Savitzky-Golay Differentiation Research Articles

Adaptive-sized residual fusion network-based segmentation of biomedical images

Medical imaging is crucial in examinations and for providing appropriate patient care. Medical images are used to diagnose and identify specific conditions. However, the diagnostic procedures are confounded by the complexity of these images, which have several overlapping characteristics. Various techniques have enhanced biomedical image segmentation, and strategies to improve the contrast and accuracy of medical images have been presented in the literature. This article presents a recurrent grasshopper herd-based optimized histogram-based contrast enhancement technique, in which sharp-edge morphological characteristics can be enhanced using inclusive Laplacian fractional order Savitzky–Golay differentiation. An adaptive-sized residual fusion network approach is used to isolate the target tissue in the biomedical image. The effectiveness of the proposed system is verified using different medical data sets and a high level of accuracy is achieved. The proposed method outperforms other state-of-the-art methodologies in terms of the scale, shape and morphological attributes of the medical image.

Surface analysis insight note: Differentiation methods applicable to noisy data for determination of sp2‐ versus sp3‐hybridization of carbon allotropes and AES signal strengths

The derivatives of the spectra are commonly used for quantification in Auger Electron Spectroscopy (AES) spectra, while the derivative of the KLL C Auger line has proven to be valuable in obtaining a measure of the relative proportions of sp2‐ and sp3‐hybridization using the D‐parameter in both AES and X‐ray Photoelectron Spectroscopy (XPS). Differentiation of X‐ray Photoelectron Spectroscopy (XPS) and Auger Electron Spectroscopy (AES) spectra by numerical means is presented and illustrated for polymeric, such as PEEK and Nylon, as well as for graphitic materials including highly ordered pyrolytic graphite and graphene oxide. The most commonly available Savitzky–Golay method is explained mathematically and developed through the case of constructing a 5‐point quadratic polynomial convolution kernel suitable for differentiating spectra of adequate signal to noise. The concept of differentiation of spectra where signal to noise is less than adequate is also developed. Two alternative strategies to Savitzky–Golay differentiation are presented, which fit curves to data that allow derivatives to be obtained where Savitzky–Golay would otherwise fail. These alternative methods involve constructing a parametric curve that fits data over the entire energy interval of interest. Derivatives of spectra are then obtained by differentiating these parametric curves directly. A comparison of results for different materials for which specific sp2‐ vs sp3‐hybridized carbon proportions are of interest is used to emphasize the importance of characterizing methods used to differentiate spectra and understanding the characteristics of instrumentation used to measure spectra. The case for using Principal Component Analysis noise reduction with C KLL spectra is made for spectra collected from a heterogeneous graphene oxide sample.

Spall Fault Quantification Method for Flight Control Electromechanical Actuator

Flight control electro-mechanical actuators (EMAs) are among the primary onboard systems that significantly influence the reliability and safety of unmanned aerial vehicles. Recent reliability studies have shown that the ball-screw element of a flight control EMA is subject to oscillating operating conditions that may initiate rapid degradation, such as fatigue spall defects. Accordingly, detecting and quantifying such faults are crucial for developing efficient fault prognostic and remaining useful life estimation capabilities. In this study, a vibration-based fault quantification method is developed to quantify the fatigue faults of a ball-screw mechanism of an EMA. The method is based on identifying the ball passing instants through a localized surface defect on the vibrational jerk rather than the vibrational acceleration measurement. The jerk is numerically determined from conventional accelerometers using a Savitzky–Golay differentiator. This method was successfully tested for ball bearings and it is adjusted in this paper for ball-screw faults. The experimental validation is investigated on a set of fault-seeded samples on NASA’s Ames Research Center Flyable Electro-Mechanical Actuator test stand.

Savitzky-Golay filter energy features-based approach to face recognition using symbolic modeling.

