Research on image compression technology based on improved SPIHT compression algorithm for power grid data

2.1 Wavelet transform

In signal analysis, we often use the time domain or frequency domain to describe the signal. Although the traditional Fourier transform can describe the characteristics of the signal well and accurately in the frequency domain space, it does not carry the information of the signal in the time domain [8]. Therefore, to study the spectral characteristics of a signal, it is necessary to obtain all the information of the signal in the time domain, and even the future information. However, in practice, we are more concerned with the spectrum of the signal over a specific period, which is not taken into account by the Fourier transform [9]. The frequency window of wavelet transform, a time node that can be adjusted, solves this problem well.

At the same time, the so-called “permissibility” condition is met.

where φ ( ω ) is the wavelet base or parent wavelet function; and φ ˆ ( ω ) is the Fourier transform of φ ( ω ) .

where a is the expansion factor and b is the translation factor.

Assuming f(x) is a square-integrable function, the continuous wavelet transform of f(x) is defined as

Obviously, with the change in parameter a, the spectrum structure, window size, and shape of the continuous wavelet will change. When the signal frequency is low, the observation window becomes larger, and vice versa, the observation window becomes smaller, so the wavelet transform has a high resolution in the time (space) domain.

According to the admissibility condition of wavelet (1), it can be obtained that

The continuous wavelet function can be reconstructed by its continuous wavelet transform to obtain the inverse transformation.

In common practical applications, the continuous wavelet transform method is not used, because the computer does not have the function of performing continuous variable operations [10]. In use, we can discretize the continuous wavelet transform to form a transformation method that can run in the computation.

Then, the discrete binary wavelet transform of f(k) is

Accordingly, the inverse discrete binary wavelet transform can be obtained.

It can be seen that with the constant change in n, the information of different signals at different periods can be analyzed. As m becomes smaller, the corresponding frequency of the signal will gradually become higher, and more details of the signal can be observed.

Multiresolution is composed of subspaces that do not overlap each other and can form nested sequences.

where h n and g n are the “two-scale relationship” between the scale function and the wavelet.

The essence of multi-resolution analysis, in a physical sense, is to decompose the original signal into components of different frequency segments according to different needs, to facilitate subsequent analysis [11].

In 1987, Mallat proposed Mallat fast wavelet algorithm based on multi-resolution analysis, which can realize signal decomposition from fine to coarse and reconstruction from coarse to fine by adjusting the scale factor.

where P N − 1 f represents an increasingly rough representation of f and Q N − 1 f represents the difference between two adjacent scale levels of the function f .

The following results are obtained:

Using equations (9), (10), and phase orthogonality, it is further obtained that

For the same reason

Define the operator

Then,

Iterating the above process over and over again, we get

Then, we can get

Iterating through this loop, we can get

This is Mallat’s algorithm. Equation (23) is called “wavelet decomposition.”

2.2 Image compression algorithm

If the Mallat multi-resolution tower decomposition algorithm is extended to the field of two-dimensional wavelet transform, the decomposition process of two-dimensional discrete wavelet transform can be completed through two steps of row processing and column processing, and the decomposition process is shown in Figure 1. First, each line of the near signal at stage j is regarded as a one-dimensional signal, which is filtered by a low-pass filter and high-pass filter h and g, respectively, and down-sampled. Then, each column of the two groups of signals obtained after line decomposition is regarded as a one-dimensional signal, and the low-pass and high-pass filters h and g are used to achieve column filtering and down-sampling processing, respectively. Finally, four groups of two-dimensional signals are obtained [11]. If we want to continue the decomposition, we need to process aj-1 by row and column again in the previous way, and so on. Finally, the Mallat multi-level tower decomposition algorithm of two-dimensional discrete wavelet transform is formed. The process of two-dimensional discrete wavelet inverse transformation is an inverse process with the decomposition process and its reconstruction process shown in Figure 1. The specific process will not be repeated here.

Figure 1

Decomposition and reconstruction process of 2D discrete wavelet transform.

The second-generation wavelet transform of the lifting format is a second-generation wavelet transform method based on the lifting format proposed by Sweldens and participated by Daubechies and others [12]. Compared with the traditional wavelet transform, the enhanced wavelet transform has the advantages of high operation speed, simple calculation, and less storage space. The promotion format mainly includes three processes: splitting, forecasting, and updating, and the reconstruction process is its inverse process.

