An air compressor operation data acquisition method and system based on the Internet of Things

Abstract

This invention relates to the field of electrical data processing technology, and more specifically, to a method and system for acquiring air compressor operation data based on the Internet of Things (IoT). The method includes: acquiring and preprocessing air compressor operation data; selecting any time point in any dimension as the target time point, and obtaining the sequence of operation data at the target time point within the nearest time window; sorting the instantaneous energy at each time point in the sequence in ascending order to obtain an instantaneous energy subsequence. This invention adaptively adjusts the standard deviation parameter in the Gaussian filtering algorithm based on global fault confidence, giving it strong smoothing capabilities for random noise data, enhancing denoising capabilities, retaining more detailed information in fault feature data, avoiding over-smoothing, and ultimately obtaining data that maximizes the elimination of noise interference while preserving true fault characteristics, providing high-quality data for subsequent high-precision condition monitoring and maintenance.

Description

Technical Field

[0001] This invention relates to the field of electrical data processing technology. More specifically, this invention relates to a method and system for acquiring operational data of an air compressor based on the Internet of Things (IoT). Background Technology

[0002] Air compressors, as key industrial equipment that converts primary energy sources such as electricity into potential energy of air, are an indispensable power source in modern industrial systems and are widely used in many fields such as machinery manufacturing, petrochemicals, textiles, and medicine. Because air compressors are high-energy-consuming devices, and their operational stability directly affects the efficiency and safety of the entire production line, real-time and accurate collection and monitoring of key operational data across multiple dimensions, such as outlet pressure, compressor temperature, motor current, and vibration, is a crucial prerequisite for achieving equipment condition monitoring, fault early warning, predictive maintenance, and energy efficiency optimization.

[0003] Before analyzing the collected raw data, denoising is usually required. Gaussian filtering is a commonly used method, which smooths noise by weighted averaging of the signal. Its key parameter is the standard deviation. However, in the complex operating environment of air compressors, the traditional fixed standard deviation Gaussian filtering method has significant drawbacks. If the standard deviation is set too small, its ability to suppress strong background noise is limited, resulting in poor denoising. If the standard deviation is set too large, it is easy to filter out the impact characteristics and periodic characteristics that characterize early faults as noise, leading to the loss of key fault information and signal distortion. Because the operating conditions of air compressors are variable, the characteristics of fault signals and noise also change dynamically. Using fixed filtering parameters makes it difficult to simultaneously meet the dual requirements of effective denoising and preservation of fault details. Summary of the Invention

[0004] This invention provides an Internet of Things-based method and system for acquiring air compressor operation data, aiming to solve the problem in related technologies that, due to the variable operating conditions of air compressors, the characteristics of fault signals and noise also change dynamically, making it difficult to simultaneously meet the dual requirements of effective noise reduction and preservation of fault details using fixed filtering parameters.

[0005] In a first aspect, the present invention provides an Internet of Things (IoT)-based method for acquiring air compressor operation data, comprising: acquiring air compressor operation data and preprocessing it; selecting any moment in any dimension as the target moment, and acquiring the sequence of operation data at the target moment within the nearest neighbor time window; sorting the instantaneous energy at each moment in the sequence in ascending order to obtain an instantaneous energy subsequence, and calculating the energy concentration index at the target moment, wherein the energy concentration index reflects the degree of energy concentration and uneven distribution within the time window in the instantaneous energy subsequence; performing envelope demodulation and cepstral analysis on the energy concentration index sequence of the nearest neighbor time window, based on the amplitude of the real cepstral sequence. The value distribution determines the dynamic threshold, and the ratio of amplitudes greater than the dynamic threshold to those not greater than the dynamic threshold is used as the cepstral harmonic factor at the target time. The entropy weighting method is applied to the sequence composed of the energy concentration index and the cepstral harmonic factor within the nearest neighbor time window to obtain the weights of the energy concentration index and the cepstral harmonic factor. A weighted sum is performed based on the energy concentration index, the cepstral harmonic factor, and their corresponding weights to obtain the global fault confidence at the target time. Based on the global fault confidence at the target time, the standard deviation of the Gaussian filtering algorithm is determined, and this standard deviation is used to denoise the running data. The standard deviation is negatively correlated with the global fault confidence. This method can adaptively adjust the standard deviation of the Gaussian filtering algorithm: when the fault confidence is high (a fault may exist), a smaller standard deviation is used to accurately preserve the characteristics of impact and periodic faults; when the confidence is low (mainly random noise), a larger standard deviation is used to effectively smooth the noise. This dynamic adjustment mechanism overcomes the limitations of fixed parameter filtering, achieving efficient noise reduction of air compressor operating data under complex and variable operating conditions, while preserving key fault information to the maximum extent, significantly improving data quality and the accuracy of subsequent fault diagnosis.