Face recognition is a well-researched domain however many issues for instance expression changes, illumination variations, and presence of occlusion in the face images seriously affect the performance of such systems. A recent survey shows that COVID-19 will also have a considerable and long-term impact on biometric face recognition systems. The work has presented two novel Savitzky–Golay differentiator (SGD) and gradient-based Savitzky–Golay differentiator (GSGD) feature extraction techniques to elevate issues related to face recognition systems. The SGD and GSGD feature descriptors are able to extract discriminative information present in different parts of the face image. In this paper, an efficient and robust person identification using symbolic data modeling approach and similarity analysis measure is devised and employed for feature representation and classification tasks to address the aforementioned issues of face recognition. Extensive experiments and comparisons of the proposed descriptors experimental results indicated that the proposed approaches can achieve optimal performance of 96–97, 92–96, 100, 84–93, and 87–96% on LFW, ORL, AR, IJB-A datasets, and newly devised VISA database, respectively.

Proposing an unbiased oxygen reduction reaction onset potential determination by using a Savitzky-Golay differentiation procedure

For proper and fair comparison of the performance of Oxygen reduction reaction (ORR) electrocatalysts an un-biased method to determine an onset potential value is needed. Here we report an easy mathematical approach based on the second derivative of linear sweep voltammetry curves, referred to as a second order discrete differentiation method (SODDM) that allows to accurately provide the onset potential. Analysis of the published results showed that the reported values might be affected by an intrinsic human error associated with the application of the most common approaches addressed as a tangent method or those relaying on a visual estimation of the onset potential based on the shape of a linear scan voltammetry (LSV) curve. We have also demonstrated that by using SODDM, electrochemical data collected on different instruments by different researchers leads to comparable results in terms of the ORR onset potential values.

Adaptive Savitzky-Golay filtering and its applications

Noise reduction is a central issue of the theory and practice of signal processing. The Savitzky-Golay (SG) smoothing and differentiation filter is widely acknowledged as a simple and efficient method for denoising. However only few book on signal processing contain this method. As is well known, the performance of the classical SG-filter depends on the appropriate setting of the window length and the polynomial degree, which should match the scale of the signal since, in the case of signals with high rate of change, the performance of the filter may be limited. This paper presents a new adaptive strategy to smooth irregular signals based on the Savitzky-Golay algorithm. The proposed technique ensures high precision noise reduction by iterative multi-round smoothing and correction. In each round the parameters dynamically change due to the results of the previous smoothing. Our study provides additional support for data compression based on optimal resolution of the signal with linear approximation. Here, simulation results validate the applicability of the novel method.

An Approach to Savitzky–Golay Differentiators

This short paper presents a new approach to the calculation of Savitzky–Golay differentiators based on symmetric differentiation. Some comparisons with the regular polynomial derivative are made in order to find differences and similarities between the two methods. The evaluation shows that the new symmetric differentiation approach presents lower noise power gain, which is an interesting characteristic for digital signal processing purposes.

A shear wave elastography method based on 2D Savitzky-Golay digital differentiator

A shear wave elastography method based on 2D Savitzky-Golay digital differentiator

Subpixel phase-based image registration using Savitzky–Golay differentiators in gradient-correlation

This paper presents a new two-step method for finding two-dimensional translational shifts with subpixel accuracy. This algorithm can measure subpixel shifts, even in images with few features, or in noisy images where many existing algorithms fail. In the first step of the algorithm (the integer part), the noise-robustness of the gradient correlation methods was improved by replacing central difference differentiators with Savitzky–Golay differentiators (SGDs). In the second step of the algorithm (the subpixel part), several modifications have been proposed to increase the accuracy and noise-robustness of phase-based methods for finding subpixel shifts. Moreover, two error metrics were introduced to quantify the output accuracy of the integer and subpixel parts of the algorithm.Comprehensive tests were conducted on 2400 standard 128 pixel × 128 pixel subimages subjected to synthetic shifts and rotations. Tests showed that the accuracy of the proposed method for finding translational shifts is of the order of a few ten-thousandths of a pixel, which is a substantial improvement over other state-of-the-art methods. For the rotation tests, the method outperformed comparable techniques. Furthermore, results showed that the proposed method generally provides better performance than other competing methods when images contained Gaussian or salt and pepper noise. The proposed method can be used in applications where high accuracy, robustness to noise, and/or computation efficiency are important.