Since CDF (9,7) enhanced wavelet transform has a linear phase, large vanishing moment, and good energy concentration, this work uses CDF (9,7) enhanced wavelet transform to decompose and reconstruct power quality signals [11].

The parameters in the figure are determined by the following:

where r 0 = h 0 − 2 h 4 h 1 / h 3 , r 1 = h 2 − h 4 − h 4 h 1 / h 3 , s 0 = h 1 − h 3 − h 3 r 0 / r 1 , t 0 = r 0 − 2 r 1 .

2.3 Power grid data compression method based on image compression

The overall scheme of the three-phase power quality data compression method proposed in this work is shown in Figure 2. First, the three-phase power quality data in the three-phase coordinate system is converted into two-phase power quality data in the two-phase coordinate system and zero-sequence power quality data by dq () transformation. When the system operates symmetrically or the three-phase unbalance is light, the value of zero-sequence power quality is about zero [12]. Therefore, some coefficients in the zero-sequence power quality data obtained by the dq () transformation will be zero, which is more conducive to the coding after wavelet transformation and can achieve a higher compression ratio. Then, to further explore the periodic redundancy of power quality data, the transformed power quality data are converted into a two-dimensional matrix according to the integer multiple periods, the signal period is taken as a row and the sampling point is taken as a column. Then, digital image smoothing technology is used to blur the two-dimensional matrix [13]. Finally, data compression is carried out based on SPIHT encoding and the DEFLATE encoding algorithm of the two-dimensional wavelet transform image. Data reconstruction is the reverse process of data compression.

Figure 2

Data compression and reconstruction.

The three-phase voltage/current in the three-phase plane coordinate system ABC can be transformed into a set of mutually orthogonal two-phase voltage/current and zero-sequence voltage/current in the two-phase plane coordinate system dq by dq () transformation.

3.1 Dependency analysis

Cyclic operation is a data structure algorithm rich in parallelism, and it always consumes most of the time during the algorithm execution [14]. The process of algorithm serialization is the process of migrating the parallelized part of the algorithm to the parallel machine for computation. Its essence is to carry out parallelism analysis on the transplanted loop body, which is one of the important components of serial algorithm parallelization. The goal is to find the loop body suitable for parallelization in the serial algorithm and keep the semantics of the parallel algorithm unchanged before and after modification, and the computation function is the same [15]. The execution efficiency of parallel algorithms is directly related to the adequacy of parallel analysis. The analysis of dependence on relation is the basis of parallelism analysis. Dependency analysis is the premise of the parallelization of serial algorithms. Only after analyzing the dependency of the algorithm can we design the corresponding parallel algorithm.

The data dependency diagram is shown in Figure 3. As can be seen from the figure, in the splitting stage, the execution of each splitting process only has a time dependence. In the prediction phase, prediction task 1 depends on the output of split task 1 and split task 2 [16]. The dependency of prediction task 1 on split task 1 is strong data dependency, and the dependency of prediction task 1 on split task 2 is weak data dependency. In the update phase, update task 2 depends on prediction task 1 and prediction task 2. The dependency of update task 2 on prediction task 2 is strong, and the dependency of prediction task 1 is weak.

Figure 3

Dependency diagram of wavelet transform and SPIHT encoded data.

Based on the above analysis of the integer 5/3 wavelet transform data dependency graph, we can eliminate the weak data dependency in the graph using data prefetching [17]. In addition, the time dependence can be eliminated directly during parallelization. Before the execution of prediction task 1, even sequence pixel 2 is prefetched, and before the execution of update task 1, odd sequence coefficient 0 is prefetched. In this way, there is no data dependence between the two tasks of wavelet coefficient transformation. In this way, the two tasks can be processed in parallel to realize the parallelization of the serial integer 5/3 wavelet transform algorithm.

