The quantum Fourier transform (QFT) is the important operation in quantum computing. It is used in many algorithms, including the Shor's quantum algorithm for finding the prime factors of integers. Color image encryption, processing and representation of quantum images are areas where the QFT is also used. In this paper, we discuss the quantum superpositions of the images and methods of calculation of the analogues of the quantum 2-D discrete Fourier transform of images. The quantum algorithms and circuits will be described, and examples of calculation of the 2-D QFT of images of 8×8 and 16×16 pixels will be given in detail.
In this paper, we describe new methods of color image enhancement alpha-rooting by the 2D quaternion discrete Fourier transform (QDFT) in the commutative algebra, or the (2,2)-model. In this model, the concept of the convolution is unique, which is very important when transforming tasks with color images into the frequency domain. Also, there are only two types of the exponential function and therefore only two QDFTs. Both these transforms can be used to reduce the convolution to operation of multiplication. Illustrative examples on color image enhancement are given. Measures of the image enhancement and selection of the best parameters of alpha-rooting are described. A comparison with the traditional non-commutative quaternion algebra is discussed and shown that the (2,2)-model is more effective in image enhancement.
Deep learning-based image enhancement is challenging in underwater and medical imaging domains, where high-quality training data is often limited. Due to water distortion, loss of color, and contrast, images captured in these settings could be of better quality, making it easier to train deep learning models effectively on large and diverse datasets. This limitation can negatively impact the performance of these models. This paper proposes an alternative approach to supervised color image enhancement to address this challenge. Specifically, the authors propose to enhance images in both the spatial and frequency domains using their two × two model quaternion image structure, which was previously proposed. The color image components plus gray or brightness are map into the grayscale image of twice size and then HE of new grays is calculated. The new colors and gray of the image are reconstructed from the new image in two × two model. The approach is tested extensively through computer simulations, demonstrating that the proposed framework achieves competitive performance in quantitative and qualitative metrics compared to state-of-the-art approaches.
In this paper we describe gradient operators that are defined along eight and more directions. For that, the model of rotations of coefficients inside the given mask is proposed. The set of such gradient operators, which is called the campus gradient is described on different examples. The 5×5 masks are considered for such operators, as the Sobel, Prewitt, Agaian gradients along 16 directions. The Nevatia-Babu, and Art-Sobel campus gradients are also described, and illustrative examples are given. The presented approach can be easier extended for large windows and even with more than 16 directions.
This paper analyzes the known method of the Fourier transforms-based alpha-rooting in image enhancement and describes a new method of alpha-rooting, by using the autocorrelation function of the image. The alpha-rooting in the frequency-domain can be described by the Taylor series, as well as in the spatial domain, by using the inverse 2-D DFT. In such a series, the alpha-rooting is the convolution of the image with the series of the autocorrelation functions. The application of the Taylor series in alpha-rooting allows to use the parameterized filters even for parameter alpha in a much larger interval than [0,1]. Examples of application of these two filters for enhancement of the grayscale images are given.
In this paper, the inverse gradient operators are considered that allow for calculating the original image from its gradients, such as gradients in the horizontal and vertical directions. Different gradient operators are considered and the image reconstruction from the gradients is presented. The presented method of image reconstruction is also applied for mean operators. Examples with images processed by different gradients are given.
In quaternion algebra the color images are processed not by separating their color components, but these components are presented and processed together as one unit. In framework of the quaternion image representation, many effective methods of processing color images can be developed, including image enhancement in spatial and frequency domains. In this paper, we unite two approaches for processing color images, by proposing a quaternion representation in quantum imaging, which includes the color images in the RGB model together with the grayscale component or brightness. The concept of quaternion two-qubit is considered and applied for image representation in each quantum pixel. The colors at each pixel are processed as one unit in quaternion representation. Other new models for quaternion image representation are also described. It is shown that a quaternion image or four component image of 𝑁 × 𝑀 pixels, can be represented by (𝑟 + 𝑠 + 2) qubits, when 𝑁 = 2𝑟 and 𝑀 = 2𝑠, 𝑟, 𝑠 > 1. The number of qubits for representing the image can be reduced to (𝑟 + 𝑠), when using the quaternion 2-qubit concept.
This paper offers a review of effective methods of image representation, or visibility images in enhancement applied to the thermal images and night vision images. The quality of images is estimated by quantitative enhancement measures, which are based on the Weber-Fechner, Michelson, and other parameterized ratio and entropy-type measures. We also apply the concept of visibility images, by using different types of gradient operators which allow for extracting and enhancing features in images. Examples of gradient visibility images, Weber-Fechner, and Michelson contrast, log and power Michelson and Weber visibility images are given. Experimental results show the effectiveness of visibility images in enhancing thermal and night vision images. Different methods of image enhancement in spatial and frequency domains are analyzed on the thermal and night vision images, by using the visibility images. Experimental results show that visibility images can be used in processing in spatial and frequency domains. Here, we mention the alpha-rooting by the classical discrete Fourier transform (DFT) and quaternion DFT, the retinex method, and new gradient based histogram equalization.
