X-ray transform - Knowlpedia

Definition

For an integrable function $f:\mathbb R^d\to\mathbb C$ , its X-ray transform is

Xf(x,\theta)=\int_{\mathbb R}f(x+t\theta)\,dt, \qquad \theta\in S^{d-1}.

It depends only on the oriented affine line represented by $(x,\theta)$ : $Xf(x+a\theta,\theta)=Xf(x,\theta)$ .

Redundant parameterization

One often restricts $x\in\theta^\perp$ to choose a unique base point. Replacing $\theta$ by $-\theta$ leaves the unoriented line integral unchanged.

Relation to the Radon transform

The X-ray transform integrates over one-dimensional affine subspaces. The classical Radon transform in $\mathbb R^d$ integrates over hyperplanes; they coincide only in dimension two after identifying lines with hyperplanes.

Role in plurisubharmonic extension

For the linewise extension used in higher-dimensional Beurling–Malliavin theory, transverse components of the Levi form are X-ray transforms of the real Hessian of the boundary weight.

References

Sigurdur Helgason, The Radon Transform, 2nd ed., Birkhäuser, 1999. DOI record.