Intrinsic dimension

In signal processing of multidimensional signals, for example in computer vision, the intrinsic dimension of the signal describes how many variables are needed to represent the signal. For a signal of "N" variables, its intrinsic dimension "M" satisfies "0 ≤ M ≤ N".

Usually the intrinsic dimension of a signal relates to variables defined in a Cartesian coordinate system. In general, however, it is also possible to describe the concept for non-Cartesian coordinates, for example, using polar coordinates.

Example

Let "f(x₁, x₂)" be a two-variable function (or signal) which is of the form

:"f(x₁,x₂)" = "g(x₁)"

for some one-variable function "g" which is not constant. This means that "f" varies, in accordance to "g", with the first variable or along the first coordinate. On the other hand, "f" is constant with respect to the second variable or along the second coordinate. It is only necessary to known the value of one, namely the first, variable in order to determine the value of "f". Hence, it is a two-variable function but its intrinsic dimension is one.

A slightly more complicated example is

:"f(x₁,x₂)" = "g(x₁ + x₂)"

"f" is still intrinsic one-dimensional, which can be seen by making a variable transformation

:x₁ + x₂ = y₁

:x₁ - x₂ = y₂

which gives

:"f(x₁,x₂)" = "g(y₁)"

Since the variation in "f" can be described by the single variable "y₁" its intrinsic dimension is one.

For the case that "f" is constant, its intrinsic dimension is zero since no variable is needed to describe variation. For the general case, when the intrinsic dimension of the two-variable function "f" is neither zero or one, it is two.

In the literature, functions which are of intrinsic dimension zero, one, or two are sometimes referred to as "i0D", "i1D" or "i2D" , respectively.

Formal definition

For an "N"-variable function "f", the set of variables can be represented as an "N"-dimensional vector x:

:"f"="f("x")" where x="(x₁, x₂, ..., x_N)"

If for some "M"-variable function "g" and "M × N" matrix A is it the case that

* for all x; "f("x")"="g("Ax")",

* "M" is the smallest number for which the above relation between "f" and "g" can be found,

then the intrinsic dimension of "f" is "M".

The intrinsic dimension is a characterization of "f", it is not an unambiguous characterization of neither "g" nor A. If the above relation is satisfied for some "f", "g", and A, it must also be satisfied for the same "f" and "g′" and A′ given by

:"g′("y")"="g("By")"

:A′=B^-1 A

where B is a non-singular "M × M" matrix, since

:"f("x")"="g′("A′x")"="g("BA′x")"="g("Ax")"

The Fourier transform of functions of low intrinsic dimension

An "N" variable function which has intrinsic dimension "M < N" has a characteristic Fourier transform. Intuitively, since this type of function is constant along one or several dimensions its Fourier transform must appear like an impulse (the Fourier transform of a constant) along the same dimension in the frequency space.

A simple example

Let "f" be a two-variable function which is i1D. This means that there exists a normalized vector n in R² and a one-variable function "g" such that

:"f("x")" = "g("n^Tx")"

for all x in R². If "F" is the Fourier transform of "f" (both are two-variable functions) it must be the case that

:"F("u")"= "G("n^Tu")" · "δ("m^Tu")"

Here "G" is the Fourier transform of "g" (both are one-variable functions), "δ" is the Dirac impulse function and m is a normalized vector in R² perpendicular to n. This means that "F" vanishes everywhere except on a line which passes through the origin of the frequency domain and is parallel to m. Along this line "F" varies according to "G".

The general case

Let "f" be an "N"-variable function which has intrinsic dimension "M", that is, there exists an "M"-variable function "g" and "M &times; N" matrix A such that

:"f("x")"="g("Ax")" for all x.

Its Fourier transform "F" can then be described as follows:

* "F" vanishes everywhere except for a subspace of dimension "M"
* The subspace "M" is spanned by the rows of the matrix A
* In the subspace, "F" varies according to "G" the Fourier transform of "g"

Generalizations

The type of intrinsic dimension described above assumes that a linear transformation is applied to the coordinates of the "N"-variable function "f" to produce the "M" variables which are necessary to represent every value of "f". This means that "f" is constant along lines, planes, or hyperplanes, depending on "N" and "M".

In a general case, "f" has intrinsic dimension "M" is there exist "M" functions "a₁", "a₂", ..., "a_M" and an "M"-variable function "g" such that

* "f("x")" = "g(a₁("x"),a₂("x"),...,a_M("x"))" for all x

* "M" is the smallest number of functions which allows the above transformation

A simple example is transforming a 2-variable function "f" to polar coordinates:

* "f(x₁,x₂)" = "g((x₁² + x₁²)^1/2)", "f" is i1D and is constant along any circle centered at the origin

* "f(x₁,x₂)" = "g(arctan(x₁ / x₁))", "f" is i1D and is constant along all rays from the origin

For the general case, a simple description of either the point sets for which "f" is constant or its Fourier transform is usually not possible.

Applications and history

The case of a two-variable signal which is i1D appears frequently in computer vision and image processing and captures the idea of local image regions which contain lines or edges. The analysis of such regions has a long history, but it was not until a more formal and theoretical treatment of such operations began that the concept of intrinsic dimension was established, even though the name has varied.

For example, the concept which here is referred to as a "image neighborhood of intrinsic dimension 1" or "i1D neighborhood" is called "1-dimensional" by Knutsson (1982), "linear symmetric" by Bigün & Granlund (1987) and "simple neighborhood" in Granlund & Knutsson (1995). The term intrinsic dimension appears to be from Zetzsche & Barth (1990).

References

* cite book
author=Hans Knutsson
title=Filtering and reconstruction in image processing
year=1982
publisher=Linköping Studies in Science and Technology, Dissertation No 88, Linköping University, Sweden

* cite conference
author=Josef Bigün
coauthors=Gösta H. Granlund
title=Optimal orientation detection of linear symmetry
booktitle=Proceedings of the International Conference on Computer Vision
year=1987
pages=pages 433—438

* cite book
author=Gösta H. Granlund
coauthors=Hans Knutsson
title=Signal Processing in Computer Vision
year=1995
publisher=Kluwer Academic Publishers

* cite journal
author=C. Zetzsche
coauthors=E. Barth
title=Fundamental limits of linear filter in the visual processing of two-dimensional signals
journal=Vision Research
year=1990
volume=30
issue=7
pages=pages 1111–1117
doi=10.1016/0042-6989(90)90120-A