# Inversion of Radon transform by incomplete data: specific case

Let $R[f](p,t)$ denote the Radon transform of smooth function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ with compact support in $\mathbb{R}^n_+$: $$R[f](p,t) = \int\limits_{x \cdot p = t} f(x) dx.$$ In field of mathematical economics $p$ and $t$ may have meaning of prices and then be positive. Hence there arises a problem of recovery of function $f(x)$ via it's Radon transforms $R[f](p,t)$ with $p \in \mathbb{R}^n_+$ and $t > 0$. I would like to know if there are some results on this topic.

I was asked about economical interpretation of the Radon transform. A part of economical models consists of models of production. The simplest model of production is so called Leontieff model. Say, in some sector one produce some product and use $n$ other resourses. To produce $1$ unit of our product we have to use $x_1$ units of first resourse, $x_2$ units of second e.t.c., in other words, if we have $(u_1,\ldots,u_n)$ resourses we can produce $\min \left( \frac{u_1}{x_{1}},\ldots,\frac{u_n}{x_n}\right)$ units of our production. This function is called Leontieff production function. In real models there exists an effect of substitution of production factors at the microlevel and we have to modify the Leontieff production function by the appropriate neoclassic production function. But we will still talk about Leontieff model. Let $p_0$ be a price of a unit of our production and $p=(p_1,\ldots,p_n)$ be a vector of prices of resourses, $\mu$ be a measure, that describes a distribution of available production powers over production technologies (any technology is given by a vector of resourses to produce a unit vector of our production, but we can use many technologies in production, see Hauthekker-Johaneson model) Then our profit function is a function $$\Pi(p,p_0) = \int (p_0 - p \cdot x)_{+} \mu(dx)$$ and it gives us a maximal profit possible (it is a result of maximisation over distributions $\mu$ allowed by existing production powers). It's second derivative is exactly the Radon transform of measure $\mu$.

• Could you expand on the relation of the Radon transform and economics? (Out of curiosity.)
– Dirk
Oct 20, 2012 at 18:31
• Of course, I modified a message. Oct 21, 2012 at 9:19
• This is well studied in the mathematics of medical imaging, you might look around there. The Radon transform is a smoothing operator with the degree of smoothing being exactly $1/2$ of a derivative, so inversion is mildly ill-posed. Oct 21, 2012 at 10:21
• Thank you Nick Alger, but can you be more exact please? I listened to a course on computer tomography (with applictions in medicine) and in this course we were speaking about reconstruction of function by incomplete data of another character: we know only integrals over any linear manifold, that doesn't intersect some convex body (Cormack-type theorems (1963-1964)). Oct 21, 2012 at 13:16
• The Radon transform smooths $H^s \mapsto H^{s+1/2}$, with eigenvalues $\lambda_i \rightarrow 0$ corresponding to increasingly oscillatory eigenvectors. If you add white noise to a radon transformed image then try to invert it, the $i'th$ components of the noise will be amplified by $1/\lambda_i$, which can be arbitrarily large. To overcome this one usually needs regularization or a prior, depending on whether you take a deterministic or probabilistic approach to the problem. The $1/2$ power smoothing is pretty weak compared to other problems. I don't know any more specifics. Oct 21, 2012 at 17:01

I second the recommendations in the comments which suggest looking at the tomography literature and at the characterization of the problem as an ill-posed inverse problem. I, in particular, recommend these two references.

The first one contains a detailed exposition of the recovery of $f$ in the special case $n=2$ (chapter 5):

That book is quite neat in the sense that it presents a very wide and rigorous exposition of the field of ill-posed inverse problems. Thus, it constitutes an excellent entry point to the field.

There are also a lot of papers in statistics on that particular problem. The second reference that I recommend is a paper which might be of particular interest to you (since it is an econometric paper that studies inversion of Radon transforms for the purpose of some non-parametric estimation problem):

Edit: I updated a link above for the actual published paper location of the second reference. A 2007 working paper version is available at the following link.