K-transform

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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.^[1] 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^[2] that the K transform exhibits global uniqueness on ${\displaystyle \mathbb {R} ^{n}}$, 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.^[3] A numerical inversion using the BFGS optimization algorithm was explored by Fichtlscherer.^[4]

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

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

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

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

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

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

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

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

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

Lai et al. generalized this proof to Riemannian manifolds.^[3]

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