[0006] Furthermore, the energy concentration index at the target time is calculated by: multiplying each element in the instantaneous energy subsequence by a linear weight related to its ranking position in the sequence to obtain a weighted instantaneous energy subsequence; summing the weighted instantaneous energy subsequences to obtain a first sum; summing the instantaneous energy subsequences again to obtain a second sum; and determining the energy concentration index based on the ratio of the first sum to the second sum. The first sum reflects the difference in energy distribution across ranking positions, indicating uneven distribution. By applying linear weights related to ranking positions to the ranked instantaneous energy sequence, data points with larger energy values ​​(typically corresponding to fault impacts) have a higher weight in the index calculation. This method can more sensitively and accurately quantify the unevenness and concentration of energy distribution in the data, thereby improving the ability to identify early, transient fault impact characteristics.

[0007] Furthermore, the method for calculating the cepstral harmonic factor includes: designating the set of cepstral amplitudes greater than a dynamic threshold in the real cepstral sequence as the over-threshold set, and the set of the remaining cepstral amplitudes as the non-over-threshold set; calculating the ratio of the average cepstral amplitude in the over-threshold set to the average cepstral amplitude in the non-over-threshold set to obtain the cepstral harmonic factor. By comparing harmonic energy with background noise energy, the periodic characteristics of the fault signal can be quantified more stably and reliably, enhancing the detection effect on faults in the development stage.

[0008] Furthermore, the standard deviation of the Gaussian filtering algorithm is determined, including: the standard deviation is negatively correlated with the reciprocal of the sum of the global fault confidence and 1, and positively correlated with the minimum standard deviation in the Gaussian filtering denoising algorithm. This method ensures that when the fault confidence changes from low to high, the standard deviation can smoothly and effectively transition from the maximum value to the minimum value, achieving precise control of the filtering intensity, thereby guaranteeing the stability and effectiveness of adaptive denoising.

[0009] Furthermore, the method for obtaining the instantaneous energy at each moment in the sequence includes: performing a square operation on each data point in the sequence to obtain the instantaneous energy at each moment.

[0010] Furthermore, the standard deviation is used to denoise the operating data, including: taking the pre-processed operating data of the air compressor as input to the Gaussian filtering algorithm, determining the standard deviation of the Gaussian filtering algorithm, and outputting the denoised operating data of the air compressor.

[0011] Furthermore, the operating data of the air compressor is collected and preprocessed, including normalizing the operating data and filling in missing values. By normalizing multi-source sensor data with different dimensions, the influence of data scale differences is eliminated, ensuring the fairness and accuracy of subsequent calculations.

[0012] Furthermore, envelope demodulation is performed on the energy concentration index sequence of the nearest neighbor time window, including: performing a Hilbert transform on the sequence of energy concentration indices within the nearest neighbor time window to obtain the envelope sequence.

[0013] Furthermore, determining the dynamic threshold based on the amplitude distribution of the real cepstral sequence includes: determining the dynamic threshold of the cepstral amplitude in the real cepstral sequence according to the Laida criterion.

[0014] In a second aspect, the present invention also provides an Internet of Things (IoT)-based air compressor operation data acquisition system, comprising a processor and a memory, wherein the memory stores a computer program, and the processor executes the computer program to implement the IoT-based air compressor operation data acquisition method described above.

[0015] Beneficial effects: By adaptively adjusting the standard deviation parameter in the Gaussian filtering algorithm based on global fault confidence, it has a strong smoothing ability for random noise data, enhances the denoising ability, retains more detailed information of fault feature data, avoids over-smoothing, and finally obtains data that can eliminate noise interference to the maximum extent while retaining the true fault characteristics, providing high-quality data for subsequent high-precision condition monitoring and maintenance. Attached Figure Description

[0016] Figure 1 This is a flowchart schematically illustrating the standard deviation in an adaptive adjustment Gaussian filtering algorithm according to an embodiment of the present invention. Detailed Implementation

[0017] The specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0018] like Figure 1 As shown, S101: Collects operating data of the air compressor.

