Radon transform

In mathematics, the Radon transform in two dimensions, named after the Austrian mathmematician Johann Radon, is the integral transform consisting of the integral of a function over straight lines. The inverse of the Radon transform is used to reconstruct images from medical computed tomography scans.

Consider the straight line defined parametrically by :(x(t),y(t)) = t ( sin alpha,-cos alpha) + s (cos alpha, sin alpha)

where "s" is the distance from the origin and alpha is the angle from the "x" axis. We define the Radon transform of a function "f" on the plane (where it is assumed that the function is continuous and vanished outside a disc of some finite radius) by:mathcal{R} [ f ] (alpha,s) = int_{-infty}^{infty} f(x(t),y(t)), dt

In "n"-dimensional ("n">2) space the Radon transform is the integral of a function over hyperplanes. The integral of a function over the set of all lines in n-dimensional space is called the "X-ray transform" (or ray transform), although it is sometimes loosely referred to as a Radon transform.

In the context of tomography the Radon transform data is often called a sinogram because the Radon transform of a Dirac delta function is a distribution with support on the graph of a sine wave. Consequently the Radon transform of a number of small objects appears graphically as a number of blurred sine waves with different amplitudes and phases.

This transform in two dimensions and three dimensions (where a function is integrated over planes) was introduced in a 1917 paper by Johann Radon, who provided formula for the inverse transform (reconstruction problem). It was later generalised, in the context of integral geometry.

The Radon transform is useful in computed axial tomography (CAT scan), electron microscopy of small particles like viruses and nuclei, reflection seismology and in the solution of hyperbolic partial differential equations.

Fourier slice theorem

The Radon transform is closely related to the Fourier transform. For a function of one variable we define the Fourier transform

:hat{f}(omega)=frac{1} (2pi)}^{1/2int f(x)e^{-ixomega },dx.

and for a function of a 2-vector mathbf{x}=(x,y)

:hat{f}(mathbf{w})=frac{1} 2piintlimits_{-infty}^{infty}intlimits_{-infty }^{infty} f(mathbf{x})e^{-imathbf{x}cdotmathbf{w,dx, dy

and for convenience define R_alpha [f] (s)= R [f] (s,alpha) as we will be taking the Fourier transform in the s variable. The Fourier slice theorem then states

:widehat{R_{alpha} [f] }(sigma)=sqrt{2pi}hat{f}(sigmamathbf{n}(alpha)).

where

:mathbf{n}(alpha)= (cos alpha,sinalpha).

This result gives an explicit inversion formula for the Radon transform, and thus shows that it is invertible on suitably chosen spaces of functions. However it is not particularly useful for numerical inversion.

Filtered back-projection

An explicit and computationally efficient inversion algorithm exists for the two-dimensional Radon transform called filtered back-projection. First consider the formal adjoint of R::R^{*} [g] (mathbf{x})=int_{alpha=0}^{2pi} g(alpha,mathbf{n}(alpha)cdot mathbf{x}),dalpha,This operator is commonly called 'back-projection' as it takes a function defined on each line in the plane and 'smears' or projects it back over the line to produce an image. Of course this adjoint is not an inverse to the Radon transform, for example the Radon transform of a small object is a function that has a thickened sine wave as its support. If this is back-projected it produces a star like image centered on the original object.

We define a ramp-filter on a function h of one variable by:widehat{H [h] }(omega)=|{w}|hat{h}({omega})(the notation is not standard) then using the Fourier slice theorem and change of variables for integration we find that for f a function of two variables, and g=R [f] :f=frac{1}{4pi}R^{*}H [g] which means that the original image f can be recovered from the 'sinogram' data g by applying a ramp filter (in the s variable) and then backprojecting.As the filtering step can be performed efficiently (for example using digital signal processing techniques) and the back projection step is simply an accumulation of values in the pixels of the image, this results in a highly efficient, and hence widely used, algorithm.

See also

*Hough transform
*Tomographic reconstruction

References and external links

*
*Frank Natterer, "The Mathematics of Computerized Tomography" (Classics in Applied Mathematics, 32), Society for Industrial and Applied Mathematics. ISBN 0-89871-493-1
*Frank Natterer and Frank Wubbeling, "Mathematical Methods in Image Reconstruction", Society for Industrial and Applied Mathematics. ISBN 0-89871-472-9
* [http://mathworld.wolfram.com/RadonTransform.html MathWorld page]


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Radon (disambiguation) — Radon is a chemical element.Radon may also refer to:*Radon, Orne, a town in France *Radon transform, a type of mathematical transform *Johann Radon, an Austrian mathematician *Rodan, known as Radon in Japanese, a fictional monster in the manner… …   Wikipedia

  • Hough transform — The Hough transform (pronEng|ˈhʌf, rhymes with tough ) is a feature extraction technique used in image analysis, computer vision, and digital image processing. [Shapiro, Linda and Stockman, George. “Computer Vision,” Prentice Hall, Inc. 2001] The …   Wikipedia

  • Funk transform — In the mathematical field of integral geometry, the Funk transform (also called Minkowski–Funk transform) is an integral transform defined by integrating a function on great circles of the sphere. It was introduced by Paul Funk in 1916, based on… …   Wikipedia

  • Johann Radon — (* 16. Dezember 1887 in Tetschen; † 25. Mai 1956 in Wien) war ein österreichischer Mathematiker. Johann Radon etwa 1920 Inhaltsverzeichnis …   Deutsch Wikipedia

  • Johann Radon — Infobox Scientist name = Johann Radon box width = 26em image width = 225px caption = birth date = 1887 12 16 birth place = Děčín, Bohemia, Austria Hungary death date = death date and age|1956|5|25|1887|12|16 death place = Vienna, Austria… …   Wikipedia

  • Abel transform — In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function f ( r ) is given by::F(y)=2int y^infty… …   Wikipedia

  • Transformada de Radon — Para otros usos de este término, véase Transformada (desambiguación). En matemáticas, la transformada de Radon bidimensional, llamada así por Johann Radon, es una transformación integral que consiste en la integral de una función sobre un… …   Wikipedia Español

  • Chirplet transform — Comparison of wave, wavelet, chirp, and chirplet In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets. Contents …   Wikipedia

  • Dirac delta function — Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention… …   Wikipedia

  • Tomographic reconstruction — The mathematical basis for tomographic imaging was laid down by Johann Radon. It is applied in Computed Tomography to obtain cross sectional images of patients. This article applies in general to tomographic reconstruction for all kinds of… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.