Generalization of stochastic-resonance-based threshold networks with Tikhonov regularization.

Injecting artificial noise into a feedforward threshold neural network allows it to become trainable by gradient-based methods and also enlarges the parameter space as well as the range of synaptic weights. This configuration constitutes a stochastic-resonance-based threshold neural network, where the noise level can adaptively converge to a nonzero optimal value for finding a local minimum of the loss criterion. We prove theoretically that the injected noise plays the role of a generalized Tikhonov regularizer for training the designed threshold network. Experiments on regression and classification problems demonstrate that the generalization of the stochastic-resonance-based threshold network is improved by the injection of noise. The feasibility of injecting noise into the threshold neural network opens up the potential for adaptive stochastic resonance in machine learning.

Signal estimation and filtering from quantized observations via adaptive stochastic resonance.

Using a gradient-based algorithm, we investigate signal estimation and filtering in a large-scale summing network of single-bit quantizers. Besides adjusting weights, the proposed learning algorithm also adaptively updates the level of added noise components that are intentionally injected into quantizers. Experimental results show that minimization of the mean-squared error requires a nonzero optimal level of the added noise. The process adaptively achieves in this way a form of stochastic resonance or noise-aided signal processing. This adaptive optimization method of the level of added noise extends the application of adaptive stochastic resonance to some complex nonlinear signal processing tasks.

Study of vibrational resonance in nonlinear signal processing.

Vibrational resonance (VR) intentionally applies high-frequency periodic vibrations to a nonlinear system, in order to obtain enhanced efficiency for a number of information processing tasks. Note that VR is analogous to stochastic resonance where enhanced processing is sought via purposeful addition of a random noise instead of deterministic high-frequency vibrations. Comparatively, due to its ease of implementation, VR provides a valuable approach for nonlinear signal processing, through detailed modalities that are still under investigation. In this paper, VR is investigated in arrays of nonlinear processing devices, where a range of high-frequency sinusoidal vibrations of the same amplitude at different frequencies are injected and shown capable of enhancing the efficiency for estimating unknown signal parameters or for detecting weak signals in noise. In addition, it is observed that high-frequency vibrations with differing frequencies can be considered, at the sampling times, as independent random variables. This property allows us here to develop a probabilistic analysis-much like in stochastic resonance-and to obtain a theoretical basis for the VR effect and its optimization for signal processing. These results provide additional insight for controlling the capabilities of VR for nonlinear signal processing. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 1)'.

Exploiting vibrational resonance in weak-signal detection.

In this paper, we investigate the first exploitation of the vibrational resonance (VR) effect to detect weak signals in the presence of strong background noise. By injecting a series of sinusoidal interference signals of the same amplitude but with different frequencies into a generalized correlation detector, we show that the detection probability can be maximized at an appropriate interference amplitude. Based on a dual-Dirac probability density model, we compare the VR method with the stochastic resonance approach via adding dichotomous noise. The compared results indicate that the VR method can achieve a higher detection probability for a wider variety of noise distributions.

Capacity of very noisy communication channels based on Fisher information.

We generalize the asymptotic capacity expression for very noisy communication channels to now include coloured noise. For the practical scenario of a non-optimal receiver, we consider the common case of a correlation receiver. Due to the central limit theorem and the cumulative characteristic of a correlation receiver, we model this channel noise as additive Gaussian noise. Then, the channel capacity proves to be directly related to the Fisher information of the noise distribution and the weak signal energy. The conditions for occurrence of a noise-enhanced capacity effect are discussed, and the capacity difference between this noisy communication channel and other nonlinear channels is clarified.

Too good to be true: when overwhelming evidence fails to convince.

Is it possible for a large sequence of measurements or observations, which support a hypothesis, to counterintuitively decrease our confidence? Can unanimous support be too good to be true? The assumption of independence is often made in good faith; however, rarely is consideration given to whether a systemic failure has occurred. Taking this into account can cause certainty in a hypothesis to decrease as the evidence for it becomes apparently stronger. We perform a probabilistic Bayesian analysis of this effect with examples based on (i) archaeological evidence, (ii) weighing of legal evidence and (iii) cryptographic primality testing. In this paper, we investigate the effects of small error rates in a set of measurements or observations. We find that even with very low systemic failure rates, high confidence is surprisingly difficult to achieve; in particular, we find that certain analyses of cryptographically important numerical tests are highly optimistic, underestimating their false-negative rate by as much as a factor of 280.

