![Ringing artifact example - original](https://upload.wikimedia.org/wikipedia/commons/7/73/Ringing_artifact_example_-_original.png)
In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, particularly
digital image processing
Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allo ...
, ringing artifacts are
artifacts that appear as spurious signals near sharp transitions in a signal. Visually, they appear as bands or "ghosts" near edges; audibly, they appear as "echos" near
transients
Transience or transient may refer to:
Music
* ''Transient'' (album), a 2004 album by Gaelle
* ''Transience'' (Steven Wilson album), 2015
* Transience (Wreckless Eric album)
Science and engineering
* Transient state, when a process variable or ...
, particularly sounds from
percussion instrument
A percussion instrument is a musical instrument that is sounded by being struck or scraped by a beater including attached or enclosed beaters or rattles struck, scraped or rubbed by hand or struck against another similar instrument. Ex ...
s; most noticeable are the
pre-echo
In audio signal processing, pre-echo, sometimes called a '' forward echo'', (not to be confused with reverse echo) is a digital audio compression artifact where a sound is heard before it occurs (hence the name). It is most noticeable in impulsiv ...
s. The term "ringing" is because the output signal oscillates at a fading rate around a sharp transition in the input, similar to a
bell
A bell is a directly struck idiophone percussion instrument. Most bells have the shape of a hollow cup that when struck vibrates in a single strong strike tone, with its sides forming an efficient resonator. The strike may be made by an inte ...
after being struck. As with other artifacts, their minimization is a criterion in
filter design
Filter design is the process of designing a signal processing filter that satisfies a set of requirements, some of which may be conflicting. The purpose is to find a realization of the filter that meets each of the requirements to a sufficient ...
.
Introduction
![High accuracy settling time measurements figure 1](https://upload.wikimedia.org/wikipedia/commons/7/76/High_accuracy_settling_time_measurements_figure_1.png)
The main cause of ringing artifacts is due to a signal being
bandlimited
Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency.
A band-limited signal is one whose Fourier transform or spectral density has bounded support.
A bandli ...
(specifically, not having high frequencies) or passed through a
low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
; this is the
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
description.
In terms of the
time domain
Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the c ...
, the cause of this type of ringing is the ripples in the
sinc function,
[, section I.6, Enhancement: Frequency Domain Techniques]
p. 16
/ref> which is the impulse response (time domain representation) of a perfect low-pass filter. Mathematically, this is called the Gibbs phenomenon
In mathematics, the Gibbs phenomenon, discovered by Available on-line at:National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. and rediscovered by , is the oscillatory behavior of the Fourier series of a piecewise continuousl ...
.
One may distinguish overshoot (and undershoot), which occurs when transitions are accentuated – the output is higher than the input – from ringing, where ''after'' an overshoot, the signal overcorrects and is now below the target value; these phenomena often occur together, and are thus often conflated and jointly referred to as "ringing".
The term "ringing" is most often used for ripples in the ''time'' domain, though it is also sometimes used for ''frequency'' domain effects:[Digital Signal Processing](_blank)
by J.S.Chitode, Technical Publications, 2008,
4 - 70
/ref>
windowing a filter in the time domain by a rectangular function causes ripples in the ''frequency'' domain for the same reason as a brick-wall low pass filter (rectangular function in the ''frequency'' domain) causes ripples in the ''time'' domain, in each case the Fourier transform of the rectangular function being the sinc function.
There are related artifacts caused by other frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
effects,
and similar artifacts due to unrelated causes.
Causes
Description
By definition, ringing occurs when a non-oscillating input yields an oscillating output: formally, when an input signal which is monotonic
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
on an interval has output response which is not monotonic. This occurs most severely when the impulse response or step response
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the out ...
of a filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
has oscillations – less formally, if for a spike input, respectively a step input (a sharp transition), the output has bumps. Ringing most commonly refers to step ringing, and that will be the focus.
Ringing is closely related to overshoot and undershoot, which is when the output takes on values higher than the maximum (respectively, lower than the minimum) input value: one can have one without the other, but in important cases, such as a low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
, one first has overshoot, then the response bounces back below the steady-state level, causing the first ring, and then oscillates back and forth above and below the steady-state level. Thus overshoot is the first step of the phenomenon, while ringing is the second and subsequent steps. Due to this close connection, the terms are often conflated, with "ringing" referring to both the initial overshoot and the subsequent rings.
