# Analysing an optics model in discrete and continuous forms

A discrete one-dimensional model of optical imaging looks like this:

$$I(r) = \sum_i e_i P(r - r_i)$$

Here, the $e_i$ are point light sources at locations $r_i$ in the object and $P$ is a point spread function that blurs each point. We can assume that $P$ is even, non-negative and has a finite extent, ie $P(x) = 0$ for $|x| > p$. The $e_i$ are all positive.

A more complex imaging process instead produces an image like this:

$$I(r)^2 = \sum_{i,j} e_i e_j P(r - r_i) P(r - r_j) \cos (r_i - r_j)$$

(The $^2$ is a consequence of the reconstruction method. In practice we normally take the square root, but per my earlier question here, I think the salient structural features of the resulting image should be unchanged by this transformation.)

Numerical simulations suggest that the latter method allows better resolution of the points when $p \sim \pi$, and is not materially different when $p \ll \pi$. (We ignore the case where $p \gg \pi$ as not physically tenable.) Intuitively, this is because the "trough" in the $\cos (r_i - r_j)$ term reduces the interaction between points at some intermediate separations.

I'd like to be able to be more analytical about this. We can readily extend the above to a continuous model with an object function $O$ in place of the $e_i$. The ordinary image becomes a convolution integral

$$I(r) = \int O(s) \: P(r - s) \; \mathrm{d}s$$

for which there are standard analytical approaches available. However, the more complex reconstruction looks something like this:

$$I(r)^2 = \int \int O(s) \; O(t) \; P(r-s) \; P(r-t) \; \cos (s - t) \;\mathrm{d}s\, \mathrm{d}t$$

This looks fairly intractable to me, and I'm not really sure where to begin with it.

So, my first question is: is there any point in trying to look at the continuous model, or should I just concentrate on the discrete? On the one hand, every single practical use case for the technique will actually be using discrete measurements, so in that sense the discrete model is more realistic. On the other, it may be that there are things one could demonstrate using the continuous model that are not available otherwise. (But even if that's the case, I'm probably not capable of doing so without at least being nudged in the right direction!)

And I guess my second question is: continuous or not, does anyone have any other suggestions of useful ways to analyse this model?

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Is $I$ a complex field value or an irradiance? Your last expression resembles something in partially coherent image theory called a transmission cross-coefficient. (See my home page; my avatar is a picture of this.) If you have any further questions about this, feel free to email me at rlgordonma at gmail dot com. – Ron Gordon Nov 9 '14 at 12:57