I will give a small background and explain the variables and the system first. I have two images which are observed and are constant and we can treat them as continuous functions and I will call them [itex]r[/itex] and [itex]f[/itex]. In my problem, I am trying to find a continuous transform (which is very non-linear) that makes [itex]f[/itex] looks like [itex]r[/itex] according to some similarity criteria or cost function. I will call this transformation function [itex]t[/itex] and I am trying to estimate its parameters [itex]w[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

So, the integral I need to compute turns out to be

[itex]

Z = \int_{-\infty}^{\infty} \exp-{\frac{\left( r(i) - f\left(t(w)\right)\right)^2}{2\sigma^2}} \, dw

[/itex]

where [itex]\sigma[/itex] is a constant. Now, given a constant linear function [itex]A[/itex], [itex]f(t(w))[/itex] is computed as:

[itex]

f(t(i, w)) = (\lceil{Aw}\rceil - Aw) * f(\lfloor{Ax}\rfloor) + (Aw - \lfloor{Aw}\rfloor) * f(\lceil{Ax}\rceil)

[/itex]

where [itex]\lceil \rceil[/itex] gives the ceiling function and [itex]\lfloor \rfloor[/itex] is the floor function. This basically means that I am using linear interpolation to make the transformation function continuous. This is because the images and the transformation are defined in the digital domain and are computed only on a uniform grid (corresponding to the pixel locations) and the transformation [itex]t[/itex] is telling me what the location of a pixel [itex]i[/itex] in image [itex]r[/itex] is in image [itex]f[/itex] through [itex]w[/itex].

Can someone tell me if I can compute such an integral? My first instinct was to use Taylor series to linearise [itex]t(w)[/itex] but it was pointed out that it is not a good idea as [itex]t(w)[/itex] is in the integral and we are integrating over [itex]w[/itex]. So the higher order terms will not cancel out and I cannot justify that approximation.

My Maths and Calculus skills are not great at all. Please do let me know if I need to generate more information about [itex]t[/itex] to be able to do this integration.

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# Can I compute this integral analytically?

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