This paper offers a new multiple signal restoration tool to solve the inverse problem, when signals are convoluted with a multiple impulse response and then degraded by an additive noise signal with multiple components. Inverse problems arise practically in all areas of science and engineering and refers to as methods of estimating data/parameters, in our case of multiple signals that cannot directly be observed. The presented tool is based on the mapping multiple signals into the quaternion domain, and then solving the inverse problem. Due to the non-commutativity of quaternion arithmetic, it is difficult to find the optimal filter in the frequency domain for degraded quaternion signals. As an alternative, we introduce an optimal filter by using special 4×4 matrices on the discrete Fourier transforms of signal components, at each frequency point. The optimality of the solution is with respect to the mean-square-root error, as in the classical theory of the signal restoration by the Wiener filter. The Illustrative example of optimal filtration of multiple degraded signals in the quaternion domain is given. The computer simulations validate the effectiveness of the proposed method.
Automatic face recognition research includes a wide range of commercial and law enforcement applications. However, only a few works focus on face recognition in the Renaissance portrait artworks which is essential to characterize the individual artists. The primary challenge of the portrait recognition is the availability of limited portraits. To cope with those issues, we develop a new class of new gradient operators for face recognition in renaissance portrait art. In mathematics, the gradient is an extension of the derivative. Gradient operators have been extensively used in many image processing and computer vision applications. The simplest examples are the Roberts, Prewitt and Sobel operators, the circular operator, the rotationally symmetric operator, and the isotropic operator. In this paper, we propose a class of gradient asymmetric and symmetric a small size (3×3 and 5×5) operators. Different examples of generated 5×5 gradient operators in the different directions are described. Extensive computer simulation is directed on 270 Renaissance portraits, including Raphael, Michelangelo, and Leonardo Da Vinci portraits. The experimental results show that the fusion of local binary patterns (LBP) and asymmetric and symmetric operators are better than traditional LBP features for face recognition, including in Renaissance portraits.
This paper presents a new method of histogram equalization for grayscale images, which is called the gradient based histogram equalization. The histogram equalization is performed on the image filtered by means of gradient operators. The proposed method is simple, fast, and the preliminary experimental examples with different images show that the method is effective for image enhancement. While preserving the range and mean intensity of the image, the new method allows for reducing the standard deviation and significantly straitening the graph of the histogram, when comparing with the traditional (or global) histogram equalization.
The proposed method is a new approach for enhancing grayscale images, when the images are map to quaternion space, and then, the quaternion based enhancement technique is used. Namely, the quaternion alpha-rooting method to enhance the so generated “quaternion” image. Currently, there are only very limited techniques to convert a grayscale image to color image, and in this article we propose a novel conversion technique which helps in easily converting a grayscale image to a color or quaternion image. In addition to that, we describe the quaternion alpha-rooting method of quaternion image enhancement. Quaternion approach of enhancement allows for processing the multi-signaled image as a single unit. The fast algorithm of quaternion discrete Fourier transforms makes the implementation of the enhancement method practically possible and effective. The results of image enhancement by the proposed method and comparison with the traditional alpha-rooting of grayscale images are described. The metric used to assess the quality of enhancement shows good values for the results of the proposed enhancement. One of the enhancement metrics is the contrast-based metric referred to as the enhancement measure estimation (EME). Other metrics used to assess the quality of the enhanced images are signal-to-noise ratio (SNR), mean-square-root error (MSRE).
The problem of color image composition from original grayscale images is considered. A few models are proposed and analyzed, which are based on the observation done on many color images that in average a proportion exists between primary colors in images. Re-coloring of grayscale images are performed in the RGB color model and the Golden, Silver, Aesthetic ratios and other rules of proportions are considered between the primary colors. The gray is used as the main color to be map into the three colors at each pixel. We also describe a parameterized model with the given ratio of re-coloring images.
KEYWORDS: Image enhancement, 3D image enhancement, Medical imaging, 3D image processing, RGB color model, Visualization, Eye models, Blood, Image processing
The proposed method is a novel image enhancement for color medical images. In this method, the 3-D medical image is transformed first to the 2-D grayscale image and then the enhancement algorithms, either in frequency domain or spatial domain, are applied to the grayscale image. This paper describes the enhancement effects on the medical images by the proposed transformation model and then the enhancement by the alpha-rooting method, for the frequency domain algorithm, and the histogram equalization, for the spatial domain enhancement algorithm. The enhancement is quantitatively measured with respect to the metric which is called the color enhancement measure estimation (CEME). The proposed method is showing good CEME values as compared to the original images.