Figure 3 shows the data dependencies during SPIHT encoding. As can be seen from the figure, because of the existence of the sequence table, different coding trees have data dependence in the coding process. To eliminate this kind of data dependency, the method of eliminating the order table can be adopted, and the removal of the order table will no longer have data dependency between various coding trees. In this way, the encoding process of different coding trees is completely independent, so that the parallelization process of serial SPIHT encoding with coding tree as parallel unit can be realized. At the same time, it can also be seen that the code stream output of each code tree is completely independent. However, the compressed streams of different code trees of the original SPIHT algorithm are output in a specific order, which requires a stream reorganization process in the subsequent stage of parallelization, so that the output streams of the parallel SPIHT encoding are the same as those of the original SPIHT algorithm.

3.2 Parallel SPIHT image compression algorithm

The parallel integer 5/3 wavelet transform is divided into three stages: preprocessing, data transformation, and post-processing. Among them, the pre-processing stage is divided into data segmentation and data expansion, whose purpose is to provide parallel data sources for parallel data transformation. The post-processing stage includes data concatenation and data recombination, through which the final two-dimensional wavelet transform coefficients can be generated.

It mainly includes two parts: data partitioning and data extension. First, the original image data is partitioned into equal rows and columns according to the image size and hardware resources, and then different expansion methods are adopted according to the location of the edge of the data block [18]. Specifically, if the edge of the data block happens to be the edge of the image, then two pixels are extended in a symmetric continuation manner. If the edge of the data block is inside the image, two pixels are taken from each of its adjacent data blocks for expansion.

The expanded data block is the basic unit for the implementation of the 5/3 wavelet transform parallel algorithm. The two-pixel extension of data blocks enables the transformation between blocks to be carried out independently, avoiding inter-block communication, which is also a prerequisite for the implementation of parallel wavelet algorithms [19].

The two-dimensional wavelet transform of each extended data block is completed by one-dimensional row wavelet transform and one-dimensional column wavelet transform in turn. First, the parallel prediction process of the odd sequence is carried out, and then the parallel update process of the dual sequence is carried out, thus completing a one-dimensional 5/3 wavelet transformation process [20]. In the process of parallel prediction and update, the prediction of each element in the odd sequence and the update of each element in the even sequence are executed in parallel.

Post-processing mainly includes two processes data concatenation and data reorganization. Data splicing is to splice the transform coefficients of each data block into wavelet transform coefficients of the same dimension as the original image in parallel. The purpose of data reorganization is to concentrate the coefficients of each frequency sub-band. After post-processing, the final 5/3 wavelet transform coefficient can be obtained for the subsequent parallel SPIHT coding process.

The parallelism of the SPIHT encoding algorithm can be analyzed from two levels. First, there is the overlap of tasks in different value coding processes, which is coarse-grained parallelism at the task level; then there is the parallelism of different code trees in specific value coding processes, which is fine-grained parallelism at the data level.

The data level parallelism in parallel SPIHT coding is encoded by using a code tree as a parallel unit. It can be seen from the coding tree that the coefficient of each high-frequency sub-band has four child nodes in the next level of the high-frequency sub-band. It should be noted that the low-frequency sub-band has no child nodes, i.e., the coefficient of the low-frequency sub-band is a coding tree with only root nodes. The data-level parallelization process of the parallel SPIHT encoding algorithm is described in detail in this work.

1. Removal of order tables: Three order tables, LIP, LIS, and LSP, were removed, and three matrices with the same dimension as the wavelet decomposition coefficients were introduced, denoted as AIP, AIS, and ASP, respectively. The purpose of this is to remove the data dependency between different coding trees so that the coding process of different coding trees can be carried out in parallel and independently. At the same time, the dynamic coding process of coefficients in the same wavelet level is transformed into a static coding process, which provides the basis for data-level parallelization.

2. A new way of transmitting position information is introduced: The original serial algorithm implicitly transmits the ranking information of the coding coefficients through the order table, but the matrix itself cannot implicitly transmit the position information. The solution is to add serial and parallel Morton scanning methods to implicitly transmit location information. Among them, the parallel Morton scan is used to generate stream packets corresponding to a single wavelet coefficient in parallel, and the serial Morton scan is used to organize the stream generated by serially encoding each wavelet coefficient and output the stream, which is equivalent to implicitly transmitting the position information of the encoding coefficient.

