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Fractional calculus is an important branch of mathematical analysis and played a fundamental role in different fields including signal processing and image processing. In this paper, we proposed a sparse super-resolution (SR) technique by using the Grünwald-Letnikov (G–L) fractional differential operator, which aims to reconstruct high-resolution images by recovering pixel information from low-resolution images. We suggested a modified fractional derivative mask based on G–L for image enhancement. The proposed method suppresses block artifacts, staircase edges, and false edges near the edges. The proposed method is very flexible and preserves detailed features. The obtained experiments demonstrated that fractional operator can nonlinearly preserve the low-frequency contour information in the smooth region and also nonlinearly improve well the high-frequency edge and texture of the image. In sparse SR images, the dictionary is constructed by texture information of the images by using integer derivatives. In this work, we computed dictionary matrices by applying the fractional masks. In fact, the extracted features based on fractional derivatives are utilized in the dictionary training procedure and sparse coding. The experimental results and analysis on natural images indicated that the proposed method achieved much better results than other algorithms in terms of both quantitative measures and visual perception. [ABSTRACT FROM AUTHOR]

Aiming at the problem that conventional rock creep constitutive models are unable to precisely describe the mechanical characteristics of the entire creep growth process, this paper proposes an FBNVP model (the fractional order differential Burgers model and the nonlinear viscoplastic model in series) that is capable of fully characterizing the creep features of deeply buried soft rocks. The fractional order differential theory is utilized to enhance the viscous element's ability to accurately depict the nonlinear growth process during the stable creep stage. The nonlinear viscoplastic element is constructed based on exponential equation to precisely describe the accelerated growth process of creep stage III. The three-dimensional constitutive equations of the model under viscoplastic conditions are accurately expressed using the Perzyna overstress function. The model is validated using indoor uniaxial and triaxial creep tests on the chlorite schist of Jinping II hydropower station. The results confirm the accuracy of both the one-dimensional and three-dimensional analytical solutions of the model, demonstrating its ability to accurately describe deeply buried soft rocks' creep characteristics and having less parameters, clear physical meaning, and wide applicability compared to conventional models. These findings provide reliable theoretical support for the creep analysis of deeply buried soft rocks. [ABSTRACT FROM AUTHOR]

Leaf water content (LWC) is one of the important indicators of crop health. It plays an important role in the physiological process of leaves, participates in almost all physiological processes of crops, and is of great significance to the survival and growth of crops. Based on the hyperspectral (350–1350 nm) and LWC data (jointing, booting, flowering, and filling periods) of winter wheat in 2020 and 2021, this work proposed to transform and process the hyperspectral data by adopting fractional order differential and continuous wavelet transform, and took a differential spectrum, wavelet coefficients, and mixed variables (differential spectrum and wavelet coefficients) as input variables of the model and adopted Gaussian process regression (GPR), classification and regression decision tree (CART), and artificial neural network (ANN) methods to estimate the LWC of wheat in different growth periods. The results indicated that fractional differential and continuous wavelet transform could highlight the spectral characteristics of winter wheat canopy and improve its correlation with LWC. The three model variables had the best estimation effect on LWC in the flowering period, and the average values of R2 were 0.86 and 0.87 in modeling and verification, which indicated that the flowering period could be used as the best estimation period for LWC. Compared with the differential spectrum and wavelet coefficients, LWC estimation based on mixed variables performed best. The average values of R2 in modeling and verification were 0.78 and 0.79. Among them, the ANN model had the highest estimation accuracy, and the R2 in modeling and verification could reach 0.92 and 0.91. This showed that fractional differential and continuous wavelet transform could effectively promote the sensitivity of spectral information to LWC and enhance the prediction ability and stability of wheat LWC. The outcomes of the present study have the potential to provide new ideas for the water monitoring of crops. [ABSTRACT FROM AUTHOR]