KEYWORDS: Image enhancement, RGB color model, Fourier transforms, Image processing, Digital image processing, CMYK color model, Signal processing, Image visualization, Color image processing, Image compression
Color image processing has attracted much interest in recent years, motivated by its use in many fields. Myriad uses include its application to object recognition and tracking, image segmentation and retrieval, image registration, multimedia systems, fashion and food industries, computer vision, entertainment, consumer electronics, production printing and proofing, digital photography, biometrics, digital artwork reproduction, industrial inspection, and biomedical applications.
The main goal of this book is to provide the mathematics of quaternions and octonions and to show how they can be used in these burgeoning areas of color image processing.
2-D quaternion discrete Fourier transform (2-D QDFT) is the Fourier transform applied to color images when the color images are considered in the quaternion space. The quaternion numbers are four dimensional hyper-complex numbers. Quaternion representation of color image allows us to see the color of the image as a single unit. In quaternion approach of color image enhancement, each color is seen as a vector. This permits us to see the merging effect of the color due to the combination of the primary colors. The color images are used to be processed by applying the respective algorithm onto each channels separately, and then, composing the color image from the processed channels. In this article, the alpha-rooting and zonal alpha-rooting methods are used with the 2-D QDFT. In the alpha-rooting method, the alpha-root of the transformed frequency values of the 2-D QDFT are determined before taking the inverse transform. In the zonal alpha-rooting method, the frequency spectrum of the 2-D QDFT is divided by different zones and the alpha-rooting is applied with different alpha values for different zones. The optimization of the choice of alpha values is done with the genetic algorithm. The visual perception of 3-D medical images is increased by changing the reference gray line.
KEYWORDS: Image processing, Color image processing, Fourier transforms, Signal processing, RGB color model, Signal generators, Image enhancement, Radon transform, 3D image enhancement, Image compression
In this paper, we present a novel concept of the quaternion discrete Fourier transform on the two-dimensional hexagonal lattice, which we call the two-dimensional hexagonal quaternion discrete Fourier transform (2-D HQDFT). The concept of the right-side 2D HQDFT is described and the left-side 2-D HQDFT is similarly considered. To calculate the transform, the image on the hexagonal lattice is described in the tensor representation when the image is presented by a set of 1-D signals, or splitting-signals which can be separately processed in the frequency domain. The 2-D HQDFT can be calculated by a set of 1-D quaternion discrete Fourier transforms (QDFT) of the splitting-signals.
KEYWORDS: Image encryption, Image processing, Fourier transforms, Digital imaging, Symmetric-key encryption, Neodymium, RGB color model, Cameras, Data processing, Binary data
In this paper, we present a novel method for encrypting and decrypting large amounts of data such as two-dimensional (2-D) images, both gray-scale and color, without the loss of information, and using private keys of varying lengths. The proposed method is based on the concept of the tensor representation of an image and splitting the 2-D discrete Fourier transform (DFT) by one-dimensional (1-D) DFTs of signals from the tensor representation, or transform. The splitting of the transform is accomplished in a three-dimensional (3-D) space, namely on the 3-D lattice placed on the torus. Each splitting-signal of the image defines the 2-D DFT along the frequency-points located on the spirals on the torus. Spirals have different form and cover the lattice on the torus in a complex form, which makes them very effective when moving data through and between the spirals, and data along the spirals. The encryption consists of several iterative applications of mapping the 3-D torus into several ones of smaller sizes, and rotates then moves the data around the spirals on all tori. The encryption results in the image which is uncorrelated. The decryption algorithm uses the encrypted data, and processes them in inverse order with an identical number of iterations. The proposed method can be extended to encrypt and decrypt documents as well as other types of digital media. Simulation results of the purposed method are presented to show the performance for image encryption.
KEYWORDS: Image encryption, Image processing, Fourier transforms, Digital imaging, Symmetric-key encryption, Neodymium, RGB color model, Cameras, Data processing, Binary data
In this paper, we present a novel method for encrypting and decrypting large amounts of data such as two-dimensional (2-D) images, both gray-scale and color, without the loss of information, and using private keys of varying lengths. The proposed method is based on the concept of the tensor representation of an image and splitting the 2-D discrete Fourier transform (DFT) by one-dimensional (1-D) DFTs of signals from the tensor representation, or transform. The splitting of the transform is accomplished in a three-dimensional (3-D) space, namely on the 3-D lattice placed on the torus. Each splitting-signal of the image defines the 2-D DFT along the frequency-points located on the spirals on the torus. Spirals have different form and cover the lattice on the torus in a complex form, which makes them very effective when moving data through and between the spirals, and data along the spirals. The encryption consists of several iterative applications of mapping the 3-D torus into several ones of smaller sizes, and rotates then moves the data around the spirals on all tori. The encryption results in the image which is uncorrelated. The decryption algorithm uses the encrypted data, and processes them in inverse order with an identical number of iterations. The proposed method can be extended to encrypt and decrypt documents as well as other types of digital media. Simulation results of the purposed method are presented to show the performance for image encryption.
