The triple correlation of an ordinary function on the real line is the integral of the product of that function with two independently shifted copies of itself:
:
The Fourier transform of triple correlation is the
bispectrum In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions.
Definitions
The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional po ...
. The triple correlation extends the concept of
autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
, which correlates a function with a single shifted copy of itself and thereby enhances its latent periodicities.
History
The theory of the triple correlation was first investigated by statisticians examining the
cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will ha ...
structure of non-
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
random processes. It was also independently studied by physicists as a tool for
spectroscopy
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
of laser beams.
Hideya Gamo in 1963 described an apparatus for measuring the triple correlation of a laser beam, and also showed how phase information can be recovered from the real part of the bispectrum—up to sign reversal and linear offset. However, Gamo's method implicitly requires the Fourier transform to never be zero at any frequency. This requirement was relaxed, and the class of functions which are known to be uniquely identified by their triple (and higher-order) correlations was considerably expanded, by the study of Yellott and Iverson (1992). Yellott & Iverson also pointed out the connection between triple correlations and the visual texture discrimination theory proposed by
Bela Julesz
Bela may refer to:
Places Asia
*Bela Pratapgarh, a town in Pratapgarh District, Uttar Pradesh, India
*Bela, a small village near Bhandara, Maharashtra, India
*Bela, another name for the biblical city Zoara
* Bela, Dang, in Nepal
* Bela, Janakpur, ...
.
Applications
Triple correlation methods are frequently used in signal processing for treating signals that are corrupted by
additive white Gaussian noise
Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
; in particular, triple correlation techniques are suitable when multiple observations of the signal are available and the signal may be translating in between the observations, e.g.,a sequence of images of an object translating on a noisy background. What makes the triple correlation particularly useful for such tasks are three properties: (1) it is invariant under translation of the underlying signal; (2) it is unbiased in additive Gaussian noise; and (3) it retains nearly all of the relevant phase information in the underlying signal. Properties (1)-(3) of the triple correlation extend in many cases to functions on an arbitrary
locally compact group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...
, in particular to the groups of rotations and rigid motions of euclidean space that arise in computer vision and signal processing.
Extension to groups
The triple correlation may be defined for any locally compact group by using the group's left-invariant
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, though ...
. It is easily shown that the resulting object is invariant under left translation of the underlying function and unbiased in additive Gaussian noise. What is more interesting is the question of uniqueness : when two functions have the same triple correlation, how are the functions related? For many cases of practical interest, the triple correlation of a function on an abstract group uniquely identifies that function up to a single unknown group action. This uniqueness is a mathematical result that relies on the
Pontryagin duality
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...
theorem, the
Tannaka–Krein duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topologic ...
theorem, and related results of Iwahori-Sugiura, and Tatsuuma. Algorithms exist for recovering bandlimited functions from their triple correlation on Euclidean space, as well as rotation groups in two and three dimensions. There is also an interesting link with
Wiener's tauberian theorem In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in or can be approximated by linear combinations ...
: any function whose translates are dense in
, where
is a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
Abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
, is also uniquely identified by its triple correlation.
References
* K. Hasselman, W. Munk, and G. MacDonald (1963), "Bispectra of ocean waves", in ''Time Series Analysis'', M. Rosenblatt, Ed., New York: Wiley, 125-139.
*
*
* R. Kakarala (1992) ''Triple correlation on groups'', Ph.D. Thesis, Department of Mathematics, University of California, Irvine.
* R. Kondor (2007), "A complete set of rotationally and translationally invariant features for images", {{arXiv, cs/0701127
Integral transforms
Fourier analysis
Signal processing
Covariance and correlation