Aliasing
Anti Aliasing
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Aliasing - Wikipedia, the free encyclopedia
... computer programming, please refer to aliasing (computing) ... Spatial aliasing in the form of a ... avoid such poor pixelizations are called anti-aliasing. ...
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Aliasing (computing) - Wikipedia, the free encyclopedia
Understanding Strict Aliasing - Mike Acton. Aliasing, pointer casts and gcc 3.3 - Informational article on NetBSD mailing list ...
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aliasing: Definition from Answers.com
aliasing n. The appearance of jagged distortions in curves and diagonal lines in computer graphics because the resolution is limited or diminished
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The Geneva Convention On The Treatment of Object Aliasing
Aliasing has been a problem in both formal verification and practical ... to the object-oriented aliasing problem have been put ... Aliasing and Objects ...
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Aliasing Demonstration Applet
Demonstration of Aliasing of a Sinusoidal Signal (Java 1.02 Version) Note ... this limit, often referred to as the folding frequency, is subject to aliasing. ...
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fractalus - Anti-aliasing and Fractals
Anti-aliasing is a technique that has long been applied to rendering 3D scenes ... This is known as aliasing-where elements (the small squares) with frequencies ...
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Aliasing and Moire patterns
An explanation of aliasing in digital images, and how to avoid the resulting Moire patterns. ... Aliasing ... Aliasing occurs when a signal is sampled at a ...
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Aliasing - PanoTools.org Wiki
Aliasing is a process that results in "jaggies" or Moire effects when an image ... Retrieved from "http://wiki.panotools.org
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sendmail Aliasing
Aliasing is a method by which mail to a specific address is redirected somewhere ... The rest of the lines are aliases, which explain a lot about how aliasing works. ...
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This article applies to signal processing, including computer graphics. For uses in computer programming, please refer to aliasing (computing).
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In statistics, signal processing, computer graphics and related disciplines, aliasing refers to an effect that causes different continuous signals to become indistinguishable (or aliases of one another) when signal sampling. It also refers to the distortion or artifact that results when a signal is sampled and reconstructed as an alias of the original signal.
When we view a digital photograph, the reconstruction (interpolation) is performed by a display or printer device, and by our eyes and our brain. If the reconstructed image differs from the original image, we are seeing an alias. An example of spatial aliasing is the Moiré pattern one can observe in a poorly pixelized image of a brick wall. Techniques that avoid such poor pixelizations are called anti-aliasing.
Temporal aliasing is a major concern in the sampling of video and Sound reproduction signals. Music, for instance, may contain high-frequency components that are inaudible to us. If we sample it with a frequency that is too low and reconstruct the music with a digital to analog converter, we may hear the low-frequency aliases of the undersampled high frequencies. Therefore, it is common practice to remove the high frequencies with a filter before the sampling is done.
Situations also exist where the low frequencies are removed (if necessary), and the high frequency components are intentionally undersampled and reconstructed as lower ones. Some digital channelizers{{cite book| last = Harris | first = Frederic J. | year = 2006 | title = Multirate Signal Processing for Communication Systems | publisher = Prentice Hall PTR | location = Upper Saddle River, NJ | isbn = 0-13-146511-2 -->exploit aliasing in this way for computational efficiency; see Sampling (signal processing)#IF/RF sampling. Signals that contain no low frequencies are often referred to as bandpass or non-baseband.
In video or cinematography, temporal aliasing results from the limited frame rate, and causes the wagon-wheel effect, whereby a spoked wheel appears to rotate too slowly or even backwards. Aliasing has changed its frequency of rotation. A reversal of direction can be described as a negative frequency.
Like the video camera, most sampling schemes are periodic; that is they have a characteristic sampling frequency in time or in space. Digital cameras provides a certain number of samples (pixels) per degree or per radian, or samples per mm in the focal plane of the camera. Audio signals are sampled (digitized) with an analog-to-digital converter, which produces a constant number of samples per second. Some of the most dramatic and subtle examples of aliasing occur when the signal being sampled also has periodic content.
