sampling theory 101

8/7/2019 Sampling Theory 101

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Sampling Theory 101Oliver Kreylos

[email protected]

Center for Image Processing and Integrated Computing (CIPIC)

Department of Computer Science

University of California, Davis

CIPIC Seminar 11/06/02 p.


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Outline

Things that typically go wrong

Mathematical model of sampling

How to do things better



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Things That Typically Go Wrong



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Where Does Sampling Occur?

Almost all data we are dealing with is discrete

Evaluation of sampled functions at arbitrarysites

Volume rendering Isosurface extraction

Ray tracing

. . .



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What Went Wrong Here?

Typical ray tracing example:



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Sampling and Reconstruction



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Undersampling?

Sampling below Nyquist rate?

Quick solution: Double sampling rate



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Things get better, but are still bad

Isnt sampling above the Nyquist ratesupposed to solve all problems?



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Sampling Theory

To understand what went wrong on the last slide,

we need a mathematical model of sampling.



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Mathematical Model of Sampling



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Choice of Domain

Process of sampling and reconstruction is

best understood in frequency domain Use Fourier transform to switch between time

and frequency domains

Function in time domain: signal

Function in frequency domain: spectrum



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Fourier Transform I

Many functions f:R R can be written as

sums of sine waves

f(x) =

a sin(x + )

= 2 frequency is angular velocity,

a is amplitude, and

is phase shift

CIPIC Seminar 11/06/02 p.1


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Fourier Transform II

Moving to complex numbers simplifies

notation:

cos(x) + i sin(x) = eix

(Euler identity) One complex coefficient ce

ix encodes bothamplitude and phase shift

Moving to integral enlarges class ofrepresentable functions



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Fourier Transform III

Now, for almost all f:

f(x) =

F()eix d

F() is the spectrum of f(x), and the aboveoperator is the inverse Fourier transform

Its inverse, the Fourier transform, is

F() =

f(x)eix dx



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Some Signals and Their Spectra

Single sine wave f(x) = sin(x):



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Sum of two sine waves

f(x) = sin(x) + 0.5 sin(2x):



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Box function:



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Wider box function:



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Triangle function:



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Comb function:



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Wider comb function:



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Convolution I

The convolution operator is a generalized

formula to express weighted averaging of aninput signal f and a weight function or filterkernel g:

(f g)(x) =

f(t) g(x t) dt

One important application of convolution isreconstructing sampled signals



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Convolution II

(f g)(x0) = f(t) g(x0 t) dt

x

f

x

g

x0x

f

x0

x

f

x0

x

f

x0

x

f*g

x0



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Convolution III

Linear interpolation can be interpreted as

convolution:

x

f

x

g

x

f

x

f



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Function Reconstruction

Constant interpolation:



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Linear interpolation:



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Catmull-Rom interpolation:



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Cubic B-spline approximation:



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The Convolution Theorem

The Convolution Theorem relates convolution

and Fourier transform:(f g) F G

Convolutions can be computed by goingthrough the frequency domain:

(f g) = IFT(FT(f) FT(g))



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Sampling in Frequency Domain

The spectrum of s is the convolution of F

and C, the spectra of f and c C is a comb function with tap distance 2/d

Sconsists of shifted copies of F, each 2/dapart



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Sampling Process - Time Domain



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Sampling Process - Frequency Domai



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Sampling in Frequency Domain

If the sampling frequency is , the replicated

copies of fs spectrum are apart If the highest-frequency component in f is f,fs spectrum covers the interval [f, f]

If f /2, then the replicated spectraoverlap!

This means, information is lost



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The Sampling Theorem

If f is a frequency-limited function with

maximum frequency f, then f must besampled with a sampling frequency largerthan 2f in order to be able to exactly

reconstruct f from its samples. This theorem is sometimes called Shannons

Theorem

2f is sometimes called Nyquist rate



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Reconstruction

The only difference between F and S is that S

is made of shifted copies of F If those extra, higher-frequency, spectra could

be removed from S, f would magically

reappear Reconstructing f means low-pass-filtering s!



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Reconstruction Process - Frequency



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Reconstruction Process - Time Domai



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Reconstruction Problem

Practical problem: The optimal reconstruction

filter, the sinc function, has infinite support Evaluating it in the time domain involves

computing a weighted average of infinitely

many samples Practical reconstruction filters must have finite

support

But: A perfect low-pass filter with finitesupport does not exist!



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The Quest For Better Filters


C l U d Fil


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Commonly Used Filters

Constant filter:


C l U d Fil


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Linear filter:


C l U d Filt


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Catmull-Rom filter:



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S li Th Filt


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Sampling Theory Filters

Truncated sinc filter:


S li Th Filt


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Lanczos filter:




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Full sinc filter:



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Sampling Theory Applications



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Function Resampling


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Function Resampling

When resampling a sampled function to adifferent grid (shifted and/or scaled), it isimportant to use a good reconstruction filter

When downsampling a function, it is

important to use a good low-pass filter to cutoff frequencies above half the new samplingfrequency


Volume Rendering


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Volume Rendering

Basic question: How many samples to takeper cell?

d


Volume Rendering


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Volume Rendering

Basic question: How many samples to takeper cell?

d

Sampling theory says 1, but integrationalong ray might demand more


Ray Tracing


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Ray Tracing

In ray tracing, the function to be sampled istypically not frequency-limited

Practical solution: Shoot multiple rays perpixel (oversampling) and sample down to

image resolution Using a good low-pass filter is important


Ray Tracing


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Ray Tracing



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Conclusions


Conclusions


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Conclusions

Knowing the mathematical background ofsampling and reconstruction helps tounderstand common problems and devisesolutions

Ad-hoc solutions usually do not work well This stuff should be taught in class!



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The End


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