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 $$f:\Omega\rightarrow\mathbb{R}^{+}_0 .$$ The K-transform of the object $$f$$ is defined as $$\mathcal{K}:L^1(\Omega,\mathbb{P}^{+}_0)\rightarrow[0,1],$$ $$\mathcal{K}f(r)\equiv\int_{L_D(r)}e^{\mathcal{P}f(l)}\,dl,$$ 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 $$\mathcal{P}$$ is the X-ray transform.

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

For $$f,g\in L^1$$ such that $$\mathcal{K}f,\mathcal{K}g\in L^\infty$$ then $$\langle\mathcal{K}f-\mathcal{K}g,f-g\rangle\leq 0$$ and the inequality is strict when $$f\neq g$$.

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

$$ \langle\mathcal{K}f-\mathcal{K}g,f-g\rangle =\int_{\mathbb{R}^n}\int_{\mathbb{S}^{n-1}} \left(e^{-\mathcal{P}f(r,\theta)}-e^{-\mathcal{P}g(r,\theta)}\right)(f-g)(r)\,d\theta\, dr$$

$$=\int_{\mathbb{S}^{n-1}}\int_{\mathbb{R}^n} \left(e^{-\mathcal{P}f(r,\theta)}-e^{-\mathcal{P}g(r,\theta)}\right)(f-g)(r)\,dr\,d\theta$$

$$=\int_{\mathbb{S}^{n-1}}\int_{\theta^\bot} \left(e^{-\mathcal{P}f(E_\theta r,\theta)}-e^{-\mathcal{P}g(E_\theta r,\theta)}\right)\int_\mathbb{R}(f-g)(E_\theta r+s\theta)\,ds\,dr_{\!H}\,d\theta$$

$$=\int_{\mathbb{S}^{n-1}}\int_{\theta^\bot} \left(e^{-\mathcal{P}f(E_\theta r,\theta)}-e^{-\mathcal{P}g(E_\theta r,\theta)}\right)\left(\mathcal{P}f(E_\theta r,\theta)-\mathcal{P}g(E_\theta r,\theta)\right)dr_{\!H}\,d\theta $$

where $$dr_{\!H}$$ is the Lebesgue measure on the hyperplane $$\theta^\bot$$. The integrand has the form $$(e^{-s}-e^{-t})(s-t)$$, which is negative except when $$s=t$$ and so $$\langle\mathcal{K}f-\mathcal{K}g,f-g\rangle<0$$ unless $$\mathcal{P}f=\mathcal{P}g$$ almost everywhere. Then uniqueness for the X-Ray transform implies that $$g=f$$ almost everywhere. $$\blacksquare$$

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 $$\int_{\mathbb{S}^{n-1}}\int_\mathbb{R}f(x+s\theta)\,ds\,d\theta$$, and therefore $$f$$ cannot be estimated from a single projection.