Automated vibration-based fault size estimation for ball bearings using Savitzky–Golay differentiators

Vibration-based fault diagnosis has been utilized as a reliable method for identifying ball bearings health since the 1970s. Recently, there has been an increased research effort to develop methods for fault quantification with the aim of estimating the fault size to allow the service life of a ball bearing to be extended beyond the detection stage. These studies have shown that the vibration signal from a localized spall (e.g. fatigue defect) in a ball bearing exhibits features corresponding to two main events, namely, the entry into and the exit from the spall. The time span between these two events is correlated with the spall size. Studies have shown that the entry into the spall is the more challenging event to identify, which often requires extensive signal processing techniques. This paper introduces an automated vibration-based technique for estimating the size of a spall in a ball bearing under axial loading conditions similar to those of linear electro-mechanical actuators. This technique is based on the extraction of the entry/exit events from the vibrational jerk, which are numerically determined from accelerometer data. The differentiation of the acceleration data to estimate jerk signal is performed using a variant of Savitzky–Golay (SG) differentiators, which provide enhancement for the detection of the entry and exit points. Sensible spall size estimations have been achieved for 24 different scenarios of fault sizes, rotor speeds and loads measured on a test rig provided by DLR (German Aerospace Center).

An alternative for the calculation of derivative spectra in the near-infrared spectroscopy

Savitzky-Golay differentiation procedure is a commonly used pre-processing technique in the near-infrared spectroscopy, being implemented in the instrumental or specialized data processing software. However, the spectra can be recorded linearly to the wavelength or energy scale depending on the available instrument, which leads to different derivative curves obtained by the Savitzky-Golay approach. The derivatives of the spectra linear to the wavelength scale tend to attenuate. A modified procedure, which avoids this problem, is described and the aim of this note is to bring this pre-processing alternative to the attention of the near-infrared specialists.

Least Squares Moving-Window Spectral Analysis

Least squares regression is proposed as a moving-windows method for analysis of a series of spectra acquired as a function of external perturbation. The least squares moving-window (LSMW) method can be considered an extended form of the Savitzky-Golay differentiation for nonuniform perturbation spacing. LSMW is characterized in terms of moving-window size, perturbation spacing type, and intensity noise. Simulation results from LSMW are compared with results from other numerical differentiation methods, such as single-interval differentiation, autocorrelation moving-window, and perturbation correlation moving-window methods. It is demonstrated that this simple LSMW method can be useful for quantitative analysis of nonuniformly spaced spectral data with high frequency noise.

Accuracy of Quasi-Static Bending Strain Measurement using Scanning Laser Doppler Vibrometry (SLDV)

This paper presents an investigation into the use of 1D scanning laser Doppler vibrometry for the measurement of quasi-static bending strain, on a beam, from measured out-of-plane deflections. To calculate the bending strain from measured deflections, a Savitzky-Golay differentiation filter is used. An experimental investigation was conducted to determine how the strain estimate would be affect by the spatial interval between measurement points, amplitude of loading, and the numerical parameters used within the Savitzky-Golay differentiation filter. A number of findings, from the investigations undertaken are presented. The bending strain measurement technique is best approximated by using a third-order polynomial in the Savitzky-Golay differentiator filter. An accurate bending strain measurement can be produced at any spatial interval, as long as the filter width spans a fixed distance. In addition, changes in applied load have no effect on the accuracy of the technique. This technique has significant application for the measurement of localised bending strain on the surface of structures or materials that are cyclically loaded and subjected to harsh environmental conditions, such as elevated temperatures.

A new technique for image enhancement using digital fractional-order Savitzky–Golay differentiator

In this manuscript, digital fractional-order Savitzky---Golay differentiator (DFOSGD) is explored in the image enhancement application. In DFOSGD, moving window is responsible for estimating the fr...

Application of Savitzky–Golay differentiation filters and Fourier functions to simultaneous determination of cefepime and the co-administered drug, levofloxacin, in spiked human plasma

The present work is concerned with simultaneous determination of cefepime (CEF) and the co-administered drug, levofloxacin (LEV), in spiked human plasma by applying a new approach, Savitzky–Golay differentiation filters, and combined trigonometric Fourier functions to their ratio spectra. The different parameters associated with the calculation of Savitzky–Golay and Fourier coefficients were optimized. The proposed methods were validated and applied for determination of the two drugs in laboratory prepared mixtures and spiked human plasma. The results were statistically compared with reported HPLC methods and were found accurate and precise.