[0019] To construct a comprehensive data acquisition network for the air compressor, multiple sensors need to be deployed in key locations. The specific deployment plan is as follows: a high-precision pressure sensor is installed at the compressor's exhaust port to collect exhaust pressure data; a piezoelectric accelerometer is installed near the compressor's bearing housing to obtain acceleration data, reflecting vibration characteristics; Hall effect sensors or current transformers are used on the motor power supply line to collect three-phase current and voltage data; thermocouples are placed on the lubrication oil circuit and motor housing surface to measure temperature data. The data acquisition interval is set to 1 second, which can be adjusted by the implementer according to actual needs.

[0020] After collecting the raw data, preprocessing is required to ensure the continuity and integrity of the data sequence, providing a high-quality data foundation for subsequent noise feature analysis and adaptive denoising. The preprocessing steps include: standardizing data from different sensors with different dimensions and value ranges using the max-min normalization method; and imputing missing values ​​in the data using linear interpolation. Since max-min normalization and linear interpolation are well-known techniques, they will not be elaborated upon here. This yields the preprocessed operating data of the air compressor.

[0021] S102: Analyze the energy difference between mechanical shock and background noise during the operation of an air compressor, and construct an energy concentration index.

[0022] When early failures such as localized spalling or pitting occur in rotating components inside an air compressor, such as bearing rolling elements or gear teeth, each rotation of the component passing over the defect location generates a brief but severe mechanical impact. This impact manifests as non-Gaussian impact pulse noise in the vibration signal, meaning a large amount of energy is released in a very short time, resulting in a significant imbalance in energy distribution. Therefore, this impact characteristic causes the signal energy to no longer fluctuate uniformly or randomly according to a Gaussian distribution, but rather to be highly concentrated at a few time points, significantly increasing the sparsity of the signal energy. Based on the above, an energy concentration index can be constructed at any given time to reflect the extreme imbalance in energy distribution.

[0023] The specific construction process includes: selecting any time as the target time; selecting a time window consisting of the target time and its nearest neighbor times, denoted as the nearest neighbor time window; the number of times contained in the nearest neighbor time window is N, which is taken as 60 in this embodiment, but can be selected according to the situation; denoting the sequence of data in the i-th dimension within the nearest neighbor time window as the window subsequence; squaring each data point in the window subsequence to obtain the instantaneous energy at each time; sorting all instantaneous energies in the window subsequence in ascending order; and denoting the sequence formed by the sorted results as the instantaneous energy subsequence.

[0024] For example, taking time t as the target time, the energy concentration index at time t is calculated as follows: .in This represents the energy concentration index of the air compressor in the i-th dimension at the t-th operating time, where N represents the number of times contained within the nearest neighbor time window. This represents the value of the k-th element in the instantaneous energy subsequence corresponding to the t-th running time in the i-th dimension. This represents the value of the j-th element in the instantaneous energy subsequence corresponding to the t-th running time in the i-th dimension. Indicates linear weights, This represents a hyperparameter, used to avoid the denominator being zero, which would prevent calculation. In this embodiment, The value is 0.1, which can be adjusted as needed. The denominator is the theoretical maximum value of the numerator, used for normalization.

[0025] The principle behind the above formula is as follows: When the air compressor is in a stable operating state, the main factor causing data deviation is noise. At this time, the energy values ​​of all data points are relatively small and close. After ascending sorting, the instantaneous energy subsequence should have a small slope and an approximately linear growth trend. Therefore, the calculated energy concentration index will be close to 0. When absolutely stable, the calculated energy concentration index is 0, that is, the operating data of the air compressor remains constant. When a severe mechanical shock occurs, it causes a huge deviation in the data. At this time, the energy of all data points is mainly concentrated at the moment of severe mechanical shock. Therefore, after ascending sorting, the moment with stronger energy has a larger weight, making the numerator approach the theoretical maximum value. Therefore, the calculated energy concentration index will be close to 1.

