CS 445 / 645 Introduction to Computer Graphics Lecture 20 Antialiasing Environment Mapping Used to model a object that reflects surrounding textures to the eye •
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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 Texture is distorted fisheye view of environment Spherical texture mapping creates texture coordinates that correctly index into this texture map Materials from NVidia http://developer.nvidia.com/object/Cube_Mapping_Paper.html Sphere Mapping Blinn/Newell Lattitude Mapping Cube Mapping Multitexturing 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 Antialiasing 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 Samples 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 Samples Samples 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 Segments Instead, quantize to many shades But what sampling algorithm is used? 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 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 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 Sampling Weighting function, W(x,y) • specifies the contribution of primitive passing through the point (x, y) from pixel center Intensity W(x,y) x Images An image is a 2D function I(x, y) that specifies intensity for each point (x, y) 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 Simplest sampling is on a grid Sample depends solely on value at grid points Point Sampling Multiply sample grid by image intensity to obtain a discrete set of points, or samples. Sampling Geometry Sampling Errors Some objects missed entirely, others poorly sampled 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 Sampling Average supersampled points All points are weighted equally Weighted Area Sampling Points in pixel are weighted differently • Flickering occurs as object moves across display Overlapping regions eliminates flicker Signal Theory Intensity Convert spatial signal to frequency domain Pixel position across scanline Example from Foley, van Dam, Feiner, and Hughes Signal Theory Represent spatial signal as sum of sine waves (varying frequency and phase shift) Very commonly used to represent sound “spectrum” 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 Examples of spatial and frequency domains 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 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 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 Samples may not align with peaks 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 • 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 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 <= k 0 elsewhere – This is called the pulse function 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 Filtering In spatial domain Sinc: sinc(x) = sin (px)/px • Note this isn’t perfect way to eliminate high frequencies – “ringing” occurs Sinc Filter Slide filter along spatial domain and compute new pixel value that results from convolution 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 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 GeForce3 Multisampling • After each pixel is rendered, write pixel value to two different places in frame buffer 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 - 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 - Multisampling No AA Multisampling GeForce3 - Multisampling • 4x Supersample Multisampling ATI Smoothvision ATI SmoothVision • Programmer selects samping pattern • Some supersampling thrown in ATI NVidia comparison ATI Radeon 8500 AA Off 2x AA NVidia GF3 Ti 500 Quincunx AA http://www.anandtech.com/video/showdoc.html?i=1562&p=4 3dfx 3dfx Multisampling • 2- or 4-frame shift and average Tradeoffs?