In practice, the data collected from a CT scanner consists of projections and thus the computation of the Radon transform is rarely needed since the data available comes in that form. In this context, one has many projections and one is looking to reconstruct the density profile f x,y that caused the projections; this is the inverse Radon transform problem. However, in order to understand how one can reconstruct a cross-section from projections, I present here both the Radon transform and its inverse. I will then discuss in more detail practical techniques available to reconstruct tomographic sections that bypass some of the difficulties of the Radon transform.
The Radon transform is a mathematical way to define a projection of a two-dimensional function f x,y along a predefined direction. The illustration below may make this more understandable. Figure 1. Geometry of the Radon transform. Left panel: multiple projections from a single image. Right panel, definition of the elements in the Radon transform.
ECE tomographic reconstruction radon transform S13 mhossain - Rhea
In the figure above a 2D function f x,y shown as a dark square with a white circular object is projected onto a line one of the views by integrating it along a direction perpendicular to the line. This results in a 1D profile R with parameters a,b or , which define the orientation of the line. In figure 2 below, is illustrated a single projection of a rotated square which is integrated along vertical lines.
This is one of many possible projections. Figure 2. Single projection of a square along the vertical direction onto the lower graph. The Radon transform consists of many such projections at many different angles of rotation. Figure 3. Same square image as in figure 2, with a complete sinogram of its Radon transform. To interpret the sinogram a representation of all the projections it is useful to visualize the projection of the square onto the horizontal axis as it is rotated. The Horizontal axis of the sinogram is the rotation angle, the vertical is the amplitude of the profile.
The Radon transform in this case was computed at 1 intervals from 0 to Leftmost in the sinogram is a boxcar profile, flat on the top and with steep slopes on both sides corresponding to integrating along the vertical lines when the square is not rotated.
The Radon Transform and Local Tomography
At 45 a white peak forms the apex of a triangle whose slope is evidenced by a gradual change from white to yellow to orange to red then to black. At 90 the profile is identical to that at 0 and Similarly at the profile is the same as that of 45 all these rotations being equivalent. Radon Transform If one uses the slope intercept form of the line equation, one can express the radon transform as follows:.
Alternatively one can use the Dirac-delta or impulse function to define the path of integration in terms of the variables x and y:.
One can also express the radon transform using the normal form of the line equation shown in figure Inverse Radon Transform The computation of the inverse Radon transform is not straightforward and more than one approach are available that attempt to circumvent some of the difficulties involved. If one used the formulation in eq 0. One can get a sense that computing the inverse Radon transform is not a trivial operation and that more often than not one will be faced with having to approximate improper integrals. Alternatively, if one used the formulation of eq.
The last term F-1  will be explained in the next section, for now accept it as one of many ways to filter the back projection in the frequency domain. Discussion and Conclusion The mathematical development presented above is intended to convey the fact that one rarely uses an analytic approach to invert the Radon transform in practice, largely because of the nature of the integrals involved in the process.
However, the study of the Radon transform and its inverse have led to some findings that lend themselves to practical solutions to reconstructing.
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One can take each 1D profile and extrude it along the line of integration to produce a 2D image. The individual extruded profiles can be added to gradually produce an approximation of the orignal image. This method is illustrated in figure 4 below. Figure 4. Consequently, the Radon transform of a number of small objects appears graphically as a number of blurred sine waves with different amplitudes and phases. The Radon transform is useful in computed axial tomography CAT scan , barcode scanners, electron microscopy of macromolecular assemblies like viruses and protein complexes , reflection seismology and in the solution of hyperbolic partial differential equations.
Concretely, the parametrization of any straight line L with respect to arc length z can always be written. It is defined by. It is also possible to generalize the Radon transform still further by integrating instead over k -dimensional affine subspaces of R n. The X-ray transform is the most widely used special case of this construction, and is obtained by integrating over straight lines. The Radon transform is closely related to the Fourier transform. We define the univariate Fourier transform here as.
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The Fourier slice theorem then states. This fact can be used to compute both the Radon transform and its inverse. The dual Radon transform is a kind of adjoint to the Radon transform. In the context of image processing, the dual transform is commonly called backprojection  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. This is a natural rotationally invariant second-order differential operator.
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The Radon transform and its dual are intertwining operators for these two differential operators in the sense that . In analyzing the solutions to the wave equation in multiple spatial dimensions, the intertwining property leads to the translational representation of Lax and Philips.
Reconstruction is an inverse problem. Roughly speaking, then, the filter makes objects more singular. The representation of a function in terms of its integrals along lines is now called its Radon transform in his honor.
The operation studied by Radon in his paper of is referred to as the inverse Radon transform. Here, I would like to explain the connection between the Radon transform and CT scans using the built-in functions for computing symbolic Radon transforms and the image processing capabilities of the Wolfram Language. I will begin by giving examples for computing the Radon transform and its inverse in closed form using RadonTransform and InverseRadonTransform , respectively.
Next, I will show how a two-dimensional image of an object can be reconstructed from its image in Radon transform space using InverseRadon. Finally, I will explain how a practical application of these ideas allows doctors to probe a diseased organ in the human body by sending x-rays at different angles through it in a CT scan machine.