[0026] S103: Calculate the cepstral harmonic factor at any given time.

[0027] The energy concentration index obtained through the above steps can identify high-energy, transient impact pulses and random noise. For complex rotating machinery such as air compressors driven by electric motors, the movement of their key internal components is highly deterministic and periodic, such as the support bearings of the main screw, the gears in the speed-increasing gearbox, and even the crank-connecting rod mechanism of a piston compressor. When a defect occurs at a certain location of these components, such as pitting on the bearing raceway, cracks on the gear surface, or uneven wear of the piston rings, each relative movement passing through the defect location will produce a physical impact or pressure surge, thereby generating an impact pulse and forming a periodic phenomenon with amplitude modulation.

[0028] Based on the above analysis, this embodiment constructs a cepstral harmonic factor based on the energy concentration index, reflecting the degree of periodic variation of extreme energy in the air compressor operating data. The specific construction process is as follows: After calculating the energy concentration index at each time point in each dimension, the sequence of energy concentration indices in the i-th dimension within the nearest neighbor time window is denoted as the nearest neighbor energy concentration index sequence. A Hilbert transform is applied to the nearest neighbor energy concentration index sequence to demodulate its envelope sequence. A fast Fourier transform is performed on the obtained envelope sequence to obtain its envelope spectrum. An inverse Fourier transform is performed on the logarithmic magnitude of the envelope spectrum to obtain the real cepstral sequence. The dynamic threshold of the cepstral magnitude in the real cepstral sequence is determined according to the Laida criterion. The set of cepstral magnitudes in the real cepstral sequence that are greater than the dynamic threshold is denoted as the overthreshold set, and the set of all elements in the real cepstral sequence other than those in the overthreshold set is denoted as the non-overthreshold set. The Hilbert transform, fast Fourier transform, inverse Fourier transform, and Laida criterion are well-known techniques and will not be described in detail here.

[0029] Following on the above, the method for calculating the cepstral harmonic factor is as follows: In the formula, The cepstral harmonic factor of the air compressor in the i-th dimension at the t-th operating time is represented by mean(), which is the average value function. Represents the set of superthresholds. This represents the non-threshold set, and 1 represents the hyperparameter, used to avoid the denominator being zero and thus unable to perform calculations. It should be noted that when the set inside the mean() function is empty, the function value is always 0.

[0030] The above formula is based on the following principle: If a component of the air compressor malfunctions, it will cause periodic amplitude modulation in the collected data. Due to the extremely transient nature of the mechanical impact process, its waveform will exhibit sharp pulse characteristics rather than a smooth sine wave. This results in a significantly larger calculated energy concentration index compared to the non-sharp pulse scenario. According to Fourier analysis, such sharp pulses will generate multiple harmonic components in the frequency domain, excluding the fundamental frequency. The energy intensity of these harmonic components is positively correlated with the severity of the mechanical impact. Therefore, if the elements in the overthreshold set are significantly larger than those in the non-overthreshold set, it indicates a higher likelihood of a defect in a component of the air compressor causing periodic amplitude modulation in the collected data, leading to a larger calculated cepstral harmonic factor.

[0031] S104: Calculate the global fault confidence of the air compressor.

[0032] The energy concentration index and cepstral harmonic factor obtained through the above steps can reflect the fault characteristics of an air compressor from the extreme uneven distribution of energy and the degree of periodic variation of extreme energy, respectively. As a complex electromechanical coupling system, the evolution of faults in an air compressor is not a single mode, but rather has composite structural characteristics. For example, when a rotating component fails, in the early stage of the fault, the data characteristics may mainly manifest as abrupt changes in energy distribution, that is, only isolated, instantaneous strong impacts. At this time, the change in the energy concentration index is relatively significant, but because the impact is not yet stable, the periodic characteristics may be very weak. As the fault develops to the middle stage, the impact becomes regular and the energy increases, and the data characteristics evolve into a fault mode in which the impact intensity and periodic structure coexist.

[0033] Based on the above analysis, this embodiment constructs a global fault confidence score based on the energy concentration index and cepstral harmonic factor to comprehensively measure the fault significance level during the operation of the air compressor. The specific construction process is as follows: the sequence composed of the energy concentration index and cepstral harmonic factor in the i-th dimension within the nearest time window is used as the input of the entropy weight method, and the output is the weight of the energy concentration index and the cepstral harmonic factor. The entropy weight method is a well-known technique and will not be described in detail here.

