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# Geometric Interpretation of KLT

Assume the N random variables in a signal vector have a normal joint probability density function:

where and are the mean vector and covariance matrix of , respectively. When , and become and , respectively, and the density function becomes the familiar single variable normal distribution:

The shape of this normal distribution in the N-dimensional space can be represented by the iso-hypersurface in the space determined by equation

where is a constant. Or, equivalently, this equation can be written as

where is another constant related to , and . In particular, with variables and , we have

Here we have assumed

As and are both positive definite, i.e.,

the quadratic equation above represents an ellipse (instead of other quadratic curves such as hyperbola and parabola) centered at . When , the quadratic equation represents an ellipsoid.

In general when , the equation represents a hyper ellipsoid in the N-dimensional space. The center and spatial distribution of this ellipsoid are determined by and , respectively.

By the KLT, the signal is completely decorrelated and its the covariance matrix becomes diagonalized:

and the isosurface equation becomes

This equation represents a standard hyper-ellipsoid in the N-dimensional space. In other words, the KLT rotates the coordinate system so that the semi-principal axes of ellipsoid associated with the normal distribution of are in parallel with ( ), the axes of the new coordinate system. Moreover, the length of the semi-principal axis parallel to is equal to .

The standardization of the ellipsoid is the essential reason why the rotation of KLT can achieve two highly desirable outcomes: (a) the decorrelation of the signal components, and (b) redistribution and compaction of the energy or information contained in the signal, as illustrated in the figure:

Next: Comparison with Other Orthogonal Up: klt Previous: KLT Optimally Compacts Signal
Ruye Wang 2013-04-23