Real-valued Counterpart Research Articles

Attention-map augmentation for hypercomplex breast cancer classification

Breast cancer is the most widespread neoplasm among women and early detection of this disease is critical. Deep learning techniques have become of great interest to improve diagnostic performance. However, distinguishing between malignant and benign masses in whole mammograms poses a challenge, as they appear nearly identical to an untrained eye, and the region of interest (ROI) constitutes only a small fraction of the entire image. In this paper, we propose a framework, parameterized hypercomplex attention maps (PHAM), to overcome these problems. Specifically, we deploy an augmentation step based on computing attention maps. Then, the attention maps are used to condition the classification step by constructing a multi-dimensional input comprised of the original breast cancer image and the corresponding attention map. In this step, a parameterized hypercomplex neural network (PHNN) is employed to perform breast cancer classification. The framework offers two main advantages. First, attention maps provide critical information regarding the ROI and allow the neural model to concentrate on it. Second, the hypercomplex architecture has the ability to model local relations between input dimensions thanks to hypercomplex algebra rules, thus properly exploiting the information provided by the attention map. We demonstrate the efficacy of the proposed framework on both mammography images as well as histopathological ones. We surpass attention-based state-of-the-art networks and the real-valued counterpart of our approach. The code of our work is available at https://github.com/ispamm/AttentionBCS.

QSAM-Net: Rain Streak Removal by Quaternion Neural Network With Self-Attention Module

Real-world images captured in remote sensing, image or video retrieval, and outdoor surveillance are often degraded due to poor weather conditions, such as rain and mist. These conditions introduce artifacts that make visual analysis challenging and limit the performance of high-level computer vision methods. In time-critical applications, it is vital to develop algorithms that automatically remove rain without compromising the quality of the image contents. This article proposes a novel approach called QSAM-Net, a quaternion multi-stage multiscale neural network with a self-attention module. The algorithm requires significantly fewer parameters by a factor of 3.98 than the real-valued counterpart and state-of-the-art methods while improving the visual quality of the images. The extensive evaluation and benchmarking on synthetic and real-world rainy images demonstrate the effectiveness of QSAM-Net. This feature makes the network suitable for edge devices and applications requiring near real-time performance. Furthermore, the experiments show that the improved visual quality of images also leads to better object detection accuracy and training speed.

Axial super-resolution optical coherence tomography via complex-valued network

Optical coherence tomography (OCT) is a fast and non-invasive optical interferometric imaging technique that can provide high-resolution cross-sectional images of biological tissues. OCT’s key strength is its depth resolving capability which remains invariant along the imaging depth and is determined by the axial resolution. The axial resolution is inversely proportional to the bandwidth of the OCT light source. Thus, the use of broadband light sources can effectively improve the axial resolution and however leads to an increased cost. In recent years, real-valued deep learning technique has been introduced to obtain super-resolution optical imaging. In this study, we proposed a complex-valued super-resolution network (CVSR-Net) to achieve an axial super-resolution for OCT by fully utilizing the amplitude and phase of OCT signal. The method was evaluated on three OCT datasets. The results show that the CVSR-Net outperforms its real-valued counterpart with a better depth resolving capability. Furthermore, comparisons were made between our network, six prevailing real-valued networks and their complex-valued counterparts. The results demonstrate that the complex-valued network exhibited a better super-resolution performance than its real-valued counterpart and our proposed CVSR-Net achieved the best performance. In addition, the CVSR-Net was tested on out-of-distribution domain datasets and its super-resolution performance was well maintained as compared to that on source domain datasets, indicating a good generalization capability.

SurReal: Complex-Valued Learning as Principled Transformations on a Scaling and Rotation Manifold.

Complex-valued data are ubiquitous in signal and image processing applications, and complex-valued representations in deep learning have appealing theoretical properties. While these aspects have long been recognized, complex-valued deep learning continues to lag far behind its real-valued counterpart. We propose a principled geometric approach to complex-valued deep learning. Complex-valued data could often be subject to arbitrary complex-valued scaling; as a result, real and imaginary components could covary. Instead of treating complex values as two independent channels of real values, we recognize their underlying geometry: we model the space of complex numbers as a product manifold of nonzero scaling and planar rotations. Arbitrary complex-valued scaling naturally becomes a group of transitive actions on this manifold. We propose to extend the property instead of the form of real-valued functions to the complex domain. We define convolution as the weighted Fréchet mean on the manifold that is equivariant to the group of scaling/rotation actions and define distance transform on the manifold that is invariant to the action group. The manifold perspective also allows us to define nonlinear activation functions, such as tangent ReLU and G -transport, as well as residual connections on the manifold-valued data. We dub our model SurReal, as our experiments on MSTAR and RadioML deliver high performance with only a fractional size of real- and complex-valued baseline models.