[0034] Following on the above, the method for calculating the global fault confidence is as follows: In the formula, This represents the global fault confidence level in the i-th dimension at time t during the operation of the air compressor. , Let represent the energy concentration index weight and cepstral harmonic factor weight at time t during the operation of the air compressor, obtained by the entropy weight method. , These represent the energy concentration index and cepstral harmonic factor at time t after normalization during the operation of the air compressor.

[0035] The above formula is based on the following principle: When an air compressor malfunctions and causes changes in operating data, the energy concentration index is more sensitive to changes in the early stages of the malfunction. As the malfunction progresses, the periodicity of the malfunction becomes more pronounced, and the sensitivity of the cepstral harmonic factor gradually increases, resulting in a higher confidence level for the final global malfunction. If the operating data only exhibits random noise interference and does not have the periodicity of the malfunction mode, the energy concentration index and the cepstral harmonic factor become insensitive, resulting in a lower confidence level for the final global malfunction.

[0036] S105: Determine the optimal standard deviation in the Gaussian filtering algorithm based on the global fault confidence, and perform noise reduction processing on the air compressor's operating data.

[0037] The global fault confidence score obtained through the above steps can reflect the significance of fault characteristics during the operation of the air compressor. Therefore, in this embodiment, the standard deviation in the Gaussian filtering algorithm is adaptively determined based on the global fault confidence score, and its calculation formula is as follows: In the formula, This represents the standard deviation when using a Gaussian filtering algorithm to denoise the data of the air compressor at time t in the i-th dimension, which is also the optimal standard deviation. and Let represent the minimum and maximum standard deviations in the Gaussian filtering denoising algorithm, respectively. This represents the global fault confidence of the air compressor in the i-th dimension at the t-th operating time.

[0038] When the global fault confidence level is high, it indicates that there is structured fault feature data that needs to be retained. In this case, a smaller standard deviation is selected to avoid the fault feature data being over-smoothed. When the global fault confidence level is low, it indicates random noise. In this case, a larger standard deviation is selected to perform stronger smoothing on the noisy data.

[0039] Finally, the preprocessed air compressor operating data collected in step S101 is used as input to the Gaussian filtering algorithm. Following the steps described above, the optimal standard deviation is adaptively determined, and the output is the denoised air compressor operating data—that is, complete, high-quality denoised data. This denoised data eliminates random Gaussian noise while retaining the impact pulses and periodic modulation characteristics closely related to equipment fault conditions, facilitating accurate subsequent analysis. The Gaussian filtering denoising process is a well-known technique and will not be elaborated upon here.

[0040] The present invention also provides an Internet of Things (IoT)-based air compressor operation data acquisition system. The system includes a processor and a memory, the memory storing computer program instructions. When the processor executes the computer program instructions, it implements the IoT-based air compressor operation data acquisition method according to the first aspect of the present invention.

[0041] The system also includes other components well known to those skilled in the art, such as communication buses and communication interfaces, the settings and functions of which are known in the art and therefore will not be described in detail here.

[0042] In this invention, the aforementioned memory can be any tangible medium containing or storing a program that can be used or combined with an instruction execution system, apparatus, or device. For example, a computer-readable storage medium can be any suitable magnetic or magneto-optical storage medium, such as Resistive Random Access Memory (RRAM), Dynamic Random Access Memory (DRAM), Static Random Access Memory (SRAM), Enhanced Dynamic Random Access Memory (EDRAM), High-Bandwidth Memory (HBM), Hybrid Memory Cube (HMC), etc., or any other medium that can be used to store desired information and can be accessed by an application, module, or both. Any such computer storage medium can be part of a device or accessible to or connected to a device. Any application or module described in this invention can be implemented using computer-readable / executable instructions stored or otherwise maintained on such a computer-readable medium.

