Scale-invariant Properties Research Articles

Compositional splines for representation of density functions

In the context of functional data analysis, probability density functions as non-negative functions are characterized by specific properties of scale invariance and relative scale which enable to represent them with the unit integral constraint without loss of information. On the other hand, all these properties are a challenge when the densities need to be approximated with spline functions, including construction of the respective spline basis. The Bayes space methodology of density functions enables to express them as real functions in the standard $$L^2$$ space using the centered log-ratio transformation. The resulting functions satisfy the zero integral constraint. This is a key to propose a new spline basis, holding the same property, and consequently to build a new class of spline functions, called compositional splines, which can approximate probability density functions in a consistent way. The paper provides also construction of smoothing compositional splines and possible orthonormalization of the spline basis which might be useful in some applications. Finally, statistical processing of densities using the new approximation tool is demonstrated in case of simplicial functional principal component analysis with anthropometric data.

Brightness Invariant Deep Spectral Super-Resolution.

Spectral reconstruction from RGB or spectral super-resolution (SSR) offers a cheap alternative to otherwise costly and more complex spectral imaging devices. In recent years, deep learning based methods consistently achieved the best reconstruction quality in terms of spectral error metrics. However, there are important properties that are not maintained by deep neural networks. This work is primarily dedicated to scale invariance, also known as brightness invariance or exposure invariance. When RGB signals only differ in their absolute scale, they should lead to identical spectral reconstructions apart from the scaling factor. Scale invariance is an essential property that signal processing must guarantee for a wide range of practical applications. At the moment, scale invariance can only be achieved by relying on a diverse database during network training that covers all possibly occurring signal intensities. In contrast, we propose and evaluate a fundamental approach for deep learning based SSR that holds the property of scale invariance by design and is independent of the training data. The approach is independent of concrete network architectures and instead focuses on reevaluating what neural networks should actually predict. The key insight is that signal magnitudes are irrelevant for acquiring spectral reconstructions from camera signals and are only useful for a potential signal denoising.

A non‐iterative transformation method for an extended Blasius problem

In this paper, we define a non‐iterative transformation method for an extended Blasius problem. The original non‐iterative transformation method, which is based on scaling invariance properties, was defined for the classical Blasius problem by Töpfer in 1912. This method allows us to solve numerically a boundary value problem by solving a related initial value problem and then rescaling the obtained numerical solution. In recent years, we have seen applications of the non‐iterative transformation method to several problems of interest. The obtained numerical results are improved by both a mesh refinement strategy and Richardson's extrapolation technique. In this way, we can be confident that the computed first six decimal places are correct.

Formation of listric normal faults by extensional duplexing: A case study from the active Langshan piedmont fault, NW China

The development of normal faults is controlled by many factors, among which mechanical layers or pre-existing fabrics play an important role in the growth of normal faults. It is widely accepted that normal faults are rarely continuous surfaces but are zones composed of fault segments, heterogeneously distributed fault rocks of various compositions, slip bands, blocks and other deformation elements. Listric normal faults, particularly those developed in metamorphic crystalline rocks or preexisting fabric-developed basements, may be composed of subparallel faults in cross-sections, linked by extensional duplex structures. Here, we choose a listric rift-bounding fault rooted in foliation developed basements, the Langshan normal fault in the northwest corner of the Ordos Block, to conduct an outcrop analysis of fault geometry and linkage behavior from cross-sectional views. The Langshan normal fault zone is composed of mixed regions of high- and low-angle segments linked by duplex structures at various scales. Mylonitic foliations play a significant role in the formation of fault segments and the evolution of the listric fault zones. Lines of evidence show that small en echelon faults initiated from preexisting foliations. Local stress perturbation between overstepping faults results in the formation of duplex structures and displays a high fracture density, which serve to the coalesce of fault segments. Consequently, the progressive localization of strain onto a large listric fault is achieved by progressive development of extensional duplexing among fault segments in cross-sections. These extensional duplexes imply a scale-invariant property, based on which we can extrapolate this pattern occurs on larger scales and is verified by seismic data. The case of Langshan piedmont fault demonstrates the possibility of a listric fault being formed by subparallel planner faults and highlights the effects of preexisting fabrics on the evolution of listric faults, especially in basement fabrics developed regions along the fault.