Efficiency in terms of both accuracy and speed is highly important in any system, especially when it comes to
image processing. The purpose of this paper is to improve an existing implementation of multi-scale retinex
(MSR) by utilizing the fast Fourier transforms (FFT) within the illumination estimation step of the algorithm
to improve the speed at which Gaussian blurring filters were applied to the original input image. In addition,
alpha-rooting can be used as a separate technique to achieve a sharper image in order to fuse its results with
those of the retinex algorithm for the sake of achieving the best image possible as shown by the values of the
considered color image enhancement measure (EMEC).
KEYWORDS: Image processing, Fourier transforms, Color image processing, Signal processing, RGB color model, Bismuth, Eye models, Signal generators, Digital video recorders, Cameras
In this paper, a general, efficient, split algorithm to compute the two-dimensional quaternion discrete Fourier transform (2-D QDFT), by using the special partitioning in the frequency domain, is introduced. The partition determines an effective transformation, or color image representation in the form of 1-D quaternion signals which allow for splitting the N × M-point 2-D QDFT into a set of 1-D QDFTs. Comparative estimates revealing the efficiency of the proposed algorithms with respect to the known ones are given. In particular, a proposed method of calculating the 2r × 2r -point 2-D QDFT uses 18N2 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.
In this paper, the concept of the two-dimensional discrete Fourier transformation (2-D DFT) is defined in the general case, when the form of relation between the spatial-points (x, y) and frequency-points (ω1, ω2) is defined in the exponential kernel of the transformation by a nonlinear form L(x, y; ω1, ω2). The traditional concept of the 2-D DFT uses the Diaphanous form xω1 +yω2 and this 2-D DFT is the particular case of the Fourier transform described by the form L(x, y; ω1, ω2). Properties of the general 2-D discrete Fourier transform are described and examples are given. The special case of the N × N-point 2-D Fourier transforms, when N = 2r, r > 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.
In this paper, the classification accuracy of galaxy images is demonstrated to be improved by enhancing the galaxy images. Galaxy images often contain faint regions that are of similar intensity to stars and the image background, resulting in data loss during background subtraction and galaxy segmentation. Enhancement darkens these faint regions, enabling them to be distinguished from other objects in the image and the image background, relative to their original intensities. The heap transform is employed for the purpose of enhancement. Segmentation then produces a galaxy image which closely resembles the structure of the original galaxy image, and one that is suitable for further processing and classification. 6 Morphological feature descriptors are applied to the segmented images after a preprocessing stage and used to extract the galaxy image structure for use in training the classifier. The support vector machine learning algorithm performs training and validation of the original and enhanced data, and a comparison between the classification accuracy of each data set is included. Principal component analysis is used to compress the data sets for the purpose of classification visualization and a comparison between the reduced and original feature spaces. Future directions for this research include galaxy image enhancement by various methods, and classification performed with the use of a sparse dictionary. Both future directions are introduced.
In this paper, the concept of partitions revealing the two-dimensional discrete Fourier transform (2-D DFT) of order q2r × q2r, where r > 1 and q is a positive odd number, is described. Two methods of calculation of the 2-D DFT are analyzed. The q2r × q2r-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 q2r × q2r-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 q2r × q2r-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.
In this paper, we consider the model of quaternion signal degradation when the signal is convoluted and an additive noise is added. The classical model of such a model leads to the solution of the optimal Wiener filter, where the optimality with respect to the mean square error. The characteristic of this filter can be found in the frequency domain by using the Fourier transform. For quaternion signals, the inverse problem is complicated by the fact that the quaternion arithmetic is not commutative. The quaternion Fourier transform does not map the convolution to the operation of multiplication. In this paper, we analyze the linear model of the signal and image degradation with an additive independent noise and the optimal filtration of the signal and images in the frequency domain and in the quaternion space.
Fast unitary transforms are widely used in different areas such as data compression, pattern recognition and image reconstruction, interpolation, linear filtering, and spectral analysis. In this paper, we analyze the general concept of rotation and processing of data around not only circles but ellipses, in general. For that, we describe and analyze the general concept of the elliptic Fourier transform which was developed by Grigoryan in 2009. The block-wise representation of the discrete Fourier transform is considered in the real space, which is effective and that can be generalized to obtain new methods in spectral analysis. The N-point Elliptic discrete Fourier transform (EDFT) uses as a basic 2 × 2 transformation the rotations around ellipses. The EDFT distinguishes well from the carrying frequencies of the signal in both real and imaginary parts. It also has a simple inverse matrix. It is parameterized and includes also the DFT. Our preliminary results show that by using different parameters, the EDFT can be used effectively for solving many problems in signal and image processing field, in which includes problems such as image enhancement, filtration, encryption and many others.