3. Adding the node sorting process: Changing the transmission mode of location information also changes the wavelet coefficient processing order of the original algorithm. It is equivalent to adding the sorting process of the nodes in the order table based on the original algorithm so that the important nodes of the high-level wavelet coefficients will always be coded preferentially than the important nodes of the low-level wavelet coefficients.

In the stage of fine coding and sequencing coding, the parallelization process based on the code tree is adopted. Parallel Morton scanning mode is adopted for parallel coding between different coding trees, and the stream packet corresponding to each wavelet coefficient is output at the same time. The whole process of unpacking is serial, and the code stream is unpacked and reassembled in serial according to the serial Morton scanning mode and the organizational structure of the code stream. Table 1 explains the structure of the stream packet. In the process of parallel SPIHT coding with specific rush values, the stream packet corresponding to each wavelet coefficient is output according to Table 1. In the process of stream organization, the stream packet is unpacked according to Table 1 and the final stream is output.

4.1 Grid data model

As can be seen from Figure 4, the power grid model mainly includes generators, transformers, LC filters, loads, transmission lines, oscilloscopes, and other components. The current, voltage, active power, reactive power, frequency, and phase angle values of the six nodes 1, 2, 3, 4, 6, 8 are monitored and output as power grid system data for the SPIHT data compression method.

Figure 4

Grid model structure.

Data signal processing and monitoring is an important technology. It is often used in conjunction with a data oscilloscope to display the entire process and results. At present, some studies are trying to combine integrated circuit technology with a digital oscilloscope, including the use of DSP and FPGA technology, to give full play to the high flexibility of hardware and software combination structure, high-speed real-time sampling updates the digital logic structure and program and has a strong advantage of universality.

Data acquisition (also known as digitization) in a digital oscilloscope consists of three main operations: sampling, quantization, and encoding. Sampling is the exponential sampling of the oscilloscope’s voltage signal within a fixed time interval, and the analog signal is discrete with time. Quantization refers to the amplitude discretization of analog signals, while encoding refers to the conversion of quantized discrete digital signals into binary data. Usually, quantization and coding are done at the same time. The digitization process of the digital oscilloscope is mainly completed by an A/D converter. In the conversion process, the sampling and holding circuit of the A/D converter ensures the accuracy of the conversion because the circuit has two working states of sampling and holding, which can collect the instantaneous value of the analog input voltage at a certain time, and keep the output voltage constant during the conversion process to ensure the smooth completion of the conversion process.

The basic principle of digital oscilloscope signal sampling follows the sampling theorem. Signal sampling methods can be divided into real-time sampling and non-real-time sampling. Non-real-time sampling can be divided into random equivalent sampling and sequential sampling. The sampling theorem is an explanation of the relationship between the sampling frequency and the signal spectrum. When the sampling frequency is more than twice the maximum frequency of the original signal, the sampled digital signal can completely retain the information in the original signal. Therefore, the determination of sampling frequency plays a very important role in the signal sampling of digital oscilloscopes. In addition, to improve the real-time sampling rate of the whole machine, time alternate parallel sampling technology is often used. The principle of this method is to concatenate multiple ADCs at a low sampling rate, and then use the phase difference of each ADC to continuously sample the original signal. Finally, all the sampled signals are arranged according to the sampling order to obtain the final sampling data result. In addition, the waveform capture rate of the digital oscilloscope is also an important index.

Waveform capture rate is the number of waveforms captured by an exponential word oscilloscope in unit time, which is related to the time of data oscilloscope acquisition and processing of waveform data. Since numerical algorithms can greatly reduce waveform capture rate in DSP processors, interpolation can shorten data processing time in hardware, to achieve the effect of improving the waveform capture rate.

4.2 Hardware implementation of SPIHT coding algorithm

SPIHT coding algorithm is a multilevel tree set split coding algorithm, it defines a special tree structure of the wavelet transform coefficient of the image and then uses the tree structure as the basis for encoding and decoding. The SPIHT coding algorithm has been introduced in the previous chapter, This study introduces the FPGA implementation process of the SPIHT algorithm encoder.

