Ringing Artifacts
In signal processing, particularly digital image processing, ringing artifacts are Artifact (error), 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 Transient (acoustics), transients, particularly sounds from percussion instruments; most noticeable are the pre-echos. The term "ringing" is because the output signal oscillates at a fading rate around a sharp transition in the input, similar to a Bell (instrument), bell after being struck. As with other artifacts, their minimization is a criterion in filter design. Introduction The main cause of ringing artifacts is due to a signal being bandlimited (specifically, not having high frequencies) or passed through a low-pass filter; this is the frequency domain description. In terms of the time domain, the cause of this type of ringing is the ripples in the sinc function,, section I.6, Enhancement: Frequency Doma ...
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Ringing Artifact Example
Ringing may mean: Vibrations * Ringing (signal), unwanted oscillation of a signal, leading to ringing artifacts * Vibration of a harmonic oscillator ** Bell ringing * Ringing (telephony), the sound of a telephone bell * Ringing (medicine), a ringing sound in the ears Other uses * Bird ringing, using numbered small metal pr plastic leg rings to track birds * Ringing (of vehicles), the illegal practice of stealing a vehicle and replacing its identification number with that of another vehicle of the same model which has been a write-off See also * Wringing (other) * Ring (other) (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

Gibbs Phenomenon
In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The Nth partial Fourier series of the function (formed by summing the N lowest constituent sinusoids of the Fourier series of the function) produces large peaks around the jump which overshoot and undershoot the function values. As more sinusoids are used, this approximation error approaches a limit of about 9% of the jump, though the infinite Fourier series sum does eventually converge almost everywhere. The Gibbs phenomenon was observed by experimental physicists and was believed to be due to imperfections in the measuring apparatus, but it is in fact a mathematical result. It is one cause of ringing artifacts in signal processing. It is named after Josiah Willard Gibbs. Description The Gibbs phenomenon is a behavior of the Fourier series of a function with a jump discontinuity and is ...
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Sine Integral
In mathematics, trigonometric integrals are a indexed family, family of nonelementary 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\,dt~. Note that the integrand \frac is the sinc function, and also the zeroth Bessel function#Spherical Bessel functions: jn.2C yn, spherical Bessel function. Since is an even function, even entire function (holomorphic over the entire complex plane), is entire, odd, and the integral in its definition can be taken along Cauchy's integral theorem, any path connecting the endpoints. By definition, is the antiderivative of whose value is zero at , and is the antiderivative whose value is zero at . Their difference is given by the Dirichlet integral, \operatorname(x) - \operatorname(x) = \int_0^\infty\frac\,dt = \frac \quad \text \quad \operatorname(x) = \frac + \operatorname(x) ~. In signal processing, t ...
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