Shift-invariant Dictionary Research Articles

A Hierarchical Fault Diagnosis Model for Planetary Gearbox with Shift-invariant Dictionary and OMPAN

Abstract Planetary gearbox has been widely applied in the mechanical transmission system, and the failure types of planetary gearbox are more and more diversified. The conventional fault diagnosis methods focus on identifying the faults in the fault library, but ignored the faults outside the fault library. However, it is impossible to build a fault library for all failure types. Targeting the problem of identifying the faults outside the fault library, a hierarchical fault diagnosis method for planetary gearbox with shift-invariant dictionary and orthogonal matching pursuit with adaptive noise (OMPAN) is proposed in this paper. By k-means singular value decomposition (K-SVD) dictionary learning method and shift-invariant strategy, a shift-invariant dictionary is constructed so that the normal modulation components of signals can be completed decomposed. OMPAN algorithm is proposed, which uses the white Gaussian noise to improve the solution method of the orthogonal matching pursuit (OMP) algorithm so that it can separate the modulation components in the signal more accurately. The fault feature extraction is developed via shift-invariant dictionary and OMPAN. A hierarchical classifier is proposed with 3 sub-classifiers so that both the faults in the fault library and the faults outside the fault library are identified. The effectiveness of the proposed hierarchical fault diagnosis method is validated by a planetary gearbox experiment. Result show that the proposed shift-invariant dictionary and OMPAN method has achieved a superior performance in highlighting fault features compared with other two sparse decomposition methods. The proposed hierarchical fault diagnosis approach has achieved a good performance both in classification of the faults in the fault library and identification of the faults outside the fault library.

Fault detection and analysis for wheelset bearings via improved explicit shift-invariant dictionary learning

The wheelset bearing is an indispensable part of the high-speed train, and monitoring its service performance is a concern of many researchers. Effective extraction of those impulse signals induced by the defects on the bearing elements is the key to fault detection and behaviour analysis. However, the presence of considerable noise and irrelevant components brings difficulties to extracting the wheelset bearing fault impulse signals from the measured vibration signals. This paper proposes an improved explicit shift-invariant dictionary learning (IE-SIDL) method to address this issue. Based on the shift-invariant characteristics of the wheelset bearing fault impulse signal in the time-domain, the circulant matrix is used to construct a shift-invariant dictionary and explicitly characterize the fault impulses at any time. To improve the efficiency of dictionary learning, a method of three flips is introduced to realize fast dictionary construction, and the frequency-domain reconstruction property of the circulant matrix is employed to quickly update the dictionary. Besides, an indicator-guided subspace pursuit (SP) method based on the sparsity of envelope spectrum (SES) is adopted for the sparse coding to improve sparse solution accuracy and adaptation. The effectiveness of the IE-SIDL method is proved through the simulated and experimental signals. The results demonstrate that the improved dictionary learning method has an excellent capacity in extracting fault impulse signal of the wheelset bearings, and the good time- and frequency-domain characteristics of the processed signals facilitate fault detection and behaviour analysis.

Sparse coefficient fast solution algorithm based on the circulant structure of a shift-invariant dictionary and its applications for machine fault diagnosis

Sparse coefficient solutions have received continuous attention as one of the core steps of sparse representation. The computational complexity of almost all sparse coefficient solution algorithms is exponentially increased with a signal dimension. In this paper, impact signals caused by localized faults are sparsely represented by a shift-invariant dictionary, which can be Fourier diagonalized based on its circulant structure. Then, the matrix–matrix and matrix–vector multiplications involved in obtaining sparse coefficients are converted into calculation forms that mainly contain the Fourier transform and its inverse to reduce computational complexity. Based on this result, fast solutions of three common sparse coefficient solution algorithms are deduced in detail. Simulation analysis shows that the deduced fast algorithms can significantly fasten running time while maintaining same accuracy as corresponding original algorithms. Localized rolling bearing faults related experiments further verify the effectiveness of the proposed sparse coefficient fast solution algorithms for machine fault diagnosis.

A sparse feature extraction method based on the shiftinvariable dictionary applied in the diagnosis of gearbox with multiple faults

Abstract The vibration signals of gearbox often contain complex components, such as the gear engagement, shafts rotation, bearings, and vibration of the box, and they brings a big challenge to fault diagnosis of gearbox. Furthermore, there are many fault types of gearbox, and the fault characteristics of them are possibly the same. For example, the pass frequency of the missing tooth or root crack in the gear on the input shaft are both equal to the rotation frequency of input shaft. In thus circumstance, traditional diagnosis methods are inapplicable, because we can’t determine which type of faults. In this paper, we proposed a dictionary learning with the shift-invariant dictionary to extract the fault features in a sparse way, and afterwards hidden Markov model (HMM) is utilized to identify the fault type from extractive features. We used the proposed method to diagnose the experimental data from a benchmark two-stage gearbox and there are nine different health conditions introduced to the pinion on the input shaft, and the diagnosis result verifies the good performance of the novel scheme in the fault diagnosis of gearbox.