There exist various implementations of the Retinex algorithm first developed by Edwin H. Land and Mcann, all of which allow for varying amounts of user control of specifications, intermediary steps and filters, and different forms of application. The purpose of this project is to study various existing algorithms implementing multiscale Retinex (and color restoration) in order to understand how they differ, their various advantages and limitations, and overall which are the most powerful methods. From this study, we attempted to improve upon an existing algorithm in order to greatly reduce processing time while still achieving a good result which we would judge using a new measure of color constancy.
KEYWORDS: Matrices, Signal processing, Tantalum, Terbium, Digital signal processing, Image enhancement, MATLAB, Image processing, Signal generators, Imaging systems
In this paper, we describe a new look on the application of Givens rotations to the QR-decomposition problem, which is similar to the method of Householder transformations. We apply the concept of the discrete heap transform, or signal-induced unitary transforms which had been introduced by Grigoryan (2006) and used in signal and image processing. Both cases of real and complex nonsingular matrices are considered and examples of performing QR-decomposition of square matrices are given. The proposed method of QR-decomposition for the complex matrix is novel and differs from the known method of complex Givens rotation and is based on analytical equations for the heap transforms. Many examples illustrated the proposed heap transform method of QR-decomposition are given, algorithms are described in detail, and MATLAB-based codes are included.
The tensor representation is an effective way to reconstruct the image from a finite number of projections, especially, when projections are limited in a small range of angles. The image is considered in the image plane and reconstruction is in the Cartesian lattice. This paper introduces a new approach for calculating the splittingsignals of the tensor transform of the discrete image f(xi, yj ) from a fine number of ray-integrals of the real image 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.
A new weighted thresholding concept is presented, which is used for the set-theoretical representation of signals,
the producing new signals containing a large number of key features that are in the original signals and the
design new morphological filters. Such representation maps many operations of non binary signal and image
processing to the union of the simple operations over the binary signals and images. The weighted thresholding
is invariant under the morphological transformations, including the basic ones, erosion and dilation. The main
idea of using the weighted thresholding is in the choice of the special level of thresholding on which we can
concentrate all our attention for the future processing. Together with arithmetical thresholding the so-called
Fibonacci levels are chosen because of many interesting properties; one of them is the effective decomposition of
the median filter. Experimental results show that the Fibonacci thresholding is much promised and can be used
for many applications, including the image enhancement, segmentation, and edge detection.
This paper presents a novel method for color image enhancement based on the discrete quaternion Fourier transform. We choose the quaternion Fourier transform, because it well-suited for color image processing applications, it processes all 3 color components (R,G,B) simultaneously, it capture the inherent correlation between the components, it does not generate color artifacts or blending , finally it does not need an additional color restoration process. Also we introduce a new CEME measure to evaluate the quality of the enhanced color images. Preliminary results show that the α-rooting based on the quaternion Fourier transform enhancement method out-performs other enhancement methods such as the Fourier transform based α-rooting algorithm and the Multi scale Retinex. On top, the new method not only provides true color fidelity for poor quality images but also averages the color components to gray value for balancing colors. It can be used to enhance edge information and sharp features in images, as well as for enhancing even low contrast images. The proposed algorithms are simple to apply and design, which makes them very practical in image enhancement.
In this paper, we describe the method of filtering the frequency components of the signals and images, by using
the discrete signal-induced heap transforms (DsiHT), which are composed by elementary rotations or Givens
transformations. 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. Due to construction of the heap transform, if the input
signal contains an additive component which is similar to the generator, this component is eliminated in the
transform of this signal, while preserving the remaining components of the signal. The energy of this component
is preserved in the first point, only. In particular case, when such component is the wave of a given frequency,
this wave is eliminated in the heap transform. Different examples of the filtration over signals and images by the
DsiHT are described and compared with the known method of the Fourier transform.
This paper describes a new approach for reconstructing images from a finite number of projections. The rayintegrals
of the image f(x, y) are transformed uniquely into the ray-sums of the discrete image fn,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°.
The reconstruction of the image f(x, y) is from a finite number of projections on the discrete Cartesian lattice N × N is described. The reconstruction is exact in the framework of the model, when image is considered as the set of N2 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 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 × 7 is described in detail.
To reconstruct the image from a finite number of projections, the concept of the point map of projections is described. Each projection is described by the corresponding set of line-integrals along a finite set of rays. The image element with its geometry is considered as a particle, or G-particle which is described by the field function. The map of each particle is considered in the form of a matrix which describes all rays passing through this particle. The concept of the field functions of particles is described as a number of rays passing through the particles with others at the same time. The consideration of the field functions for these G-particles leads to a representation of the image by the field functions, and this representation allows for reconstructing the image from its projections. The reconstruction fn,m of the image f(x, y) on the 64×64 and 128×128 Cartesian lattices by the method of G-particles is demonstrated on the images with random rectangles.
