Comparametric equation
Encyclopedia
A comparametric equation is an equation
that describes a parametric
relationship between a function
and a dilated
version of the same function, where the equation does not involve the parameter
. For example, ƒ(2t) = 4ƒ(t) is a comparametric equation, when we define g(t) = f(2t), so that we have g = 4ƒ no longer contains the parameter, t. The comparametric equation g = 4ƒ has a family of solutions, one of which is ƒ = t2.
To see that ƒ = t2 is a solution, we merely substitute back in: g = ƒ(2t) = (2t)2 = 4t2 = 4ƒ, so that g = 4ƒ.
Comparametric equations arise naturally in signal processing
when we have multiple measurements of the same phenomenon, in which each of the measurements was acquired using a different sensitivity. For example, two or more differently exposed pictures of the same subject matter give rise to a comparametric relationship, the solution of which is the response function of the camera, image sensor, or imaging system.
Comparametric equations have been used in many areas of research, and have many practical applications to the real world. They are used in radar
, microphone array
s, and have been used in processing crime scene video in homicide
trials in which the only evidence against the accused was video recordings of the murder.
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
that describes a parametric
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
relationship between a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
and a dilated
Dilation (mathematics)
In mathematics, a dilation is a function f from a metric space into itself that satisfies the identityd=rd \,for all points where d is the distance from x to y and r is some positive real number....
version of the same function, where the equation does not involve the parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
. For example, ƒ(2t) = 4ƒ(t) is a comparametric equation, when we define g(t) = f(2t), so that we have g = 4ƒ no longer contains the parameter, t. The comparametric equation g = 4ƒ has a family of solutions, one of which is ƒ = t2.
To see that ƒ = t2 is a solution, we merely substitute back in: g = ƒ(2t) = (2t)2 = 4t2 = 4ƒ, so that g = 4ƒ.
Comparametric equations arise naturally in signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
when we have multiple measurements of the same phenomenon, in which each of the measurements was acquired using a different sensitivity. For example, two or more differently exposed pictures of the same subject matter give rise to a comparametric relationship, the solution of which is the response function of the camera, image sensor, or imaging system.
Comparametric equations have been used in many areas of research, and have many practical applications to the real world. They are used in radar
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
, microphone array
Microphone array
A microphone array is any number of microphones operating in tandem. There are many applications:*Systems for extracting voice input from ambient noise *Surround sound and related technologies...
s, and have been used in processing crime scene video in homicide
Homicide
Homicide refers to the act of a human killing another human. Murder, for example, is a type of homicide. It can also describe a person who has committed such an act, though this use is rare in modern English...
trials in which the only evidence against the accused was video recordings of the murder.
Related concepts
- Parametric equationParametric equationIn mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
- Functional equationFunctional equationIn mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
- Contraction mappingContraction mappingIn mathematics, a contraction mapping, or contraction, on a metric space is a function f from M to itself, with the property that there is some nonnegative real number k...
or dilation mapping