This subject I have been actively avoiding to write on. Not only because it’s a lot to write. All you bloggers out there know how tough it is sitting down at the end of the day, after 8-to-fuckall hours on the computer and type (or god forbid – think!) for a few more. But I need to scrape the ol’e think box, so here we are.
Before we actually start imaging, there a fair bit of background material to overview. I’ll try to keep the proofs as intuitive as possible, because all of this is hardly the main issue. Spoiler, the entire issue here is a couple of Fourier transforms (FT), in the end. But before writing them down, we need to know what they actually mean. Here, we are going to start with simple field propagation expressions, and then make them even simpler.
Propagation 101
Let us introduce the wave equation.
This can be valid for a plethora of propagating fields: Acoustic, Electric, Magnetic, etc. Here, n is the refractive index in area of interest (a.k.a Object Domain, OD), c0 is free-space propagating speed, u is the scalar field in discussion and J are all the active sources inside the OD. In propagation problems it is customary to decompose the field in the OD to the incident and scattered field, as such
If you got this intuitively at first, you are probably lying and you learned this before. The rest of us need to go through the following stage: Each of the parts, incident and total field, can be depicted with a separate wave equation. The incident field applies
and here, I must say that to this day, I think once or twice while seeing this. We claim that the incident field is only affected by the basic refractive index. If this perplexed you, understand that the explanation is actually the exact opposite. This is the basic demand. That there is a background propagating field “attacking” the OD, that applies the wave equation. The demand isn’t the decomposition of the total field, but the background propagating incident field. Now we can write the total wave equation, that is exactly what was written initially, but with the total field
Subtracting the two equations from one another obtains
For simplicity reasons, it is customary to define here the Object Function, denoted here by
The last wave equation here can be resolved with a Huygens integral within the entire OD, using a Green’s function
This is the infamous Lippmann-Schwinger equation. Up until now, this is basic diffraction theory. Now let’s go hard core math mode!
Let’s describe the scattered field from before using an operator notation
Here we refer to G as the propagating operator over the specific medium O. Switching sides a bit, this can also be phrased by
Where I is the identity operator. Bamboozled yet? Here comes the worse part you will need to handle here: We assume that the propagating operator has unique solution, hence invertible. It is a bit irresponsible to not at least show an outline of a proof, but here we will be satisfied that this is a characteristic of the wave equation. If it has a unique solution, it must have an inverse operator. So assuming that a solution for
exists, there is a unique inversion operator. This is is excellent, as it provides a relation between the well known incident field, to the total, hence scattered field! Cool right?
NO.
But up to this very point everything was very cool, in a nerdy, boring sort-of way. We have no knowledge on the inverse operator, even though we did write it in an extremely elegant fashion. So there you have a terrible introduction to the fact that imaging problems are inverse problems.
Born Series and A Speedy Introduction To Born Approximation
Again, without proof, we expand the inverse operator to something similar to a Maclaurin series.
that translates to the extremely readable integral series
It is important to at least partially understand this series before performing the final simplifying step. So the first element of the series is simply the incident field. The second considers the effect of each point in the OD (r‘) on the sampled point, r. The third also considers the coupling of two points inside the OD on each other.

So it is plain to see that if we keep only the first and second elements in the series, the problem is not only simpler, but Linear. So let’s find some excuses as to why it can be done. The first commonly used one is the possibility that the medium is weakly scattering. This means that the Object Function is relatively close to 0, hence making the integral negligible. This is hardly enough, but you can read further in the article referenced at the bottom in order to have a more rigorous explanation.
But what if the object is not weakly scattering? Well, basically you are screwed. But! We neglect the third+ elements anyway, knowing that it will cause artefacts to appear while building the image. There are some approaches to attempt to overcome these limitations, hopefully I’ll discuss them later on.
So neglecting the unwanted integrals, we remain with the Born approximated scattering operator, that we will use to define the algorithms
This can be used a myriad of algorithms: Computerized tomography, Diffraction tomography, Radar imaging, etc. In these lessons I’ll only discuss Computerized Tomography (CT), that uses X-Ray frequencies in order to image various body parts.
On the limitations of First-Order Born Approx.
Slaney, Malcolm, Avinash C. Kak, and Lawrence E. Larsen. “Limitations of imaging with first-order diffraction tomography.” IEEE transactions on microwave theory and techniques 32.8 (1984): 860-874.