We discuss the concept of the direction image multiresolution, which is derived as a property of the 2-D discrete Fourier transform, when it splits by 1-D transforms. The N×N image, where N is a power of 2, is considered as a unique set of splitting-signals in paired representation, which is the unitary 2-D frequency and 1-D time representation. The number of splitting-signals is 3N−2, and they have different durations, carry the spectral information of the image in disjoint subsets of frequency points, and can be calculated from the projection data along one of 3N/2 angles. The paired representation leads to the image composition by a set of 3N−2 direction images, which defines the directed multiresolution and contains periodic components of the image. We also introduce the concept of the resolution map, as a result of uniting all direction images into log2 N series. In the resolution map, all different periodic components (or structures) of the image are packed into a N×N matrix, which can be used for image processing in enhancement, filtration, and compression
This paper presents a novel approach to compose discrete unitary transforms that are induced by input signals
which are considered to be generators of the transforms. Properties and examples of such transforms, which we
call 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.
KEYWORDS: Transform theory, Fourier transforms, Control systems, Signal processing, Computer engineering, Image processing, Imaging systems, Electronic imaging, Current controlled current source, Binary data
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 signal or image. The Fourier transform is split into a minimum set of
short transforms. By means of the paired transform, the 2-D signal is represented as a set of 1-D signals which
carry the spectral information of the signal at disjoint sets of frequency-points. The paired transform-based
2-D DFT involves a few operations of multiplication that can be approximated by integer transforms. Such
one-point transforms with one control bit are applied for calculating the 2-D DFT. 24 real multiplications and
24 control bits are required to perform the 8x8-point RiDFT, and 264 real multiplications and 168 control bits
for the 16 x 16-point 2-D RiDFT of real inputs. The computational complexity of the proposed 2-D RiDFTs is
comparative with the complexity of the fast 2-D DFT.
In this paper, we focus on the effective representation of the image, which is called the paired representation
and reduces the image to the set of independent 1-D signals and splits the 2-D DFT into a minimal number of
1-D DFTs. The paired transform is a frequency and time representation of the image. Splitting-signals carry
the spectral information in disjoint subsets of frequencies, which allows for enhancing the image by processing
splitting-signals separately and changing the resolution of periodic structures composing 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 using the Fourier transform technique.
The image is reconstructed directly from the splitting-signals which can be calculated from projection data.
The same inverse formula can be used for image enhancement, such as the known method of α-rooting. A new
concept of direction images is introduced, that define the decomposition of the image by directions.
The concept of the 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 R2N. This representation is not integer, and is based on the matrix 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.
This paper introduces a new class of discrete unitary transformations, the so-called discrete Haar-type heap
transformations (DHHT) which are induced by input signals and use the path similar to the traditional Haar
transformation. These transformations are fast and performed by simple rotations, can be composed for any
order, and their complete systems of basis functions represent themselves variable waves that are generated by
signals. The 2r-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.
This paper describes a new class of discrete heap transforms which are unitary energy-preserving transforms
and induced by input signals. These transforms have a simple form of composition and fast algorithms for any
size of processed signals. We consider the heap transforms, defined by two-dimensional elementary rotations, as
satisfying the given decision equations. The main feature of each heap transform is the corresponding system of
basis functions, which represent themselves a family of interactive waves which are moving in the field generated
by the input signal. Properties and examples of heap transforms, which we also call discrete signal-induced heap
transforms, are described in detail.
KEYWORDS: Image enhancement, Image processing, Wavelets, Denoising, Image filtering, Wavelet transforms, Fourier transforms, Signal processing, Visualization, Signal to noise ratio
In this paper, an effective realization of the α-rooting method of image enhancement by splitting-signals is
proposed. The splitting-signals completely determine the image and split its spectrum by disjoint subsets of
frequencies. Image enhancement is reduced to processing separate splitting-signals. We focus on processing
only one specified splitting-signal, to achieve effective image enhancement that in many cases exceeds the
enhancement by known a-rooting and wavelet methods. An effective realization of enhancement of image
(N × N) is achieved by using one coefficient, instead of N/2 such coefficients for splitting-signals in the split
α-rooting and N × N in traditional α-rooting. The proposed method does not require Fourier transforms, its
realization is performed with N multiplications. The processing of the splitting-signal leads to the change of
the image along the parallel lines by N different values, which leads to the concept of directional images and
their application in enhancing the image along directions. A novel method of combining paired transform (pre-step
of SMEME (spectral spatial maximum exclusive mean) filter) by wavelet transforms is proposed. While
denoising directional clutters, the most corrupted splitting-signal is estimated and found, depending on the
angle of long-waves.
In this paper, a novel transform-based method of reconstruction of three-dimensional (3-D) positron emission tomography (PET) images is proposed. The proposed method is based on the concept of the non-traditional tensor form of representation of the 3-D image with respect to the 3-D discrete Fourier transform (DFT). Such representation uses a minimal number of projections. The proposed algorithms are described in detail for an image (N × N × 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.