If one has a linear time invariant
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defin ...
(LTI) filter, then one can understand the filter and ringing in terms of the impulse response (the time domain view), or in terms of its Fourier transform, the frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...
(the frequency domain view). Ringing is a ''time'' domain artifact, and in filter design
Filter design is the process of designing a signal processing filter that satisfies a set of requirements, some of which may be conflicting. The purpose is to find a realization of the filter that meets each of the requirements to a sufficient ...
is traded off with desired frequency domain characteristics: the desired frequency response may cause ringing, while reducing or eliminating ringing may worsen the frequency response.
sinc filter
The central example, and often what is meant by "ringing artifacts", is the ideal ( brick-wall) low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
, the sinc filter
In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given cutoff frequency, without affecting lower frequencies, and has linear phase response. The filter's impulse response is a sinc functi ...
. This has an oscillatory impulse response function, as illustrated above, and the step response – its integral, the sine integral
In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions.
Sine integral
The different sine integral definitions are
\operatorname(x) = \int_0^x\frac\,dt
\operatorname(x) = -\int_x^\infty\frac ...
– thus also features oscillations, as illustrated at right.
These ringing artifacts are not results of imperfect implementation or windowing: the ideal low-pass filter, while possessing the desired frequency response, necessarily causes ringing artifacts in the ''time'' domain.
Time domain
In terms of impulse response, the correspondence between these artifacts and the behavior of the function is as follows:
* impulse undershoot is equivalent to the impulse response having negative values,
* impulse ringing (ringing near a point) is precisely equivalent to the impulse response having oscillations, which is equivalent to the derivative of the impulse response alternating between negative and positive values,
* and there is no notion of impulse overshoot, as the unit impulse is assumed to have infinite height (and integral 1 – a Dirac delta function), and thus cannot be overshot.
Turning to step response,
the step response is the integral of the impulse response; formally, the value of the step response at time ''a'' is the integral of the impulse response. Thus values of the step response can be understood in terms of ''tail'' integrals of the impulse response.
Assume that the overall integral of the impulse response is 1, so it sends constant input to the same constant as output – otherwise the filter has gain, and scaling by gain gives an integral of 1.
* Step undershoot is equivalent to a tail integral being negative, in which case the magnitude of the undershoot is the value of the tail integral.
* Step overshoot is equivalent to a tail integral being greater than 1, in which case the magnitude of the overshoot is the amount by which the tail integral exceeds 1 – or equivalently the value of the tail in the other direction, since these add up to 1.
* Step ringing is equivalent to tail integrals alternating between increasing and decreasing – taking derivatives, this is equivalent to the impulse response alternating between positive and negative values. Regions where an impulse response are below or above the ''x''-axis (formally, regions between zeros) are called lobes, and the magnitude of an oscillation (from peak to trough) equals the integral of the corresponding lobe.
The impulse response may have many negative lobes, and thus many oscillations, each yielding a ring, though these decay for practical filters, and thus one generally only sees a few rings, with the first generally being most pronounced.
Note that if the impulse response has small negative lobes and larger positive lobes, then it will exhibit ringing but not undershoot or overshoot: the tail integral will always be between 0 and 1, but will oscillate down at each negative lobe. However, in the sinc filter, the lobes monotonically decrease in magnitude and alternate in sign, as in the alternating harmonic series
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions:
\sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots.
The first n terms of the series sum to approximately \ln n + \gamma, wher ...
, and thus tail integrals alternate in sign as well, so it exhibits overshoot as well as ringing.
Conversely, if the impulse response is always nonnegative, so it has no negative lobes – the function is a probability distribution – then the step response will exhibit neither ringing nor overshoot or undershoot – it will be a monotonic function growing from 0 to 1, like a cumulative distribution function. Thus the basic solution from the time domain perspective is to use filters with nonnegative impulse response.
Frequency domain
The frequency domain perspective is that ringing is caused by the sharp cut-off in the rectangular passband
A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
in the frequency domain, and thus is reduced by smoother roll-off
Roll-off is the steepness of a transfer function with frequency, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a passband and a stopband. It is most typically app ...
