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U. Glaeser

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where m is the derivative order. The order of derivative necessary to center a mode at a particular frequency<br />

is therefore<br />

© 2002 by CRC Press LLC<br />

(28.38)<br />

The Wigner Distribution<br />

The previous examples indicate that a local frequency representation need not have an orthogonal basis.<br />

In fact, it need not even be linear. The Wigner distribution was introduced by Eugene Wigner in 1932<br />

[16] for use in quantum mechanics (in 1-D). In 2-D, the Wigner distribution can be written as<br />

(28.39)<br />

where the asterisk denotes complex conjugation. The Wigner distribution is real valued, so does not have<br />

an explicit phase component (as seen in, e.g., the Fourier transform). A number of discrete approximations<br />

to this distribution (sometimes referred to as pseudo-Wigner distributions) have also been formulated.<br />

Spatial/Scale Representations (Wavelets)<br />

Scale is a concept that has proven very powerful in many applications, and may under some circumstances<br />

be considered as fundamental as frequency. Given a set of (1-D) functions<br />

(28.40)<br />

where the indices j and k correspond to dilation (change in scale) and translation, respectively, a signal<br />

decomposition<br />

(28.41)<br />

emphasizes the scale (or resolution) characteristics of the signal (specified by j) at specific points along<br />

x (specified by k), yielding a multiresolution description of the signal.<br />

A class of functions W jk(x) that have proven extremely useful are referred to as wavelets. A detailed<br />

discussion of wavelets is beyond the scope of this chapter (see [17–19] for excellent treatments of this<br />

topic); however, an important aspect of any representation (including wavelets) is the resolution of the<br />

representation, and how it can be measured.<br />

Resolution<br />

m ( Ωmσ) 2<br />

=<br />

Wf ( x, y, ξx, ξy) f x α<br />

+ -- , y<br />

2<br />

β<br />

⎛ + -- ⎞ ∗ α<br />

f x – -- , y<br />

⎝ 2 ⎠ 2<br />

β<br />

∞ ∞<br />

=<br />

⎛ – -- ⎞<br />

∫ ∫<br />

e<br />

⎝ 2 ⎠<br />

– ∞ – ∞<br />

Wjk( x)<br />

W 2 j = ( x– k)<br />

∑∑<br />

f( x)<br />

= bjkWjk( x)<br />

j<br />

−j2π(αξx + βξy )<br />

dα dβ<br />

In dealing with joint representations, resolution is a very important issue. It arises in a number of ways.<br />

In discussing the Gabor representation, it was noted that the functions minimized the uncertainty<br />

inequalities, e.g.,<br />

(28.42)<br />

Note that it is the product that is minimized. Arbitrarily high resolution cannot be achieved in both domains<br />

simultaneously, but can be traded between the two domains at will. The proper balance depends on the<br />

application. It should be noted that the “effective width” measures ∆x, ∆ξ x, etc. (normalized second moment<br />

measures) are not the only way to define resolution. For example, the degree of energy concentration could<br />

k<br />

∆x ∆ξ x<br />

1<br />

4π<br />

----- ≥

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