Classifications

• • H—ELECTRICITY
  • H03—ELECTRONIC CIRCUITRY
  • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
  • H03M7/00—Conversion of a code where information is represented by a given sequence or number of digits to a code where the same, similar or subset of information is represented by a different sequence or number of digits
  • H03M7/30—Compression; Expansion; Suppression of unnecessary data, e.g. redundancy reduction
  • H03M7/3059—Digital compression and data reduction techniques where the original information is represented by a subset or similar information, e.g. lossy compression
  • H03M7/3062—Compressive sampling or sensing
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L65/00—Network arrangements, protocols or services for supporting real-time applications in data packet communication
  • H04L65/60—Network streaming of media packets
• • H—ELECTRICITY
  • H03—ELECTRONIC CIRCUITRY
  • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
  • H03M7/00—Conversion of a code where information is represented by a given sequence or number of digits to a code where the same, similar or subset of information is represented by a different sequence or number of digits
  • H03M7/30—Compression; Expansion; Suppression of unnecessary data, e.g. redundancy reduction
  • H03M7/3059—Digital compression and data reduction techniques where the original information is represented by a subset or similar information, e.g. lossy compression
  • H03M7/3064—Segmenting
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L43/00—Arrangements for monitoring or testing data switching networks
  • H04L43/08—Monitoring or testing based on specific metrics, e.g. QoS, energy consumption or environmental parameters
  • H04L43/0876—Network utilisation, e.g. volume of load or congestion level

Definitions

• the present invention is directed to a method for compressed sensing of streaming data and to the means for performing the same. More specifically, some embodiments include a method of compressed sensing of streaming data that employs a recursive algorithm for performing compressed sensing on streaming data, and an apparatus or system or computer program product capable of performing the method for compressed sensing of streaming data.
• the signals of interest can be represented sparsely by using few coefficients in an appropriately selected orthonormal basis.
• the Fourier basis is used for bandlimited signals or wavelet bases for piecewise continuous signals, such as images. While a small number of coefficients in the respective bases are enough to represent such signals, the Nyquist/Shannon sampling theorem suggests a sampling rate that is at least twice the signal bandwidth. Such a sampling rate is known in the art as the Nyquist rate. In many cases, the indicated sampling rate is much higher than the sufficient number of coefficients.
• the present invention proposes in accordance with one of its embodiments a computer-implemented method for sensing streaming data, comprising recursively sampling an input stream of data using overlapping windowing to obtain at least one previous measurement regarding said input data stream, and employing said at least one previous measurement for obtaining a subsequent measurement.
• a system for sensing streaming data including a plurality of modules, each module comprising a computer readable medium having thereon computer executable instructions for recursively sampling an input stream of data using overlapping windowing to obtain at least one previous measurement regarding said input data stream, and employing said at least one previous measurement for obtaining a subsequent measurement.
• a computer readable storage medium having recorded thereon a computer program for sensing streaming data, the computer program comprising an algorithm capable of recursively sampling an input stream of data using overlapping windowing to obtain at least one previous measurement regarding said input data stream, and employing said at least one previous measurement for obtaining a subsequent measurement.
• sensing of streaming data is a compressed sensing of streaming data
• the method for sensing streaming data employs a recursive scheme for performing sampling.
• the step of employing at least one previous measurement for obtaining a subsequent measurement comprises processing an input stream of data sampled during the step of recursive sampling via recursive estimation.
• the method and means proposed by the invention involve inputting information regarding the data stream regarding a previous estimate obtained during a previous estimate obtention step, the previous estimate obtention step being prior to the recursive sampling step.
• the method further comprises performing count estimation based on information obtained during a data stream support detection step and calculating a least squares estimation (LSE) value for a data stream support set based on data obtained during said recursive estimation step.
• LSE least squares estimation
• the method comprises in accordance with another embodiment of the invention an averaging step, wherein the calculated least squares estimation value, the count estimation value, and the previous estimate to calculate an averaged value are averaged to obtain an averaged value. The averaged value is employed to obtain a new estimate for the streaming data.
