Tomographical transformation

Rallabandi, V.P.S. Roy, P.K (2008). Stochastic resonance-based tomographic transform for image enhancement of brain lesions. Jour. Comp. Asst. Tomography, vol.28 issue 9, pp.966-974. 

For liquid sprays, the droplet size varies at different radial and axial directions from the nozzle. The time-averaged measurement and data analysis procedures described above cannot provide information about the local structure of the droplet size distribution. Several techniques have been developed to transform ordinary laser diffraction measurements into spatially resolved local measurements along the radial directions of the spray. The data from the measurements at different radial directions are then processed using either a deconvolution method with optical extinction and scattering coefficients [45] or a tomographical transformation method [46,47], yielding pointwise droplet size and liquid concentration distribution as well as all mean diameters of practical interest. 

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]. 

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. 

Complete information about the specimen would be available only by tomographic methods with a stepwise rotation of the sample (see e.g. Schroer, 2006) or using inherent symmetry properties of the sample. Under the assumption of fibre symmetry of the stretched specimen around the tensile axis, from the slices through the squared FT-structure the three-dimensional squared FT-structure in reciprocal space can be reconstructed and hence also the projection of the squared FT-structure in reciprocal space. The Fourier back-transformation of the latter delivers slices through the autocorrelation function of the initial structure. Stribeck pointed out that the chord distribution function (CDF) as Laplace transform of the autocorrelation function can be computed from the scattering intensity l(s) simply by multiplying I(s) by the factor L(s) = prior to the Fourier back-... 

An important feature in the formulation of the tomographic reconstruction process is the assumption of linearity attached to the various operations involved. This leads to the concept of a spatially invariant point-spread function that is a measure of the performance of a given operation. In practice the transformation associated with an operation involves the convolution of the input wiA the point-spread function in the spatial domain to provide the output. This is recognised as a cumbersome mathematical process and leads to an alternative representation that describes the input in terms of sinusoidal functions. The associated transformation is now conducted in the frequency domain and with the transfer function described by the Fourier integral. In discussing these principles, the functions in the spatial domain and the frequency domain are considered to be continuous in their respective independent variables. Howevm, for practical applications the relevant processes involve dis te and finite data sampling. This has a significant effect on the accuracy of the rccon struction, and in this respect certain conditions are imposed on the type and amount of data in order to improve the result. 

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