Sampling and Reconstruction - TU Wien
Transcript of Sampling and Reconstruction - TU Wien
![Page 1: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/1.jpg)
Institute of Computer Graphics and Algorithms
Vienna University of Technology
Sampling and Reconstruction
Peter Rautek, Eduard Gröller, Thomas Theußl
![Page 2: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/2.jpg)
Motivation
Theory and practice of sampling and reconstructionAliasing
Understanding the problem Handling the problem
Examples:Photography/VideoRenderingComputed TomographyCollision detection
Eduard Gröller, Thomas Theußl, Peter Rautek 1
![Page 3: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/3.jpg)
OverviewIntroductionTools for Sampling and Reconstruction
Fourier TransformConvolution (dt.: Faltung)Convolution TheoremFiltering
SamplingThe mathematical model
Reconstruction Sampling TheoremReconstruction in Practice
Eduard Gröller, Thomas Theußl, Peter Rautek 2
![Page 4: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/4.jpg)
Image Data
Eduard Gröller, Thomas Theußl, Peter Rautek 3
Demo - Scan Lines- Sampling
http://www.cs.brown.edu/exploratories/freeSoftware/repository/edu/brown/cs/exploratories/applets/sampling/introduction_to_sampling_java_browser.html
![Page 5: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/5.jpg)
Image Storage and Retrieval
Eduard Gröller, Thomas Theußl, Peter Rautek 4
![Page 6: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/6.jpg)
Sampling Problems
Eduard Gröller, Thomas Theußl, Peter Rautek 5
![Page 7: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/7.jpg)
OverviewIntroductionTools for Sampling and Reconstruction
Fourier TransformConvolution (dt.: Faltung)Convolution TheoremFiltering
SamplingThe mathematical model
Reconstruction Sampling TheoremReconstruction in Practice
Eduard Gröller, Thomas Theußl, Peter Rautek 6
![Page 8: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/8.jpg)
Sampling Theory
Relationship between signal and samples View image data as signals Signals are plotted as intensity vs. spatial
domain Signals are represented as sum of sine waves
- frequency domain
Eduard Gröller, Thomas Theußl, Peter Rautek 7
![Page 9: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/9.jpg)
Square Wave Approximation
Eduard Gröller, Thomas Theußl, Peter Rautek 8
![Page 10: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/10.jpg)
Fourier Series
Eq1:
Eq2:
Eq3:
Euler’s identity: Eduard Gröller, Thomas Theußl, Peter Rautek 9
1
0 )cos(2
)f(n
nn xnAax
xixeix sincos
1
0 )sin()cos(2
)f(n
nn xnbxnaax
n
xinnecx )f(
![Page 11: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/11.jpg)
Fourier Transform
Link between spatial and frequency domain
Eduard Gröller, Thomas Theußl, Peter Rautek 10
dex
dxex
xi
i
ωπ2
ωxπ2
)ωF()f(
)f()ωF(
![Page 12: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/12.jpg)
Box & Tent
Eduard Gröller, Thomas Theußl, Peter Rautek 11
![Page 13: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/13.jpg)
Square Wave & Scanline
Eduard Gröller, Thomas Theußl, Peter Rautek 12
![Page 14: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/14.jpg)
Fourier Transform
Yields complex functions for frequency domain
Extends to higher dimensions Complex part is phase information - usually
ignored in explanation and visual analysis
Alternative: Hartley transformWavelet transform
Eduard Gröller, Thomas Theußl, Peter Rautek 13
![Page 15: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/15.jpg)
Discrete Fourier Transform
For discrete signals (i.e., sets of samples)
Complexity for N samples: DFT O(N²)Fast FT (FFT): O(N log N)
Eduard Gröller, Thomas Theußl, Peter Rautek 14
N2-1-N
0ω
N21-N
0ω
)xf(N1)ωF(
)ωF()f(
xj
xj
e
ex
Demo - Fast Fourier Transform- Different Frequencies
http://www.cs.brown.edu/exploratories/freeSoftware/repository/edu/brown/cs/exploratories/applets/fft1DApp/1d_fast_fourier_transform_java_browser.html
![Page 16: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/16.jpg)
OverviewIntroductionTools for Sampling and Reconstruction
Fourier TransformConvolution (dt.: Faltung)Convolution TheoremFiltering
SamplingThe mathematical model
Reconstruction Sampling TheoremReconstruction in Practice
Eduard Gröller, Thomas Theußl, Peter Rautek 15
![Page 17: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/17.jpg)
Convolution
Operation on two functions Result: Sliding weighted average of a function
The second function provides the weights
Eduard Gröller, Thomas Theußl, Peter Rautek 16
dgxfxgf )()())((
Demo - Copying (Dirac Pulse)- Averaging (Box Filter)
http://www.cs.brown.edu/exploratories/freeSoftware/repository/edu/brown/cs/exploratories/applets/convolution/convolution_java_browser.html
![Page 18: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/18.jpg)
Convolution - Examples
Eduard Gröller, Thomas Theußl, Peter Rautek 17
![Page 19: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/19.jpg)
Convolution Theorem
The spectrum of the convolution of two functions is equivalent to the product of the transforms of both input signals, and vice versa.