Multiscale imaging of plants: current approaches and challenges.

We review a set of recent multiscale imaging techniques, producing high-resolution images of interest for plant sciences. These techniques are promising because they match the multiscale structure of plants. However, the use of such high-resolution images is challenging in the perspective of their application to high-throughput phenotyping on large populations of plants, because of the memory cost for their data storage and the computational cost for their processing to extract information. We discuss how this renews the interest for multiscale image processing tools such as wavelets, fractals and recent variants to analyse such high-resolution images.

Double-maximum enhancement of signal-to-noise ratio gain via stochastic resonance and vibrational resonance.

This paper studies the signal-to-noise ratio (SNR) gain of a parallel array of nonlinear elements that transmits a common input composed of a periodic signal and external noise. Aiming to further enhance the SNR gain, each element is injected with internal noise components or high-frequency sinusoidal vibrations. We report that the SNR gain exhibits two maxima at different values of the internal noise level or of the sinusoidal vibration amplitude. For the addition of internal noise to an array of threshold-based elements, the condition for occurrence of stochastic resonance is analytically investigated in the limit of weak signals. Interestingly, when the internal noise components are replaced by high-frequency sinusoidal vibrations, the SNR gain displays the vibrational multiresonance phenomenon. In both considered cases, there are certain regions of the internal noise intensity or the sinusoidal vibration amplitude wherein the achieved maximal SNR gain can be considerably beyond unity for a weak signal buried in non-Gaussian external noise. Due to the easy implementation of sinusoidal vibration modulation, this approach is potentially useful for improving the output SNR in an array of nonlinear devices.

Stochastic resonance with colored noise for neural signal detection.

We analyze signal detection with nonlinear test statistics in the presence of colored noise. In the limits of small signal and weak noise correlation, the optimal test statistic and its performance are derived under general conditions, especially concerning the type of noise. We also analyze, for a threshold nonlinearity-a key component of a neural model, the conditions for noise-enhanced performance, establishing that colored noise is superior to white noise for detection. For a parallel array of nonlinear elements, approximating neurons, we demonstrate even broader conditions allowing noise-enhanced detection, via a form of suprathreshold stochastic resonance.

Weak-periodic stochastic resonance in a parallel array of static nonlinearities.

This paper studies the output-input signal-to-noise ratio (SNR) gain of an uncoupled parallel array of static, yet arbitrary, nonlinear elements for transmitting a weak periodic signal in additive white noise. In the small-signal limit, an explicit expression for the SNR gain is derived. It serves to prove that the SNR gain is always a monotonically increasing function of the array size for any given nonlinearity and noisy environment. It also determines the SNR gain maximized by the locally optimal nonlinearity as the upper bound of the SNR gain achieved by an array of static nonlinear elements. With locally optimal nonlinearity, it is demonstrated that stochastic resonance cannot occur, i.e. adding internal noise into the array never improves the SNR gain. However, in an array of suboptimal but easily implemented threshold nonlinearities, we show the feasibility of situations where stochastic resonance occurs, and also the possibility of the SNR gain exceeding unity for a wide range of input noise distributions.

Local-feature-based similarity measure for stochastic resonance in visual perception of spatially structured images.

For images, stochastic resonance or useful-noise effects have previously been assessed with low-level pixel-based information measures. Such measures are not sensitive to coherent spatial structures usually existing in images. As a result, we show that such measures are not sufficient to properly account for stochastic resonance occurring in visual perception. We introduce higher-level similarity measures, inspired from visual perception, and based on local feature descriptors of scale invariant feature transform (SIFT) type. We demonstrate that such SIFT-based measures allow for an assessment of stochastic resonance that matches the visual perception of images with spatial structures. Constructive action of noise is registered in this way with both additive noise and multiplicative speckle noise. Speckle noise, with its grainy appearance, is particularly prone to introducing spurious spatial structures in images, and the stochastic resonance visually perceived and quantitatively assessed with SIFT-based measures is specially examined in this context.

Study of time reversibility/irreversibility of cardiovascular data: theoretical results and application to laser Doppler flowmetry and heart rate variability signals.

