Logistics Checkpoint 1 all graded Intro to Sampling Theory Grades - PDF document

Logistics  Checkpoint 1 all graded Intro to Sampling Theory  Grades / comments on mycourses  Project Proposals due tonight  Please place in mycourses dropbox  Questions? Plan for today Approaches to modeling in CG  Sampling Theory  How does one describe reality?  Empirical -- Use measured data  Fixed model  Procedural Modeling  Physical simulation  Heuristic Sampling Theory Sampling in modeling  The world is continuous  Like it or not, CG is discrete.  We work using a discrete array of pixels  We use discrete values for color  We use discrete arrays and subdivisions for specifying textures and surfaces  Process of going from continuous to Digibotics Laser Scanner 3D Scanner discrete is called sampling. 1

Sampling in reflectance Sampling in modeling modeling goniometer Surface subdivision Sampling in textures Sampling in textures Point sampled textures Filtered textures www.sharkyextreme.com www.sharkyextreme.com Sampling in image generation Sampling Theory  Signal - function that conveys information  Audio signal (1D - function of time)  Image (2D - function of space)  Continuous vs. Discrete  Continuous - defined for all values in range  Discrete - defined for a set of discrete points in range. 2

Sampling Theory Sampling Theory  Point Sampling  start with continuous signal  calculate values of signal at discrete, evenly spaced points (sampling)  convert back to continuous signal for display or output (reconstruction) Foley/VanDam Sampling Theory Sampling Theory  Sampling Rate = number of samples per unit  Sampling can be described as creating a set of values representing a function = 1 evaluated at evenly spaced samples f � f n f ( i ) i 0 , 1 , 2 , , n K = � =  Example -- CD Audio Δ = interval between samples = range / n.  sampling rate of 44,100 samples/sec  Δ = 1 sample every 2.26x10 -5 seconds Issues: Sampling Theory  Important features of a scene may be  Rich mathematical foundation for missed sampling theory  If view changes slightly or objects move  Hope to give an “intuitive” notion of slightly, objects may move in and out of these mathematical concepts visibility.  To fix, sample at a higher rate, but how high does it need to be? 3

Sampling Theory Sampling Theory  Spatial vs frequency domains  Most well behaved functions can be described as a sum of sin waves (possibly offset) at various frequencies  Frequency specturm - a function by the contribution (and offset) at each frequency is describing the function in the frequency domain  Higher frequencies equate to greater detail Foley/VanDam Sampling Theory Sampling Theory  Interactive wave builder  Annoying audio applet 1  Fourier Synthesis -- Graphic Equalizer  http://www.sunsite.ubc.ca/LivingMathe  http://www.phy.ntnu.edu.tw/ntnujava/vie matics/V001N01/UBCExamples/Fourier/f wtopic.php?t=33 ourier.html Sampling Theory Sampling Theory  Nyquist Theorum  Example -- CD Audio  A signal can be properly reconstructed if the signal  sampling rate of 44,100 samples/sec is sampled at a frequency (rate) that is greater  Δ = 1 sample every 2.26x10 -5 seconds than twice the highest frequency component of the signal.  Using Nyquist Theorem  Said another way, if you have a signal with highest frequency component of f h , you need at  CDs can accurately reproduce sounds with lease 2f h samples to represent this signal frequencies as high as 22,050 Hz. accurately. 4

Sampling Theory Sampling Theory  Aliasing  Aliasing - example  Failure to follow the Nyquist Theorum results in aliasing .  Aliasing is when high frequency components of a signal appear as low frequency due to inadequate sampling.  In CG:  Jaggies (edges) Foley/VanDam(628)  Textures  Missed objects High frequencies masquerading as low frequencies Sampling Theory Anti-Aliasing  Annoying Audio Applet 2  What to do in an aliasing situation  Aliasing  Increase your sampling rate (supersampling)  Decrease the frequency range of your signal  http://ptolemy.eecs.berkeley.edu/eecs20/week13/ (Filtering) aliasing.html  How do we determine the contribution of each frequency on our signal? Fourier analysis Sampling Theory  Fourier Transforms  Given f(x) we can generate a function f(x) F(u) which indicates how much Inverse contribution each frequency u has on Fourier F(u) Fourier Transform the function f. Transform  F(u) is the Fourier Transform f(x)  Fourier Transform has an inverse 5