• the method proposed by the present invention further comprises the step of analyzing a computational complexity of the compressed sensing of streaming data and estimating an error degree of the method for sensing streaming data.
• the method further comprises obtaining convergence in an iterative optimization algorithm to decode a new window, the obtaining step comprising leveraging an overlapping window structure employed by the step of overlapping windowing and a signal estimate regarding the previous window.
• the method further yet comprises averaging signal estimates obtained from a plurality of windows, performing support set detection, and signal amplitude estimation.
• a voting scheme for robust support estimation in the presence of a high measurement noise may be as well applied in accordance with the present invention.
• FIG. 1 is a block diagram of the method for sensing streaming data in accordance with one embodiment of the present invention
• Fig. 2 represents the average processing time for recursive compressed sensing versus a 'naive approach' over a single time window
• Fig. 3 represents the results of support set estimation using LASSO
• Fig. 4 represents error plots for a) averaged estimates, b) debiased and averaged estimates and c) estimates obtained by voting and averaging, and
• Fig. 5 represents error plots for averaged LASSO estimate and 'voting and averaging' on streaming data.
• aspects of the present invention may be embodied as a system, method or computer program product. Accordingly, aspects of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a "circuit,” "module” or “system.” Furthermore, aspects of the present invention may take the form of a computer program product embodied in one or more computer readable medium/media (i.e., data storage medium/media) having computer readable program code recorded thereon.
• computer readable medium/media i.e., data storage medium/media
• the computer readable medium may be a computer readable signal medium or a computer readable storage medium.
• a computer readable storage medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any suitable combination of the foregoing.
• a computer readable storage medium i.e., data storage medium, may be any tangible medium that can contain, or store a program for use by or in connection with an instruction execution system, apparatus, or device.
• a computer readable signal medium may include a propagated data signal with computer readable program code embodied therein, for example, in baseband or as part of a carrier wave. Such a propagated signal may take any of a variety of forms, including, but not limited to, electro-magnetic, optical, or any suitable combination thereof.
• a computer readable signal medium may be any computer readable medium that is not a computer readable storage medium and that can communicate, propagate, or transport a program for use by or in connection with an instruction execution system, apparatus, or device.
• Program code embodied on a computer readable medium may be transmitted using any appropriate medium, including but not limited to wireless, wireline, optical fiber cable, RF, etc., or any suitable combination of the foregoing.
• Computer program code for carrying out operations for aspects of the present invention may be written in any combination of one or more programming languages, including an object oriented programming language such as Java, Smalltalk, C++ or the likes and conventional procedural programming languages, such as the "C" programming language or similar programming languages.
• the program code may be executed entirely on the user's computer, partly on the user's computer, as a stand-alone software package, partly on the user's computer and partly on a remote computer or entirely on the remote computer or server.
• the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be made to an external computer (for example, through the Internet using an Internet Service Provider).
• LAN local area network
• WAN wide area network
• Internet Service Provider an Internet Service Provider
• These computer program instructions may also be stored in a computer readable medium that can direct a computer, other programmable data processing apparatus, or other devices to function in a particular manner.
• the computer program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other devices to cause a series of operational steps to be performed on the computer, other programmable apparatus or other devices to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide processes for implementing the functions/acts specified in the flowchart and/or block diagram block or blocks.
• x t is used to denote the i th entry of vector x, and a; is used to denote the i th column of matrix A.
• the i th sample vector such as the i th window of the input stream, or the 1 TH sampling matrix is denoted by superscript (e.g., x ( ' ] or A ( ' ).
• ⁇ S ⁇ is indicative for the cardinality of a set S.
• a vector x is sparse if and only if
• Random matrices such as Gaussian, Bernoulli, randomly selected rows from DFT matrices are known to have been used as matrices for compressed sensing in the literature, since they satisfy the restricted isometry property with high probability.
