Aliasing and Anti-aliasing Filters - TI training
training ti com/sites/default/files/docs/adcs-aliasing-and-anti-aliasing-filters-presentation-quiz pdf
d) It depends on the anti-aliasing filter 2 Applying an input signal that exceeds the Nyquist frequency will ______ a) Cause an erroneous “alias” signal to
Anti-Aliasing Filters Reduce Errors in Data Acquisition
pfinc com/wp-content/uploads/2021/06/Filtering_and_Anti-Aliasing pdf
The aliases of a given frequency in the signal of interest, fa, lying in the interval from DC to FS/2 are nFS ±fa The fre- quencies in this interval are
ANTI-ALIASING - MBBM-VAS
www mbbm-vas com/fileadmin/MBBM-VAS/Publications/White_Paper___Use_Cases/PAK_KHC_Anti-Aliasing_1605_EN pdf
ANTI-ALIASING Aliasing occurs when frequency components which are higher than the Nyquist frequency, are visible in the sampled signal level gradient
Basic Signal Processing: Sampling, Aliasing, Antialiasing
graphics stanford edu/courses/cs148-11-fall/lectures/sampling pdf
Pat Hanrahan, Fall 2011 Key Concepts Frequency space Filters and convolution Sampling and the Nyquist frequency Aliasing and Antialiasing
Lesson 7: Anti-Aliasing Filtering
www engr siu edu/staff/spezia/NewWeb438B/Lecture 20Notes/Lesson 207_et438b pdf
25 avr 2016 Design Procedure 1 ) Determine the acceptable level of signal gain, Av, at sampling frequency fs 2 ) Use the formula below to determine
Sampling, Aliasing and Anti aliasing filter
meeng technion ac il/wp-content/uploads/2015/09/5 Digitizing pdf
Its goal is to eliminate, before sampling, all frequencies in the signal that are, at least , above the Nyquist frequency and therefore avoid aliasing • Note
BBM 413 Fundamentals of Image Processing Frequency Domain
web cs hacettepe edu tr/~erkut/bbm413 f15/slides/09-frequency-part2 pdf
Anti-aliasing Solutions: • Sample more often • Get rid of all frequencies that are greater than half the new sampling frequency – Will lose information
Time-domain aliasing and anti-aliasing effects in differentiating a
www matec-conferences org/articles/matecconf/ pdf /2018/69/matecconf_cscc2018_05005 pdf
mode with ideal anti-aliasing filtering (AAF) with a cut-off at the Nyquist frequency We disclosed that regardless sampling frequency the error from AAF is
Lecture 8: Reconstruction and Aliasing
smartdata ece ufl edu/eee5502/2019_fall/media/2019_eee5502_slides08 pdf
Anti-Aliasing We need to sample twice as fast as the maximum frequency ?s > 2?max Question: What is happening when I multiply in frequency?