[0043] The above-described embodiments are merely illustrative of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the patent for these embodiments. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

Claims

1. An air compressor operation data acquisition method based on the Internet of Things, characterized in that, include: Collect and preprocess the operating data of the air compressor; Select any moment in any dimension as the target moment, and obtain the sequence of the running data of the target moment within the nearest time window; sort the instantaneous energy of each moment in the sequence in ascending order to obtain the instantaneous energy subsequence, and calculate the energy concentration index of the target moment. The energy concentration index reflects the degree of energy concentration and the uneven distribution of energy in the instantaneous energy subsequence within the time window. The energy concentration index at the target time is calculated, including: Each element in the instantaneous energy subsequence is multiplied by a linear weight related to its sorting position in the sequence to obtain a weighted instantaneous energy subsequence; The weighted instantaneous energy subsequences are summed to obtain a first sum, and the instantaneous energy subsequences are summed to obtain a second sum. The energy concentration index is determined based on the ratio of the first sum to the second sum. The first sum reflects the difference in energy in the sequence with respect to the sorting position, reflecting the uneven distribution. Envelope demodulation and cepstral analysis are performed on the energy concentration index sequence of the nearest time window. A dynamic threshold is determined based on the amplitude distribution of the real cepstral sequence, and the ratio of amplitudes greater than the dynamic threshold to those not greater than the dynamic threshold is used as the cepstral harmonic factor at the target time. The calculation method for the cepstral harmonic factor includes: The set of cepstral amplitudes greater than the dynamic threshold in the real cepstral sequence is denoted as the overthreshold set, and the set of the remaining cepstral amplitudes is denoted as the non-overthreshold set. The cepstral harmonic factor is obtained by calculating the ratio of the average cepstral amplitude in the overthreshold set to the average cepstral amplitude in the non-overthreshold set. The entropy weighting method is applied to the sequence composed of the energy concentration index and cepstral harmonic factor within the nearest neighbor time window to obtain the weights of the energy concentration index and the cepstral harmonic factor; a weighted sum is then performed based on the energy concentration index, the cepstral harmonic factor, and their corresponding weights to obtain the global fault confidence at the target time, satisfying: ； This represents the global fault confidence level in the i-th dimension at time t during the operation of the air compressor. , Let represent the energy concentration index weight and cepstral harmonic factor weight at time t during the operation of the air compressor, obtained by the entropy weight method. , These represent the energy concentration index and cepstral harmonic factor at time t after normalization during the operation of the air compressor, respectively. Based on the global fault confidence at the target time, the standard deviation of the Gaussian filtering algorithm is determined, and this standard deviation is used to denoise the operating data. The standard deviation is negatively correlated with the global fault confidence. Determining the standard deviation of the Gaussian filtering algorithm includes: The standard deviation and global fault confidence are negatively correlated with the reciprocal of the sum of 1 and positively correlated with the minimum standard deviation in the Gaussian filtering denoising algorithm.

2. The method for collecting air compressor operation data based on the Internet of Things according to claim 1, characterized in that, The methods for obtaining the instantaneous energy at each moment in this sequence include: The instantaneous energy at each time step is obtained by squaring each data point in the sequence.

3. The method for collecting air compressor operation data based on the Internet of Things according to claim 1, characterized in that, The standard deviation is used to denoise the operating data, including: The preprocessed air compressor operating data is used as input to the Gaussian filtering algorithm. The standard deviation of the Gaussian filtering algorithm is determined, and the output is the denoised air compressor operating data.

4. The method for collecting air compressor operation data based on the Internet of Things according to claim 1, characterized in that, Collect and preprocess the operating data of the air compressor, including: The running data is normalized, and missing values ​​in the running data are filled in.

5. The method for collecting air compressor operation data based on the Internet of Things according to claim 1, characterized in that, Envelope demodulation of the energy concentration index sequence of the nearest time window includes: The Hilbert transform is applied to the sequence of energy concentration indices within the nearest neighbor time window to obtain the envelope sequence.

6. The method for collecting air compressor operation data based on the Internet of Things according to claim 1, characterized in that, Determining the dynamic threshold based on the amplitude distribution of the real cepstral sequence includes: The dynamic threshold of the cepstral amplitude in the real cepstral sequence is determined according to the Laida criterion.

7. An Internet of Things-based air compressor operation data acquisition system, comprising a processor and a memory, characterized in that, The memory stores a computer program, and the processor executes the computer program to implement the Internet of Things-based air compressor operation data acquisition method as described in any one of claims 1-6.

Images