Time irreversibility can be qualitatively defined as the degree of a signal for temporal asymmetry. Recently, a time irreversibility characterization method based on entropies of positive and negative increments has been proposed for experimental signals and applied to heart rate variability (HRV) data (central cardiovascular system (CVS)). The results led to interesting information as a time asymmetry index was found different for young subjects and elderly people or heart disease patients. Nevertheless, similar analyses have not yet been conducted on laser Doppler flowmetry (LDF) signals (peripheral CVS). We first propose to further investigate the above-mentioned characterization method. Then, LDF signals, LDF signals reduced to samples acquired during ECG R peaks (LDF_R(ECG) signals) and HRV recorded simultaneously in healthy subjects are processed. Entropies of positive and negative increments for LDF signals show a nonmonotonic pattern: oscillations--more or less pronounced, depending on subjects--are found with a period matching the one of cardiac activity. However, such oscillations are not found with LDF_R(ECG) nor with HRV. Moreover, the asymmetry index for LDF is markedly different from the ones of LDF_R(ECG) and HRV. The cardiac activity may therefore play a dominant role in the time irreversibility properties of LDF signals.

Assessing spatial resolution versus sensitivity from laser speckle contrast imaging: application to frequency analysis.

For blood perfusion monitoring, laser speckle contrast (LSC) imaging is a recent non-contact technique that has the characteristic of delivering noise-like speckled images. To exploit LSC images for quantitative physiological measurements, we developed an approach that implements controlled spatial averaging to reduce the detrimental impact of the noise and improve measurement sensitivity. By this approach, spatial resolution and measurement sensitivity can be traded-off in a flexible way depending on the quantitative prospect of the study. As an application, detectability of the cardiac activity from LSC images of forearm using power spectrum analysis is studied through the construction of spatial activity maps offering a window on the blood flow perfusion and its regional distribution. Comparisons with results obtained with signals of laser Doppler flowmetry probes are performed.

Fisher information as a metric of locally optimal processing and stochastic resonance.

The origins of Fisher information are in its use as a performance measure for parametric estimation. We augment this and show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing of a weak signal in additive white noise, we demonstrate that the Fisher information determines (i) the maximum output signal-to-noise ratio for a periodic signal; (ii) the optimum asymptotic efficacy for signal detection; (iii) the best cross-correlation coefficient for signal transmission; and (iv) the minimum mean square error of an unbiased estimator. This unifying picture, via inequalities on the Fisher information, is used to establish conditions where improvement by noise through stochastic resonance is feasible or not.

Fisher-information condition for enhanced signal detection via stochastic resonance.

Various situations where a signal is enhanced by noise through stochastic resonance are now known. This paper contributes to determining general conditions under which improvement by noise can be a priori decided as feasible or not. We focus on the detection of a known signal in additive white noise. Under the assumptions of a weak signal and a sufficiently large sample size, it is proved, with an inequality based on the Fisher information, that improvement by adding noise is never possible, generically, in these conditions. However, under less restrictive conditions, an example of signal detection is shown with favorable action of adding noise.

Multiscale analysis of microvascular blood flow: a multiscale entropy study of laser Doppler flowmetry time series.

Processes regulating the cardiovascular system (CVS) are numerous. Each possesses several temporal scales. Their interactions lead to interdependences across multiple scales. For the CVS analysis, different multiscale studies have been proposed, mostly performed on heart rate variability signals (HRV) reflecting the central CVS; only few were dedicated to data from the peripheral CVS, such as laser Doppler flowmetry (LDF) signals. Very recently, a study implemented the first computation of multiscale entropy for LDF signals. A nonmonotonic evolution of multiscale entropy with two distinctive scales was reported, leading to a markedly different behavior from the one of HRV. Our goal herein is to confirm these results and to go forward in the investigations on origins of this behavior. For this purpose, 12 LDF signals recorded simultaneously on the two forearms of six healthy subjects are processed. This is performed before and after application of physiological scales-based filters aiming at isolating previously found frequency bands linked to physiological activities. The results obtained with signals recorded simultaneously on two different sites of each subject show a probable central origin for the nonmonotonic behavior. The filtering results lead to the suggestion that origins of the distinctive scales could be dominated by the cardiac activity.

Joint acquisition-processing approach to optimize observation scales in noisy imaging.