Scale invariance in a nonvibrating magnetic granular system

A nonvibrating magnetic granular system is studied by using a time series approach. The system consists of steel balls confined inside a circular wall that surrounds a glass plate. Kinetic energy is provided to the particles by the application of an external vertical time-dependent magnetic field of different amplitudes. We carried out a characterization of the system dynamics through the measurement of the correlations present in the time series of positions, in the x-direction, of each particle. In particular, by performing Fourier spectral analysis, we find that the time series are fractal and scale invariant, in such a way that the corresponding Fourier power spectra follow a power law P(f)propto 1/f^beta, with 0<beta <2.5. More specifically, we find that the values of beta, and therefore the strength of the correlations, increase as the magnetic field also increases. In this way, the present system constitutes an experimental model to generate correlated random walks. Additionally, we show how the introduction of a constant magnetic field breaks down this scale invariance property in the positions of each particle. Finally, we confirm the above results by applying detrended fluctuation analysis.

Multi-scale invariant fields: estimation and prediction

Extending the concept of multi-selfsimilar random field, we study multi-scale invariant (MSI) fields with component-wise discrete scale invariant properties. Assuming scale parameters as λ i > 1, i = 1, …, d and the parameter space as (1, ∞) d , the first scale rectangle is referred to the rectangle (1, λ 1) ×⋯× (1, λ d ). We show that the covariance functions of the sampled Markov MSI field are characterized by the variances and covariances of samples inside the first scale rectangle. As an example of an MSI field, a two-dimensional simple fractional Brownian sheet (sfBss) is demonstrated. In addition, real data for precipitation in the area of Brisbane, Australia, covering a period of two days (25 and 26 January 2013), are examined. We show that precipitation in this area demonstrates MSI properties, and estimate it as a simple MSI field with stationary increments inside scale intervals. This structure enables us to predict the precipitation in terms of surface and time. We apply the mean absolute percentage error as a measure for the accuracy of the predictions.

Beyond the Standard Models with cosmic strings

We examine which information on the early cosmological history can be extracted from the potential measurement by third-generation gravitational-wave observatories of a stochastic gravitational wave background (SGWB) produced by cosmic strings. We consider a variety of cosmological scenarios breaking the scale-invariant properties of the spectrum, such as early long matter or kination eras, short intermediate matter and inflation periods inside a radiation era, and their specific signatures on the SGWB . This requires to go beyond the usually-assumed scaling regime, to take into account the transient effects during the change of equation of state of the universe. We compute the time evolution of the string network parameters and thus the loop-production efficiency during the transient regime, and derive the corresponding shift in the turning-point frequency. We consider the impact of particle production on the gravitational-wave emission by loops. We estimate the reach of future interferometers LISA, BBO, DECIGO, ET and CE and radio telescope SKA to probe the new physics energy scale at which the universe has experienced changes in its expansion history. We find that a given interferometer may be sensitive to very different energy scales, depending on the nature and duration of the non-standard era, and the value of the string tension. It is fascinating that by exploiting the data from different GW observatories associated with distinct frequency bands, we may be able to reconstruct the full spectrum and therefore extract the values of fundamental physics parameters.

Impact of the global financial crisis on the crude oil market

This study examines the effect of the 2008 global financial crisis on the crude oil market. We use the Hurst exponent, Shannon entropy, and the scaling exponent to characterize the changes in the oil market properties (i.e., efficiency, long-term equilibrium, and collective phenomena) caused by the financial crisis. Although volatility in the oil market remained the same after the crisis, we find that the crisis altered the scale-invariant property of the oil market, and it also negatively influenced market properties in terms of efficiency and long-term equilibrium. Accordingly, we propose (1) to introduce tight market regulation requesting a transaction report for sizable trades and (2) to incentivize investors to provide accurate information.