Sampling sinusoidal functions Sinusoids are an important type of periodic function, because realistic signals are often modeled as the summation of many sinusoids of different frequencies and different amplitudes. Understanding what aliasing does to the individual sinusoids is a big help in understanding what happens to their sum.
image:AliasingSines.png
Here a plot depicts a set of samples whose sample-interval is 1.0 and two (of many) different sinusoids that could have produced the samples. The sample-rate in this case is f_s\, = 1.0. For instance, if the interval is 1 second, the rate is 1 sample per second. 9 cycles of the red sinusoid and 1 cycle of the blue sinusoid span an interval of 10. The respective sinusoid frequencies are f_\mathrm{red}\, = 0.9 and f_\mathrm{blue}\, = 0.1.
In general, when a sinusoid of frequency f\, is sampled with frequency f_s,\, the resulting samples are indistinguishable from those of another sinusoid of frequency f_\mathrm{image}(N) = |f - Nf_s|\, for any integer N\, (with f_\mathrm{image}(0) = f\, being the actual signal frequency). Most reconstruction techniques produce the minimum of these frequencies, so it is often important that f_\mathrm{image}(0)\, be the unique minimum. A sufficient condition for that is f_s/2 > f,\, where f_s/2\, is commonly called the Nyquist frequency.
In our graphic example, the Nyquist condition is satisfied if the original signal is the blue sinusoid (f = f_\mathrm{blue}\,). But if f = f_\mathrm{red},\, the lowest image frequency is: :f_\mathrm{image}(1) = |0.9 - 1.0| = 0.1 = f_\mathrm{blue}.\,
- A reconstruction technique that constructs the lowest possible frequency from the samples will reproduce the blue sinusoid instead of the red one.
- We note that -0.1\, is also an image frequency, but since there is no way to distinguish a sinusoid of frequency -f\, from one of frequency f,\, all aliases can be described in terms of just positive frequencies.
Sample frequency
When the condition f_s/2 > f\, is met for the highest frequency component of the original signal, then it is met for all the frequency components, a condition known as the Nyquist theorem. That is typically approximated by filtering the original signal to attenuate high frequency components before it is sampled. They still generate low-frequency aliases, but at very low amplitude levels, so as not to cause a problem. A filter chosen in anticipation of a certain sample frequency is called an anti-aliasing filter. The filtered signal can subsequently be reconstructed without significant additional distortion, for example by the Whittaker–Shannon interpolation formula.
The Nyquist criterion presumes that the frequency content of the signal being sampled has an upper bound. Implicit in that assumption is that its duration has no upper bound. Similarly, the Whittaker–Shannon interpolation formula assumes instantaneous sampling and an interpolation filter with an unrealizable frequency response. These assumptions comprise a mathematical model that is only an idealized approximation, at best, to any realistic situation. The conclusion, that perfect reconstruction is possible, is mathematically correct for the model, but only an approximation for the real samples and the real signal.
Complex signal representation Complex signals are signals whose samples are complex numbers, and the concept of negative frequency is necessary for such signals. In that case, the frequencies of the aliases are given by just: f_\mathrm{image}(N) = f - Nf_s.\, Therefore, as f\, increases from f_s/2\, to f_s,\, the image closest to 0 moves from -f_s/2\, up to 0.
Folding Real-valued sinusoids have the same negative-frequency aliases as complex ones. The absolute value operator, |f - Nf_s|,\, is possible because there is always an equivalent sinusoid with a positive frequency. Therefore, as f\, increases from f_s/2\, to f_s,\, an image moves from f_s/2\, down to 0. This creates a local symmetry about the frequency f_s/2.\, For example, a frequency component at 0.6 f_s\, has a "mirror" image at 0.4 f_s.\, That effect is commonly referred to as folding. And another name for f_s/2\, (the Nyquist frequency) is folding frequency.