On learning with shift-invariant structures

In this paper, we describe new results and algorithms, based on circulant matrices, for the task of learning shift-invariant components from training data. We deal with the shift-invariant dictionary learning problem which we formulate using circulant and convolutional matrices (including unions of such matrices), define optimization problems that describe our goals and propose efficient ways to solve them. Based on these findings, we also show how to learn a wavelet-like dictionary from training data. We connect our work with various previous results from the literature and we show the effectiveness of our proposed algorithms using synthetic as well as real ECG signals and images.

Enhanced image approximation using shifted rank-1 reconstruction

Low rank approximation has been extensively studied in the past. It is most suitable to reproduce rectangular like structures in the data. In this work we introduce a generalization using "shifted" rank-$ 1 $ matrices to approximate $ \mathit{\boldsymbol{{A}}}\in \mathbb{C}^{M\times N} $. These matrices are of the form $ S_{\mathit{\boldsymbol{{\lambda}}}}(\mathit{\boldsymbol{{u}}}\mathit{\boldsymbol{{v}}}^*) $ where $ \mathit{\boldsymbol{{u}}}\in \mathbb{C}^M $, $ \mathit{\boldsymbol{{v}}}\in \mathbb{C}^N $ and $ \mathit{\boldsymbol{{\lambda}}}\in \mathbb{Z}^N $. The operator $ S_{\mathit{\boldsymbol{{\lambda}}}} $ circularly shifts the $ k $-th column of $ \mathit{\boldsymbol{{u}}}\mathit{\boldsymbol{{v}}}^* $ by $ \lambda_k $.These kind of shifts naturally appear in applications, where an object $ \mathit{\boldsymbol{{u}}} $ is observed in $ N $ measurements at different positions indicated by the shift $ \mathit{\boldsymbol{{\lambda}}} $. The vector $ \mathit{\boldsymbol{{v}}} $ gives the observation intensity. This model holds for seismic waves that are recorded at $ N $ sensors at different times $ \mathit{\boldsymbol{{\lambda}}} $. Other examples are a car that moves through a video changing its position $ \mathit{\boldsymbol{{\lambda}}} $ in each of the $ N $ frames, or non-destructive testing based on ultrasonic waves that are reflected by defects inside the material.The main difficulty of the above stated problem lies in finding a suitable shift vector $ \mathit{\boldsymbol{{\lambda}}} $. Once the shift is known, a simple singular value decomposition can be applied to reconstruct $ \mathit{\boldsymbol{{u}}} $ and $ \mathit{\boldsymbol{{v}}} $. We propose a greedy method to reconstruct $ \mathit{\boldsymbol{{\lambda}}} $. By using the formulation of the problem in Fourier domain, a shifted rank-$ 1 $ approximation can be calculated in $ O(NM\log M) $. Convergence to a locally optimal solution is guaranteed. Furthermore, we give a heuristic initial guess strategy that shows good results in the numerical experiments.We validate our approach in several numerical experiments on different kinds of data. We compare the technique to shift-invariant dictionary learning algorithms. Furthermore, we provide examples from application including object segmentation in non-destructive testing and seismic exploration as well as object tracking in video processing.

Shift Invariant Dictionary Learning for Human Action Recognition

Sparse representation is an emerging topic among researchers. The method to represent the huge volume of dense data as sparse data is much needed for various fields such as classification, compression and signal denoising. The base of the sparse representation is dictionary learning. In most of the dictionary learning approaches, the dictionary is learnt based on the input training signals which consumes more time. To solve this issue, the shift-invariant dictionary is used for action recognition in this work. Shift-Invariant Dictionary (SID) is that the dictionary is constructed in the initial stage with shift-invariance of initial atoms. The advantage of the proposed SID based action recognition method is that it requires minimum training time and achieves highest accuracy.