Iron oxide is the main form of iron present in soils, and its accumulation and migration activities reflect the leaching process and the degree of weathering development of the soil. Therefore, it is important to have information on the iron oxide content of soils. However, due to the overlapping characteristic spectra of iron oxide and organic matter in the visible-near infrared, appropriate spectral transformation methods are important. In this paper, we first used conventional spectral transformation (continuum removal, CR; standard normal variate, SNV; absorbance, log (1/R)), continuous wavelet transform (CWT), and fractional order differential (FOD) transform to process original spectra (OS). Secondly, competitive adaptive reweighted sampling (CARS) was used to extract characteristic wavelengths. Finally, two regression models (backpropagation neural network, BPNN; support vector regression (SVR) were used to predict the content of iron oxide. The results show that the FOD can significantly improve the correlation with iron oxide compared with the CR, SNV, log (1/R) and CWT; the baseline drift and overlapping peaks decrease with increasing the order of FOD; the CARS algorithm based on 50th averaging can select more stable characteristic wavelengths; the FOD achieves better results regardless of the modelling method, and the model based on 0.5-order differential has the best prediction performance (R2 = 0.851, RMSE = 5.497, RPIQ = 3.686). [ABSTRACT FROM AUTHOR]

Iron oxide is the main form of iron present in soils, and its accumulation and migration activities reflect the leaching process and the degree of weathering development of the soil. Therefore, it is important to have information on the iron oxide content of soils. However, due to the overlapping characteristic spectra of iron oxide and organic matter in the visible-near infrared, appropriate spectral transformation methods are important. In this paper, we first used conventional spectral transformation (continuum removal, CR; standard normal variate, SNV; absorbance, log (1/R)), continuous wavelet transform (CWT), and fractional order differential (FOD) transform to process original spectra (OS). Secondly, competitive adaptive reweighted sampling (CARS) was used to extract characteristic wavelengths. Finally, two regression models (backpropagation neural network, BPNN; support vector regression (SVR) were used to predict the content of iron oxide. The results show that the FOD can significantly improve the correlation with iron oxide compared with the CR, SNV, log (1/R) and CWT; the baseline drift and overlapping peaks decrease with increasing the order of FOD; the CARS algorithm based on 50th averaging can select more stable characteristic wavelengths; the FOD achieves better results regardless of the modelling method, and the model based on 0.5-order differential has the best prediction performance (R2 = 0.851, RMSE = 5.497, RPIQ = 3.686).

In order to extract more texture and detail information from a given image, a memristor cell neural network is proposed by replacing the state resistances of the neurons as the memristors. The characteristic of memristor could be maintained in the whole information processing of the cell neural network so that the historical state information could be used and the information processing ability of the network could be enhanced. Furthermore, based on the fractional order calculus theory, a fractional order control template is designed for the memristor cell neural network to enhance the middle- and high-frequency information and retain more low-frequency texture information in the image edge extraction. The network could be used to extract edge information from various kinds of images. The simulation results show that the edge images extracted by this method have more complete and clear contour information and more abundant texture detail information. Both the average gradient and the information entropy of the edge images could be significantly improved.

Leaf area index (LAI) is highly related to crop growth, and the traditional LAI measurement methods are field destructive and unable to be acquired by large-scale, continuous, and real-time means. In this study, fractional order differential and continuous wavelet transform were used to process the canopy hyperspectral reflectance data of winter wheat, the fractional order differential spectral bands and wavelet energy coefficients with more sensitive to LAI changes were screened by correlation analysis, and the optimal subset regression and support vector machine were used to construct the LAI estimation models for different growth stages. The precision evaluation results showed that the LAI estimation models constructed by using wavelet energy coefficients combined with a support vector machine at the jointing stage, fractional order differential combined with support vector machine at the booting stage, and wavelet energy coefficients combined with optimal subset regression at the flowering and filling stages had the best prediction performance. Among these, both flowering and filling stages could be used as the best growth stages for LAI estimation with modeling and validation R2 of 0.87 and 0.71, 0.84 and 0.77, respectively. This study can provide technical reference for LAI estimation of crops based on remote sensing technology.

A non-destructive identification method was developed here based on dropout deep belief network in multi-spectral data of ancient ceramic. A fractional differential algorithm was proposed to enhance the spectral details by making use of the difference between the first and second-order differential pre-process spectral data. An unsupervised multi-layer restricted Boltzmann machine (RBM) was employed to extract some high-level features during pre-training. Some weight and bias values trained by RBM were used to initialize a back propagation (BP) neural network. The RBM deep belief network was fine-tuned by the BP neural network to promote the initiative performance of network training, which helped to overcome local optimal limitation of the network due to the random initializing weight parameter. The dropout strategy has been put forward into the RBM network to solve the over-fitting of small sample spectral data. The experimental results show that the proposed method has excellent recognition performance of the ceramics by comparisons with some other ones.