Eduard Gröller, Thomas Theußl, Peter Rautek 18
2121
2121
ffFFFFff
![Page 20: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/20.jpg)
Example - Low-Pass
Low-pass filtering performed on Mandrill scanline
Spatial domain: convolution with sincfunctionFrequency domain: cutoff of high frequencies - multiplication with box filter
Sinc function corresponds to box function and vice versa!
Eduard Gröller, Thomas Theußl, Peter Rautek 19
![Page 21: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/21.jpg)
Low-Pass in Spatial Domain 1
Eduard Gröller, Thomas Theußl, Peter Rautek 20
![Page 22: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/22.jpg)
Low-Pass in Spatial Domain 2
Eduard Gröller, Thomas Theußl, Peter Rautek 21
![Page 23: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/23.jpg)
OverviewIntroductionTools for Sampling and Reconstruction
Fourier TransformConvolution (dt.: Faltung)Convolution TheoremFiltering
SamplingThe mathematical model
Reconstruction Sampling TheoremReconstruction in Practice
Eduard Gröller, Thomas Theußl, Peter Rautek 22
![Page 24: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/24.jpg)
Sampling
The process of sampling is a multiplication of the signal with a comb function
The frequency response is convolved with a transformed comb function.
Eduard Gröller, Thomas Theußl, Peter Rautek 23
)(comb)()( T xxfxfs
)(comb)()( T1 FFs
![Page 25: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/25.jpg)
Base Functions
Eduard Gröller, Thomas Theußl, Peter Rautek 24
Comb FunctionDirac Pulse
![Page 26: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/26.jpg)
FT of Base Functions
Comb function:
Eduard Gröller, Thomas Theußl, Peter Rautek 25
ω)(comb)(comb T1T x
![Page 27: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/27.jpg)
OverviewIntroductionTools for Sampling and Reconstruction
Fourier TransformConvolution (dt.: Faltung)Convolution TheoremFiltering
SamplingThe mathematical model
Reconstruction Sampling TheoremReconstruction in Practice
Eduard Gröller, Thomas Theußl, Peter Rautek 26
![Page 28: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/28.jpg)
Reconstruction
Recovering the original functionfrom a set of samples
Sampling theorem Ideal reconstruction
Sinc function Reconstruction in practice
Eduard Gröller, Thomas Theußl, Peter Rautek 27
![Page 29: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/29.jpg)
Definitions
A function is called band-limited if it contains no frequencies outside the interval [-u,u]. u is called the bandwidth of the function
The Nyquist frequency of a function is twice its bandwidth, i.e., w = 2u
Eduard Gröller, Thomas Theußl, Peter Rautek 28
![Page 30: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/30.jpg)
Sampling Theorem
A function f(x) that isband-limited andsampled above the Nyquist frequency
is completely determined by its samples.
Eduard Gröller, Thomas Theußl, Peter Rautek 29
![Page 31: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/31.jpg)
Sampling at Nyquist Frequency
Eduard Gröller, Thomas Theußl, Peter Rautek 30
![Page 32: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/32.jpg)
Sampling Below Nyquist Frequency
Eduard Gröller, Thomas Theußl, Peter Rautek 31
Demo - Under sampling- Nyquist Frequency
http://www.cs.brown.edu/exploratories/freeSoftware/repository/edu/brown/cs/exploratories/applets/nyquist/nyquist_limit_java_browser.html
![Page 33: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/33.jpg)
Ideal Reconstruction
Replicas in frequency domain must not overlap
Multiplying the frequency response with a box filter of the width of the original bandwidth restores original
Amounts to convolution with Sinc function
Eduard Gröller, Thomas Theußl, Peter Rautek 32
![Page 34: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/34.jpg)
Sinc Function
Infinite in extent Ideal reconstruction filter FT of box function
Eduard Gröller, Thomas Theußl, Peter Rautek 33
00
1)sinc( πx
x sin
xx
ifif
x
![Page 35: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/35.jpg)
Sinc & Truncated Sinc
Eduard Gröller, Thomas Theußl, Peter Rautek 34
![Page 36: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/36.jpg)
Reconstruction: Examples
Sampling and reconstruction of the Mandrill image scanline signal
with adequate sampling rate with inadequate sampling rate demonstration of band-limiting
With Sinc and tent reconstruction kernels
Eduard Gröller, Thomas Theußl, Peter Rautek 35
![Page 37: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/37.jpg)
Adequate Sampling Rate
Eduard Gröller, Thomas Theußl, Peter Rautek 36
![Page 38: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/38.jpg)
Adequate Sampling Rate
Eduard Gröller, Thomas Theußl, Peter Rautek 37
![Page 39: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/39.jpg)
Inadequate Sampling Rate
Eduard Gröller, Thomas Theußl, Peter Rautek 38
![Page 40: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/40.jpg)
Inadequate Sampling Rate
Eduard Gröller, Thomas Theußl, Peter Rautek 39
![Page 41: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/41.jpg)
Band-Limiting a Signal
Eduard Gröller, Thomas Theußl, Peter Rautek 40
![Page 42: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/42.jpg)
Band-Limiting a Signal
Eduard Gröller, Thomas Theußl, Peter Rautek 41
![Page 43: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/43.jpg)
Reconstruction in Practice
Problem: which reconstruction kernel should be used?
