Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections. The mathematical basis for tomographic imaging was laid down by Johann Radon. A notable example of applications is the reconstruction of computed tomography (CT) where cross-sectional images of patients are obtained in non-invasive manner. Recent developments have seen the Radon transform and its inverse used for tasks related to realistic object insertion required for testing and evaluating computed tomography use in airport security. This article applies in general to reconstruction methods for all kinds of tomography, but some of the terms and physical descriptions refer directly to the reconstruction of X-ray computed tomography.
Introducing formula
The projection of an object, resulting from the tomographic measurement process at a given angle θ {\displaystyle \theta } , is made up of a set of line integrals (see Fig. 1). A set of many such projections under different angles organized in 2D is called a sinogram (see Fig. 3). In X-ray CT, the line integral represents the total attenuation of the beam of X-rays as it travels in a straight line through the object. As mentioned above, the resulting image is a 2D (or 3D) model of the attenuation coefficient. That is, we wish to find the image μ ( x , y ) {\displaystyle \mu (x,y)} . The simplest and easiest way to visualise the method of scanning is the system of parallel projection, as used in the first scanners. For this discussion we consider the data to be collected as a series of parallel rays, at position r {\displaystyle r} , across a projection at angle θ {\displaystyle \theta } . This is repeated for various angles. Attenuation occurs exponentially in tissue:
I = I 0 exp ( − ∫ μ ( x , y ) d s ) {\displaystyle I=I_{0}\exp \left({-\int \mu (x,y)\,ds}\right)}
where μ ( x , y ) {\displaystyle \mu (x,y)} is the attenuation coefficient as a function of position. Therefore, generally the total attenuation p {\displaystyle p} of a ray at position r {\displaystyle r} , on the projection at angle θ {\displaystyle \theta } , is given by the line integral:
p θ ( r ) = ln ( I I 0 ) = − ∫ μ ( x , y ) d s {\displaystyle p_{\theta }(r)=\ln \left({\frac {I}{I_{0}}}\right)=-\int \mu (x,y)\,ds}
Using the coordinate system of Figure 1, the value of r {\displaystyle r} onto which the point ( x , y ) {\displaystyle (x,y)} will be projected at angle θ {\displaystyle \theta } is given by:
x cos θ + y sin θ = r {\displaystyle x\cos \theta +y\sin \theta =r\ }
So the equation above can be rewritten as
p θ ( r ) = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) δ ( x cos θ + y sin θ − r ) d x d y {\displaystyle p_{\theta }(r)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\delta (x\cos \theta +y\sin \theta -r)\,dx\,dy}
where f ( x , y ) {\displaystyle f(x,y)} represents μ ( x , y ) {\displaystyle \mu (x,y)} and δ ( ) {\displaystyle \delta ()} is the Dirac delta function. This function is known as the Radon transform (or sinogram) of the 2D object. The Fourier Transform of the projection can be written as
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