CODC: a Copula-based model to identify differential coexpression

Differential coexpression has recently emerged as a new way to establish a fundamental difference in expression pattern among a group of genes between two populations. Earlier methods used some scoring techniques to detect changes in correlation patterns of a gene pair in two conditions. However, modeling differential coexpression by means of finding differences in the dependence structure of the gene pair has hitherto not been carried out. We exploit a copula-based framework to model differential coexpression between gene pairs in two different conditions. The Copula is used to model the dependency between expression profiles of a gene pair. For a gene pair, the distance between two joint distributions produced by copula is served as differential coexpression. We used five pan-cancer TCGA RNA-Seq data to evaluate the model that outperforms the existing state of the art. Moreover, the proposed model can detect a mild change in the coexpression pattern across two conditions. For noisy expression data, the proposed method performs well because of the popular scale-invariant property of copula. In addition, we have identified differentially coexpressed modules by applying hierarchical clustering on the distance matrix. The identified modules are analyzed through Gene Ontology terms and KEGG pathway enrichment analysis.

Characterizing reticulation in online social networks during disasters

Online social network has become a new form of infrastructure for communities in spreading situational information in disasters. Developing effective interventions to improve the network performance of information diffusion is essential for people to rapidly retrieve information in coping with disasters and subsequent disruptions. Existing studies have investigated multiple aspects of online social networks in stationary situations and a separate manner. However, the networks are dynamic and different properties of the networks are co-related in the evolving disaster situations. In particular, disaster events motivate people to communicate online, create and reinforce their connections, and lead to a dynamic reticulation of the online social networks. To understand the relationship among these elements, we proposed an Online Network Reticulation (ONR) framework to examine four modalities (i.e., enactment, activation, reticulation, and network performance) in the evolution of online social networks to analyze the interplays among disruptive events in disasters, user activities, and information diffusion performance on social media. Accordingly, we examine the temporal changes in four elements for characterization of reticulation: activity timing, activity types (post, share, reply), reticulation mechanism (creation of new links versus reinforcement of existing links), and structure of communication instances (self-loop, converging, and reciprocal). Finally, the aggregated effects of network reticulation, using attributed network-embedding approach, are examined in the average latent distance among users as a measure of network performance for information propagation. The application of the proposed framework is demonstrated in a study of network reticulation on Twitter for a built environment disruption event during 2017 Hurricane Harvey in Houston. The results show that the main underlying mechanism of network reticulation in evolving situations was the creation of new links by regular users. The main structure for communication instances was converging, indicating communication instances driven by information-seeking behaviors in the wake of a disruptive event. With the evolution of the network, the proportion of converging structures to self-loop and reciprocal structures did not change significantly, indicating the existence of a scale-invariance property for network structures. The findings demonstrate the capability of the proposed online network reticulation framework for characterizing the complex relationships between events, activities, and network performance in online social networks during disasters.

A robust digital image watermarking scheme based on bat algorithm optimization and SURF detector in SWT domain

This paper presents an optimized robust digital image watermarking scheme based on Stationary Wavelet Transform (SWT) using Bat optimization Algorithm (BA) and Speed-Up Robust Feature (SURF). The proposed scheme applies high-frequency coefficients of the SWT of the host image in the BA framework to optimize watermark strength factors in the embedding process, considering relevant attacks. On the final step of this process, the SURF detector is employed on the watermarked image for getting point features used for geometric distortion correction. For watermark extracting, the primary step is to correct probable geometrical distortions, utilizing the SURF rotation and scaling invariance property, and the procedure goes on by executing the reverse of embedding phase steps. For evaluating the capabilities of the proposed algorithm, different types of image processing operations such as Gaussian filtering, scaling, rotation and salt and pepper, Poison, speckle, and Gaussian noise, have been used as attacks. According to the experimental results, the proposed combination of techniques exhibits an overall superior performance in both imperceptibility and robustness metrics in various situations compared to state-of-the-art and relevant methods.