A novel complex-valued convolutional neural network for medical image denoising

Several applications of complex-valued networks have been reported for computer vision tasks like image processing and classification. However, complex-valued convolutional neural networks are yet to be explored for medical image denoising. In this paper, a novel complex-valued convolutional neural network-based model, termed as CVMIDNet, is investigated for medical image denoising. Unlike traditional approaches, which learn clean images from noisy images, the proposed model utilizes residual learning, which learns noise from noisy images and then subtracts it from noisy images so as to obtain clean images. To assess the denoising performance of CVMIDNet, standard image quality metrics, namely, peak signal to noise ratio and the structural similarity index, have been used for five different additive white Gaussian noise levels in chest X-ray images. Chest X-ray denoising performance of CVMIDNet was compared with four recent state-of-the-art models, namely, Block-Matching and 3D (BM3D) filtering, DnCNN, Feature-guided Denoising Convolutional Neural Network (FDCNN), and deep CNN with residual learning. Additionally, it is also benchmarked against its real-valued counterpart, termed RVMIDNet. In all the reported performance investigations, CVMIDNet was found to be superior. For instance, for a Gaussian noise level of σ = 15, peak signal to noise ratio and structural similarity index values achieved by the CVMIDNet are 37.2010 and 0.9227, respectively, against the 36.2292 and 0.9086, 36.3203 and 0.9139, 35.0995 and 0.9005, 36.1830 and 0.8968, 34.2436 and 0.8874 achieved by BM3D filtering, DnCNN, RVMIDNet, FDCNN, and deep CNN with residual learning, respectively. Therefore, based on the presented investigations, it is concluded that CVMIDNet is a potential deep learning model for medical image denoising.

Despeckling Polarimetric SAR Data Using a Multistream Complex-Valued Fully Convolutional Network

A Polarimetric Synthetic Aperture Radar (PolSAR) sensor is able to collect images in different polarization states, making it a rich source of information for target characterization. PolSAR images are inherently affected by speckle. Therefore, before deriving ad hoc products from the data, the polarimetric covariance matrix needs to be estimated by reducing speckle. In recent years, deep learning based despeckling methods have started to evolve from single channel SAR images to PolSAR images. To this aim, deep learning based approaches separate the real and imaginary components of the complex-valued covariance matrix and use them as independent channels in a standard convolutional neural networks. However, this approach neglects the mathematical relationship that exists between the real and imaginary components, resulting in sub-optimal output. Here, we propose a multi-stream complex-valued fully convolutional network to reduce speckle and effectively estimate the PolSAR covariance matrix. To evaluate the performance of CV-deSpeckNet, we used Sentinel-1 dual polarimetric SAR images to compare against its real-valued counterpart, that separates the real and imaginary parts of the complex covariance matrix. CV-deSpeckNet was also compared against the state of the art PolSAR despeckling methods. The results show CV-deSpeckNet was able to be trained with a fewer number of samples, has a higher generalization capability and resulted in a higher accuracy than its real-valued counterpart and state-of-the-art PolSAR despeckling methods. These results showcase the potential of complex-valued deep learning for PolSAR despeckling.

An optical neural chip for implementing complex-valued neural network

Complex-valued neural networks have many advantages over their real-valued counterparts. Conventional digital electronic computing platforms are incapable of executing truly complex-valued representations and operations. In contrast, optical computing platforms that encode information in both phase and magnitude can execute complex arithmetic by optical interference, offering significantly enhanced computational speed and energy efficiency. However, to date, most demonstrations of optical neural networks still only utilize conventional real-valued frameworks that are designed for digital computers, forfeiting many of the advantages of optical computing such as efficient complex-valued operations. In this article, we highlight an optical neural chip (ONC) that implements truly complex-valued neural networks. We benchmark the performance of our complex-valued ONC in four settings: simple Boolean tasks, species classification of an Iris dataset, classifying nonlinear datasets (Circle and Spiral), and handwriting recognition. Strong learning capabilities (i.e., high accuracy, fast convergence and the capability to construct nonlinear decision boundaries) are achieved by our complex-valued ONC compared to its real-valued counterpart.

Finite-time synchronization of fully complex-valued networks with or without time-varying delays via intermittent control

This paper explores finite-time (F-T) synchronization for a type of fully complex-valued (C-V) networks with or without delays by proposing intermittent control schemes but without using the ordinary separation technique. Above all, the vector signum function in complex field is proposed as the generalization of real-valued counterpart, which is critical to the control design in complex field and the construction of Lyapunov functions. To realize F-T synchronization, several complex-valued intermittent controllers, only dependent of the information of controlled nodes, are designed on the addressed complex-valued networks, which are different from the design on the divided real-valued subsystems. In the theoretical analysis, some nontrivial Lyapunov functionals and functions are constructed by virtue of the proposed signum function and some inequalities are developed in complex domain to establish criteria of F-T synchronization. Finally, some numerical examples are performed to reveal the effectiveness of the derived results and the practicality of the proposed control strategies.