Sampling Theory Sampling Theory  The Fourier transform is defined as:  The Inverse Fourier transform is defined as: � � i � 2 ut i � 2 ut F ( u ) f ( t ) e dt f ( t ) F ( u ) e du � = � = � � � � � Note: the Fourier Transform is defined in the complex plane Signals in the frequency domain Sampling Theory  http://www.falstad.com/fourier/  How do we calculate the Fourier Transform?  Use Mathematics  For discrete functions, use the Fast Fourier Transform algorithm (FFT)  Break… Anti-Aliasing Supersampling  Increase your sampling rate  What to do in an aliasing situation  Examples  Increase your sampling rate  Resolution in images (supersampling)  Number of subdivisions in modeling  Decrease the frequency range of your  Number of sample points signal (Filtering)  Number of rays per pixel.  May not always have this luxury 6

Supersampling Anti-aliasing -- Filtering  Can filter the transform to remove offending high frequencies - partial solution to anti-aliasing  Removes high component frequencies from a signal.  Removing high frequencies results in removing detail from the signal.  Can be done in the frequency or spatial domain Wikipedia Filtering – Frequency Domain Getting rid of High Frequencies Original Spectrum  Filtering -- Frequency domain  Place function into frequency domain F(u) Low-Pass Filter  Simple multiplication with box filter S(u), aka pulse function , band(width) limiting or low-pass filter. Spectrum with Filter 1 , when k u k � � � � S ( u ) = � 0 , elsewhere � Filtered Spectrum  Suppress all frequency components above some specified cut-off point k Foley/VanDam(631) Getting Rid of High Convultion sinc Function Frequencies  Filtering -- Spatial Domain  Convolving with a sinc function in the spatial domain is the same as using a box filter in the  Convolution (* operator) - equivalent to multiplying two Fourier transforms frequency domain FT → � h ( x ) f ( x ) g ( x ) f ( ) g ( x ) d = � = � � � � � � � Taking a weighted average of the neighborhood around each point of f , weighted by g (the ← FT -1 convolution or filter kernel ) centered at that point. Foley/VanDam (634) 7

Filtering using Convolution Convolution Original Spectrum  Joy of Convolution applet Sinc Filter http://www.jhu.edu/~signals/convolve/index. Spectrum with Filter html value of filtered signal Filtered Spectrum Foley/VanDam (633) Sampling Theory Sampling Theory  Anti-aliasing -- Filtering  2D Sampling  Removes high component frequencies from  Images are examples of sampling in 2- a signal. dimensions.  Removing high frequencies results in  2D Fourier Transforms provides strength of removing detail from the signal. signals at frequencies in the horizontal and vertical directions  Can be done in the frequency or spatial domain Sampling Theory Sampling Theory  2D Aliasing  2D Fourier Transform � � i 2 ( ux vy ) F ( u , v ) f ( x , y ) e � � + dxdy = � � � � � � aliased image anti-aliased image Foley/VanDam 8

Sampling Theory Sampling Theory  Filtering - Convolution in 2D     Castleman Castleman Sampling Theory Sampling Theory  Filtering – Convolution with images  Filtering – Convolution in frequency domain Image 2D FFT Filter out Filtered high 2D FFT frequencies Castleman Castleman Other Anti-aliasing Methods Anti-Aliasing  Applet  Pre-filtering - filtering at object precision before calculating pixel’s sample  Post-filtering - supersampling (as we’ve seen) http://www.nbb.cornell.edu/neurobio/land/Ol  Adaptive supersampling - sampling rate is dStudentProjects/cs490- varied, applied only when needed (changes, 96to97/anson/AntiAliasingApplet/index.html edges, small items)  Stochastic supersampling - places samples at stocastically determined positions rather than regular grid 9

Sampling Theory Sampling Theory  Summary  Further Reading  Digital images are discrete with finite  Foley/VanDam – Chapter 14 resolution…the world is not.  Digital Image Processing by Kenneth  Spatial vs. Frequency domain Castleman  Nyquist Theorum  Glassner, Unit II (Book 1)  Convolution and Filtering  2D Convolution & Filtering  Questions? Remember  Class Web Site:  http://www.cs.rit.edu/~jmg/cgII  Any questions? 10