Measuring human capital divergence in a growing economy

The stylized fact that the fraction of workers who are college graduates appears to increase more in US cities where the initial share is larger has attracted significant attention. Furthermore, more educated cities appear to grow faster. These two trends could portend the divergence of cities by skill, with low-skill workers segregated in slow-growing or declining cities. This paper compares measures of skill divergence and finds that relative measures, which have the property of scale invariance, show no divergence for the period from 1970 to 2010. In addition, the relation between skill intensity and city growth appears to be concave, so that differences in the growth rate of skill intensity across cities may diminish over time as the average college share of the country rises.

Adaptive Neuro-Fuzzy Inference System Based Impulse Denoising

Removing impulse noise from images is a critical issue in image processing because it may occur frequently during acquisition or transmission of images. We propose an anfis based impulse denoising algorithm to preserve the intrinsic geometric details of an image. The main target of this project is to restore the features of an image without losing any information from the degraded image. This method is more suitable to preserve the features of an image with scale invariant properties of an image. Here we are performing the training the noisy image with ANFIS and testing the image to retain the features of an original image.

A new fractal model for porous media based on low-field nuclear magnetic resonance

Fractal geometry has been successfully applied to the study of structures and transport properties of porous media due to its scaleinvariant property. A new fractal probability density (FPD) model is developed to characterize the pore structures of porous media in this paper. Mechanical press filtration (MPF) is performed on pre-dewatered sludge samples in order to achieve different porous structures. The pore structures and water contents of the dewatered porous sludge cakes are then measured by low-field nuclear magnetic resonance (NMR). The pore fractal dimensions are determined with the proposed FPD model based on NMR results. It has been found that the pore fractal dimension increases with porosity under fixed pore size range. The value of the pore fractal dimension of porous samples is larger than 2.6, which indicates highly heterogeneous pore structures. The present work may provide insight on the multiscale structures of porous media based on NMR technique.

THE FRACTAL DIMENSION-BASED DESCRIPTOR OF THE MICROVASCULAR PATTERN ON THE HISTOLOGICAL SECTION

The microvascular pattern in the histological section, i.e. the point-pattern composed of capillaries perpendicular to the plane of section, contains information about the three-dimensional structure of the capillary network. Histological processing is followed by the shrinkage of tissue of uncertain magnitude. In order to obtain relevant information, the scale-independent analysis is necessary. We used an approach based on the Minkowski cover of measured set. The true fractal dimension of the point pattern is obviously of zero, but the artificial result of the algorithm can be related to the complexity of shape. We fitted the log-log plot by the modified rounded ramp function and the slope of the oblique part was used as the fractal based descriptor. We demonstrated on histological samples of the heart that this fractal-based parameter has the property of scale and rotation invariance.

&amp;lt;i&amp;gt;π&amp;lt;/i&amp;gt;-theorem generalization of the ice-age theory

Abstract. Analyzing a dynamical system describing the global climate variations requires, in principle, exploring a large space spanned by the numerous parameters involved in this model. Dimensional analysis is traditionally employed to deal with equations governing physical phenomena to reduce the number of parameters to be explored, but it does not work well with dynamical ice-age models, because, as a rule, the number of parameters in such systems is much larger than the number of independent dimensions. Physical reasoning may, however, allow us to reduce the number of effective parameters and apply dimensional analysis in a way that is insightful. We show this with a specific ice-age model (Verbitsky et al., 2018), which is a low-order dynamical system based on ice-flow physics coupled with a linear climate feedback. In this model, the ratio of positive-to-negative feedback is effectively captured by a dimensionless number called the “V number”, which aggregates several parameters and, hence, reduces the number of governing parameters. This allows us to apply the central theorem of the dimensional analysis, the π theorem, efficiently. Specifically, we show that the relationship between the amplitude and duration of glacial cycles is governed by a property of scale invariance that does not depend on the physical nature of the underlying positive and negative feedbacks incorporated by the system. This specific example suggests a broader idea; that is, the scale invariance can be deduced as a general property of ice age dynamics if the latter are effectively governed by a single ratio between positive and negative feedbacks.

Multifractal processes: Definition, properties and new examples

We investigate stochastic processes possessing scale invariance properties which we refer to as multifractal processes. The examples of such processes known so far do not go much beyond the original cascade construction of Mandelbrot. We provide a new definition of the multifractal process by generalizing the definition of the self-similar process. We establish general properties of these processes and show how existing examples fit into our setting. Finally, we define a new class of examples inspired by the idea of Lamperti transformation. Namely, for any pair of infinitely divisible distribution and a stationary process one can construct a multifractal process.

