BUILT-IN MATHEMATICA SYMBOL

KarhunenLoeveDecomposition

gives the Karhunen-Loeve transform

of the numerical arrays

and the transformation matrix m, returning the result in the form

.

KarhunenLoeveDecomposition

uses the inverse of the matrix m for transforming the

.

MORE INFORMATION

KarhunenLoeveDecomposition works with lists of arbitrary numerical commensurate arrays.

KarhunenLoeveDecomposition also works with arbitrary images.

The inner product of m and gives .

The total variance of the is the same as the total variance of the .

The are given in order of decreasing variance.

The rows of the transformation matrix m returned by KarhunenLoeveDecomposition are the eigenvectors of the covariance matrix formed from the arrays .

KarhunenLoeveDecomposition effectively computes the inverse Karhunen-Loeve transformation. If the length of is less than the size of m, missing components are assumed to be zero.

With an option setting True, KarhunenLoeveDecomposition shifts the datasets so that their means are zero.

EXAMPLES

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Basic Examples (2)

Karhunen-Loeve decomposition of two datasets:

Principal component decomposition of RGB color channels:

Karhunen-Loeve decomposition of two datasets:

In[1]:=

Out[1]=

Principal component decomposition of RGB color channels:

In[1]:=

Out[1]=

Scope (5)

Principal components of two grayscale images:

Karhunen-Loeve decomposition of three matrix-valued datasets:

Principal components of a list of color images:

Specify the transformation matrix:

Use a transformation matrix and lesser datasets:

Options (1)

Karhunen-Loeve decomposition with datasets shifted to mean zero:

Applications (3)

Enhance the color contrast of an RGB image:

Reconstruct a multichannel image from 1, 2, or 3 components:

Transform a list of pictorial faces:

Show the residual images when using only the first three components:

Properties & Relations (5)

The Karhunen-Loeve decomposition preserves the total variance:

Relation to PrincipalComponents:

A setting

True is equivalent to subtracting the mean from the input data:

Normalizing by the square root of the number of datasets better preserves the input dynamics:

KarhunenLoeveDecomposition normally returns images with data type

: