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 , 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]

Definition[edit]

Let an object be a function of compact support that maps into the positive real numbers

The K-transform of the object is defined as
where is the set of all lines originating at a point and terminating on the single-pixel detector , and is the X-ray transform.

Proof of global uniqueness[edit]

Let be the X-ray transform transform on and let be the non-linear operator defined above. Let be the space of all Lebesgue integrable functions on , and be the essentially bounded measurable functions of the dual space. The following result says that is a monotone operator.

For such that then

and the inequality is strict when .

Proof. Note that is constant on lines in direction , so , where denotes orthogonal projection on . Therefore:

where is the Lebesgue measure on the hyperplane . The integrand has the form , which is negative except when and so unless almost everywhere. Then uniqueness for the X-Ray transform implies that almost everywhere.

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

Applications[edit]

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 and the true mass , and therefore cannot be estimated from a single projection.

References[edit]

  1. ^ a b Kemp, R. Scott; et al. (August 2, 2016). "Physical cryptographic verification of nuclear warheads". Proceedings of the National Academy of Sciences. 113 (31): 8618–8623. Bibcode:2016PNAS..113.8618K. doi:10.1073/pnas.1603916113. PMC 4978267. PMID 27432959.
  2. ^ Kemp, R. Scott; et al. (August 2, 2016). "Supporting information: physical cryptographic verification of nuclear warheads" (PDF). Proceedings of the National Academy of Sciences. 113 (31): SI-5. doi:10.1073/pnas.1603916113. PMC 4978267. PMID 27432959. Retrieved 22 Feb 2021.
  3. ^ a b Lai, Ru-Yu; Uhlmann, Gunther; Zhai, Jian; Zhou, Hanming (2021). "Single pixel X-ray transform and related inverse problems". arXiv:2112.13978 [math.AP].
  4. ^ Fichtlscherer, Christopher (19 August 2020). "The K-Transform". K-Transform Tomography: Applications in Nuclear Verification (MSc). University of Hamburg.