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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 $\theta^\perp$.