I have a function $$ f(x) = \int_{0}^{\infty} \exp\left\{-\iint_{R(x,y)} g(x',y') \, dx' \, dy'\right\} \, dy, $$ where $R(x,y) \subset \mathbb{R}^2$ is a given family of closed regions in the $(x', y')$-plane, parametrized by $(x, y)$.
Suppose $f$ is observed/measured. I want to draw conclusions about $g$. When can we solve for $g$ uniquely? When can we partially recover $g$? How do a priori bounds or statistical assumptions on $f$ translate into properties of $g$?
I understand this problem might be rather general, but I am wondering what is the best field to approach this type of analysis. Below are some thoughts and a simple example.
Thoughts:
- This problem appears related to integral transforms, where $f(x)$ is an integral over transformed values of $g(x', y')$. Examples include the Radon transform and its attenuated variants.
- If $R(x, y)$ represents line segments or curves, this resembles problems in tomography (e.g., X-ray or optical tomography).
- The exponential in the integral suggests a connection to attenuation problems, seen in exponential integral geometry.
- Uniqueness and inversion depend on the geometry of $R(x, y)$. Sufficient coverage or diversity in $R(x, y)$ might help fully recover $g$.
- If $f(x)$ is noisy or incomplete, techniques from ill-posed problems (e.g., Tikhonov regularization) could help estimate $g$.
- Statistical or bounding assumptions on $f(x)$ might guide how $g$ is constrained or regularized, possibly via Bayesian inference methods.
I might just be lacking exactly the type of approach one would need for this specific problem. Any ideas?
A simple example:
If $ g(x, y) = g $ is a constant, and $R(x, y) = \{(x', y') : 0 \leq y' \leq y, \, x-y' \leq x' \leq x+y'\}$ (often known as a past light cone), the integral simplifies as follows $$ f(x)=\int_0^\infty \exp(-g \cdot y^2) \, dy = \frac12 \sqrt{\frac{\pi}{g}} $$ which makes the inversion trivial.
What if $g(x,y)\equiv g(x)$, only $x$-dependent? In this case, one would get $$ f(x) = \int_{0}^{\infty} \exp\left\{-\int_0^y\int_{x-y'}^{x+y'} g(x') \, dx' \, dy'\right\} \, dy $$ Which seems to be injective. What properties of $g$ can I infer from $f$? Is is invertible?