Newtonian Program Analysis via Tensor Product

Recently, Esparza et al. generalized Newton’s method—a numerical-analysis algorithm for finding roots of real-valued functions—to a method for finding fixed-points of systems of equations over semirings. Their method provides a new way to solve interprocedural dataflow-analysis problems. As in its real-valued counterpart, each iteration of their method solves a simpler “linearized” problem. One of the reasons this advance is exciting is that some numerical analysts have claimed that “‘all’ effective and fast iterative [numerical] methods are forms (perhaps very disguised) of Newton’s method.” However, there is an important difference between the dataflow-analysis and numerical-analysis contexts: When Newton’s method is used in numerical-analysis problems, commutativity of multiplication is relied on to rearrange an expression of the form “ a * X * b + c * X * d ” into “( a * b + c * d )* X .” Equations with such expressions correspond to path problems described by regular languages. In contrast, when Newton’s method is used for interprocedural dataflow analysis, the “multiplication” operation involves function composition and hence is non-commutative: “ a * X * b + c * X * d ” cannot be rearranged into “( a * b + c * d )* X .” Equations with such expressions correspond to path problems described by linear context-free languages (LCFLs). In this article, we present an improved technique for solving the LCFL sub-problems produced during successive rounds of Newton’s method. Our method applies to predicate abstraction, on which most of today’s software model checkers rely.

Newtonian program analysis via tensor product

Recently, Esparza et al. generalized Newton's method -- a numerical-analysis algorithm for finding roots of real-valued functions---to a method for finding fixed-points of systems of equations over semirings. Their method provides a new way to solve interprocedural dataflow-analysis problems. As in its real-valued counterpart, each iteration of their method solves a simpler ``linearized'' problem. One of the reasons this advance is exciting is that some numerical analysts have claimed that ```all' effective and fast iterative [numerical] methods are forms (perhaps very disguised) of Newton's method.'' However, there is an important difference between the dataflow-analysis and numerical-analysis contexts: when Newton's method is used on numerical-analysis problems, multiplicative commutativity is relied on to rearrange expressions of the form ``c*X + X*d'' into ``(c+d) * X.'' Such equations correspond to path problems described by regular languages. In contrast, when Newton's method is used for interprocedural dataflow analysis, the ``multiplication'' operation involves function composition, and hence is non-commutative: ``c*X + X*d'' cannot be rearranged into ``(c+d) * X.'' Such equations correspond to path problems described by linear context-free languages (LCFLs). In this paper, we present an improved technique for solving the LCFL sub-problems produced during successive rounds of Newton's method. Our method applies to predicate abstraction, on which most of today's software model checkers rely.

Revisiting the RIP of Real and Complex Gaussian Sensing Matrices Through RIV Framework

In this paper, we aim to revisit the restricted isometry property of real and complex Gaussian sensing matrices. We do this reconsideration via the recently introduced restricted isometry random variable (RIV) framework for the real Gaussian sensing matrices. We first generalize the RIV framework to the complex settings and illustrate that the restricted isometry constants (RICs) of complex Gaussian sensing matrices are smaller than their real-valued counterpart. The reasons behind the better RIC nature of complex sensing matrices over their real-valued counterpart are delineated. We also demonstrate via critical functions, upper bounds on the RICs, that complex Gaussian matrices with prescribed RICs exist for larger number of problem sizes than the real Gaussian matrices.

Fully complex-valued radial basis function networks: Orthogonal least squares regression and classification

We consider a fully complex-valued radial basis function (RBF) network for regression and classification applications. For regression problems, the locally regularised orthogonal least squares (LROLS) algorithm aided with the D-optimality experimental design, originally derived for constructing parsimonious real-valued RBF models, is extended to the fully complex-valued RBF (CVRBF) network. Like its real-valued counterpart, the proposed algorithm aims to achieve maximised model robustness and sparsity by combining two effective and complementary approaches. The LROLS algorithm alone is capable of producing a very parsimonious model with excellent generalisation performance while the D-optimality design criterion further enhances the model efficiency and robustness. By specifying an appropriate weighting for the D-optimality cost in the combined model selecting criterion, the entire model construction procedure becomes automatic. An example of identifying a complex-valued nonlinear channel is used to illustrate the regression application of the proposed fully CVRBF network. The proposed fully CVRBF network is also applied to four-class classification problems that are typically encountered in communication systems. A complex-valued orthogonal forward selection algorithm based on the multi-class Fisher ratio of class separability measure is derived for constructing sparse CVRBF classifiers that generalise well. The effectiveness of the proposed algorithm is demonstrated using the example of nonlinear beamforming for multiple-antenna aided communication systems that employ complex-valued quadrature phase shift keying modulation scheme.

Complex associative memory

We previously proposed to use complex numbers in neural networks to take into consideration the phase differences among impulse trains. In the present paper, we propose a complex associative memory in which input and output patterns are composed of the values 1 and − 1 and weights, and hence the membrane potentials, take complex values. The learning rule of the complex weights is a direct extension of that of the complex perceptron that we previously proposed. We also propose the “sigmoidal learning” for the real-valued associative memory, which replaces the signature function in the orthogonal learning rule with the sigmoid function that yields weight matrices different from those obtained by the generalized inverse method. The complex associative memory yielded better performance than would be predicted by the increase of the degree of freedom due to complexification. It also showed some interesting characteristics in its dynamics and in the distribution of its weights, which are quite distinguished from those of the real-valued counterpart.