Genuine Sinc function unusable in practiceTruncated Sinc often sub-optimal
Various approximations exist; none is optimal for all purposes
Eduard Gröller, Thomas Theußl, Peter Rautek 42
![Page 44: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/44.jpg)
Tasks of Reconstruction Filters
Remove the extraneous replicas of the frequency response
Retain the original undistorted frequency response
Eduard Gröller, Thomas Theußl, Peter Rautek 43
![Page 45: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/45.jpg)
Used Reconstruction Filters
Nearest neighbour Linear interpolation Symmetric cubic filters Windowed Sinc
More sophisticated ways of truncating the Sincfunction
Eduard Gröller, Thomas Theußl, Peter Rautek 44
![Page 46: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/46.jpg)
Box & Tent Responses
Eduard Gröller, Thomas Theußl, Peter Rautek 45
![Page 47: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/47.jpg)
Windowed Sinc Responses
Eduard Gröller, Thomas Theußl, Peter Rautek 46
![Page 48: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/48.jpg)
Sampling & Reconstruction Errors
Aliasing: due to overlap of original frequency response with replicas - information loss
Truncation Error: due to use of a finite reconstruction filter instead of the infinite Sincfilter
Non-Sinc error: due to use of a reconstruction filter that has a shape different from the Sincfilter
Eduard Gröller, Thomas Theußl, Peter Rautek 47
![Page 49: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/49.jpg)
Interpolation - Zero Insertion
Operates on series of n samplesTakes advantage of DFT properties
Algorithm:Perform DFT on seriesAppend zeros to the sequencePerform the inverse DFT
Eduard Gröller, Thomas Theußl, Peter Rautek 48
![Page 50: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/50.jpg)
Zero Insertion - Properties
Preserves frequency spectrum Original signal has to be sampled above
Nyquist frequency Values can only be interpolated at evenly
spaced locations The whole series must be accessible, and it is
always completely processed
Eduard Gröller, Thomas Theußl, Peter Rautek 49
![Page 51: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/51.jpg)
Zero Insertion - Original Series
Eduard Gröller, Thomas Theußl, Peter Rautek 50
![Page 52: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/52.jpg)
Zero Insertion - Interpolation
Eduard Gröller, Thomas Theußl, Peter Rautek 51
![Page 53: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/53.jpg)
ConclusionSampling
Going from continuous to discrete signalMathematically modeled with a multiplication with comb functionSampling theorem: How many samples are needed
Reconstruction Sinc is the ideal filter but not practicableReconstruction in practiceAliasing
Eduard Gröller, Thomas Theußl, Peter Rautek 52
![Page 54: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/54.jpg)
Summary (1)
Eduard Gröller, Thomas Theußl, Peter Rautek 53
![Page 55: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/55.jpg)
Summary (2)
Eduard Gröller, Thomas Theußl, Peter Rautek 54
![Page 56: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/56.jpg)
Summary (3)
Eduard Gröller, Thomas Theußl, Peter Rautek 55
![Page 57: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/57.jpg)
Sampling and Reconstruction
References:Computer Graphics: Principles and Practice, 2nd Edition, Foley, vanDam, Feiner, Hughes, Addison-Wesley, 1990What we need around here is more aliasing, Jim Blinn, IEEE Computer Graphics and Applications, January 1989Return of the Jaggy, Jim Blinn, IEEE Computer Graphics and Applications, March 1989Demo Applets, Brown University, Rhode Island, USAhttp://www.cs.brown.edu/exploratories/freeSoftware/home.html
Eduard Gröller, Thomas Theußl, Peter Rautek 56
![Page 58: Sampling and Reconstruction - TU Wien](https://reader030.fdocuments.us/reader030/viewer/2022013001/61cbd67d03c081274b143ff2/html5/thumbnails/58.jpg)
Questions
Eduard Gröller, Thomas Theußl, Peter Rautek 57