In this paper, applications of the tensor and paired representations of an image are presented for image enhancement. The proposed methods are based on the fact that the 2-D image can be represented by a set of 1-D "independent" signals that split the 2-D discrete Fourier transform (DFT) of the image into different groups of frequencies. Each splitting-signal carries information of the spectrum in a specific group. Rather than enhance the image by traditional methods of the Fourier transform (or other transforms), splitting-signals can be processed separately and the 2-D DFT of the processed image can be defined by 1-D DFTs of new splitting-signals. The process of splitting-signals related to the paired representation is very effective, because of no redundancy of spectral information carrying by ifferent splitting-signals. The effectiveness of such approach is illustrated through processing the image by the a-rooting method of enhancement. Images can be enhanced by processing only a few splitting-signals, to achieve enhancement that in many cases exceeds the enhancement by the α-rooting method and other known methods. The selection of such splitting-signals is described.
The integer-to-integer discrete cosine and other unitary transforms become popular in recent years in such applications as lossless image coding, mobile computing, filter banks, and other areas. In this paper, we present new matrix representations of the reversible integer discrete cosine transforms (IDCT) that are based on the canonical representation and floor function. A new concept of the kernel integer discrete cosine transform is introduced, that allows us to reduce the calculation of the IDCT of type II to the kernel IDCT with a fewer operations of multiplication and floor function. The application of the kernel IDCT is described for calculation of the eight-point IDCT of type II, when seven multiplications and seven floor functions can be saved. The parameterized two-point DCT of type IV and its particular case that requires two operations of multiplication, four additions, and two floor functions are presented. The golden two-point DCT that minimizes the error of the cosine transform approximation by the IDCT is also considered. Application of the kernel DCT for calculating the eight-point IDCT results in the saving of twelve multiplications and twelve floor functions, when considered the decomposition of the transform by the Walsh-Hadamard transform.
A new concept of weighted thresholding is considered and a new set-theoretical representation of signals and images is described, that can be used for design of new nonlinear and morphological filters. Such representation maps many operations of nonbinary signal and image processing to the union of simple operations over the binary signals. The weighted thresholding is invariant under the morphological transformation, including such basic operations as the erosion and dilation. The main idea of using the weighted thresholding is in the choice of a special few levels of thresholding on which we can process the signals and images. We focus on the arithmetical weighted thresholding, but other thresholding, including the geometrical, probability-based, and the so-called Fibonacci series based thresholding, are also considered. Properties of these kinds of thresholding are described. Experimental results show that the weighted thresholding is very promising and can be used for many applications, such as image enhancement and edge detection.
In this paper a general parameterized integer-to-integer discrete cosine transform (DCT) is introduced. The parameter of the transform relates to the operation of floor function. The traditional method of integer transforms is based on the use of floor function by adding number 0.5 in each stage of the integer transform calculation. We consider the integer DCT with other parameters to be chosen in an optimal way. The optimality is with respect to the minimal mean-square-error of the integer DCT compared with the original DCT with the float-point multiplication. And while achieving the minimum error we are able to reconstruct the inputs exactly. Examples for the 2-, 4-, and 8-point integer reversible and inverse DCTs of types II and IV are analyzed in detail and optimal parameters for estimation of these transforms by integer DCTs are defined.
In this paper, a new method of image enhancement is introduced. The method is based on the tensor (or vectorial) representation of the two-dimensional image with respect to the Fourier transform. In this representation, the image is defined as a set of one-dimensional (1-D) image-signals that split the Fourier transform into a set of 1-D transforms. As a result, the problem of the image enhancement is reduced to the 1-D processing the splitting signals.
The splitting of the image yields a simple model for the image enhancement, when by using only a few image-signals it is possible to achieve the image enhancement that is comparative to the known class of the frequency domain based parametric image enhancement algorithms, that are used widely for the object detection and visualization. A quantitative measure of image enhancement that is related to the Weber's law of the human visual system is considered. Based on the quantitative measure the best parameters for image enhancement can be found for each image-signal to be processed separately. Examples of image-signals and their contributions in process of enhancement of an image 256×256 are given.
A new concept of ga-cross sections by arbitrary homothetic curves ga 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.
This paper presents results of the automatic counting of illuminated spheres, where the random Boolean model depends on certain distributions of parameters. The problem is to estimate the number of randomly sized spheres in a region of 3D space by taking a set of parallel slices and using the slice intersections with the spheres to form the estimate. The simulation software is developed in the framework of the MATLAB-based Graphical User Interface, which generates spheres and allows visualization of spheres as well as their 2 D projections onto slices, which themselves appear as ordinary images. The dynamic interface provides manipulation of all parameters of the model, including the sampling rate (number of slices), sphere-size, location and intensity distributions, overlapping of spheres, and parameters of noise. That allows us to analyze the illuminated sphere counting along different ranges of various model parameters with respect to the sampling rate, specially for cases when random spheres may intersect and noise may exist.
