K-transform

In mathematics, the K transform (also called the Single-Pixel X-ray Transform) is an integral transform introduced by R. Scott Kemp and Ruaridh Macdonald in 2016. The transform allows the structure of a N-dimensional inhomogeneous object to be reconstructed from scalar point measurements taken in the volume external to the object.

Gunther Uhlmann proved that the K transform exhibits global uniqueness on ${\mathbb{ℝ}}^n$, meaning that different objects will always have a different K transform. This uniqueness arises by the use of a monotone, nonlinear transform of the X-ray transform. By selecting the exponential function for the monotone nonlinear function, the behavior of the K transform coincides with attenuation of particles in matter as described by the Beer–Lambert law, and the K transform can therefore be used to perform tomography of objects using a low-resolution single-pixel detector.

An inversion formula based on a linearization was offered by Lai et al., who also showed that the inversion is stable under certain assumptions. A numerical inversion using the BFGS optimization algorithm was explored by Fichtlscherer.

Definition

Let an object $f$ be a function of compact support that maps into the positive real numbers $$f:\mathrm{\Omega }\to {\mathbb{ℝ}}_0^{+}.$$ The K-transform of the object $f$ is defined as $${𝒦}:L^1(\mathrm{\Omega },{\mathbb{\mathbb{P}}}_0^{+})\to [0,1],$$ $${𝒦}f(r)\equiv \underset{L_D(r)}{\int }{e}^{{𝒫}f(l)}dl,$$ where $L_D(r)\equiv L(r)\cap L(D)$ is the set of all lines originating at a point $r$ and terminating on the single-pixel detector $D$, and ${𝒫}$ is the X-ray transform.

Proof of global uniqueness

Let ${𝒫}f$ be the X-ray transform transform on ${\mathbb{ℝ}}^n$ and let ${𝒦}$ be the non-linear operator defined above. Let $L^1$ be the space of all Lebesgue integrable functions on ${\mathbb{ℝ}}^n$ , and $L^{\mathrm{\infty }}$ be the essentially bounded measurable functions of the dual space. The following result says that $-{𝒦}$ is a monotone operator.

For $f,g\in L^1$ such that ${𝒦}f,{𝒦}g\in L^{\mathrm{\infty }}$ then $$⟨{𝒦}f-{𝒦}g,f-g⟩\le 0$$ and the inequality is strict when $f\ne g$.

Proof. Note that ${𝒫}f(r,\theta )$ is constant on lines in direction $\theta$, so ${𝒫}f(r,\theta )={𝒫}f(E_{\theta }r,\theta )$, where $E_{\theta }$ denotes orthogonal projection on ${\theta }^{\mathrm{\perp }}$. Therefore:

$$⟨{𝒦}f-{𝒦}g,f-g⟩=\underset{{\mathbb{ℝ}}^n}{\int }\underset{{\mathbb{𝕊}}^{n-1}}{\int }(e^{-{𝒫}f(r,\theta )}-e^{-{𝒫}g(r,\theta )})(f-g)(r)d\theta dr$$

$$=\underset{{\mathbb{𝕊}}^{n-1}}{\int }\underset{{\mathbb{ℝ}}^n}{\int }(e^{-{𝒫}f(r,\theta )}-e^{-{𝒫}g(r,\theta )})(f-g)(r)drd\theta$$

$$=\underset{{\mathbb{𝕊}}^{n-1}}{\int }\underset{{\theta }^{\mathrm{\perp }}}{\int }(e^{-{𝒫}f(E_{\theta }r,\theta )}-e^{-{𝒫}g(E_{\theta }r,\theta )})\underset{\mathbb{ℝ}}{\int }(f-g)(E_{\theta }r+s\theta )dsd{r}_{H}d\theta$$

$$=\underset{{\mathbb{𝕊}}^{n-1}}{\int }\underset{{\theta }^{\mathrm{\perp }}}{\int }(e^{-{𝒫}f(E_{\theta }r,\theta )}-e^{-{𝒫}g(E_{\theta }r,\theta )})({𝒫}f(E_{\theta }r,\theta )-{𝒫}g(E_{\theta }r,\theta ))d{r}_{H}d\theta$$

where $d{r}_H$ is the Lebesgue measure on the hyperplane ${\theta }^{\mathrm{\perp }}$. The integrand has the form $(e^{-s}-e^{-t})(s-t)$, which is negative except when $s=t$ and so $⟨{𝒦}f-{𝒦}g,f-g⟩<0$ unless ${𝒫}f={𝒫}g$ almost everywhere. Then uniqueness for the X-Ray transform implies that $g=f$ almost everywhere. $◼$

Lai et al. generalized this proof to Riemannian manifolds.

Applications

The K transform was originally developed as a means of performing a physical one-time pad encryption of a physical object. The nonlinearity of the transform ensures the there is no one-to-one correspondence between the density $f$ and the true mass $\underset{{\mathbb{𝕊}}^{n-1}}{\int }\underset{\mathbb{ℝ}}{\int }f(x+s\theta )dsd\theta$, and therefore $f$ cannot be estimated from a single projection.