• Examples of matrices satisfying the restricted isometry property are: a) n random vectors sampled from the m-dimensional unit sphere,
• matrix A satisfies a prescribed 5 k for any k ⁇ m/log ⁇ ) with a probability larger or equal than 1 - 2e ⁇ 3 ⁇ 4m , where Ci and C2 are constants that only depend on S k .
• Recursively is employed to indicate that the procedure is performed by repeating items in a self-similar way.
• the term refers to a method of defining functions in which the function being defined is applied within its own definition. Specifically, this defines an infinite number of instances (function values), using a finite expression that for some instances may refer to other instances, but in such a way that no loop or infinite chain of references can occur.
• the term is also used more generally to describe a process of repeating objects in a self- similar way. Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be "recursive".
• a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation. When another function or waveform/data-sequence is multiplied by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window”.
• window functions include spectral analysis, filter design, and beamforming. In typical applications, the window functions used are non-negative smooth "bell-shaped" curves, though rectangle, triangle, and other functions can be used.
• window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, and, more specifically, that the function goes sufficiently rapidly toward zero.
• a common practice is to subdivide it into smaller sets and window them individually. To mitigate the "loss" at the edges of the window, the individual sets may overlap in time.
• the present invention proposes a method and means to recover x when m « n.
• the system denoted with equation (1) is an underdetermined linear system.
• the main result is that if x is k-sparse and k ⁇ c m l log( « / k), a solution to this undetermined linear system is possible.
• problem Po is in general NP-hard, this problem has to be 'approximated' by tractable methods.
• Two convex optimization problems are used for recovering sparse vectors from linear measurements.
• a first optimization problem is referred to as "Basis Pursuit” wherein solving problem Po is equivalent to solving the li minimization problem BP: minimize
• subject to Ax y for all k-sparse vectors x, if A satisfies the restricted isometry property (RIP) with
• BP Basis Pursuit
• a second optimization problem is referred to as the Least Absolute Selection and Shrinkage Operator (LASSO).
• LASSO Least Absolute Selection and Shrinkage Operator
• Basis Pursuit may be applied, namely the Basis Pursuit Denoising (BPDN), best known as LASSO in the statistics literature, as: minimize
• BPDN Basis Pursuit Denoising
• the theorem states that the reconstruction error is upper bounded by the sum of two terms: the first is the error due to model mismatch, and the second is proportional to the measurement noise variance.
• the key is that the assumption on the isometry constant is satisfied with high probability by matrices obtained from random vectors sampled from unit sphere, random Gaussian matrices and random Bernoulli matrices if m ⁇ C k log( « / k) where C is a constant depending on each instance.
• the support set of the k-sparse signal may be detected as follows:
• the algorithm outputs a subset of columns of A, by iteratively selecting the column minimizing the residual error of approximating y by projecting to the linear span of already selected columns. It is shown that OMP recovers k-sparse signals from noiseless measurements if the mutual coherence of the measurement matrix A satisfies u(A) ⁇ — -— .
• ISTA is a proximal gradient method
• FISTA is an accelerated proximal gradient method
• SALSA is an application of the alternative direction method of multipliers.
• the signal of interest is an infinite sequence, , as is the case when dealing with streaming data.
• the ith window taken from the streaming signal is defined as
• a method for efficiently sampling and recovering streaming data is proposed.
• Such a method is a recursive compressed sensing method, and it will be described in detail bellow.
• the method of the invention exhibits low complexity in both the sampling and estimation parts, which makes its algorithms suitable for an efficient online implementation.
• a dynamical model with input in R is represented as
• ⁇ xj is sufficiently sparse in each window, namely if x (!)
• the signal overlap between successive windows is leveraged, consisting of recursive sampling (an encoding step) and recursive estimation (a decoding step) as follows:
• a sequence of sensing matrices A ( recursively is chosen as: where and P is the permutation matrix:
• the noisy measurements are noted as
• ⁇ ( ⁇ ) needs to satisfy the restricted isometry property.