Impact of Aliasing on Generalization in Deep Convolutional Networks
openaccess thecvf com/content/ICCV2021/papers/Vasconcelos_Impact_of_Aliasing_on_Generalization_in_Deep_Convolutional_Networks_ICCV_2021_paper pdf
Under this second assumption, we claim that because aliasing can leak frequencies across the entire spectrum, an anti-aliased model has the potential to improve
ANTI-ALIASING Aliasing occurs when frequency components which are higher than the Nyquist frequency, are visible in the sampled signal level gradient
When would it be faster to apply the filter in the frequency domain? Page 17 Sampling CS148 Lecture 13 Pat Hanrahan, Fall 2011
their bandwidth to the one required by the available sampling frequency, or adopt Impact of the frequency response of a real anti-aliasing filter on the signal
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113362_3sampling.pdf CS148: Introduction to Computer Graphics and Imaging
Basic Signal Processing:
Sampling, Aliasing, Antialiasing
No Jaggies
CS148 Lecture 13Pat Hanrahan, Fall 2011
Key Concepts
Frequency space
Filters and convolution
Sampling and the Nyquist frequency
Aliasing and Antialiasing
Frequency Space
sin2 -x
CS148 Lecture 13Pat Hanrahan, Fall 2011
Sines and Cosines
cos2 x
Frequencies
cos4 -x cos2 fx f=1 f=1 T f=2
CS148 Lecture 13Pat Hanrahan, Fall 2011
cos2 πx
Euler's Formula
Odd (-x)
Therefore
Hence, use complex exponentials for sines/cosines
Recall Complex Exponentials
CS148 Lecture 13Pat Hanrahan, Fall 2011
e jx = cosx+jsinx e -jx = cos-x+jsin-x= cosx-jsinx cosx=e jx +e -jx 2 sinx=e jx -e -jx 2 j
CS148 Lecture 13Pat Hanrahan, Fall 2011
Constant
Spatial DomainFrequency Domain
sin(2-/32)x
CS148 Lecture 13Pat Hanrahan, Fall 2011
Spatial DomainFrequency Domain
Frequency = 1/32; 32 pixels per cycle
sin(2?/16)x
CS148 Lecture 13Pat Hanrahan, Fall 2011
Spatial DomainFrequency Domain
sin(2/16)y
CS148 Lecture 13Pat Hanrahan, Fall 2011
Spatial DomainFrequency Domain
sin(2?/32)x×sin(2?/16)y
CS148 Lecture 13Pat Hanrahan, Fall 2011
Spatial DomainFrequency Domain
e r 2 / 16 2
CS148 Lecture 13Pat Hanrahan, Fall 2011
Spatial DomainFrequency Domain
e ?r 2 / 32
2
CS148 Lecture 13Pat Hanrahan, Fall 2011
Spatial DomainFrequency Domain
e -x 2 / 32
2 ×e -y 2 / 16 2
CS148 Lecture 13Pat Hanrahan, Fall 2011
Spatial DomainFrequency Domain
Rotate 45
CS148 Lecture 13Pat Hanrahan, Fall 2011
Spatial DomainFrequency Domain
e -x 2 / 32
2 ×e -y 2 / 16 2
Filtering
CS148 Lecture 13Pat Hanrahan, Fall 2011
My Humble Frequencies
Spatial DomainFrequency Domain
CS148 Lecture 13Pat Hanrahan, Fall 2011
Remove Low Frequencies (Edges)
Spatial DomainFrequency Domain
CS148 Lecture 13Pat Hanrahan, Fall 2011
Remove High Frequencies (Blur)
Spatial DomainFrequency Domain
CS148 Lecture 13Pat Hanrahan, Fall 2011
Remove Low and High Frequencies
Spatial DomainFrequency Domain
CS148 Lecture 13Pat Hanrahan, Fall 2011
Remove Low and High Frequencies
Spatial DomainFrequency Domain
Filters = Convolution
CS148 Lecture 13Pat Hanrahan, Fall 2011
Convolution
130421
12 1 * 1 + 3 * 2 = 7
CS148 Lecture 13Pat Hanrahan, Fall 2011
Convolution
7
130421
12
CS148 Lecture 13Pat Hanrahan, Fall 2011
Convolution
3 * 1 + 0 * 2 = 3
130421
12 73
CS148 Lecture 13Pat Hanrahan, Fall 2011
Convolution
0 * 1 + 4 * 2 = 8
130421
12 738
CS148 Lecture 13Pat Hanrahan, Fall 2011
Convolution Theorem
A ? lter can be implemented in the spatial domain using convolution A ? lter can also be implemented in the frequency domain
Convert image to frequency domain
Convert
? lter to frequency domain
Multiply
? lter times image in frequency domain
Convert result to the spatial domain
CS148 Lecture 13Pat Hanrahan, Fall 2011
Box Filter
11 11
CS148 Lecture 13Pat Hanrahan, Fall 2011
Box Filter = Low-Pass Filter
Spatial DomainFrequency Domain
CS148 Lecture 13Pat Hanrahan, Fall 2011
Wider Filters, Lower Frequencies
Spatial DomainFrequency Domain
CS148 Lecture 13Pat Hanrahan, Fall 2011
Size of Filter
As a ? lter is localized in space, it spreads out in frequency
Conversely, as a
? lter is localized in frequency, it spreads out in space
A box
? lter is very localized in space; it has in ? nite extent in frequency space
CS148 Lecture 13Pat Hanrahan, Fall 2011
Ef ? ciency?
