Automated classification systems have allowed for the rapid development of digital large sky surveys. Such systems increase the independence of human intervention in the analysis stage of star and galaxy classification. Artificial neural networks, hierarchical classifiers and ensembles of classifiers have been used as the methods of classification in these systems. This paper investigates the development of an automated classification system for galaxies in astronomical images based on the method of sparse representation. The dependency of classification based on image enhancement by the alpha-rooting, heap-, and paired-transforms is secondarily investigated.

This paper introduces a novel method of image reconstruction from a finite number of projections by processing the image along parallel rays. The geometry from the image plane is transferred to the Cartesian lattice by means of using the original image's line-integrals to calculate the line-sums of the discrete image to be reconstructed. Such a transformation of geometry allows for the 2D discrete paired transform, whose complete set of functions is defined by directions, to be effectively used in the exact reconstruction of the original image. The model of image reconstruction is described, and both examples and experimental results of implementation of the proposed method are provided for reconstruction on the Cartesian lattice of size 2(r) × 2(r), where r ≥ 2.

This paper discusses the decomposition of the image by direction images, which is based on the concept of the tensor representation and its advanced form, the paired representation. The 2-D image is considered of the size N×N, where N is prime, a power of two, and a power of odd primes. The tensor and paired representations in the frequency-and-time domain define the image as a set of 1-D signals, which we call splitting-signals. Each of such splitting-signals is calculated as the sum of the image along the parallel lines, and it defines the direction image as a component of the original image. The unique decomposition of the image by direction images is described, and formulas for the inverse tensor and paired transforms are given. These formulas can be used for image reconstruction from projections, when splitting-signals or their direction images are calculated directly from the projection data. The number of required projections is uniquely defined by the tensor representation of the image.

This paper discusses the paired transform-based method of reconstruction of 2-D images from their projections. The complete set of basic functions of the 2-D discrete paired transform are defined by specific directions, i.e. the transform is directional and can be calculated from the projection data. A simple formula is presented for image reconstruction without calculating the 2-D discrete Fourier transform in the case, when the size of image is Lr× Lr, when L is prime. The image reconstruction is described by the discrete model that is used in the series expansion methods of image reconstruction. The proposed method of reconstruction has been implemented and successfully applied for modeled images on Cartesian grid of sizes up to 256×256.

This paper describes the 2-D reversible integer discrete Fourier transform (RiDFT), which is based on the concept of the paired representation of the 2-D image, which is referred to as the unique 2-D frequency and 1-D time representation. The 2-D DFT of the image is split into a minimum set of short transforms, and the image is represented as a set of 1-D signals. The paired 2-DDFT involves a few operations of multiplication that can be approximated by integer transforms, such as one-point transforms with one control bit. 24 control bits are required to perform the 8×8-point RiDFT, and 264 control bits for the 16×16-point 2-D RiDFT of real inputs. The fast paired method of calculating the 1-D DFT is used. The computational complexity of the proposed 2-D RiDFTs is comparative with the complexity of the fast 2-D DFT.

This paper presents a novel concept of the iV-point elliptic DFT of type II (EDFT-II), by considering and generalizing the iV-point DFT in the real space R2N. In the definition of such Fourier transformation, the block-wise representation of the matrix of the DFT is reserved and the Givens transformations for multiplication by the twiddle coefficients are substituted by other basic transformations. The elliptic transformations are defined by different iVth roots of the identity matrix 2 × 2, whose groups of motion move the point (1, 0) around ellipses. The elliptic DFTs of type II are parameterized by two vector-parameters, exist for any order N, and differ from the class of elliptic DFT of type I whose basic transformations are defined by the elliptic matrix cos(¿)I + sin(¿)R, where R is such a matrix that R2= -I and I is the identity matrix 2 × 2. Examples of application of the proposed iV-block EDFT-II in signal and image processing are given.

To solve the problem of image reconstruction from projections, we describe an effective representation of the image by the paired transform which is a frequency and time representation of the image. We present a new effective formula for the inverse 2-D paired transform, which can be used for solving the algebraic system of equations with measurement data for image reconstruction without calculating the Fourier transforms. The image is reconstructed directly from the paired image-signals which can be calculated from projection data. A new concept of direction images is introduced, that define the decomposition of the image by directions. The proposed method of image reconstruction has been implemented and applied for modeled images of sizes up to 256 × 256.

In this paper, we discuss a concept of the mixed Fourier transformation, when signals are transformed to the time-frequency domain, where the difference between the time and frequency is disappeared. Both cases of continuous-time and discrete-time signals are considered, and properties of the mixed transformations are described. The concept of mixed Fourier transform includes the fractional power of the discrete Fourier transform (DFT) which is referred to as a discrete version of the fractional Fourier transform. Mixed Fourier transformations can be used for calculation of roots of the Fourier and identity transformations. Examples of such root transforms for signal processing and image encryption are given.

In this paper, the concept of the discrete unitary transforms induced by given signals or input signals is modified and developed. The basic transformations composing such transforms are parameterized and the energy (or its partial part) of signals is transferred to one of their components in different paths. We focus on the case when all basic transformations themselves represent Givens rotations of the inputs or modified inputs. The consideration of these transformations is of much interest in the unique angular forms, which allows us to consider signals and their transforms in the very compact region of 2¿. Applications of proposed angular transforms for image enhancement are described.

