**Publications**(56)

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^{๐}and ๐ = 2

^{๐ }, ๐, ๐ > 1. The number of qubits for representing the image can be reduced to (๐ + ๐ ), when using the quaternion 2-qubit concept.

^{r }ร 2

^{r}-point 2-D QDFT uses 18

*N*

^{2}less multiplications than the well-known column-row method and method of calculation based on the symplectic decomposition. The proposed algorithm is simple to apply and design, which makes it very practical in color image processing in the frequency domain.

^{r}ร q2

^{r}, where r > 1 and

*q*is a positive odd number, is described. Two methods of calculation of the 2-D DFT are analyzed. The q2

^{r}ร q2

^{r}-point 2-D DFT can be calculated by the traditional column-row method with 2(q2r) 1-D DFTs, and we propose the fast algorithm which splits each 1-D DFT by the short transforms by means of the fast paired transforms. Another effective algorithm of calculation of the q2

^{r}ร q2

^{r}-point 2-D DFT is based on the tensor or paired representations of the image when the image is represented as a set of 1-D signals which define the 2-D transform in the different subsets of frequency-points and they all together cover the complete set of frequencies. In this case, the splittings of the q2

^{r}ร q2

^{r}-point 2-D DFT are performed by the 2-D discrete tensor or paired transforms, respectively, which lead to the calculation with a minimum number of 1-D DFTs. Examples of the transforms and computational complexity of the proposed algorithms are given.

*(x, y)*and frequency-points (ω

_{1}, ω

_{2}) is defined in the exponential kernel of the transformation by a nonlinear form

*L*(

*x, y*; ω

_{1}, ω

_{2}). The traditional concept of the 2-D DFT uses the Diaphanous form

*xω*and this 2-D DFT is the particular case of the Fourier transform described by the form

_{1}+yω_{2}*L*(

*x, y*; ω

_{1}, ω

_{2}). Properties of the general 2-D discrete Fourier transform are described and examples are given. The special case of the

*N ร N*-point 2-D Fourier transforms, when

*N = 2*> 1, is analyzed and effective representation of these transforms is proposed. The proposed concept of nonlinear forms can be also applied for other transformations such as Hartley, Hadamard, and cosine transformations.

^{r}, r*f(x*from a fine number of ray-integrals of the real image

_{i},_{yj })*f(x, y).*The properties of the tensor transform allows for calculating a large part of the 2-D discrete Fourier transform in the Cartesian lattice and obtain high quality reconstructions, even when using a small range of projections, such as [0°, 30°) and down to [0°, 20°). The experimental results show that the proposed method reconstructs images more accurately than the known method of convex projections and filtered backprojection.

*f*(

*x, y*) are transformed uniquely into the ray-sums of the discrete image

*f*

_{n,m}on the Cartesian lattice. This transformation allows for calculating the tensor representation of the discrete image, when the image is considered as the sum of direction images, or splitting-signals carrying the spectral information of the image at frequency-points of different subsets that cover the Cartesian lattice. These subsets are intersected and this property of redundancy is used to reduce the angular range of projections. The proposed approach is presented for parallel projections and the continuous model. Preliminary results show very good results of image reconstruction when the angular range scanned is 27° and down to 10°.

*f*of the image

_{n,m}*f(x, y)*on the 64ร64 and 128ร128 Cartesian lattices by the method of G-particles is demonstrated on the images with random rectangles.

*f(x, y)*is from a finite number of projections on the discrete Cartesian lattice

*N*ร

*N*is described. The reconstruction is exact in the framework of the model, when image is considered as the set of

*N*cells, or image elements with constant intensity each. Such reconstruction is achieved because of the following two facts. Each basis function of the tensor transformation is determined by the set of parallel rays, and, therefore, the components of the tensor transform can be calculated by ray-sums. These sums can be determined from the ray-integrals, and we introduce here the concept of geometrical, or

^{2}*G*-rays to solve this problem. The examples of image reconstruction by the proposed method are given, and the reconstruction on the Cartesian lattice 7 ร 7 is described in detail.

*the discrete heap transforms*are given. The transforms are fast, because of a simple form of decomposition of their matrices, and they can be applied for signals of any length. Fast algorithms of calculation of the direct and inverse heap transforms do not depend on the length of the processed signals. In this paper, we demonstrate the applications of the heap transforms for transformation and reconstruction of one-dimensional signals and two-dimensional images. The heap transforms can be used in cryptography, since the generators can be selected in different ways to make the information invisible; these generators are keys for recovering information. Different examples of generating and applying heap transformations over signals and images are considered.

*N*-point DFT is generalized, by considering it in the real space (not complex). The multiplication by twiddle coefficients is considered in matrix form; as the Givens transformation. Such block-wise representation of the matrix of the DFT is effective. The transformation which is called the

*T*-generated

*N*-block discrete transform, or

*N*-block T-GDT is introduced. For each

*N*-block T-GDT, the inner product is defined, with respect to which the rows (and columns) of the matrices

*X*are orthogonal. By using different parameterized matrices

*T*, we define metrics in the real space of vectors. The selection of the parameters can be done among only the integer numbers, which leads to integer-valued metric. We also propose a new representation of the discrete Fourier transform in the real space

*R*. This representation is not integer, and is based on the matrix

^{2N}*C*(2x2) which is not a rotation, but a root of the unit matrix. The point (1, 0) is not moving around the unite circle by the group of motion generated by C, but along the perimeter of an ellipse. The

*N*-block C-GDT is therefore called the

*N*-block elliptic FT (EFT). These orthogonal transformations are parameterized; their properties are described and examples are given.

^{r}-point discrete Haar transform is the particular case of the proposed transformations, when the generator is the constant sequence {1, 1, 1, ..., 1}. These transformations can be used in many applications and improve the results of the Haar transformation. As an example, the approximation of signals in the simple compression process, when truncating the coefficients of the discrete Haar-type heap transform is illustrated.

*discrete signal-induced heap transforms*, are described in detail.

*N*ร

*N*ร

*N*), where

*N*is a power of two. The paired transform is defined completely by projections along the discrete grid nested on the image domain. The measurement data set containing specified projections of the 3-D image are generated according to the tensor representation and the proposed algorithm is tested on the data. The algorithm for selecting a required number of projections is described. This algorithm allows the user to select the projections that contain the maximum information and automatically selects the rest of the projections, so that there is no redundancy in the spectral information of the projections.

*g*cross sections by arbitrary homothetic curves

_{a}-*g*that generalizes the traditional horizontal cross sections used in the mathematical morphology is considered. On the base of these kinds of cross sections, the corresponding set representations in the form of umbrae, as well as the function processing transformations such as dilation and erosion are given. Main properties of these transformations are described.

_{a}
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