Radon transformation

At first, in order to use some standard results from the theory of the Radon transform, we restrict the analysis to 2-D tensor fields whose elements belong to either the space of rapidly decreasing C° functions or the space of compactly supported C°° functions. Thus, some of the detailed issues associated with the boundary conditions are avoided. 

The following properties of the Radon transform can be readily verified... 

After a simple modification of the standard formula, the inverse Radon transform can be written as... 

If A is invertible for all m and oj then the complete Radon transform of the tensor field is easily obtained from the inner product measurements using... 

In the special case when a = 0 these measurements can be named as longitudinal, transversally longitudinal, and transverse measurement, respectively. These potentials can be reconstructed directly using the inverse Radon transform (5). 

Using the equilibrium equations of the elasticity theory enables one to reduce these integrals to the ordinary Radon transform [1]. 

Grangeat P. Mathematical framework of cone beam three-dimensional reconstruction via the first derivative of the Radon transform.. Math. Methods in Tomography, V.1947 of Springer Lecturre Notes in Math-cs, Springer-Verlag, Berlin, 1991, p.66-97. 

Typical tomographic 2D-reconstruction, like the filtered backprojection teelinique in Fan-Beam geometry, are based on the Radon transform and the Fourier slice theorem [6]. 

Another efficient and practical method for exact 3D-reconstruction is the Grangeat algorithm [11]. First the derivative of the three-dimensional Radon transfomi is computed from the Cone-Beam projections. Afterwards the 3D-Object is reconstructed from the derivative of the Radon transform. At present time this method is not available for spiral orbits, instead two perpendicular circular trajectories are suitable to meet the above sufficiency condition. 

Fig. 2. Description of the functions represented in the mathematical definition of the Radon and inverse Radon transforms. See text.

Keywords radar radar imaging tomography high resolution synthetic aperture radar interferometry polarimetry Radon transform projection slice theorem backprojection. 

Directional detection will be a fantastic means of recognizing the presence of a dark stream through the Solar system. The recoil distribution due to WIMPs in a stream is very much different from the recoil distribution due a Maxwellian velocity distribution. The corresponding Radon transforms that appear in Eq. (16.44) are for a stream of velocity V,... 

The Radon transform permits reconstruction of a 2-D slice of an object from a complete set of its line integrals. Reconstruction is performed on a 3-D object array consisting of the two spatial coordinates (x, y) in the illuminated slice and one momentum transfer (or angular) coordinate, q. This has to be calculated from the distance of an object voxel (x, y) from the detector, d, and the vertical distance, a, of the corresponding detector pixel from the central detector row. From Eq. 13, the calculation of q for all object positions along a ray results in curved trajectories described by ... 

Reconstruction of images from tomographic methods are performed using the reverse Radon transform (Herman, 1980) which uses the series of angular projections to reconstruct images. The resulting data set can be displayed as a rotating three-dimensional movie or resliced in any direction to display a series of tw o-dimensional slices. 

The first point has not been clearly recognized or appreciated in the early days of the method, so terms like radon transformation aad filtered backprojection were introduced [Herl, Manl], In practical realizations of image reconstruction from projections, however, numerical frers must, indeed, be used [Herl]. 

Projections are therefore relatively easUy obtained, but the following reconstruction stage is more challenging. Formally this involves the inverse Radon transform [14, 15] - computing the three-dimensional spectrum S(Fi,F2J 3) starting from all the recorded projections. Inverse problems of this kind are notoriously tricky to solve but an NMR spectrum is a favourable case because the target spectrum comprises discrete resonances sparsely distributed in three dimensions rather than a continuum of absorption. There are two general approaches to this problem -deterministic and statistical [16]. 

Deans SR (1983) The Radon transform and some of its applications. Wiley, New York... 

Key words 7-rays Imaging - Earth Occultation, Radon Transform, Maximum Entropy... 

The Maximum Entropy Method is used to perform the image reconstruction. The size of the FOV can be chosen to be between 5°x5 and 40 x40 , depending on the required angular resolution, available computer memory, and other computing limitations. The forward transform from the image space to the data space is composed of two separate transforms, as shown in Figure 1. The first one is a curved Radon transform. A numerical calculation of the curved Radon transform is implemented since no analytical form is... 

Figure 1 Formulation and Filtering of the Earth Occultation Transform, in terms Radon transform, step trans- form and Butterworth high-pass fil-ter. The raw detector counting rate J is also filtered by the same filter to match the filtered step function. A real occultation step after the filter-i ing looks similar to the bottom fig-j ure except the transition width from the maximum to the minimum is fi- nite ( seconds), and thus the recon-4 structed image of a discrete source in 1 the sky will have a finite angular size I This is the angular res-...

A low energy GRB telescope based on the use of silicon strip detectors and 1-D coded masks is proposed. It has the ability to locate a GRB rapidly with accuracy better than 2. It therefore should permit rapid follow-up observations to be made at other wavelengths and may enhance the probability of making an identification with counterparts in the optical waveband. The system can be extended to higher energies by the use of a second, thicker layer of silicon strip detectors. The telescope can also be used as an all sky X-ray monitor, by utilising the satellite orbit precession and Radon transforms [6]. 

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