In this paper, we discuss the application of the two-dimensional paired representation for processing medical images. This representation leads to the effective solution of the discrete as well as continuous model of image reconstruction from their projections, and to the image enhancement. These two applications can be combined in order to receive high quality images. The method of paired representation of two-dimensional (2D) images is considered with respect to the 2D discrete Fourier transform (DFT). Basis functions of the paired transformation are defined completely by parallel projections. At the same time, the paired representation describes the image as a set of short 1D real signals (splitting-signals) which completely determine the 2D DFT of the image at disjoint subsets of frequency-points. The image enhancement procedure is thus can be reduced to processing splitting-signals and such process requires only a few spectral components of the image. For instance, the traditional alpha-rooting method of image enhancement can be fulfilled through processing the splitting-signals defined for only 3N - 2 frequency-points, when the image has the size N times N, and N is a power of two. It is shown, that processing one or a few splitting signals leads to a high image enhancement.

In the tensor representation, a two-dimensional (2-D) image is represented uniquely by a set of one-dimensional (1-D) signals, so-called splitting-signals, that carry the spectral information of the image at frequency-points of specific sets that cover the whole domain of frequencies. The image enhancement is thus reduced to processing splitting-signals and such process requires a modification of only a few spectral components of the image, for each signal. For instance, the alpha-rooting method of image enhancement can be fulfilled through processing separately a maximum of 3N/2 splitting-signals of an image (N x N), where N is a power of two. In this paper, we propose a fast implementation of the a-rooting method by using one splitting-signal of the tensor representation with respect to the discrete Fourier transform (DFT). The implementation is described in the frequency and spatial domains. As a result, the proposed algorithms for image enhancement use two 1-D N-point DFTs instead of two 2-D N x N-point DFTs in the traditional method of alpha-rooting.

The method of tensor representation of an image with respect to the Fourier transform and its application for image enhancement is described. The method is based on the fact that a two-dimensional (2D) image can be represented by a set of 1D signals that split the 2D Fourier transform of the image into different groups of frequencies. Each splitting-signal carries information of the spectrum in a specific group. The processing of the image is reduced to processing splitting-signals. The effectiveness of such an approach is illustrated through processing the image by the /spl alpha/-rooting method of enhancement. We propose to enhance the image by processing only one or a few splitting-signals, to achieve image enhancement which, in many cases, can exceed enhancement by the /spl alpha/-rooting method. The selection of such splitting-signals is described.

In this paper, the integral Fourier transform is described as a specific wavelet-like transform with a fully scalable modulated window, but not all possible translations. The transform is defined by sinusoidal waves of half periods. A geometrical locus of frequency-time points for the proposed wavelet-like transform is derived and the function coverage is described and compared with the short-time Fourier transform as well as with the wavelet transform.

An efficient split algorithm for calculating the one-dimensional discrete Hartley transforms, by using a special partitioning in the frequency domain, is introduced. The partition determines a fast paired transform that splits the 2/sup r/-point unitary Hartley transform into a set of 2/sup r-n/-point odd-frequency Hartley transforms, n=1:r. A proposed method of calculation of the 2/sup r/-point Hartley transform requires 2/sup r-1/(r-3)+2 multiplications and 2/sup r-1/(r+9)-r/sup 2/-3r-6 additions.

A method of the vectorial representation for splitting the calculation of the nonseparable three-dimensional discrete Hartley transform (3D DHT) into a set of one-dimensional (1D) DHTs is presented. The method is based on the vectorial, or tensor form of representation of a 3D image and can be applied to any orders of the transform. The case of the N/spl times/N&N-point DHT, for N=2/sup r/ (r>1), is considered in detail. The number of multiplications required for calculating the 3D DHT by the method of vectorial representation equals 7[8/sup r-1/(r-3)+4/sup r-1/] that is the best estimate of all existent algorithms. The improvement of this method, the recurrent algorithm, that reduces about 1.6 times the number of multiplications is also described.

In this paper, an effective method of discrete image reconstruction from its projections is introduced. The method is based on the vector and paired representations of the two-dimensional (2D) image with respect to the 2D discrete Fourier transform. Such representations yield algorithms for image reconstruction by a minimal number of attenuation measurements in certain projections. The proposed algorithms are described in detail for an N x N image, when N = 2(r), r > 1. The inverse formulas for image reconstruction are given. The efficiency of the algorithms is expressed in the fact that they require a minimal number of multiplications, or can be implemented without such at all. The problem of discrete image reconstruction is also considered in three-dimensional (3D) space, namely on the 3D torus, where the reconstruction is performed by means of the nonlinear projections that are integral over 3D spirals on the torus.

This paper presents a new class of the "frequency domain"-based signal/image enhancement algorithms including magnitude reduction, log-magnitude reduction, iterative magnitude and a log-reduction zonal magnitude technique. These algorithms are described and applied for detection and visualization of objects within an image. The new technique is based on the so-called sequency ordered orthogonal transforms, which include the well-known Fourier, Hartley, cosine, and Hadamard transforms, as well as new enhancement parametric operators. A wide range of image characteristics can be obtained from a single transform, by varying the parameters of the operators. We also introduce a quantifying method to measure signal/image enhancement called EME. This helps choose the best parameters and transform for each enhancement. A number of experimental results are presented to illustrate the performance of the proposed algorithms.

In this paper, three algorithms based on the method of vector and paired transforms for dividing the computation of the nonseparable two-dimensional discrete Hartley transform (2-D DHT) into the "minimal" number of the one-dimensional (1-D) DHT's are presented. The computational complexity of the proposed method is analyzed, and the comparative estimates revealing the efficiency of the proposed algorithms with respect to the known algorithms are given.