Pretreating near infrared spectra with fractional order Savitzky–Golay differentiation (FOSGD)

Abstract With the aid of Riemann–Liouville fractional calculus theory, fractional order Savitzky–Golay differentiation (FOSGD) is calculated and applied to pretreat near infrared (NIR) spectra in order to improve the performance of multivariate calibrations. Similar to integral order Savitzky–Golay differentiation (IOSGD), FOSGD is obtained by fitting a spectral curve in a moving window with a polynomial function to estimate its coefficients and then carrying out the weighted average of the spectral curve in the window with the coefficients. Three NIR datasets including diesel, wheat and corn datasets were utilized to test this method. The results showed that FOSGD, which is easy to compute, is a general method to obtain Savitzky–Golay smoothing, fractional order and integral order differentiations. Fractional order differentiation computation to the NIR spectra often improves the performance of the PLS model with smaller RMSECV and RMSEP than integral order ones, especially for physical properties of interest, such as density, cetane number and hardness.

Multifilter phase imaging with partially coherent light.

A novel quantitative phase imaging method is shown to estimate phase accurately over a wide range of length scales using Köhler illumination from an extended incoherent source. The method is based on estimating the longitudinal intensity derivative in the transport-of-intensity equation via convolution with multiple Savitzky-Golay differentiation filters and generalizes methods previously developed for coherent imaging to the practical scenario of partially coherent imaging. The resulting noise and resolution performance are evaluated via numerical simulation and demonstrated experimentally using a blazed transmission grating as well as a single-mode fiber as test objects.

Correcting NIR spectra of dimethyl fumarate in milk measured for different brands and in different dates

The calibration transfer was used to correct near-infrared spectra (NIR) of dimethyl fumarate (DMF) for their variations resulting from differences of sample brand and measurement date. Because the concentration of DMF in milk is very small, an enrichment method called fluidized bed enrichment was used to concentrate the analyte before NIR measurement. For correcting spectra between different brands and measurement dates, the spectra of Bright dairy measured on 2/5/2011 were assigned as primary spectra. And the spectra of Yili dairy measured on 2/5/2011 and Bright dairy measured on 17/5/2011 were set as secondary spectra for changes in brand and measurement date, respectively. Two spectral pretreatment methods were applied to preprocess spectra: multiple scattering correction (MSC) and the combination of MSC and differentiation (Savitzky–Golay differentiation). After spectral pretreatment, two calibration transfer methods including canonical correlation analysis (CCA) and piecewise direct standardization were used to correct the secondary spectra. The results showed that although the variations in both brand and measurement date can cause secondary spectra to deviate from primary spectra, the CCA under the combination of MSC and differentiation can correct the secondary spectra to use the model of primary spectra with high accuracy.

Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter - theory and applications

Several existing strategies for estimating the axial intensity derivative in the transport-of-intensity equation (TIE) from multiple intensity measurements have been unified by the Savitzky-Golay differentiation filter--an equivalent convolution solution for differentiation estimation by least-squares polynomial fitting. The different viewpoint from the digital filter in signal processing not only provides great insight into the behaviors, the shortcomings, and the performance of these existing intensity derivative estimation algorithms, but more important, it also suggests a new way of improving solution strategies by extending the applications of Savitzky-Golay differentiation filter in TIE. Two novel methods for phase retrieval based on TIE are presented--the first by introducing adaptive-degree strategy in spatial domain and the second by selecting optimal spatial frequencies in Fourier domain. Numerical simulations and experiments verify that the second method outperforms the existing methods significantly, showing reliable retrieved phase with both overall contrast and fine phase variations well preserved.

Fractional Differential-Based Approach for CT Image Enhancement

In this paper, a fractional differential-based approach for CT image enhancement is introduced. This approach uses a group of fractional differential masks, which are generalized from the one-dimensional digital fractional order Savitzky-Golay differentiator, to process the image and a max-saturation strategy is designed to enhance these processed images. Some experiments are used to assess the performance of the proposed fractional differential-based image enhancing algorithm, and the results demonstrate that the proposed enhancing method is able to achieve a good tradeoff between the feature enhancement and texture preservation.