• the restricted isometry property is satisfied with high probability for the product of a random matrix A ⁇ and any fixed matrix.
• a (l) is expressed as: ninimize
• where the input signal is given by x (l) Ocr .
• the problem that needs to be solved is how to find a recursive update for a ⁇ l+l) based on so as to have a good initial estimate for accelerated convergence in recursive estimation, as E ⁇ !+1)
• the Fourier basis is of particular interest for ⁇ since an efficient update rule can be derived for such basis.
• the Recursive Sampling for Fourier Basis is as follows:
• the least absolute selection and shrinkage operator yields a biased estimator as it maps R m ⁇ R n where m ⁇ n. If the overlaps between windows are utilized by averaging the LASSO estimates directly, the reconstruction error does not go to zero due to bias.
• least squares estimation LSE is an unbiased estimator for an overdetermined system; as shown above in this document, LASSO can be used for estimating the support set of the signal on which LSE can be subsequently applied. Based on these observations, a two-step estimation procedure is proposed for recovering the sampled signal to reduce the estimation error. First, the LASSO estimates x w are obtained, which are then used as input to a de- biasing algorithm.
• a block diagram of the method for sensing streaming data in accordance with one embodiment of the present invention is represented in figure 1.
• a method according to this embodiment comprises the step of recursively sampling an input stream of data using the steps of overlapping windowing to obtain at least a previous measurement, and employing said previous measurement for obtaining a subsequent measurement. More precisely, the method comprises the step of recursive sampling 102 of an input stream of data. The data sampled during the step of recursive sampling 102 is processed via a recursive estimation step 104. Information regarding a previous estimate, obtained during a previous estimate obtention step 116 previous to the recursive sampling is as well imputed during the step of recursive estimation 104.
• the data obtained during the step of recursive estimation 104 is utilized during the step of support detection 106, as described above in connection with step 6 of the recursive sampling algorithm.
• the information processed as the result of the support detection step may be used for count estimation, as shown in step 1 10 of figure 1 and as corresponding to step 7 of the recursive sampling algorithm.
• the information obtained in step 106 is utilized during the step 108 of calculating the LSE on the support set, as described above in connection with step 8 of the recursive sampling algorithm.
• the LSE on the support set and the estimation count are averaged in step 1 12, as described above in connection with step 9 of the recursive sampling algorithm.
• the previous estimate obtained during the step 1 16 is as well averaged in step 1 12.
• a new estimate is obtained in step 1 14, as described above in connection with step 10 of the recursive sampling algorithm.
• the method according to one such embodiment is as well capable of analyzing the computational complexity and estimation error of the method.
• FIG. 1 The block diagram of figure 1 representing one embodiment of the method may be extended to a method comprising variable overlap between successive windows.
• a e R"TM denotes the sampling matrix, i denotes the window index, and ⁇ ; is the sampling efficiency, that is the ratio of total samples taken until time (n + i) to the number of entries sensed. For one window, the sampling efficiency is m/n since sampling matrix is A e R mxn .
• n + (i - l) T elements have been recovered while having taken im samples.
• the asymptotic sampling efficiency is:
• the recursive sampling approach is asymptotically equivalent to taking one sample for each time instance.
• the benefit of such approach lies in noise suppression.
• the obtained two step algorithm with voting may be represented within the support detection block 106 in Figure 1.
• the sequence containing the votes is defined as ⁇ v ; ⁇ and the number of times an index i is used in the least squares estimation (LSE) as ⁇ L X ⁇ .
• LSE least squares estimation
• the ⁇ v ; ⁇ and the ⁇ L X ⁇ are set to zero.
• votes on the positions that are in the set are added, the set being indicated by S ( ' ⁇ as (i) ⁇ — v s n +1 + 1 > where the subscript S, !) + i is used to translate the indices within the window to global indices on the streaming data.
