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# Unitary and Orthogonal Transforms

A square matrix ( for the ith column vector of ) is unitary if its inverse is equal to its conjugate transpose, i.e., . In particular, if a unitary matrix is real , then and it is orthogonal. Both the column and row vectors ( ) of a unitary or orthogonal matrix are orthogonal (perpendicular to each other) and normalized (of unit length), or orthonormal, i.e., their inner product satisfies:

These orthonormal vectors can be used as the basis vectors of the n-dimensional vector space.

Any unitary (orthogonal) matrix can define a unitary (orthogonal) transform of a vector :

or in tabular form:

• Forward transform:

The first equation above is the forward transform and can be written in component form as:

This is the transform coefficient representing the projection of vector onto the ith column vector of the transform matrix (for the ith basis of the n-dimensional space).

• Inverse transform:

The second equation is the inverse transform and can be written as:

which represents the signal vector as a linear combination (weighted sum) of the column vectors of the transform matrix . Geometrically, is a vector in the n-dimensional space spanned by the orthonormal vectors as the bases, and each coefficient (coordinate) is the projection of onto the corresponding basis vector .

As the n-dimensional space can be spanned by the column vectors of any n by n unitary (orthogonal) matrix, a vector in the space can be represented by any of such matrices, each defining a different transform.

Examples:

• The identical matrix is orthogonal and defines a special orthogonal transform . Here the ith column or row vector of is with the ith element equal 1 and all others 0. The inverse transform is

In this special case, the signal and its transform are identical.

• When the (m,n)th component of matrix is , the corresponding transform is discrete Fourier transform. The nth column vector of the transform matrix represents a sinusoid of a frequency , and the corresponding complex coordinate represents the magnitude and phase of this nth frequency component. As any two orthogonal coordinate systems can be related by a rotation, Fourier transform represents the signal in a rotated coordinate system.

An orthogonal ( unitary) transform (where ) can be interpreted geometrically as the rotation of the vector about the origin, or equivalently, the representation of the same vector in a rotated coordinate system. A orthogonal (unitary) transform does not change the vector's length:

as . This is the Parseval's relation. If is interpreted as a signal, then its length represents the total energy or information contained in the signal, which is preserved during any unitary transform.

Next: Principal Component Analysis (PCA) Up: Image_Processing Previous: Transform Coding and JPEG
Ruye Wang 2007-11-15