This paper presents a simulation toolbox for counting illuminated 3D bodies. The current model is limited to illuminated balls but can be extended to other illuminated bodies. Upon simulation of a set of balls in space, horizontal slices are taken to provide a stack of 2D gray- scale images. Based on these images, a morphological algorithm estimates the number of balls. Construction of the toolbox has been motivated by the need to count spots in FISH images to test for elevated gene copy numbers. The toolbox facilitates analysis of various algorithm parameters based on the distribution of the 3D bodies. These include the number of slices and various settings for the morphological filters composing the algorithm.
Fluorescent in situ hybridization (FISH) is an excellent method for detection of gene copy number alterations in cancer and other diseases. A limitation of the technology is the tedious, inaccurate and often highly subjective spot counting. For a number of reasons, automation of FISH spot, counting has not been accomplished. FISH signals are often at different focal planes, resulting in interfering out-of- focus light. This paper describes current progress towards automated FISH spot counting, with particular reference to the previous technical limitations.
The optimal binary window filter is the binary conditional expectation of the pixel value in the ideal image given the set of observations in the window. This filter is typically designed from pairs of ideal and observation images, and the filter used in practice is the resulting estimate of the optimal filter, not the optimal filter itself. For large windows, design is hampered by the exponentially growing number of window observations. This paper discusses two types of prior information that can facilitate design for large windows: design by adapting a given (prior) filter known to work fairly well, and Bayesian design resulting from assuming the conditional probabilities determining the optimal filter satisfy prior distributions reflecting the possible states of nature. A second problem is that a filter must be applied in imaging environments different from the one in which it is designed. This results in the robustness problem: how well does a filter designed for one environment perform in a changed environment? This problem is studied by considering the ideal and observed images to be determined by distributions whose parameters are random and possess prior distributions. Then, based on these prior distributions determining the design conditions, we can evaluate filter performance across the various states. Moreover, a global filter can be designed that tends to maintain performance across states, albeit, at the cost of some increase in error relative to specific states.
KEYWORDS: Image filtering, Optimal filtering, Data modeling, Image processing, Binary data, Digital filtering, Medical diagnostics, Medical imaging, Electrical engineering, Logic
Optimal binary filters estimate an ideal random set by means of an observed random set. By parameterizing the ideal and observation random sets, one can examine the robustness of filter design relative to parameter states. This paper addresses the question as to which states possess the most robust optimal filters. Based on the prior distribution of the states, a measure of robustness is defined for each state and the state possessing maximal robustness is determined. The paper focuses on sparse noise, for which an analytic formulation of robustness is known. It proposes a parametric model from which to approximate robustness by estimating model parameters from image data.
An optimal binary image filter is an operator defined on an observed random set (image) and the output random set estimates some ideal (uncorrupted) random set with minimal error. Assuming the probability law of the ideal process is determined by a parameter vector, the output law is also determined by a parameter vector, and this latter law is a function of the input law and a degradation operator producing the observed image from the ideal image. The robustness question regards the degree to which performance of an optimal filter degrades when it is applied to an image process whose law differs (not too greatly) form the law of the process for which it is optimal. The present paper examines robustness of the optimal translation-invariant binary filter for restoring images degraded by sparse salt-and-pepper noise. An analytical model is developed in terms of prior probabilities of the signal and this model is used to compute a robustness surface.
A new concept of a mixed median filter that generalizes the traditional median filter is considered. These kinds of median filters unite the various filters using in signal and image processing, including the weighted median filters, order statistic filters, dilation and erosion of functions by the set. By means of the mixed median filters, the corresponding minimizing conditions for the order statistics are given. The threshold decomposition properties of the mixed median filter is considered to show that unlike stack filters the firsts are determined by different Boolean functions at the threshold levels. Representation of the mixed median filter and its basis statistical properties are studied.
This paper presents two new models of image restoration under consideration the linear- invariant system of image formation, which is described by the convolution type Fradholm integral equation of the first kind. The models come to the preliminary restoration of the noise imposed on the image when the last is formed. The corresponding approximate solutions of the restored image are describe and the theoretical comparative estimates are given. Also in the framework of these models the well-known inverse and Wiener filters are analyzed and the new so-called noise-homomorphic filters are considered. The best approximation of the true image in the sense of the mean-root-square error is obtained and its main properties are considered. It is shown that this approximation is better than the Wiener estimate obtained in the classical model of image restoration.
The theory of a quasilinear space generalizing the linear space is considered, and, on the basis of it, a concept of quasilinear systems is introduced, which allows us to construct, unify, and classify wide classes of nonlinear systems, such as median filters, morphological and homomorphic systems.
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