, as discussed below.[Microscope Image Processing](_blank)
by Qiang Wu, Fatima Merchant, Kenneth Castleman,
p. 71
/ref>
Solutions
Solutions depend on the parameters of the problem: if the cause is a low-pass filter, one may choose a different filter design, which reduces artifacts at the expense of worse frequency domain performance. On the other hand, if the cause is a band-limited signal, as in JPEG, one cannot simply replace a filter, and ringing artifacts may prove hard to fix – they are present in JPEG 2000 and many audio compression codecs (in the form of pre-echo
In audio signal processing, pre-echo, sometimes called a '' forward echo'', (not to be confused with reverse echo) is a digital audio compression artifact where a sound is heard before it occurs (hence the name). It is most noticeable in impulsiv ...
), as discussed in the examples
Example may refer to:
* '' exempli gratia'' (e.g.), usually read out in English as "for example"
* .example, reserved as a domain name that may not be installed as a top-level domain of the Internet
** example.com, example.net, example.org, e ...
.
Low-pass filter
If the cause is the use of a brick-wall low-pass filter, one may replace the filter with one that reduces the time domain artifacts, at the cost of frequency domain performance. This can be analyzed from the time domain or frequency domain perspective.
In the time domain, the cause is an impulse response that oscillates, assuming negative values. This can be resolved by using a filter whose impulse response is non-negative and does not oscillate, but shares desired traits. For example, for a low-pass filter, the Gaussian filter
In electronics and signal processing mainly in digital signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response would have infinite impulse respons ...
is non-negative and non-oscillatory, hence causes no ringing. However, it is not as good as a low-pass filter: it rolls off in the passband, and leaks in the stopband: in image terms, a Gaussian filter "blurs" the signal, which reflects the attenuation of desired higher frequency signals in the passband.
A general solution is to use a window function
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the int ...
on the sinc filter, which cuts off or reduces the negative lobes: these respectively eliminate and reduce overshoot and ringing. Note that truncating some but not all of the lobes eliminates the ringing beyond that point, but does not reduce the amplitude of the ringing that is not truncated (because this is determined by the size of the lobe), and increases the magnitude of the overshoot if the last non-cut lobe is negative, since the magnitude of the overshoot is the integral of the ''tail,'' which is no longer canceled by positive lobes.
Further, in practical implementations one at least truncates sinc, otherwise one must use infinitely many data points (or rather, all points of the signal) to compute every point of the output – truncation corresponds to a rectangular window, and makes the filter practically implementable, but the frequency response is no longer perfect.
In fact, if one takes a brick wall low-pass filter (sinc in time domain, rectangular in frequency domain) and truncates it (multiplies with a rectangular function in the time domain), this convolves the frequency domain with sinc (Fourier transform of the rectangular function) and causes ringing in the ''frequency'' domain, which is referred to as ''ripple
Ripple may refer to:
Science and technology
* Capillary wave, commonly known as ripple, a wave traveling along the phase boundary of a fluid
** Ripple, more generally a disturbance, for example of spacetime in gravitational waves
* Ripple (electri ...
.'' In symbols, The frequency ringing in the stopband is also referred to as side lobe
In antenna engineering, sidelobes are the lobes (local maxima) of the far field radiation pattern of an antenna or other radiation source, that are not the ''main lobe''.
The radiation pattern of most antennas shows a pattern of "''lobes'' ...
s. Flat response in the passband is desirable, so one windows with functions whose Fourier transform has fewer oscillations, so the frequency domain behavior is better.
Multiplication in the time domain corresponds to convolution in the frequency domain, so multiplying a filter by a window function corresponds to convolving the Fourier transform of the original filter by the Fourier transform of the window, which has a smoothing effect – thus windowing in the time domain corresponds to smoothing in the frequency domain, and reduces or eliminates overshoot and ringing.
In the frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
, the cause can be interpreted as due to the sharp (brick-wall) cut-off, and ringing reduced by using a filter with smoother roll-off. This is the case for the Gaussian filter, whose magnitude Bode plot is a downward opening parabola (quadratic roll-off), as its Fourier transform is again a Gaussian, hence (up to scale) – taking logarithms yields
In electronic filters, the trade-off between frequency domain response and time domain ringing artifacts is well-illustrated by the Butterworth filter: the frequency response of a Butterworth filter slopes down linearly on the log scale, with a first-order filter having slope of −6 dB per octave, a second-order filter –12 dB per octave, and an ''n''th order filter having slope of dB per octave – in the limit, this approaches a brick-wall filter. Thus, among these the, first-order filter rolls off slowest, and hence exhibits the fewest time domain artifacts, but leaks the most in the stopband, while as order increases, the leakage decreases, but artifacts increase.