The nanodevices often work in a thermal environment and lead to the viscoelastic and thermoelastic coupling which becomes more complicated when further coupled with the size effects. A spatial-temporal fractional order model is proposed to study the dynamic behavior of thermoelastic nanoplate in the present work. The integer order differential is extended to the fractional order differential with the integer order differential as a special case. This extension makes the modeling of small-scale effects of mechanical behavior and generalized heat conduction more flexible. Non-local strain gradient elasticity and non-local heat conduction are used to describe the spatial non-local effects while the fractional order differential to describe the complicated history-dependent effects or the temporal nonlocal effects. The standard solid viscoelastic model and generalized heat conduction model with the fractional order differentials lead to the fractional-order differential governing equations. The fractional order differential equation is solved by the Laplace transform method, and the analytical solution of the response is expressed in terms of the Mittag-Leffler function. In order to verify the reliability of the analytical solution, the numerical solution is also provided for comparison. Based on the numerical results of dynamic response under the step load, the effects of thermoelastic fractional order parameters, viscoelastic fractional order parameters, and small-scale parameters are discussed. The new spatial-temporal fractional order model is a natural extension of the traditional integer order model which can be recovered from the present model. • A new nonlocal thermoelastic model of micro-plate with temporal fraction order derivative is proposed. • The viscoelastic dissipation and the thermoelastic dissipation are both considered and compared. • The complex natural frequency due to viscoelastic dissipation and the thermoelastic dissipation is investigated. • The different influences of viscoelastic and thermoelastic fraction orders on the dynamic response are clarified. • The nonlocal effects of mechanical and heat conduction behaviors are both considered. [ABSTRACT FROM AUTHOR]

It is meaningful to develop the automation segmentation of rice planthopper pests based on imaging technology in precision agriculture. However, rice planthopper images affected by light and complicated backgrounds in open rice fields make the segmentation difficult. This study proposed a segmentation approach of rice planthopper images based on the Markov random field to conduct effective segmentation. First, fractional order differential was introduced into the extraction process of image texture features to gain complete texture information of rice planthopper images. Observation data modeling was established by a combination of image color features and texture features to overcome the disadvantages of insufficient image texture information. Finally, the improved potential function models, the neighborhood relationship between the pixel labels, and the attributes of pixels were defined. The segmentation results were assessed by quantitative evaluation. The experiments showed that the proposed improved approach in the study was more robust, especially with the changes in the illumination condition. This approach can effectively improve segmentation accuracy and promote vision segmentation results of rice planthopper images.

In recent years, the fractional order derivative has been introduced for image enhancement. It was proved that the medical image enhancement method based on the fractional order derivative has better effect than the method based on the integral order calculus. However, a priori information such as texture surrounding a pixel is normally ignored by the traditional fractional differential operators with the same value in the eight directions. To address the above problem, this paper presents a new medical image enhancement method by taking the merits of fractional differential and directional derivative. The proposed method considers the surrounding information (such as the image edge, clarity and texture information) and structural features of different pixels, as well as the directional derivative of each pixel in constructing the masks. By proposing this method, it can not only improve the high frequency information, but also improve the low frequency information of the image. Ultimately, it enhances the texture information of the image. Extensive experiments on four kinds of medical image demonstrate that the proposed algorithm is in favor of preserving more texture details and superior to the existing fractional differential algorithms on medical image enhancement. [ABSTRACT FROM AUTHOR]

In the process of image enhancement, the optimal value of the fractional order was often chosen by the experience or a large number of experiments, it couldn't realize the adaptation, and it did not give full play to the excellent characteristics of the fractional differential. This paper proposed an adaptive fractional order differential image enhancement algorithm based on 3 parameters related to image texture, which were the local gradient of the image, the information entropy and the variance. Moreover, this proposed algorithm was applied to some related medical images. Based on the quantitative assessment criteria of texture analysis, such as information entropy and average gradient, this paper analyzed and compared the enhanced image. Experimental results show that compared with other algorithms, adaptive fractional order differential algorithm can not only enhance the image edge and texture, hut also keep smooth regions of details and gain better visual effect. [ABSTRACT FROM AUTHOR]

