Radon Transform

An Integral Transform whose inverse is used to reconstruct images from medical CT scans. A technique for using Radon transforms to reconstruct a map of a planet's polar regions using a spacecraft in a polar orbit has also been devised (Roulston and Muhleman 1997).

The Radon transform can be defined by

$R(p, \tau)[f(x, y)]=\int_{-\infty}^\infty{} f(x, \tau+px)\,dx$
$= \int_{-\infty}^\infty{} \int_{-\infty}^\infty{} f(x, y)\delta[y-(\tau+px)]\,dy\,dx\equiv U(p, \tau),\quad$	(1)

where $p$ is the Slope of a line and $\tau$ is its intercept. The inverse Radon transform is

$$f(x,y)={1\over 2\pi} \int_{-\infty}^\infty {d\over dy} H[U(p, y-px)]\,dp,$$ (2)

where $H$ is a Hilbert Transform. The transform can also be defined by

$$R'(r, \alpha)[f(x, y)]=\int_{-\infty}^\infty{}\int_{-\infty}^\infty{}f(x,y) \delta(r-x\cos\alpha-y\sin\alpha)\,dx\,dy,$$ (3)

where $r$ is the Perpendicular distance from a line to the origin and $\alpha$ is the Angle formed by the distance Vector.

Using the identity

$${\mathcal F} [R[f(\omega, \alpha)]] = {\mathcal F}^2[f(u,v)],$$ (4)

where ${\mathcal F}$ is the Fourier Transform, gives the inversion formula

$$f(x, y)=c\int_0^\pi \int_{-\infty}^\infty {\mathcal F}[R[f(\... ...a\vert e^{i\omega(x\cos\alpha+y\sin\alpha)}\,d\omega\,d\alpha.$$ (5)

The Fourier Transform can be eliminated by writing

$$f(x, y)=\int_0^\pi \int_{-\infty}^\infty R[f(r, \alpha)]W(r, \alpha, x, y)\,dr\,d\alpha,$$ (6)

where $W$ is a Weighting Function such as

$$W(r, \alpha, x, y)=h(x\cos\alpha+y\sin\alpha-r) = {\mathcal F}^{-1}[\vert\omega\vert].$$ (7)

Nievergelt (1986) uses the inverse formula

$$f(x, y)={1\over \pi} \lim_{c\to 0} \int_0^\pi \int_{-\infty}... ...ty R[f(r+x\cos\alpha+y\sin\alpha, \alpha)]G_c(r)\,dr\,d\alpha,$$ (8)

where

$$G_c(r) = \cases{ {1\over \pi c^2} & for $\vert r\vert\leq c... ...{1-{1\over\sqrt{1-c^2/r^2}}}\right)& for $\vert r\vert>c$.\cr}$$ (9)

Ludwig's Inversion Formula expresses a function in terms of its Radon transform. $R'(r, \alpha)$ and $R(p, \tau)$ are related by

$$p = \cot\alpha\qquad \tau=r\csc\alpha$$ (10)

$$r = {\tau\over 1+p^2} \qquad \alpha=\cot^{-1} p.$$ (11)

The Radon transform satisfies superposition

$$R(p, \tau)[f_1(x,y)+f_2(x,y)]=U_1(p, \tau)+U_2(p, \tau),$$ (12)

linearity

$$R(p, \tau) [af(x, y)] = aU(p, \tau),$$ (13)

scaling

$$R(p, \tau)\left[{f\left({{x\over a}, {y\over b}}\right)}\right]= \vert a\vert U\left({p {a\over b}, {\tau\over b}}\right),$$ (14)

Rotation, with $R_\phi$ Rotation by Angle $\phi$

$$R(p, \tau)[R_\phi f(x,y)] = {1\over \vert\cos\phi+p\sin\phi\... ...\phi\over 1+p\tan\phi},{\tau\over \cos\phi+p\sin\phi}}\right),$$ (15)

and skewing

$$R(p, \tau)[f(ax+by,cx+dy)] ={1\over \vert a+bp\vert} U\left[{{c+dp\over a+bp}, \tau {d-b(c+bd)\over a+bp}}\right]$$ (16)

(Durrani and Bisset 1984).

The line integral along $p, \tau$ is

$$I=\sqrt{1+p^2}\, U(p, \tau).$$ (17)

The analog of the 1-D Convolution Theorem is

$$R(p, \tau) [f(x, y)*g(y)]=U(p, \tau)*g(\tau),$$ (18)

the analog of Plancherel's Theorem is

$$\int_{-\infty}^\infty U(p, \tau)\,d\tau=\int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y)\,dx\,dy,$$ (19)

and the analog of Parseval's Theorem is

$$\int_{-\infty}^\infty R(p, \tau) [f(x, y)]^2\,d\tau = \int_{-\infty}^\infty \int_{-\infty}^\infty f^2(x, y)\,dx\,dy.$$ (20)

If $f$ is a continuous function on $\Bbb{C}$, integrable with respect to a plane Lebesgue Measure, and

$$\int_l f\,ds=0$$ (21)

for every (doubly) infinite line $l$ where $s$ is the length measure, then $f$ must be identically zero. However, if the global integrability condition is removed, this result fails (Zalcman 1982, Goldstein 1993).

See also Tomography

References

Radon Transforms

Anger, B. and Portenier, C. Radon Integrals. Boston, MA: Birkhäuser, 1992.

Armitage, D. H. and Goldstein, M. ``Nonuniqueness for the Radon Transform.'' Proc. Amer. Math. Soc. 117, 175-178, 1993.

Deans, S. R. The Radon Transform and Some of Its Applications. New York: Wiley, 1983.

Durrani, T. S. and Bisset, D. ``The Radon Transform and its Properties.'' Geophys. 49, 1180-1187, 1984.

Esser, P. D. (Ed.). Emission Computed Tomography: Current Trends. New York: Society of Nuclear Medicine, 1983.

Gindikin, S. (Ed.). Applied Problems of Radon Transform. Providence, RI: Amer. Math. Soc., 1994.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1979.

Helgason, S. The Radon Transform. Boston, MA: Birkhäuser, 1980.

Kunyansky, L. A. ``Generalized and Attenuated Radon Transforms: Restorative Approach to the Numerical Inversion.'' Inverse Problems 8, 809-819, 1992.

Nievergelt, Y. ``Elementary Inversion of Radon's Transform.'' SIAM Rev. 28, 79-84, 1986.

Rann, A. G. and Katsevich, A. I. The Radon Transform and Local Tomography. Boca Raton, FL: CRC Press, 1996.

Robinson, E. A. ``Spectral Approach to Geophysical Inversion Problems by Lorentz, Fourier, and Radon Transforms.'' Proc. Inst. Electr. Electron. Eng. 70, 1039-1053, 1982.

Roulston, M. S. and Muhleman, D. O. ``Synthesizing Radar Maps of Polar Regions with a Doppler-Only Method.'' Appl. Opt. 36, 3912-3919, 1997.

Shepp, L. A. and Kruskal, J. B. ``Computerized Tomography: The New Medical X-Ray Technology.'' Amer. Math. Monthly 85, 420-439, 1978.

Strichartz, R. S. ``Radon Inversion--Variation on a Theme.'' Amer. Math. Monthly 89, 377-384 and 420-423, 1982.

Zalcman, L. ``Uniqueness and Nonuniqueness for the Radon Transform.'' Bull. London Math. Soc. 14, 241-245, 1982.