Generalized Cross-correlation
- From: "Jon Slaughter" <Jon_Slaughter@xxxxxxxxxxx>
- Date: Thu, 26 Nov 2009 11:48:14 -0600
I'm trying to generalize the cross-correlation to an arbitrary set of parameters rather than just translation.
http://senduit.com/79043a
I need some form of "regularization" to make sense of the correlation spectrum. The example at the bottom of the pdf gives an example.
With the standard cross-correlation we see that shifting does not the extrema of the "kernel". Similarly, for the "fourier kernel" e^(it), the frequency parameter w does not modify the magnitude either but just creats more extrema all with the same magnitude. For other kernels this is not so.
Suppose K(t;p) is kernel that makes "sense" in whatever way. Let Q(t;p1,p2) = e^(p2)*K(t;p1). Here, because of the p2 depenency, the correlation interpretation of the p-spectrum will generally not make much sense. That is, the spectrum will, in general, grow with p2 even if there is no intuitive correlation.
For the standard cross-correlation we get a consistent view because the shifting parameter does not effect the kernel negatively(talking about interpretation). I'm not sure how to handle this in general. Simply requiring that max_(p_k) K(x; p) be indepenent of p does not seem good enough? (basicaly scaling is bad while shifting in time, frequency or any other non-scaling parameter is ok)
Any Ideas?
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