Tensor transform based method of image reconstruction by projections
Methods of the Fourier transform are widely used for practical applications of image reconstruction from projections, such as the computerized tomography. We mention the well known methods of back-projection and methods based on the Fourier slice theorem, which requires a crude interpolation when transforming the Fourier projections from the polar grid to the traditional Cartesian grid. The solution of this complex problem is very important in medical diagnoses, where projections data for reconstructing two- and three-dimensional images are obtained by means of the roentgen radiation with an investigated part of the body.
In this work, we analyze solutions of the problem of reconstruction of the discrete image on the Cartesian grid from projections of the image on the spatial domain, which are based on the concept of the two-dimensional discrete tensor transformation. In the framework of the constructed model, we show a way of using the line-integrals of the image, or real projections data for exact reconstructing the discrete image. The model of image reconstruction proposed in this research is described for the cases, when the size of the Cartesian grid are primes and power of two.
The problem we focus on is formulated as follows. For a given image f(x, y) on the bounded region (such as the square [0, 1]×[0, 1]) and the N ×N Cartesian grid placed on the region, reconstruct exactly the discrete image fn,m from the line-integrals of the image f(x, y) calculated in a finite number of projections. The solution of this problem is based on the new approach proposed by Grigoryan, which allows to transfer uniquely the geometry of the projections from the image plane to the geometry of projections onto the Cartesian grid. This transformation allows calculating the tensor representation of the discrete image, where the image is described by one-dimensional splitting-signals carrying the spectral information about the image at frequency-points of different subsets covering the Cartesian lattice. When the size of the image is a power of two, these subsets are intersected and this property can be used effectively for solution of the well known problem of image reconstruction from limited angle range projections. Our preliminary results show that the proposed method of reconstruction is more accurate than the known projections onto convex sets algorithm. In addition, the simulations of our algorithm demonstrate good reconstructions when the projections are within a limited angular range. The proposed method of image reconstruction is robust relative to the additive signal-independent noise in projection data.