Benefits
While ringing artifacts are generally considered undesirable, the initial overshoot (haloing) at transitions increases acutance
In photography, acutance describes a subjective perception of sharpness that is related to the edge contrast of an image. Acutance is related to the amplitude of the derivative of brightness with respect to space. Due to the nature of the hu ...
(apparent sharpness) by increasing the derivative across the transition, and thus can be considered as an enhancement.
Related phenomena
Overshoot
Another artifact is overshoot (and undershoot), which manifests itself not as rings, but as an increased jump at the transition. It is related to ringing, and often occurs in combination with it.
Overshoot and undershoot are caused by a negative tail – in the sinc, the integral from the first zero to infinity, including the first negative lobe. While ringing is caused by a following ''positive'' tail – in sinc, the integral from the second zero to infinity, including the first non-central positive lobe.
Thus overshoot is ''necessary'' for ringing, but can occur separately: for example, the 2-lobed Lanczos filter has only a single negative lobe on each side, with no following positive lobe, and thus exhibits overshoot but no ringing, while the 3-lobed Lanczos filter exhibits both overshoot and ringing, though the windowing reduces this compared to the sinc filter or the truncated sinc filter.
Similarly, the convolution kernel used in bicubic interpolation is similar to a 2-lobe windowed sinc, taking on negative values, and thus produces overshoot artifacts, which appear as halos at transitions.
Clipping
Following from overshoot and undershoot is clipping
Clipping may refer to:
Words
* Clipping (morphology), the formation of a new word by shortening it, e.g. "ad" from "advertisement"
* Clipping (phonetics), shortening the articulation of a speech sound, usually a vowel
* Clipping (publications) ...
.
If the signal is bounded, for instance an 8-bit or 16-bit integer, this overshoot and undershoot can exceed the range of permissible values, thus causing clipping.
Strictly speaking, the clipping is caused by the combination of overshoot and limited numerical accuracy, but it is closely associated with ringing, and often occurs in combination with it.
Clipping can also occur for unrelated reasons, from a signal simply exceeding the range of a channel.
On the other hand, clipping can be exploited to conceal ringing in images. Some modern JPEG codecs, such as mozjpeg
libjpeg is a free library with functions for handling the JPEG image data format. It implements a JPEG codec (encoding and decoding) alongside various utilities for handling JPEG data.
It is written in C and distributed as free software toget ...
and ISO libjpeg, use such a trick to reduce ringing by deliberately causing overshoots in the IDCT results. This idea originated in a mozjpeg patch.
Ringing and ripple
In signal processing and related fields, the general phenomenon of time domain oscillation is called ringing, while frequency domain oscillations are generally called ripple
Ripple may refer to:
Science and technology
* Capillary wave, commonly known as ripple, a wave traveling along the phase boundary of a fluid
** Ripple, more generally a disturbance, for example of spacetime in gravitational waves
* Ripple (electri ...
, though generally not "rippling".
A key source of ripple in digital signal processing is the use of window function
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the int ...
s: if one takes an infinite impulse response
Infinite impulse response (IIR) is a property applying to many linear time-invariant systems that are distinguished by having an impulse response h(t) which does not become exactly zero past a certain point, but continues indefinitely. This is in ...
(IIR) filter, such as the sinc filter, and windows it to make it have finite impulse response
In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse ...
, as in the window design method
In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse ...
, then the frequency response of the resulting filter is the convolution of the frequency response of the IIR filter with the frequency response of the window function. Notably, the frequency response of the rectangular filter is the sinc function (the rectangular function and the sinc function are Fourier dual to each other), and thus truncation of a filter in the time domain corresponds to multiplication by the rectangular filter, thus convolution by the sinc filter in the frequency domain, causing ripple. In symbols, the frequency response of is In particular, truncating the sinc function itself yields in the time domain, and in the frequency domain, so just as low-pass filtering (truncating in the frequency domain) causes ''ringing'' in the time domain, truncating in the time domain (windowing by a rectangular filter) causes ''ripple'' in the frequency domain.
