Physical optics takes into account the wave nature of light. This becomes important when the size of the lens becomes small (eg pinholes) ... there's diffraction around edges and pixels receive contributions from many points in space.
Geometric optics lets you model using ray-tracing, reflections, and Snell's law refraction.
Physical optics uses tools such as Fourier transforms, convolutions, and sinc functions.
Understanding a simple lens system? Geometric optics is your friend. Building an astronomical telescope? Check into physical optics.
Is this correct from a physics standpoint? Feynman had a lot to say about light in his book (QED https://www.amazon.com/QED-Strange-Theory-Light-Matter/dp/06...) and never framed it this way. At one point he remarked that the explanations in the book were related to diffraction, and the explanation there was very different from the Fourier transform.
That statement is a bit muddled, let me unpack it. The infinitely many overlaps thing can be expressed mathematically as the convolution of the image with a function that's one where the aperture is open, and zero where it's not. The thing about the Fourier transform is actually related to a different phenomenon. When the slit is really small, you start getting diffraction effects. The diffraction bands are approximately the Fourier transform of the slit function. However that is not significant unless the slit is extraordinarily tiny.