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

Proof of global uniqueness

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

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

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

[math]\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 }[/math]

[math]\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 }[/math]

[math]\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 }[/math]

[math]\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 }[/math]

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

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 [math]\displaystyle{ f }[/math] and the true mass [math]\displaystyle{ \int_{\mathbb{S}^{n-1}}\int_\mathbb{R}f(x+s\theta)\,ds\,d\theta }[/math], and therefore [math]\displaystyle{ f }[/math] cannot be estimated from a single projection.