Point and Line Spread Functions

Question

For optical systems, is there a way to estimate its point-spread function from its line-spread function?

In particular, I wish to estimate the encircled energy.

Answer 1

quick and dirt answer: take the (minimal) width of the line and build a circle with that as diameter. here is your PSF.

Answer 2

Here is the analytical solution using Abel transform.

Let's have $LSF(x), x\in\mathbb{R}$ so that it is symmetric $LSF(x) = LSF(-x)$ and for transform to work$$\lim_{x \to \infty} \frac{LSF(x)}{x} = 0.$$

Let's assume we are looking for radially symmetric function $PSR(r)$, such that $PSF(x,y) = PSR(\sqrt{x^2+y^2})$ and $$LSF(x) = \int_{-\infty}^{\infty} PSF(x,y) dy.$$

Thus $$LSF(x) = 2 \int_{0}^{\infty} PSR(\sqrt{x^2+y^2}) dy$$ and upon substitution $r=\sqrt{x^2+y^2}$ we have $$dr = \frac{y dy}{\sqrt{x^2+y^2}} = \frac{\sqrt{r^2 - x^2} dy}{r}$$ so that $$dy = \frac{r}{\sqrt{r^2-x^2}} dr $$ and $$LSF(x) = 2 \int_{x}^{\infty} PSR(r) \frac{r}{\sqrt{r^2-x^2}} dr,$$ the last equation has a structure of the Abel transform and thus $$PSR(r) = -\frac{1}{\pi} \int_{r}^{\infty} \frac{d LSF(x)}{dx} \frac{1}{\sqrt{x^2-r^2}} dx.$$

Here is the method using Fourier transform in 2D

Fourier transform of PSF is called optical transfer function $$OTF(\xi_x,\xi_y) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} PSF(x,y) e^{-2 \pi i (x \xi_x + y \xi_y)} dx dy$$ and there is also Fourier transform of LSF $$\mathcal{F}(LSF)(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} LSF(x) e^{-2 \pi i x \xi} dx $$

now we have $$OTF(\xi_x,0) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} PSF(x,y) dy e^{-2 \pi i (x \xi_x)} dx = \frac{1}{2\pi} \int_{-\infty}^{\infty} LSF(x) e^{-2 \pi i (x \xi_x)} dx$$ and also $$OTF(\xi_x,0) = \frac{1}{\sqrt{2\pi}}\mathcal{F}(LSF)(\xi_x)$$ and if you believe that $OTF(\xi_x,\xi_y)$ is radially symmetric then $$OTF(\xi_x,\xi_y) = \frac{1}{\sqrt{2\pi}}\mathcal{F}(LSF)(\sqrt{\xi_x^2 + \xi_y^2}).$$ Now we can sample OTF using that formula and perform inverse 2D Fourier transform. By this kind of slice theorem we obtain the solution. Note that I might use different scaling of Fourier transform.

Now for the case of non-symmetric functions, Marchand 1965 might be of interest. In Marchand 1964 there can be found somehow more heavy-weight derivation of the first method but the reduction of the second method, which then does not need double integral by means of integration using Bessel function of the first kind. In general first method might be of interest for those having some smooth fit of LSF, the second method might be useful in case of sampled LSF.