Slice-Based Online Convolutional Dictionary Learning

Convolutional dictionary learning (CDL) aims to learn a structured and shift-invariant dictionary to decompose signals into sparse representations. While yielding superior results compared to traditional sparse coding methods on various signal and image processing tasks, most CDL methods have difficulties handling large data, because they have to process all images in the dataset in a single pass. Therefore, recent research has focused on online CDL (OCDL) which updates the dictionary with sequentially incoming signals. In this article, a novel OCDL algorithm is proposed based on a local, slice-based representation of sparse codes. Such representation has been found useful in batch CDL problems, where the convolutional sparse coding and dictionary learning problem could be handled in a local way similar to traditional sparse coding problems, but it has never been explored under online scenarios before. We show, in this article, that the proposed algorithm is a natural extension of the traditional patch-based online dictionary learning algorithm, and the dictionary is updated in a similar memory efficient way too. On the other hand, it can be viewed as an improvement of existing second-order OCDL algorithms. Theoretical analysis shows that our algorithm converges and has lower time complexity than existing counterpart that yields exactly the same output. Extensive experiments are performed on various benchmarking datasets, which show that our algorithm outperforms state-of-the-art batch and OCDL algorithms in terms of reconstruction objectives.

Impulsive component extraction using shift-invariant dictionary learning and its application to gear-box bearing early fault diagnosis

The impulsive components induced by bearing faults are key features for assessing gear-box bearing faults. However, because of heavy background noise and the interferences of other vibrations, it is difficult to extract these impulsive components caused by faults, particularly early faults, from the measured vibration signals. To capture the high-level structure of impulsive components embedded in measured vibration signals, a dictionary learning method called shift-invariant K-means singular value decomposition (SI-K-SVD) dictionary learning is used to detect the early faults of gear-box bearings. Although SI-K-SVD is more flexible and adaptable than existing methods, the improper selection of two SI-K-SVD-related parameters, namely, the number of iterations and the pattern lengths, has an adverse influence on fault detection performance. Therefore, the sparsity of the envelope spectrum (SES) and the kurtosis of the envelope spectrum (KES) are used to select these two key parameters, respectively. SI-K-SVD with the two selected optimal parameter values, referred to as optimal parameter SI-K-SVD (OP-SI-K-SVD), is proposed to detect gear-box bearing faults. The proposed method is verified by both simulations and an experiment. Compared to the state-of-the-art methods, namely, empirical model decomposition, wavelet transform and K-SVD, OP-SI-K-SVD has better performance in diagnosing the early faults of a gear-box bearing.

Impulse detection using a shift-invariant dictionary and multiple compressions

Impulse components in vibration signals are regarded as a symptom of localized damage in rotary machines. When influenced by excessive noise, it is difficult to detect these impulses, especially from under-sampled measurements. In this paper, a method called Multi-Compressive Matching Pursuit (MCMP) is developed for impulse component detection. A shift-invariant dictionary is constructed to sparsely represent the impulse components caused by localized faults. Impulse features are enhanced by integrating multi-channel compressive samples using the sum-proxy strategy. The impulse components are then sparsely reconstructed from noisy compressive samples. The performance of the proposed method is compared with the Compressive Sampling Matching Pursuit (CoSaMP) method using numerical simulations. The advantage of the proposed method is further validated through detecting localized faults on gears and roller bearings. The simulation and experimental results also prove that the proposed method can effectively reconstruct the impulse components under slowly time-varying conditions.

Detection of particle contaminants in rolling element bearings with unsupervised acoustic emission feature learning

The detection of contaminants in the lubricant of rolling element bearings using acoustic emission signals is a challenging problem, in particular at high rotational speeds. This problem calls for new analysis methods beyond the conventional amplitude- and frequency-based methods. Feature learning is successfully used in the machine learning field to characterize complex signals. Here we use an unsupervised feature learning approach to distinguish acoustic emission signals. We investigate the repetition rates of features identified with shift-invariant dictionary learning and find that the signature of contaminated lubricant is significantly stronger than the effect on conventional condition indicators like the RMS and the enveloped RMS at rotational speeds above 300 rpm and up to 3000 rpm.