In imaging, the choice of an observation scale is conventionally settled by the operator in charge of the image acquisition, who is left alone with tuning the framing and zooming parameters of the imaging system. In a somewhat decoupled manner, the operator in charge of processing the data has access to the images after their acquisition, and seeks to extract information from the observed scene. This Letter proposes a manifestation of the interest of an alternative joint acquisition-processing approach. We demonstrate with quantitative informational measures how the choice of an observation scale can be directly related to the performance of the final information processing task. Illustrations are given with various tools from statistical information theory with possible applications of practical interest to any noisy imaging domains.

Multifractal analysis of heart rate variability and laser Doppler flowmetry fluctuations:comparison of results from different numerical methods.

To contribute to the understanding of the complex dynamics in the cardiovascular system (CVS), the central CVS has previously been analyzed through multifractal analyses of heart rate variability (HRV) signals that were shown to bring useful contributions. Similar approaches for the peripheral CVS through the analysis of laser Doppler flowmetry (LDF) signals are comparatively very recent. In this direction, we propose here a study of the peripheral CVS through a multifractal analysis of LDF fluctuations, together with a comparison of the results with those obtained on HRV fluctuations simultaneously recorded. To perform these investigations concerning the biophysics of the CVS, first we have to address the problem of selecting a suitable methodology for multifractal analysis, allowing us to extract meaningful interpretations on biophysical signals. For this purpose, we test four existing methodologies of multifractal analysis. We also present a comparison of their applicability and interpretability when implemented on both simulated multifractal signals of reference and on experimental signals from the CVS. One essential outcome of the study is that the multifractal properties observed from both the LDF fluctuations (peripheral CVS) and the HRV fluctuations (central CVS) appear very close and similar over the studied range of scales relevant to physiology.

Rényi entropy measure of noise-aided information transmission in a binary channel.

This paper analyzes a binary channel by means of information measures based on the Rényi entropy. The analysis extends, and contains as a special case, the classic reference model of binary information transmission based on the Shannon entropy measure. The extended model is used to investigate further possibilities and properties of stochastic resonance or noise-aided information transmission. The results demonstrate that stochastic resonance occurs in the information channel and is registered by the Rényi entropy measures at any finite order, including the Shannon order. Furthermore, in definite conditions, when seeking the Rényi information measures that best exploit stochastic resonance, then nontrivial orders differing from the Shannon case usually emerge. In this way, through binary information transmission, stochastic resonance identifies optimal Rényi measures of information differing from the classic Shannon measure. A confrontation of the quantitative information measures with visual perception is also proposed in an experiment of noise-aided binary image transmission.

Generalized fractal dimensions of laser Doppler flowmetry signals recorded from glabrous and nonglabrous skin.

PURPOSE: The technique of laser Doppler flowmetry (LDF) is commonly used to have a peripheral view of the cardiovascular system. To better understand the microvascular perfusion signals, the authors herein propose to analyze and compare the complexity of LDF data recorded simultaneously in glabrous and nonglabrous skin. Glabrous zones are physiologically different from the others partly due to the presence of a high density of arteriovenous anastomoses. METHODS: For this purpose, a multifractal analysis based on the partition function and generalized fractal dimensions computation is proposed. The LDF data processed are recorded simultaneously on the right and left forearms and on the right and left hand palms of healthy subjects. The signal processing method is first tested on a multifractal binomial measure. The generalized fractal dimensions of the normalized LDF signals are then estimated. Furthermore, for the first time, the authors estimate the generalized fractal dimensions from a range of scales corresponding to factors influencing the microcirculation flow (cardiac, respiratory, myogenic, neurogenic, and endothelial). RESULTS: Different multifractal behaviors are found between normalized LDF signals recorded in the forearms and in the hand palms of healthy subjects. Thus, the variations in the estimated generalized fractal dimensions of LDF signals recorded in the hand palms are higher than those of LDF signals recorded in the forearms. This shows that LDF signals recorded in glabrous zones may be more complex than those recorded in nonglabrous zones. Furthermore, the results show that the complexity in the hand palms could be more important at scales corresponding to the myogenic control mechanism than at the other studied scales. CONCLUSIONS: These findings suggest that the multifractality of the normalized LDF signals is different on glabrous and nonglabrous skin. This difference may rely on the density of arteriovenous anastomoses and differences in nerve supply or biochemical properties. This study provides useful information for an in-depth understanding of LDF data and a more detailed knowledge of the peripheral cardiovascular system.