• threshold ⁇ 2 e Z + on the number of votes ⁇ vj the indices that have been voted sufficiently many times to be accepted as non-zeros are found and they are stored in
• the threshold ⁇ 2 is chosen so that
• the overdetermined least squares problem is solved based on these indices,
• A is the matrix obtained by extracting columns of indexed by the set . Then in order to perform averaging of the least squares estimates, the number of recoveries is incremented for the entries used in LSE procedure as
• the first condition is:
• v (! is defined as the Lasso estimate. It is obtained by least squares applied on the support of Lasso, v (,) .
• the first window is sampled by A (0) x (0) . This requires 0 ⁇ mn) basic operations, such as additions and multiplications.
• the total complexity of sampling is 0 ⁇ mn) + 0 ⁇ mz)i, an average complexity of ⁇ ) for recursive sampling.
• the other contribution to computational complexity stems from the iterative solver.
• the expected complexity attributes to the iterative solver which can be calculated as the number of operations in each iteration times the expected number of iterations for convergence. While the former depends on the particular algorithm the later is a function of the distance of the starting point to the optimal solution, which is bound to the case of using recursive estimation as follows:
• the expected number of iterations for convergence of algorithms where cost function decays sublinearly is for noiseless measurements and for i.i.d. measurement noise.
• n 6pn
• Figure 3 shows the resulting curves for the two methods, obtained by randomly generating the input signal 20 times for each value of m and averaging the resulting detection rates and false positives.
• the detection rate behaves similarly in both methods, the false positives can be reduced significantly by properly adjusting the threshold on the resulting LASSO estimates.
• > 3.34 , threshold ⁇ ⁇ 0.01 , 0.10, and 1.00.
• the circle markers depict detection rate, and square markers depict false positive rate.
• the LASSO method can be used together with a voting strategy and least squares estimation to yield an unbiased estimator.
• Figure 4 shows the comparison of performance of a) single LASSO estimates, b) averaged estimates, c) voting strategy, and d) debiasing and averaging.
• FIG. 4 is a representation of error plots for a) averaged estimates, b) debiased and averaged estimates and c) estimates obtained by voting and averaging.
• FIG. 5 the figure illustrates error plots for averaged LASSO estimate and 'voting and averaging' on streaming data.
• Figure 5 shows a comparison between the window reconstruction error obtained with averaged LASSO estimates and 'voting and averaging' algorithm on streaming data.
• voting gives a reconstruction error that shows jumps due to waiting for ⁇ 2 votes to be collected to use an entry in LSE.
• the error drops instantly to lower values than simply averaging the LASSO estimates.
• what is proposed in accordance with one embodiment is an efficient method for recursive sampling and iterative recovery pertinent to compressed sensing on streaming data.
• the method leverages signal overlaps between successive processing windows in obtaining a faster convergence speed for the estimates of the signal while achieving estimation variance reduction in the presence of noise.
• a two step estimation procedure to approximate an unbiased estimator of the signal based on LASSO where a) support detection is performed by solving LASSO, and b) signal estimation is obtained by solving ordinary least squares on the estimated support set.
• the computational complexity of the algorithm is O(mn) where m is the number of samples taken and n is the window length. The convergence time is shown by experiments to be appropriate for online implementation on streaming data.
• embodiments include a method for compressed sensing of streaming data that involves performing a recursive scheme for performing compressed sensing on streaming data, that is as well capable of analyzing the computational complexity and estimation error of the method.
• the input stream of data is sampled recursively via overlapping windowing while making use of the previous measurement in obtaining the next one.
• the signal estimate from the previous window is utilized, in order to achieve faster convergence in an iterative optimization algorithm, to decode the new window.
• the estimation accuracy is enhanced by averaging signal estimates obtained from multiple windows.
• a two step estimation procedure is proposed in accordance with one embodiment comprising support set detection and signal amplitude estimation.
• one embodiment includes a voting scheme for robust support estimation in the presence of high measurement noise.
• simulation results obtained while employing the means according to one embodiment for compressed sensing of streaming data show a speed up of ten times with respect to applying traditional compressed sensing on a stream of data, while obtaining significantly lower reconstruction error under mild conditions on the signal magnitudes and the noise level.