Fault detection of a wheelset bearing in a high-speed train using the shock-response convolutional sparse-coding technique

The wheelset bearing is an essential mechanical component of a high-speed train, and its fault detection is of great importance for ensuring the safety of high-speed train service. However, the strong nonlinear and non-stationary modulation characteristics of the impulses induced by bearing faults, noises and interferences make fault detection of the wheelset bearing difficult and complex. The impulses induced by bearing faults have convolutional sparse characteristics, and can be described as the convolution of shock responses and time location coefficients. This paper presents a new fault detection method based on the shock-response convolutional sparse-coding technique (SRCSCT) to resolve the two inverse problems related to convolution characteristics. One of the problems is that shock responses are learned directly from vibration signals via the shift-invariant dictionary learning (SIDL) of SRCSCT; the other is that the time location coefficients are inferred from vibration signals via shift-invariant sparse coding (SISC) of SRCSCT. Shock responses can reflect the high-level structures of impulses embedded in vibration signals. Time location coefficients can be used to determine the time of impulse occurrence. The convolutions of learned shock responses and inferred corresponding time location coefficients are used to extract impulses submerged in vibration signals and are convenient for post-processing methods, such as Hilbert envelope analysis. Both the simulated signals and real vibration signals collected from a wheelset bearing test bench are used to verify the proposed method. The results show that SRCSCT not only effectively detects faults of the wheelset bearing but also accurately characterizes the fault dynamic behaviours of the bearing defects.

Scalable Online Convolutional Sparse Coding.

Convolutional sparse coding (CSC) improves sparse coding by learning a shift-invariant dictionary from the data. However, most existing CSC algorithms operate in the batch mode and are computationally expensive. In this paper, we alleviate this problem by online learning. The key is a reformulation of the CSC objective so that convolution can be handled easily in the frequency domain, and much smaller history matrices are needed. To solve the resultant optimization problem, we use the alternating direction method of multipliers (ADMMs), and its subproblems have efficient closed-form solutions. Theoretical analysis shows that the learned dictionary converges to a stationary point of the optimization problem. Extensive experiments are performed on both the standard CSC benchmark data sets and much larger data sets such as the ImageNet. Results show that the proposed algorithm outperforms the state-of-the-art batch and online CSC methods. It is more scalable, has faster convergence, and better reconstruction performance.

Imposing shift-invariance using Flexible Structure Dictionary Learning (FSDL)

Sparse representation and Dictionary learning have attracted a lot of research attention in the last couple of decades and have provided state of the art results in many fields such as denoising, classification, inpainting and compression. However, applying general dictionary learning such as Method of Optimal Directions and Recursive Least Squares Dictionary Learning Algorithm can be computationally expensive, due to the large amount of free variables to be learned. Also sometimes the signal class has obvious repetitive structure which could benefit from a structured dictionary. One way to deal with these shortcomings is to impose a structure on the dictionary itself, for example the dictionary can be sparse or the atoms can be shift-invariant. In practice, imposing a structure means limiting the number of free variables. There are many examples of structured dictionaries such as double sparsity model or shift-invariant dictionaries. We have recently proposed a closed form solution to impose arbitrary structures onto a dictionary, called Flexible Structure Dictionary Learning. In this paper, we use this method to impose shift-invariant structure when training a dictionary. This structure allows us to not only simplify the original solution and make it computationally feasible to be used for large signals but also extend the concept of shift-invariance to include variable sized shifts in different atoms. The proposed dictionary update step finds all the free variables in all the atoms jointly, whereas some shift-invariant structured dictionaries in the recent literature, update one atom at a time. We have compared our proposed method with a general dictionary learning method and another shift-invariant method. Results show that signal approximation can be a promising application.

Fault Diagnosis for a Wind Turbine Generator Bearing via Sparse Representation and Shift-Invariant K-SVD

It is always a primary challenge in fault diagnosis of a wind turbine generator to extract fault character information under strong noise and nonstationary condition. As a novel signal processing method, sparse representation shows excellent performance in time–frequency analysis and feature extraction. However, its result is directly influenced by dictionary, whose atoms should be as similar with signal's inner structure as possible. Due to the variability of operation environment and physical structure in industrial systems, the patterns of impulse signals are changing over time, which makes creating a proper dictionary even harder. To solve the problem, a novel data-driven fault diagnosis method based on sparse representation and shift-invariant dictionary learning is proposed. The impulse signals at different locations with the same characteristic can be represented by only one atom through shift operation. Then, the shift-invariant dictionary is generated by taking all the possible shifts of a few short atoms and, consequently, is more applicable to represent long signals that in the same pattern appear periodically. Based on the learnt shift-invariant dictionary, the coefficients obtained can be sparser, with the extracted impulse signal being closer to the real signal. Finally, the time–frequency representation of the impulse component is obtained with consideration of both the Wigner–Ville distribution of every atom and the corresponding sparse coefficient. The excellent performance of different fault diagnoses in a fault simulator and a wind turbine proves the effectiveness and robustness of the proposed method. Meanwhile, the comparison with the state-of-the-art method is illustrated, which highlights the superiority of the proposed method.