Historical usage Historically the term aliasing evolved from radio engineering because of the action of superheterodyne receivers. When the receiver shifts multiple signals down to lower frequencies, from radio frequency to Intermediate frequency by heterodyning, an unwanted signal, from an RF frequency equally far from the local oscillator (LO) frequency as the desired signal, but on the wrong side of the LO, can end up at the same IF frequency as the wanted one. If it is strong enough it can interfere with reception of the desired signal. This unwanted signal is known as an image or alias of the desired signal.
More examples Online "live" example The qualitative effects of aliasing can be heard in the following audio demonstration. Six sawtooth waves are played in succession, with the first two sawtooths having a fundamental frequency of 440 Hz (A4), the second two having fundamental frequency of 880 Hz (A5), and the final two at 1760 Hz (A6). The sawtooths alternate between bandlimited (non-aliased) sawtooths and aliased sawtooths and the sampling rate is 22.05 kHz. The bandlimited sawtooths are synthesized from the sawtooth waveform's Fourier series such that no harmonics above the Nyquist frequency are present.
The aliasing distortion in the lower frequencies is increasingly obvious with higher fundamental frequencies, and while the bandlimited sawtooth is still clear at 1760 Hz, the aliased sawtooth is degraded and harsh with a buzzing audible at frequencies lower than the fundamental. Note that the audio file has been coded using Ogg Vorbis codec, and as such the audio is somewhat degraded.
- media:Sawtooth-aliasingdemo.ogg {440 Hz bandlimited, 440 Hz aliased, 880 Hz bandlimited, 880 Hz aliased, 1760 Hz bandlimited, 1760 Hz aliased}
Direction finding A form of spatial aliasing can also occur in antenna arrays or microphone arrays used to estimate the direction of arrival of a wave signal, as in geophysical exploration by seismic waves. Waves must be sampled at more than two points per wavelength, or the wave arrival direction becomes ambiguous.
See also
- Anti-aliasing
- Wagon-wheel effect
- Sinc filter
- Sinc function
- Temporal aliasing
- Nyquist–Shannon sampling theorem
- Whittaker–Shannon interpolation formula
- Nyquist rate
- Nyquist frequency
- Kell factor
References External links
- Frequency Aliasing Demonstration by Burton MacKenZie using stop frame animation and a clock.
- Your Calculator is Wrong Video from YouTube, includes some information about aliasing toward the end.
aliasing from FOLDOC
aliasing. 1. < jargon > When several different identifiers refer to the same object. The term is very general and is used in many contexts. See alias, aliasing bug, anti-aliasing.
anti-aliasing from FOLDOC
anti-aliasing < graphics > A technique used on a grey-scale or colour bitmap display to make diagonal edges appear smoother by setting pixels near the edge to intermediate colours ...
Anti-Aliasing
Creating graphics for the Web: Anti-aliasing . One of the most important techniques in making graphics and text easy to read and pleasing to the eye on-screen is anti-aliasing.
Aliasing Demonstration Applet
Note This page contains a Java 1.02 applet: it should work with Netscape Navigator (version 2.0 or later) or Internet Explorer 3.0.
Aliasing Demonstration Applet
Note This is a Java 1.1 applet. It should work with HotJava 1.1, Internet Explorer 4.0 or Netscape Navigator 4.0 with the Java 1.1 upgrade patch applied.
Aliasing - Wikipedia, the free encyclopedia
In statistics, signal processing, computer graphics and related disciplines, aliasing refers to an effect that causes different continuous signals to become indistinguishable (or ...
Aliasing (computing) - Wikipedia, the free encyclopedia
In computing, aliasing describes a situation in which a data location in memory can be accessed through different symbolic names in the program.
Definition: aliasing from Online Medical Dictionary
The Online Medical Dictionary is a searchable dictionary of definitions from medicine, science and technology.
PhotoNotes.org Dictionary - Aliasing
The PhotoNotes.org Dictionary of Film and Digital Photography. An extensive dictionary or glossary of photography terminology.
Aliasing - Wikimedia Commons
Aliasing is a phenomenon in analog to digital conversion in which the frequency of the converted signal is lower than that of the original signal.