• means for sensing streaming data are as well proposed.
• Said means for sensing streaming data comprise means for recursively sampling an input stream of data, and means for employing previous measurements for obtaining a subsequent measurement.
• the means for recursively sampling an input stream of data are capable of using the steps of overlapping windowing to obtain at least a previous measurement.
• One embodiment includes ranging with ultra wideband signals. For example, a device may continuously monitor return pulses to regularly transmitted spikes (the signal emitted is periodic). Generally, the device receives a main echo, plus added multiple echoes. If the device is moving, the main echo just slightly changes from one period to another period, as do the multiple echoes. This is not a periodic signal, just an almost periodic signal, and a sliding window algorithm will naturally track these shifting echoes. In such embodiments, echo data from an acoustic sensor is thereby transformed into range data.
• biomedical sample data may be transformed according to methods and systems described above.
• samples from one or more electrical sensors configured to receive electrical signals from a human body may be processed according to embodiments disclose herein to reconstruct electrocardiographic or electroencephalographic signals.
• the reconstructed signal may be matched to signatures characteristic of one or more diagnostic conditions.
• sample data from a camera, photoelectric, or other pixel array sensor may be processed as described herein.
• an apparatus may be able to improve power efficiency or increase effective sensitivity of the sensor.
• environmental sensor data such as temperature, wind speed, wind direction, precipitation may be processed as described herein.
• the means of the present invention may be implemented as software means, hardware means or a combination thereof.
• embodiments disclosed herein may be implemented or performed with an electronic device or circuit such as a general purpose processor, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA) or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein.
• DSP digital signal processor
• ASIC application specific integrated circuit
• FPGA field programmable gate array
• a general purpose processor may be a microprocessor, but in the alternative, the processor may be any conventional processor, controller, microcontroller, or state machine.
• a processor may also be implemented as a combination of computing devices, e.g., a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration.
• a software module may reside in RAM memory, flash memory, ROM memory, EPROM memory, EEPROM memory, registers, hard disk, a removable disk, a CD-ROM, or any other form of storage medium known in the art.
• An exemplary storage medium is coupled to the processor such the processor can read information from, and write information to, the storage medium.
• the storage medium may be integral to the processor.
• the processor and the storage medium may reside in an ASIC.
• the ASIC may reside in a user terminal.
• the processor and the storage medium may reside as discrete components in a user terminal.
• an apparatus for performing compressed sensing of streaming data comprises a recursive sampler arranged for recursively sampling an input stream of data using overlapping windowing to obtain at least one previous measurement regarding the input data stream, and a unit employing the at least one previous measurement for obtaining a subsequent measurement.
• the apparatus proposed in accordance with one embodiment of the present invention comprises a recursive sampler that is based on a recursive scheme for performing sensing.
• the apparatus for performing compressed sensing of streaming data of the present invention also comprises a processing unit for processing she input stream of data sampled during the step of recursive sampling via recursive estimation.
• the apparatus for performing compressed sensing of streaming data also comprises storing means for delivering the inputting information regarding the data stream of a previous estimate obtained during a previous estimate obtention step.
• the apparatus for performing compressed sensing of streaming data of the present invention comprises a counter arranged for performing count estimation based on information obtained during a data stream support detection step.
• the apparatus of the present invention may also comprise a calculator for calculating a least squares estimation (LSE) value for a data stream support set based on data obtained during said recursive estimation step.
• LSE least squares estimation
• a processing unit for averaging the calculated least squares estimation value, the count estimation value, and the previous estimate to calculate an averaged value to obtain an averaged value is as well comprised by the apparatus proposed in accordance with one embodiment of the present invention.
• An estimator for estimating an error degree of the method for sensing streaming data is as well envisioned to be comprised by the apparatus proposed in accordance with the present invention.
• a processing unit for averaging signal estimates obtained from a plurality of windows may be as well comprised by the apparatus proposed by the present invention.

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