K-transform

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 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.

References

References

1. (August 2, 2016). "Physical cryptographic verification of nuclear warheads". Proceedings of the National Academy of Sciences.
2. (August 2, 2016). "Supporting information: physical cryptographic verification of nuclear warheads". Proceedings of the National Academy of Sciences.