The SPIHT encoder implemented in this paper is controlled by a state machine, the initial state is IDLE, and the SPIHT encoder does not work. When the DWT transformation is completed, i.e., when dwt_done is 1, the next state is entered, i.e., the INIT state is initialized. When the initialization is complete, the LIP entry and LIS table are filled, and init_done is pulled higher, the LIP state is entered. After the scanning of the LIP entry is completed, the lip_done signal is sent, the LIS state is entered, and the scanning of the LIS entry is started. The final EXOU state is the fine scanning state after LIS scanning is completed. If the encoding is completed, the IDLE state will be returned. If not, the threshold value will be reduced by one, and the LIP state will be returned for the next stage of encoding scanning. The implementation of each state in the state machine is explained in detail.

The initialization involves initializing the threshold value Tn and filling the LIP and LIS entries. Since the LSP is empty during initialization, it is not processed. The initial threshold is set here to 8, and subsequent updates to the threshold are placed in the Fine Scan section. There are two storage media options for LIP and LIS entries, one is RAM and the other is FIFO. However, scanning entries are performed according to the sequence of storage in the table. Deletion operations are often performed during scanning. We hope that subsequent data can be automatically replenished after deletion so that subsequent data scanning is not affected. If RAM is used to store LIP and LIS entries, then when any entry in the table is deleted, subsequent data will need to be moved forward by one address. This undoubtedly increases the difficulty and complexity of the system control. The FIFO (First In, First Out) structure is just in line with the working logic of LIP and LIS entries, each reads a data, that is, the default deletion of the data, and the subsequent data automatically complete. If you do not want to delete the current data, add the data at the end of the entry. The writing of data is also stored in sequence. Therefore, FIFO is the most suitable entry for LIS and LIP. As for the critical data stored by LSP, it is only increased, and it needs to be read in cycles, so RAM is more suitable.

Compared with the previous modules, the structure of the LIS sorting module is more complex, including the processing of two different types of entries, namely, Type A and Type B, and each type of entry processing is different. Its main work is to scan the child node and the grandson node judges the importance and deals with the results of different important judgments accordingly.

4.3 Simulation analysis

First, the original power grid data obtained from the output of the simulation model is read. To facilitate the acquisition of experimental data, the power grid system running time within 0–1 s is set in the Simulink simulation model. It can be seen from the experimental results that there are about 13,485 × 257 power grid data outputs within 0–1 s. To adapt the SPIHT compression method, this study converts the data within 1 s into several 256 × 256 matrix data blocks. It can be seen from the calculation that a 256 × 256 matrix data block is about 0.02 s of acquired power grid data. The converted 256 × 256 2D power grid data obtained every 002 s is applied to the classical 2D SPIHT image processing compression method introduced in this study to compress the power grid data (Table 2).

Experimental data show that both the SPIHT compression algorithm and the improved SPIHT algorithm have a compression effect on the two-dimensional network data, and the peak signal to noise ratio (PSNR) of the improved SPIHT algorithm is higher than that of the original SPIHT algorithm at each bit rate. The experimental results show that the improved SPIHT algorithm can compress the grid data to obtain better grid image quality.

Information entropy is used to evaluate the randomness and unpredictability of pixel distribution. To evaluate the correlation between the plaintext image and the ciphertext image, 5,000 pairs of adjacent pixels are randomly selected from the plaintext image and the ciphertext image in different directions, and the corresponding r _xy is calculated. Figure 5 shows the correlation distribution diagram of the R channel of the plaintext Lena image and the corresponding ciphertext image.

Figure 5

Correlation distribution of adjacent pixels in image R channel.

As shown in Figure 5, the horizontal, vertical, and diagonal directions of the R channel in the plaintext Lena image are all positively correlated, basically conforming to the y = x function relationship. The points of adjacent pixels in the plaintext image are centrally distributed near the x-line. The points of adjacent pixels of the ciphertext image are distributed in the entire two-dimensional plane, indicating that the algorithm greatly eliminates the r _xy of adjacent pixels of the plaintext image, and there is no correlation between adjacent pixels of the ciphertext image.

By setting the initial value and conducting a MATLAB simulation experiment, the trajectory diagram of the random sequence is obtained, as shown in Figure 6. It shows that the output sequence directly generated by the SPIHT compression algorithm has strong randomness and its value changes without regularity.

Figure 6

Trajectory diagram and time change of random sequence.