Examples
JPEG
JPEG compression can introduce ringing artifacts at sharp transitions, which are particularly visible in text.
This is a due to loss of high frequency components, as in step response ringing.
JPEG uses 8×8 blocks, on which the discrete cosine transform (DCT) is performed. The DCT is a Fourier-related transform
This is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinusoidal and are therefore strongly localized i ...
, and ringing occurs because of loss of high frequency components or loss of precision in high frequency components.
They can also occur at the edge of an image: since JPEG splits images into 8×8 blocks, if an image is not an integer number of blocks, the edge cannot easily be encoded, and solutions such as filling with a black border create a sharp transition in the source, hence ringing artifacts in the encoded image.
Ringing also occurs in the wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
-based JPEG 2000.
JPEG and JPEG 2000 have other artifacts, as illustrated above, such as blocking ("jaggies
"Jaggies" is the informal name for artifacts in raster images, most frequently from aliasing, which in turn is often caused by non-linear mixing effects producing high-frequency components, or missing or poor anti-aliasing filtering prior to samp ...
") and edge busyness (" mosquito noise"), though these are due to specifics of the formats, and are not ringing as discussed here.
Some illustrations:
Baseline JPEG and JPEG2000 Artifacts Illustrated
Pre-echo
In audio signal processing
Audio signal processing is a subfield of signal processing that is concerned with the electronic manipulation of audio signals. Audio signals are electronic representations of sound waves— longitudinal waves which travel through air, consist ...
, ringing can cause echoes to occur before and after transients
Transience or transient may refer to:
Music
* ''Transient'' (album), a 2004 album by Gaelle
* ''Transience'' (Steven Wilson album), 2015
* Transience (Wreckless Eric album)
Science and engineering
* Transient state, when a process variable or ...
, such as the impulsive sound from percussion instrument
A percussion instrument is a musical instrument that is sounded by being struck or scraped by a beater including attached or enclosed beaters or rattles struck, scraped or rubbed by hand or struck against another similar instrument. Ex ...
s, such as cymbals (this is ''impulse'' ringing). The (causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
) echo after the transient is not heard, because it is masked by the
transient, an effect called temporal masking
In audio signal processing, auditory masking occurs when the perception of one sound is affected by the presence of another sound.Gelfand, S.A. (2004) ''Hearing – An Introduction to Psychological and Physiological Acoustics'' 4th Ed. New York, ...
. Thus only the ( anti-causal) echo before the transient is heard, and the phenomenon is called pre-echo
In audio signal processing, pre-echo, sometimes called a '' forward echo'', (not to be confused with reverse echo) is a digital audio compression artifact where a sound is heard before it occurs (hence the name). It is most noticeable in impulsiv ...
.
This phenomenon occurs as a compression artifact in audio compression algorithms that use Fourier-related transforms
This is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinusoidal and are therefore strongly localized i ...
, such as MP3
MP3 (formally MPEG-1 Audio Layer III or MPEG-2 Audio Layer III) is a coding format for digital audio developed largely by the Fraunhofer Society in Germany, with support from other digital scientists in the United States and elsewhere. Origin ...
, AAC, and Vorbis
Vorbis is a free and open-source software project headed by the Xiph.Org Foundation. The project produces an audio coding format and software reference encoder/decoder (codec) for lossy audio compression. Vorbis is most commonly used in conj ...
.
Similar phenomena
Other phenomena have similar symptoms to ringing, but are otherwise distinct in their causes. In cases where these cause circular artifacts around point sources, these may be referred to as "rings" due to the round shape (formally, an annulus), which is unrelated to the "ringing" (oscillatory decay) frequency phenomenon discussed on this page.
Edge enhancement
Edge enhancement
Edge enhancement is an image processing filter that enhances the edge contrast of an image or video in an attempt to improve its acutance (apparent sharpness).
The filter works by identifying sharp edge boundaries in the image, such as the e ...
, which aims to increase edges, may cause ringing phenomena, particularly under repeated application, such as by a DVD player followed by a television. This may be done by ''high''-pass filtering, rather than low-pass filtering.
