Radon transforms on a class of cones with fixed axis direction

Abstract

Integral transforms which map functions on R 3 onto their integrals on circular cones having fixed axis direction and variable opening angle are introduced and studied as generalizations of the known Radon transform. Besides their intrinsic mathematical interest, they serve as backbone support to emission imaging based on Compton scattered radiation, the way the standard Radon transform does for emission imaging based on non-scattered radiation. In this work, we establish its basic properties and prove analytically its invertibility. Formulae to express it in terms of the standard Radon transform (or vice versa) are given. We also discuss some extensions as applications.

On the V‐Line Radon Transform and Its Imaging Applications

“…Then (3) becomes $\begin{array}{c}\tilde{g}\left(q,\omega \right)=\underset{0}{\overset{\infty }{true\int }}\frac{dr}{r}\mathrm{\hspace{0.17em}\hspace{0.17em}}\tilde{f}\left(q,r\cos \hspace{0.17em}\hspace{0.17em}\omega \right)2\cos \hspace{0.17em}\hspace{0.17em}\left(2\pi qr\sin \hspace{0.17em}\omega \right).\end{array}$ After changing to variables z and t and defining $\tilde{G}false(q,tfalse)=\tilde{g}false(q,\omega false)$ with $\tilde{F}false(q,zfalse)=\tilde{f}false(q,zfalse)/z$, one finds that, after going to Fourier space, the V-line Radon transform appears as a Fourier-cosine transform $\begin{array}{c}\tilde{G}\left(q,t\right)=\underset{0}{\overset{\infty }{true\int }}dz\hspace{0.17em}\tilde{F}\left(q,z\right)2\cos \hspace{0.17em}\hspace{0.17em}\left(2\pi qzt\right).\end{array}$ Let us point out that, for the conical Radon transform (CRT), passage to partial Fourier transform has led to a Hankel transform (or Fourier-Bessel transform) in which the kernel is a Bessel function J 0 [8]. Here the role of the Bessel function J 0 is played by a cosine function.…”

The Mathematical Foundations of 3D Compton Scatter Emission Imaging

“…This can be rewritten under the form of a Fredholm integral equation of the first kind with a delta function kernel concentrated on the sheet of a circular cone [31] $\begin{array}{ll}{\widehat{f}}_1(x_D,y_D,\omega )& \\ \hspace{1em}=\iint dxdydz{\mathcal{K}}_1(x_D,y_D,\omega |x,y,z)f(x,y,z)\mathrm{,}& \end{array}$ with $\begin{array}{ll}{\mathcal{K}}_1(x_D,y_D,\omega |x,y,z)& \\ \hspace{1em}=\delta (\cos \omega \sqrt{{(x-x_D)}^2+{(y-y_D)}^2}-z\sin \omega )\mathrm{.}& \end{array}$ …”

Exact Inversion of the Cone Transform Arising in an Application of a Compton Camera Consisting of Line Detectors

“…Regarding the data space, due to the high volume of data obtained from the planar detectors (5 dimensions), one tries to use only partial data for efficiency in implementations (see, e.g., [2,5,6,13,21,25,28]), but it requires considerable costs to store unnecessary data and this also causes additional costs to retrieve the target data. The Compton camera usually carries considerable noise in applications and utilizing full 5 dimensional data is advised to obtain accurate numerical results (see, e.g., [1]) although significant computational time and resources should be required.…”

“…Several inversion formulas for various types of cone transforms were derived in [2,5,6,11,12,13,16,19,21,24,25,28,31,34]. The cone transform with planar vertex positions and a fixed central axis was studied in [5,6,16,21,24,25,34].…”

“…The cone transform with planar vertex positions and a fixed central axis was studied in [5,6,16,21,24,25,34]. Haltme ier derived an exact backprojection-type inversion formula for the n-dimensional cone transform which integrates a given n-dimensional function over all conical surfaces having vertices on a hyperplane and a central axis orthogonal to this hyperplane in [13].…”