Instantaneous frequency and amplitude of complex signals based on quaternion Fourier transform

The ideas of instantaneous amplitude and phase are well understood for signals with real-valued samples, based on the analytic signal which is a complex signal with one-sided Fourier transform. We extend these ideas to signals with complex-valued samples, using a quaternion-valued equivalent of the analytic signal obtained from a one-sided quaternion Fourier transform which we refer to as the hypercomplex representation of the complex signal. We present the necessary properties of the quaternion Fourier transform, particularly its symmetries in the frequency domain and formulae for convolution and the quaternion Fourier transform of the Hilbert transform. The hypercomplex representation may be interpreted as an ordered pair of complex signals or as a quaternion signal. We discuss its derivation and properties and show that its quaternion Fourier transform is one-sided. It is shown how to derive from the hypercomplex representation a complex envelope and a phase. A classical result in the case of real signals is that an amplitude modulated signal may be analysed into its envelope and carrier using the analytic signal provided that the modulating signal has frequency content not overlapping with that of the carrier. We show that this idea extends to the complex case, provided that the complex signal modulates an orthonormal complex exponential. Orthonormal complex modulation can be represented mathematically by a polar representation of quaternions previously derived by the authors. As in the classical case, there is a restriction of non-overlapping frequency content between the modulating complex signal and the orthonormal complex exponential. We show that, under these conditions, modulation in the time domain is equivalent to a frequency shift in the quaternion Fourier domain. Examples are presented to demonstrate these concepts

Color Image Analysis by Quaternion-Type Moments

International audienceIn this paper, by using the quaternion algebra, the conventional complex-type moments (CTMs) for gray-scale images are generalized to color images as quaternion-type moments (QTMs) in a holistic manner. We first provide a general formula of QTMs from which we derive a set of quaternion-valued QTM invariants (QTMIs) to image rotation, scale and translation transformations by eliminating the influence of transformation parameters. An efficient computation algorithm is also proposed so as to reduce computational complexity. The performance of the proposed QTMs and QTMIs are evaluated considering several application frameworks ranging from color image reconstruction, face recognition to image registration. We show they achieve better performance than CTMs and CTM invariants (CTMIs). We also discuss the choice of the unit pure quaternion influence with the help of experiments. appears to be an optimal choice

Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images

In the present work hypercomplex spectral methods of the processing and analysis of images are introduced. The thesis is divided into three main chapters. First the quaternionic Fourier transform (QFT) for 2D signals is presented and its main properties are investigated. The QFT is closely related to the 2D Fourier transform and to the 2D Hartley transform. Similarities and differences of these three transforms are investigated with special emphasis on the symmetry properties. The Clifford Fourier transform is presented as nD generalization of the QFT. Secondly the concept of the phase of a signal is considered. We distinguish the global, the local and the instantaneous phase of a signal. It is shown how these 1D concepts can be extended to 2D using the QFT. In order to extend the concept of global phase we introduce the notion of the quaternionic analytic signal of a real signal. Defining quaternionic Gabor filters leads to the definition of the local quaternionic phase. The relation between signal structure and local signal phase, which is well-known in 1D, is extended to 2D using the quaternionic phase. In the third part two application of the theory are presented. For the image processing tasks of disparity estimation and texture segmentation there exist approaches which are based on the (complex) local phase. These methods are extended to the use of the quaternionic phase. In either case the properties of the complex approaches are preserved while new features are added by using the quaternionic phase

Spinor Fourier Transform for Image Processing

We propose in this paper to introduce a new spinor Fourier transform for both grey-level and color image processing. Our approach relies on the three following considerations: mathematically speaking, defining a Fourier transform requires to deal with group actions; vectors of the acquisition space can be considered as generalized numbers when embedded in a Clifford algebra; the tangent space of the image surface appears to be a natural parameter of the transform we define by means of so-called spin characters. The resulting spinor Fourier transform may be used to perform frequency filtering that takes into account the Riemannian geometry of the image. We give examples of low-pass filtering interpreted as diffusion process. When applied to color images, the entire color information is involved in a really non marginal process

Comparative Analysis of Techniques Used to Detect Copy-Move Tampering for Real-World Electronic Images

Evolution of high computational powerful computers, easy availability of several innovative editing software package and high-definition quality-based image capturing tools follows to effortless result in producing image forgery. Though, threats for security and misinterpretation of digital images and scenes have been observed to be happened since a long period and also a lot of research has been established in developing diverse techniques to authenticate the digital images. On the contrary, the research in this region is not limited to checking the validity of digital photos but also to exploring the specific signs of distortion or forgery. This analysis would not require additional prior information of intrinsic content of corresponding digital image or prior embedding of watermarks. In this paper, recent growth in the area of digital image tampering identification have been discussed along with benchmarking study has been shown with qualitative and quantitative results. With variety of methodologies and concepts, different applications of forgery detection have been discussed with corresponding outcomes especially using machine and deep learning methods in order to develop efficient automated forgery detection system. The future applications and development of advanced soft-computing based techniques in digital image forgery tampering has been discussed

Comparative Analysis of Techniques Used to Detect Copy-Move Tampering for Real-World Electronic Images

Evolution of high computational powerful computers, easy availability of several innovative editing software package and high-definition quality-based image capturing tools follows to effortless result in producing image forgery. Though, threats for security and misinterpretation of digital images and scenes have been observed to be happened since a long period and also a lot of research has been established in developing diverse techniques to authenticate the digital images. On the contrary, the research in this region is not limited to checking the validity of digital photos but also to exploring the specific signs of distortion or forgery. This analysis would not require additional prior information of intrinsic content of corresponding digital image or prior embedding of watermarks. In this paper, recent growth in the area of digital image tampering identification have been discussed along with benchmarking study has been shown with qualitative and quantitative results. With variety of methodologies and concepts, different applications of forgery detection have been discussed with corresponding outcomes especially using machine and deep learning methods in order to develop efficient automated forgery detection system. The future applications and development of advanced soft-computing based techniques in digital image forgery tampering has been discussed

On The Potential of Image Moments for Medical Diagnosis

Medical imaging is widely used for diagnosis and postoperative or post-therapy monitoring. The ever-increasing number of images produced has encouraged the introduction of automated methods to assist doctors or pathologists. In recent years, especially after the advent of convolutional neural networks, many researchers have focused on this approach, considering it to be the only method for diagnosis since it can perform a direct classification of images. However, many diagnostic systems still rely on handcrafted features to improve interpretability and limit resource consumption. In this work, we focused our efforts on orthogonal moments, first by providing an overview and taxonomy of their macrocategories and then by analysing their classification performance on very different medical tasks represented by four public benchmark data sets. The results confirmed that convolutional neural networks achieved excellent performance on all tasks. Despite being composed of much fewer features than those extracted by the networks, orthogonal moments proved to be competitive with them, showing comparable and, in some cases, better performance. In addition, Cartesian and harmonic categories provided a very low standard deviation, proving their robustness in medical diagnostic tasks. We strongly believe that the integration of the studied orthogonal moments can lead to more robust and reliable diagnostic systems, considering the performance obtained and the low variation of the results. Finally, since they have been shown to be effective on both magnetic resonance and computed tomography images, they can be easily extended to other imaging techniques