Special functions
Many special functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined b ...
exhibit oscillatory decay, and thus convolving with such a function yields ringing in the output; one may consider these ringing, or restrict the term to unintended artifacts in frequency domain signal processing.
Fraunhofer diffraction
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance (a distance satisfying Fraunhofer ...
yields the Airy disk
In optics, the Airy disk (or Airy disc) and Airy pattern are descriptions of the best- focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light. The Airy disk is of importance in physics, ...
as point spread function, which has a ringing pattern.
The Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
of the first kind, which is related to the Airy function
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent solut ...
, exhibits such decay.
In cameras, a combination of defocus
In optics, defocus is the aberration in optical systems, aberration in which an image is simply out of focus (optics), focus. This aberration is familiar to anyone who has used a camera, videocamera, microscope, telescope, or binoculars. Opti ...
and spherical aberration can yield circular artifacts ("ring" patterns). However, the pattern of these artifacts need not be similar to ringing (as discussed on this page) – they may exhibit oscillatory decay (circles of decreasing intensity), or other intensity patterns, such as a single bright band.
Interference
Ghosting is a form of television interference where an image is repeated. Though this is not ringing, it can be interpreted as convolution with a function, which is 1 at the origin and ε (the intensity of the ghost) at some distance, which is formally similar to the above functions (a single discrete peak, rather than continuous oscillation).
Lens flare
In photography, lens flare
A lens flare happens when light is scattered or flared in a lens system, often in response to a bright light, producing a sometimes undesirable artifact in the image. This happens through light scattered by the imaging mechanism itself, for ex ...
is a defect where various circles can appear around highlights, and with ghosts throughout a photo, due to undesired light, such as reflection and scattering off elements in the lens.
Visual illusions
Visual illusions can occur at transitions, as in Mach bands
Mach bands is an optical illusion named after the physicist Ernst Mach. It exaggerates the contrast between edges of the slightly differing shades of gray, as soon as they contact one another, by triggering edge-detection in the human visual s ...
, which perceptually exhibit a similar undershoot/overshoot to the Gibbs phenomenon.
See also
* Artifact (error)
In natural science and signal processing, an artifact or artefact is any error in the perception or representation of any information introduced by the involved equipment or technique(s).
Computer science
In ''computer science'', digital art ...
* Digital artifact
Digital artifact in information science, is any undesired or unintended alteration in data introduced in a digital process by an involved technique and/or technology.
Digital artifact can be of any content types including text, audio, video, ...
* sinc filter
In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given cutoff frequency, without affecting lower frequencies, and has linear phase response. The filter's impulse response is a sinc functi ...
* Brick-wall filter
In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given cutoff frequency, without affecting lower frequencies, and has linear phase response. The filter's impulse response is a sinc functi ...
* Chromatic aberration
In optics, chromatic aberration (CA), also called chromatic distortion and spherochromatism, is a failure of a lens to focus all colors to the same point. It is caused by dispersion: the refractive index of the lens elements varies with the w ...
* Ghosting (television)
In television, a ghost is a replica of the transmitted image, offset in position, that is superimposed on top of the main image. It is often caused when a TV signal travels by two different paths to a receiving antenna, with a slight difference ...
* Gibbs phenomenon
In mathematics, the Gibbs phenomenon, discovered by Available on-line at:National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. and rediscovered by , is the oscillatory behavior of the Fourier series of a piecewise continuousl ...
* Low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
* Pre-echo
In audio signal processing, pre-echo, sometimes called a '' forward echo'', (not to be confused with reverse echo) is a digital audio compression artifact where a sound is heard before it occurs (hence the name). It is most noticeable in impulsiv ...
* Purple fringing
In photography (particularly digital photography), purple fringing (sometimes called PF) is the term for an unfocused purple or magenta "ghost" image on a photograph. This optical aberration is generally most visible as a coloring and lightening ...
References
* {{citation
, title=Signal analysis: time, frequency, scale, and structure
, first1=Ronald L.
, last1=Allen
, first2=Duncan W.
, last2=Mills
, publisher=Wiley-IEEE
, year=2004
, url=https://books.google.com/books?id=ZmyKvXQmQwIC
, isbn=978-0-471-23441-8
Signal processing
Computer graphic artifacts