rate

Nyquist Rate

  • The lower bound on the sampling rate equals twice the highest frequency component in the image’s spectrum

  • This lower bound is the Nyquist Rate

Band limited signals

Band-limited Signals

  • If you know a function contains no components of frequencies higher than x

    • Band-limited implies original function will not require any ideal functions with frequencies greater than x

    • This facilitates reconstruction

Flaws with nyquist rate

Flaws with Nyquist Rate

  • Samples may not align with peaks

Flaws with nyquist rate1

Flaws with Nyquist Rate

  • When sampling below Nyquist Rate, resulting signal looks like a lower-frequency one

    • With no knowledge of band-limits, samples could have been derived from signal of higher frequency

Low pass filtering

Low-pass Filtering

  • We know we are limited in the resolution of our screen

  • We want the screen (sampling grid) to have twice the resolution of the signal (image) we want to display

  • How can we reduce the high-frequencies of the image?

    • Low-pass filter

    • Band-limits the image

Low pass filtering1

Low-pass Filtering

  • In frequency domain

    • If signal is F(u)

    • Just chop off parts of F(u) in high frequencies using a second function, G(u)

      • G(u) == 1 when –k <= u <= k0 elsewhere

      • This is called the pulse function

Low pass filtering2

Low-pass Filtering

  • In spatial domain

    • Multiplying two Fourier transforms in the spatial domain corresponds exactly to performing an operation called convolution in the spatial domain

    • f(x) * g(x) = h(x)  the convolution of f with g…

      • The value of h(x) at x is the integral of the product of f(x) with the filter g(x) such that g(x) is centered at x

    • The pulse (frequency) == sinc (spatial)

Low pass filtering3

Low-pass Filtering

  • In spatial domain

  • Sinc: sinc(x) = sin (px)/px

    • Note this isn’t perfect way to eliminate high frequencies

      • “ringing” occurs

Sinc filter

Sinc Filter

  • Slide filter along spatial domain and compute new pixel value that results from convolution

Bilinear filter

Bilinear Filter

  • Sometimes called a tent filter

  • Easy to compute

    • just linearly interpolate between samples

  • Finite extent and no negative values

  • Still has artifacts

Sampling pipeline

Sampling Pipeline

How is this done today full screen antialiasing

How is this done today?Full Screen Antialiasing

  • Nvidia GeForce2

    • OpenGL: render image 400% larger and supersample

    • Direct3D: render image 400% - 1600% larger

  • Nvidia GeForce3

    • Multisampling but with fancy overlaps

      • Don’t render at higher resolution

      • Use one image, but combine values of neighboring pixels

      • Beware of recognizable combination artifacts

        • Human perception of patterns is too good



  • Multisampling

    • After each pixel is rendered, write pixel value to two different places in frame buffer

Geforce3 multisampling

GeForce3 - Multisampling

  • After rendering two copies of entire frame

    • Shift pixels of Sample #2 left and up by ½ pixel

    • Imagine laying Sample #2 (red) over Sample #1 (black)

Geforce3 multisampling1

GeForce3 - Multisampling

  • Resolve the two samples into one image by computing average between each pixel from Sample 1 (black) and the four pixels from Sample 2 (red) that are 1/ sqrt(2) pixels away

Geforce3 multisampling2

GeForce3 - Multisampling

  • No AA Multisampling

Cs 445 645 introduction to computer graphics

GeForce3 - Multisampling

  • 4x Supersample Multisampling

Ati smoothvision

ATI Smoothvision

  • ATI SmoothVision

    • Programmer selects samping pattern

    • Some supersamplingthrown in

Ati nvidia comparison

ATI NVidia comparison

  • ATI Radeon 8500 NVidia GF3 Ti 500 AA Off 2x AA Quincunx AA


Cs 445 645 introduction to computer graphics


  • 3dfx Multisampling

    • 2- or 4-frame shift and average

  • Tradeoffs?

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