• Quantitative estimation of wheat stripe rust severity using hyperspectral data. • The hyperspectral data is processed by fractional order differential to enhance spectral information and extract effective information about wheat stripe rust. • Downscaling and modeling of hyperspectral data using Gaussian process approach. • The best model constructed appears at order 1.2, R2 = 0.912, RMSE = 0.097, MAE = 0.071. Wheat stripe rust is the main cause of yield loss in winter wheat. For nondestructive monitoring of wheat stripe rust by remote sensing, a high-precision stripe rust monitoring model can be constructed so that field management can be performed rationally and environmental damage from large doses of pesticides and chemicals can be avoided. Fractional order differential (FOD) equations can be used to flexibly control the differential step size, which enhances the spectral information and reduces the background noise. In this study, the canopy hyperspectral data under the influence of wheat stripe rust were evaluated by FOD, and the disease severity level (SL) of stripe rust in the field was analyzed by evaluating the polar difference, coefficient of variation, and correlation coefficient. Subsequently, the spectral features associated with stripe rust were screened using significance tests and Gaussian process regression sigma (GPR sigma) analysis methods. Then, models for the various wheat stripe rust severity levels were established by Gaussian process regression (GPR) with different data inputs. The results showed that the 0.8–1.4 differential order could effectively improve the correlation between spectral bands and disease severity, and the optimal correlation between the 1.2 order differential spectra and wheat stripe rust severity improved by 15% compared with the original reflectance spectra. The GPR sigma band analysis method screens only 8 bands at order 1.2, the R2 between the model-predicted SL and the measured SL is improved by 13% compared to the original reflectance spectrum, and the RMSE and MAE are reduced by 34% and 39%, respectively. Compared with the significance test method, GPR sigma band analysis selected fewer bands, constructed models with higher accuracy and was more suitable for the construction of models for estimating the severity of wheat stripe rust disease. [ABSTRACT FROM AUTHOR]

An orthotropic higher-order shear micro-plate model is established with consideration the small-scale effect and the thermoelastic coupling effects to study the dispersion and attenuation of flexural waves in the present work. The nonlocal elasticity and the strain gradient elasticity are incorporated to lead the nonlocal strain gradient model to reflect the size effect. The generalized thermoelasticity (non-Fourier heat conduction) is introduced to reflect the history-dependent feature of heat conduction. Furthermore, the integer order derivatives are replaced by the fractional derivatives of Caputo-type to get a more refined model which can extend the range of application of model and provide more exquisite description of the spatial nonlocal behavior and temporal memory behavior at small scale. In addition, the nonlocal characteristic length and the memory characteristic length as new small scale parameters are first introduced in the present model to focus on describing the short-range nonlocal and the short-term memory behavior. A detailed parameter study is performed and the effects of nonlocal characteristic length, strain gradient characteristic length, the memory characteristic length and the spatiotemporal fractional order on the dispersion and attenuation of thermoelastic coupled flexural waves are discussed based on the numerical results. [ABSTRACT FROM AUTHOR]

Time series prediction is an important topic in the field of data analytics for real industrial production. However, the time series from real system usually has strong nonstationarity, which affects the generalization ability of the prediction model. An improved forecasting technique, named as fractional stochastic configuration networks (FSCN), is proposed for the prediction of nonstationary time series. FSCN is built on the basis of traditional stochastic configuration network by introducing fractional differential operator. The fractional differential technique avoids the challenge of infinite variance caused by modeling nonstationary series and the over-difference problem caused by traditional integer order difference. First, several data analysis methods are introduced to find the tendency, periodicity and probability density distribution characteristics hidden in the raw industrial data. Hurst exponent is calculated to determine the order of fractional difference to eliminate the nonstationarity of the raw data. Then FSCN network is constructed to model and forecast the sequential data. An explicit prediction uncertainty is derived to provide the confidence interval for the FSCN prediction. The proposed method is tested on a nonstationary time series benchmark dataset and a real cooling system. The experiment result demonstrates that it has a good potential prediction performance compared with several traditional prediction methods. • Stochastic configuration network is applied for nonstationary time series prediction. • Fractional difference is introduced to improve the stochastic configuration network. • The confidence interval is calculated directly with the prediction uncertainty. • The proposed methodology is verified on a real supermarket cooling system. [ABSTRACT FROM AUTHOR]

