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The X-ray transform is a transform with applications in computed tomography that maps an integrable function to its line integrals.

Definition

Given a positive integer n\ge 2, and a function f\in \mathcal L^1\left(\mathbb R^n\right), the X-ray transform of f is a function Pf defined on the space \mathbb R^n \times S^{n-1}, by Pf\left(x,\theta\right)=\int_\mathbb R f\left(x+t\theta\right)\;dt. Here, x represents a position in the space for which f is defined, whereas \theta represents the direction of the line. Some may restrict the domain of Pf so that x\in\theta^\perp, as integration by substitution can show that Pf\left(x,\theta\right) = Pf\left(y,\theta\right), if y is the (unique) orthogonal projection of x onto \theta^\perp.

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