Complex signal analysis for planetary gearbox fault diagnosis via shift invariant dictionary learning

Planetary gearbox vibration signals are intricate due to the unique structure, complex kinetics and background noise interference. Moreover, they have time shift invariant feature because of the periodical rotation of the mechanical components. Reliable feature extraction from such signals is the key to success in planetary gearbox fault diagnosis. The recently proposed shift invariant K-means singular value decomposition (SI-K-SVD) dictionary learning method offers a good approach to analyze such signals. This method requires neither a priori knowledge on signals to construct any analytic dictionary, nor a large number of sample signals to form a training dictionary. It is therefore data-driven in nature, and highly flexible and adaptive to represent signals. In this paper, such properties of the SI-K-SVD are exploited to extract the latent constituent components of complex signals and use them to represent signals sparsely, thus suppressing background noise and revealing the true vibration patterns. To apply the SI-K-SVD method to planetary gearbox signals, the criteria for determining the key parameters used in the SI-K-SVD, i.e., the length of pattern and number of decompositions, are suggested. The method has been illustrated by analyzing numerically simulated signals. It has been favorably compared with the frequently used matching pursuit (MP) and K-means singular value decomposition (K-SVD) methods. Its effectiveness in real planetary gearbox fault diagnosis has been validated by analysis of lab experimental signals of a planetary gearbox. The results show that both localized and distributed gear faults can be diagnosed successfully.

Detection and diagnosis of bearing faults using shift-invariant dictionary learning and hidden Markov model

Many existing signal processing methods usually select a predefined basis function in advance. This basis functions selection relies on a priori knowledge about the target signal, which is always infeasible in engineering applications. Dictionary learning method provides an ambitious direction to learn basis atoms from data itself with the objective of finding the underlying structure embedded in signal. As a special case of dictionary learning methods, shift-invariant dictionary learning (SIDL) reconstructs an input signal using basis atoms in all possible time shifts. The property of shift-invariance is very suitable to extract periodic impulses, which are typical symptom of mechanical fault signal. After learning basis atoms, a signal can be decomposed into a collection of latent components, each is reconstructed by one basis atom and its corresponding time-shifts. In this paper, SIDL method is introduced as an adaptive feature extraction technique. Then an effective approach based on SIDL and hidden Markov model (HMM) is addressed for machinery fault diagnosis. The SIDL-based feature extraction is applied to analyze both simulated and experiment signal with specific notch size. This experiment shows that SIDL can successfully extract double impulses in bearing signal. The second experiment presents an artificial fault experiment with different bearing fault type. Feature extraction based on SIDL method is performed on each signal, and then HMM is used to identify its fault type. This experiment results show that the proposed SIDL-HMM has a good performance in bearing fault diagnosis.

Learning Recurrent Waveforms Within EEGs.

We demonstrate an algorithm to automatically learn the time-limited waveforms associated with phasic events that repeatedly appear throughout an electroencephalogram. To learn the phasic event waveforms we propose a multiscale modeling process that is based on existing shift-invariant dictionary learning algorithms. For each channel, waveforms at different temporal scales are learned based on the assumption that only a few waveforms occur in any window of the time-series, but the same waveforms reoccur throughout the signal. Once the waveforms are learned, the timing and amplitude of the phasic event occurrences are estimated using matching pursuit. To summarize the waveforms learned across multiple channels and subjects, we analyze their frequency content, their similarity to Gabor-Morlet wavelets, and perform shift-invariant k-means to cluster the waveforms. A prototype waveform from each cluster is then tested for differential spatial patterns between different motor imagery conditions. On multiple human EEG datasets, the learned waveforms capture key characteristics of signals they were trained to represent, with a consistency in waveform morphology and frequency content across multiple training sections and initializations. On multichannel datasets, the spatial amplitude patterns of the waveforms are also consistent and can be used to distinguish different modalities of motor imagery. We explored a methodology that can be used for modeling the recurrent waveforms in EEG traces. The methodology automatically identifies the most frequent phasic event waveforms in EEG, which could then be used as features for automatic evaluation and comparison of EEG during sleep, pathology, or mentally engaging tasks.

Fast orthogonal sparse approximation algorithms over local dictionaries

In this work we present a new greedy algorithm for sparse approximation called LocOMP. LocOMP is meant to be run on local dictionaries made of atoms with much shorter supports than the signal length. This notably encompasses shift-invariant dictionaries and time–frequency dictionaries, be they monoscale or multiscale. In this case, very fast implementations of Matching Pursuit are already available. LocOMP is almost as fast as Matching Pursuit while approaching the signal almost as well as the much slower Orthogonal Matching Pursuit.