Model fitting by least squares

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## Smoothing the denominator spectrum

Equation (1) gives us one way to divide by zero. Another way is stated by the equation

 (6)

where the strange notation in the denominator means that the spectrum there should be smoothed a little. Such smoothing fills in the holes in the spectrum where zero-division is a danger, filling not with an arbitrary numerical value but with an average of nearby spectral values. Additionally, if the denominator spectrum is rough, the smoothing creates a shorter autocorrelation function.

Both divisions, equation (1) and equation (6), irritate us by requiring us to specify a parameter, but for the latter, the parameter has a clear meaning. In the latter case we smooth a spectrum with a smoothing window of width, say which this corresponds inversely to a time interval over which we smooth. Choosing a numerical value for has not such a simple interpretation.

We jump from simple mathematical theorizing towards a genuine practical application when I grab some real data, a function of time and space from another textbook. Let us call this data and its 2-D Fourier transform . The data and its autocorrelation are in Figure 1.

antoine10
Figure 1.
2-D data (right) and a quadrant of its autocorrelation (left). Notice the longest nonzero time lag on the data is about 5.5 sec which is the latest nonzero signal on the autocorrelation.

The autocorrelation of is the inverse 2-D Fourier Transform of . Autocorrelations satisfy the symmetry relation . Figure 2 shows only the interesting quadrant of the two independent quadrants. We see the autocorrelation of a 2-D function has some resemblance to the function itself but differs in important ways.

Instead of messing with two different functions and to divide, let us divide by itself. This sounds like but we will watch what happens when we do the division carefully avoiding zero division in the ways we usually do.

Figure 2 shows what happens with

 (7)

and with
 (8)

From Figure 2 we notice that both methods of avoiding zero division give similar results. By playing with the and the smoothing width the pictures could be made even more similar. My preference, however, is the smoothing. It is difficult to make physical sense of choosing a numerical value for . It is much easier to make physical sense of choosing a smoothing window. The smoothing window is in space, but Fourier transformation tells us its effect in space.

antoine11
Figure 2.
Equation (7) (left) and equation (8) (right). Both ways of dividing by zero give similar results.

 Model fitting by least squares

Next: Imaging Up: HOW TO DIVIDE NOISY Previous: Damped solution

2011-07-17