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Sampling Theory Sampling Theory • The world is continuous • Like it or not, images are discrete. Intro to Sampling Theory – 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 discrete is called sampling. Sampling Theory Sampling Theory • Signal - function that conveys information • Point Sampling – Audio signal (1D - function of time) – start with continuous signal – Image (2D - function of space) – calculate values of signal at discrete, evenly • Continuous vs. Discrete spaced points (sampling) – Continuous - defined for all values in range – convert back to continuous signal for display or output (reconstruction) – Discrete - defined for a set of discrete points in range. Sampling Theory Sampling Theory • Sampling can be described as creating a set of values representing a function evaluated at evenly spaced samples fn = f (i∆) i = 0,1,2,K,n ∆ = interval between samples = range / n. Foley/VanDam 1 Sampling Theory Issues: • Sampling Rate = number of samples per unit • Important features of a scene may be missed 1 • If view changes slightly or objects move f = slightly, objects may move in and out of ∆ visibility. • Example -- CD Audio • To fix, sample at a higher rate, but how high – sampling rate of 44,100 samples/sec does it need to be? – ∆ = 1 sample every 2.26x10-5 seconds Sampling Theory Sampling Theory • Rich mathematical foundation for sampling • Spatial vs frequency domains theory – Most well behaved functions can be described • Hope to give an “intuitive” notion of these as a sum of sin waves (possibly offset) at mathematical concepts 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 Sampling Theory Sampling Theory • Nyquist Theorum – A signal can be properly reconstructed if the signal is sampled at a frequency (rate) that is greater than twice the highest frequency component of the signal. – Said another way, if you have a signal with highest frequency component of fh, you need at lease 2fh samples to represent this signal accurately. Foley/VanDam 2 Sampling Theory Sampling Theory • Example -- CD Audio • Aliasing – Failure to follow the Nyquist Theorum results in – sampling rate of 44,100 samples/sec aliasing. – ∆ = 1 sample every 2.26x10-5 seconds – Aliasing is when high frequency components of a signal appear as low frequency due to inadequate • Using Nyquist Theorem sampling. – CDs can accurately reproduce sounds with • In CG: frequencies as high as 22,050 Hz. – Jaggies (edges) –Textures – Missed objects Sampling Theory Sampling Theory • Aliasing - example • Annoying Audio Applet – http://ptolemy.eecs.berkeley.edu/eecs20/week13/aliasin g.html Foley/VanDam(628) High frequencies masquerading as low frequencies Anti-Aliasing Fourier analysis • What to do in an aliasing situation • Given f(x) we can generate a function F(u) – Increase your sampling rate (supersampling) which indicates how much contribution – Decrease the frequency range of your signal each frequency u has on the function f. (Filtering) • F(u) is the Fourier Transform • How do we determine the contribution of • Fourier Transform has an inverse each frequency on our signal? 3 Sampling Theory Sampling Theory • Fourier Transforms • The Fourier transform is defined as: f(x) ∞ Inverse −i2πut Fourier F(u) Fourier F(u) = f (t)e dt Transform Transform ∫ −∞ f(x) Note: the Fourier Transform is defined in the complex plane Sampling Theory Sampling Theory • The Inverse Fourier transform is defined as: • How do we calculate the Fourier Transform? ∞ – Use Mathematics f (t) = F(u)ei2πut du – For discrete functions, use the Fast Fourier ∫ Transform algorithm (FFT) −∞ • Can filter the transform to remove offending high frequencies - partial solution to anti- aliasing Anti-aliasing -- Filtering Getting rid of High Frequencies • Removes high component frequencies from • Filtering -- Frequency domain a signal. – Place function into frequency domain F(u) – Simple multiplication with box filter S(u), aka pulse • Removing high frequencies results in function, band(width) limiting or low-pass filter. removing detail from the signal. • Can be done in the frequency or spatial ⎧1, when − k ≤ u ≤ k S(u) = ⎨ domain ⎩0, elsewhere – Suppress all frequency components above some specified cut-off point k 4 Filtering – Frequency Domain Getting Rid of High Frequencies Original Spectrum • Filtering -- Spatial Domain – Convolution (* operator) - equivalent to multiplying Low-Pass Filter two Fourier transforms ∞ Spectrum with Filter h(x) = f (x)∗ g(x) = ∫ f (τ )g(x −τ )dτ −∞ Taking a weighted average of the neighborhood Filtered Spectrum around each point of f, weighted by g (the convolution or filter kernel) centered at that point. Foley/VanDam(631) Filtering using Convolution Convultion sinc Function Original Spectrum • Convolving with a sinc function in the spatial domain is the same as using a box filter in the frequency Sinc Filter domain FT→ Spectrum with Filter value of filtered signal Filtered Spectrum ←FT-1 Foley/VanDam (634) Foley/VanDam (633) Convolution Sampling Theory • Joy of Convolution applet • Anti-aliasing -- Filtering – Removes high component frequencies from a signal. http://www.jhu.edu/~Esignals/convolve/index.html – Removing high frequencies results in removing detail from the signal. – Can be done in the frequency or spatial domain 5 Sampling Theory Sampling Theory • 2D Sampling • 2D Aliasing – Images are examples of sampling in 2- dimensions. – 2D Fourier Transforms provides strength of signals at frequencies in the horizontal and vertical directions aliased image anti-aliased image Foley/VanDam Sampling Theory Sampling Theory • 2D Fourier Transform ∞ ∞ ÆÆÆÆ F(u,v) = ∫ ∫ f (x, y)e−i2π (ux+vy)dxdy −∞ −∞ Castleman Sampling Theory Sampling Theory • Filtering - Convolution in 2D • Filtering – Convolution with images Castleman Castleman 6 Sampling Theory Other Anti-aliasing Methods • Filtering – Convolution in frequency domain • Pre-filtering - filtering at object precision before calculating pixel’s sample • Post-filtering - supersampling (as we’ve seen) • Adaptive supersampling - sampling rate is varied, applied only when needed (changes, edges, small items) Image 2D FFT Filter out Filtered high 2D FFT • Stochastic supersampling - places samples at frequencies stocastically determined positions rather than regular Castleman grid Anti-Aliasing Sampling Theory • Applet •Summary http://www.nbb.cornell.edu/neurobio/land/OldStu – Digital images are discrete with finite resolution…the world is not. dentProjects/cs490- – Spatial vs. Frequency domain 96to97/anson/AntiAliasingApplet/index.html – Nyquist Theorum – Convolution and Filtering – 2D Convolution & Filtering – Questions? Sampling Theory Remember • Further Reading • Class Web Site: – Foley/VanDam – Chapter 14 – http://www.cs.rit.edu/~jmg/cgII – Digital Image Processing by Kenneth • Any questions? Castleman – Glassner, Unit II (Book 1) 7.
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