Degrees of Freedom for Half-Duplex and Full-Duplex Multi-User Cognitive Radios

The degrees of freedom (DoF) of multi-user cognitive radios (CR) under half-duplex (HD) and full-duplex (FD) are studied in this work. Due to the fact that FD adopted multiple cognitive users (CUs) suffer not only self-interference (SI) and primary-user interference (PUI) but also multi-user interference (MUI) from the other CUs, the DoF metric is constrained by various interference factors and shows scale invariance properties with respect to the temporal, frequency and spatial dimensions. In this regard, the average DoF is characterized in both spatio and temporal dimensions using limitation calculation (LC) and interference alignment (IA) methods, under interweave, underlay, overlay and hybrid CR patterns. Due to the multi-user specified impacts, the achievable DoF of CU is constrained by the maximum transmit power of the other CUs, and the HD and FD tradeoff for multi-user CR is determined by both MUI and time-antenna splitting. Theoretical analysis indicates that the secondary user power and secondary user number only inflict impacts on LC and IA based average DoF respectively. Moreover, because of distributed CR processing, the average DoF for MU CR is upper-bounded by that of single-user CR. Simulation results validate the effects of MUI, SI, PUI and user number on the average DoF. It shows that IA achieves higher DoF than that of LC for multi-user CR, indicating that the DoF loss due to MUI can be reduced if interference mitigation is available.

New Set of Non-separable Orthogonal Invariant Moments for Image Recognition

It is known that the rotation, scaling and translation invariant property of image moments has a high significance in image recognition. For this reason, the seven invariant moments presented by Hu are widely used in the field of image analysis. These moments are of finite order; therefore, they do not comprise a complete set of image descriptors. For this reason, we introduce in this paper another series of invariant moments of infinite order, which are based on normalized central moments. The non-orthogonal property of these moments causes the redundancy of information. To overcome this problem, we propose a new construction technique of non-separable orthogonal polynomials in two variables based on a recurrence formula and we present a new set of orthogonal moments, which are invariant to translation, scaling and rotation. The presented approaches are tested in several well-known computer vision datasets including moment’s invariability, image retrieval and classification of objects, this latter based on fuzzy K-means clustering algorithm. The performance of these invariant moments for classification and image retrieval is compared with some recent invariant moments such as invariants of multi-channel orthogonal radial-substituted Chebyshev moments, invariants of quaternion radial-substituted Chebyshev moments, invariants of rotational moments in Radon space and Legendre–Fourier moments in Radon space. The experimental results made using four databases of images, namely Columbia Object Image Library (COIL-20) database, MPEG7-CE shape database, COIL-100 database and ORL database, show that our orthogonal invariant moments have done better than the other descriptors tested.

The scale-invariant space for attention layer in neural network

Recently, the Transformer architecture has achieved state-of-the-art performance in many natural language processing tasks. One key component in the Transformer architecture is the attention layer, which captures the relation between tokens. In this paper, we show that the weight of the attention layer has scale-invariant property, i.e. the output is invariant to a rescaling of weights. However, optimization algorithms in the vector space of weight such as SGD are not scaling invariant. This mismatch will potentially hurt the optimization process. To solve the mismatch, we seek a new parameter space for attention layer that is both scale-invariant and can sufficiently represent the output of attention, so that we can employ optimization algorithms in the scale-invariant parameter space. To achieve this goal, we first show that the output of the attention layer can be represented using scale-invariant variables, which is called paths. Then, we define basis paths which are an independent subset of all paths and are sufficient to represent all other paths. We prove that the Scale-Invariant (SI) space for the attention layer is composed of the basis path. Finally, we design an Attention Basis Path Identification(ABPI) Method to identify the basis paths and propose optimizing the attention layer directly in its SI space. Several experiments on benchmark datasets show that we can obtain more effective neural networks with the attention layer by optimizing the attention layer directly in its SI space.