New Orthogonal Transforms for Signal and Image Processing

In the paper, orthogonal transforms based on proposed symmetric, orthogonal matrices are created. These transforms can be considered as generalized Walsh–Hadamard Transforms. The simplicity of calculating the forward and inverse transforms is one of the important features of the presented approach. The conditions for creating symmetric, orthogonal matrices are defined. It is shown that for the selection of the elements of an orthogonal matrix that meets the given conditions, it is necessary to select only a limited number of elements. The general form of the orthogonal, symmetric matrix having an exponential form is also presented. Orthogonal basis functions based on the created matrices can be used for orthogonal expansion leading to signal approximation. An exponential form of orthogonal, sparse matrices with variable parameters is also created. Various versions of orthogonal transforms related to the created full and sparse matrices are proposed. Fast computation of the presented transforms in comparison to fast algorithms of selected orthogonal transforms is discussed. Possible applications for signal approximation and examples of image spectrum in the considered transform domains are presented.

Color image watermarking using spatio-chromatic complex Hadamard transform in sequency domain

Purpose Watermarking technique is one of the significant methods in which carrier signal hides digital information in the form of watermark to prevent the authenticity of the stakeholders by manipulating different coefficients as watermark in time and frequency domain to sustain trade-off in performance parameters. One challenging component among others is to maintain the robustness, to limit perceptibility with embedding information. Transform domain is more popular to achieve the required results in color image watermarking. Variants of complex Hadamard transform (CHT) have been applied for gray image watermarking, and it has been proved that it has better performance than other orthogonal transforms. This paper is aimed at analyzing the performance of spatio-chromatic complex Hadamard transform (Sp-CHT) that is proposed as an application of color image watermarking in sequency domain (SD). Design/methodology/approach In this paper, color image watermarking technique is designed and implemented in SD using spatio-chromatic – conjugate symmetric sequency – ordered CHT. The color of a pixel is represented as complex number a*+jb*, where a* and b* are chromatic components of International Commission on Illumination (CIE) La*b* color space. The embedded watermark is almost transparent to human eye although robust against common signal processing attacks. Findings Based on the results, bit error rate (BER) and peak signal to noise ratio are measured and discussed in comparison of CIE La*b* and hue, saturation and value color model with spatio-chromatic discrete Fourier transform (Sp-DFT), and results are also analyzed with other discrete orthogonal transforms. It is observed from BER that Sp-CHT has 8%–12% better performance than Sp-DFT. Structural similarity index has been measured at different watermark strength and it is observed that presented transform performs better than other transforms. Originality/value This work presents the details and comparative analysis of two orthogonal transforms as color image watermarking application using MATLAB software. A finding from this study demonstrates that the Complex Hadamard transform is the competent candidate that can be replaced with DFT in many signal processing applications.

Discrete orthogonal transforms on lattices of integer elements of quadratic fields

In this paper, we introduce a new class of discrete orthogonal transforms (DОT) defined on lattices of integer elements of quadratic fields. The method of synthesis of such transforms essentially uses the specifics of the representation of integer quadratic elements in the so-called quasi-canonical number systems. This article, which presents the results of the first part of the author's research, deals exclusively with problems related to binary number systems in quadratic fields. We also consider the issues of synthesis of fast algorithms of the introduced and the possibility of their application to the analysis of fractal (or self-similar) objects. We also consider the issues of synthesis of fast algorithms of the introduced methods and the possibility of their application for the analysis of fractal (or self-similar) objects.

Hierarchical Cubical Tensor Decomposition through Low Complexity Orthogonal Transforms

In this work, new approaches are proposed for the 3D decomposition of a cubical tensor of size N × N × N for N = 2n through hierarchical deterministic orthogonal transforms with low computational complexity, whose kernels are based on the Walsh-Hadamard Transform (WHT) and the Complex Hadamard Transform (CHT). On the basis of the symmetrical properties of the real and complex Walsh-Hadamard matrices are developed fast computational algorithms whose computational complexity is compared with that of the famous deterministic transforms: the 3D Fast Fourier Transform, the 3D Discrete Wavelet Transform and the statistical Hierarchical Tucker decomposition. The comparison results show the lower computational complexity of the offered algorithms. Additionally, they ensure the high energy concentration of the original tensor into a small number of coefficients of the so calculated transformed spectrum tensor. The main advantage of the proposed algorithms is the reduction of the needed calculations due to the low number of hierarchical levels compared to the significant number of iterations needed to achieve the required decomposition accuracy based on the statistical methods. The choice of the 3D hierarchical decomposition is defined by the requirements and limitations related to the corresponding application area.