Leaf area index (LAI) is highly related to crop growth, and the traditional LAI measurement methods are field destructive and unable to be acquired by large-scale, continuous, and real-time means. In this study, fractional order differential and continuous wavelet transform were used to process the canopy hyperspectral reflectance data of winter wheat, the fractional order differential spectral bands and wavelet energy coefficients with more sensitive to LAI changes were screened by correlation analysis, and the optimal subset regression and support vector machine were used to construct the LAI estimation models for different growth stages. The precision evaluation results showed that the LAI estimation models constructed by using wavelet energy coefficients combined with a support vector machine at the jointing stage, fractional order differential combined with support vector machine at the booting stage, and wavelet energy coefficients combined with optimal subset regression at the flowering and filling stages had the best prediction performance. Among these, both flowering and filling stages could be used as the best growth stages for LAI estimation with modeling and validation R2 of 0.87 and 0.71, 0.84 and 0.77, respectively. This study can provide technical reference for LAI estimation of crops based on remote sensing technology. [ABSTRACT FROM AUTHOR]

A non-destructive identification method was developed here based on dropout deep belief network in multi-spectral data of ancient ceramic. A fractional differential algorithm was proposed to enhance the spectral details by making use of the difference between the first and second-order differential pre-process spectral data. An unsupervised multi-layer restricted Boltzmann machine (RBM) was employed to extract some high-level features during pre-training. Some weight and bias values trained by RBM were used to initialize a back propagation (BP) neural network. The RBM deep belief network was fine-tuned by the BP neural network to promote the initiative performance of network training, which helped to overcome local optimal limitation of the network due to the random initializing weight parameter. The dropout strategy has been put forward into the RBM network to solve the over-fitting of small sample spectral data. The experimental results show that the proposed method has excellent recognition performance of the ceramics by comparisons with some other ones. [ABSTRACT FROM AUTHOR]

The possibility of quantitative inversion of salinized soil moisture content (SMC) from Zhuhai-1 hyperspectral imagery and the application effect of fractional order differentially optimized spectral indices were discussed, which provided new research ideas for improving the accuracy of hyperspectral remote sensing inversion. The hyperspectral data from indoor and Zhuhai-1 remote sensing imagery were resampled to the same spectral scale. The soil hyperspectral data were processed by fractional order differential preprocessing method and optimized spectral indices method, and the Pearson correlation coefficient (PCC/r) analysis was made with SMC data. The sensitive optimized spectral indices were used to establish the ground hyperspectral estimation model, and a variety of modeling methods were used to select the best SMC inversion model. The results were as follows: the maximum one-dimensional r between SMC and the 466–938 nm band was −0.635, the maximum one-dimensional r with the 0.5-order absorbance spectrum was 0.665, and the maximum two-dimensional r with the difference index (DI) calculated by the 0.5-order absorbance spectrum was ±0.72. The maximum three-dimensional r with the triangle vegetation index (TVI) calculated from the 0.5-order absorbance spectrum reached 0.755, which exceeded the one-dimensional r extreme value of 400–2400 nm. The TreeNet gradient boosting machine (TGBM) regression model had the highest modeling accuracy, with a calibration coefficient of determination (R2C) = 0.887, calibration root mean square error (RMSEC) = 2.488%, standard deviation (SD) = 6.733%, and r = 0.942. However, the partial least squares regression (PLSR) model had the strongest predictive ability, with validation coefficient of determination (R2V) = 0.787, validation root mean square error (RMSEV) = 3.247%, and relative prediction deviation (RPD) = 2.071. The variable importance in projection (VIP) method could not only improve model efficiency but also increased model accuracy. R2C of the optimal PLSR model was 0.733, RMSEC was 3.028%, R2V was 0.805, RMSEV was 3.100%, RPD was 1.976, and Akaike information criterion (AIC) was 151.050. The three-band optimized spectral indices with fractional differential pretreatment could to a certain extent break through the limitation of visible near-infrared spectrum in SMC estimation due to the lack of shortwave infrared spectra, which made it possible to quantitatively retrieve saline SMC on the basis of Zhuhai-1 hyperspectral imagery. [ABSTRACT FROM AUTHOR]