When would it be faster to apply the
? lter in the spatial domain?
When would it be faster to apply the
? lter in the frequency domain?
Sampling
CS148 Lecture 13Pat Hanrahan, Fall 2011
Image Generation = Sampling
Evaluating a function at a point is sampling
for( int x = 0; x < xmax; x++ ) for( int y = 0; y < ymax; y++ ) Image[x][y] = f(x,y);
Rasterization is equivalent to evaluating the
function inside(triangle,x,y)
CS148 Lecture 13Pat Hanrahan, Fall 2011
Sampling Causes Jaggies
Retort, by Don Mitchell
Staircase pattern or jaggies
CS148 Lecture 13Pat Hanrahan, Fall 2011
Sampling in Computer Graphics
Artifacts due to sampling -
Aliasing
Jaggies - sampling in space
Wagon wheel effect - sampling in time
Temporal strobing - sampling in space-time
Moire - sampling texture coordinates
Sparkling highlights - sampling normals
Preventing these artifacts -
Antialiasing
Aliasing
Wagon Wheel Effect
http://www.michaelbach.de/ot/mot_wagonWheel/
CS148 Lecture 13Pat Hanrahan, Fall 2011
"Aliases"
These two sine waves are indistinguishable
Indistinguishable frequencies are called "aliases"
CS148 Lecture 13Pat Hanrahan, Fall 2011
Nyquist Frequency
De ? nition: The Nyquist frequency is ½ the sampling frequency (1/Ts)
Frequencies above the Nyquist frequency appear
as aliases
No aliases appear if the function being sampled
has no frequencies above the Nyquist frequency
Antialiasing
CS148 Lecture 13Pat Hanrahan, Fall 2011
Antialiasing
Simple idea:
Remove frequencies above the Nyquist
frequency before sampling
How? Filtering before sampling
CS148 Lecture 13Pat Hanrahan, Fall 2011
Pre ? ltering by Computing Coverage
A 1 pixel box
? lter removes frequencies whose period is less than or equal to 1 pixel
Original
Filtered
CS148 Lecture 13Pat Hanrahan, Fall 2011
Point- vs. Area-Sampled
PointArea
Checkerboard sequence by Tom Duff
CS148 Lecture 13Pat Hanrahan, Fall 2011
Antialiasing
JaggiesPre?lter
CS148 Lecture 13Pat Hanrahan, Fall 2011
Antialiasing vs. Blurred Aliases
Blurred JaggiesPre?lter
CS148 Lecture 13Pat Hanrahan, Fall 2011
Things to Remember
Signal processing
Frequency domain vs. spatial domain
Filters in the frequency domain
Filters in the spatial domain = convolution
Sampling and aliasing
Image generation involves sampling
May also sample geometry, motion, ...
Nyquist frequency is
½ the sampling rate
Frequencies above the Nyquist frequency
appear as other frequencies - aliases
Antialiasing - Filter before sampling
Extra Slides
Supersampling
CS148 Lecture 13Pat Hanrahan, Fall 2011
Approximate a box
? lter by taking more samples and averaging them together
Supersampling
4 x 4 supersampling
CS148 Lecture 13Pat Hanrahan, Fall 2011
Point-sampling vs. Super-sampling
Point4x4 Super-sampled
Checkerboard sequence by Tom Duff
CS148 Lecture 13Pat Hanrahan, Fall 2011
Area-Sampling vs. Super-sampling
Exact Area4x4 Super-sampled
Frequency Documents PDF, PPT , Doc