Noise reduction method, device, and program

Abstract

The present invention reduces a noise component included in measurement data. This noise reduction method comprises: a step S1 for acquiring a measurement signal Y; a step S2 for extracting sample data y N of a plurality of points N from the measurement signal Y; a step S3 for creating a multi-component matrix H from the sample data y N of the plurality of points N; a step S4 for performing singular value decomposition on the matrix H to obtain a singular value σ of the matrix H; a step S5 for, in an equation represented by a sum of a first term pertaining to the total sum of singular values σ and a second term represented by the product of a normalization parameter λ and a term pertaining to the difference between the measurement signal Y and a restored signal X, obtaining, by Bayesian-optimization of the normalization parameter λ, the normalization parameter λ that minimizes the difference Z-δ between a standard deviation Z of an absolute value of the difference between the restored signal X and the measurement signal Y and a standard deviation δ of noise included in the measurement signal Y; and a step S6 for generating the restored signal X on the basis of a singular value σ when the normalization parameter λ is optimized.

Description

Noise reduction method, apparatus, and program

[0001] This invention relates to a method, apparatus, and program for reducing noise in measurement signals.

[0002] Cutting-edge scientific research is closely intertwined with measurement technologies that can visualize the scientific background of the subject of study, and the development of measurement technologies supports new and important discoveries and the opening up of new research areas. To measure signals and elucidate natural phenomena through that measurement data, highly accurate analysis and processing are necessary.

[0003] On the other hand, noise components are always introduced into measurements. Therefore, in measurement methods with low measurement sensitivity, the presence of noise introduced during measurement makes subsequent analysis difficult. Various methods have been proposed to remove noise components contained in measurement data.

[0004] For example, Patent Document 1 discloses a method for improving the signal-to-noise ratio (S / N ratio) of spectral data. In the method of Patent Document 1, the main components in the measurement data are identified by determining the correlation between each set of data points S1(dn) to SM(dn) with the same ordinal number dn. Set S is spectral data, and the correlation between data points with the same ordinal number dn is determined for each of the M spectral data SMs before integration. Each of the M spectral data SMs is recorded in a memory device during the observation process.

[0005] Japanese Patent Publication No. 2011-242295

[0006] Improving measurement sensitivity remains a crucial theme in measurement technology, not just for measurement methods with low sensitivity. There is a need for methods applicable to various measurement techniques to reduce noise components contained in measurement data.

[0007] The present invention aims to reduce noise components contained in measurement data.

[0008] The present invention, for solving the above problems, includes, for example, the following embodiments: (1) A signal acquisition step of acquiring a measurement signal Y, and a sample data y of multiple points N from the measurement signal Y.Ｎ A sampling step to extract the sample data y of multiple points N, and Ｎ A noise reduction method comprising: a pluralization step of creating a pluralized matrix H from; a singular value decomposition step of performing singular value decomposition on the matrix H to obtain the singular values ​​σ of the matrix H; an optimization step of obtaining the regularization parameter λ that minimizes (Z-δ), which is the difference between the standard deviation Z of the absolute value of the difference between the restored signal X and the measured signal Y and the standard deviation δ of the noise contained in the measured signal Y, in a formula expressed as the sum of a first term relating to the sum of the singular values ​​σ, a second term expressed as the product of a term relating to the difference between the measured signal Y and the restored signal X and a regularization parameter λ, by Bayesian optimization of the regularization parameter λ; and a signal restoration step of generating the restored signal X based on the singular values ​​σ when the regularization parameter λ is optimized. (Item 2) The noise reduction method according to Item 1, wherein the formula is expressed as (Equation 1) below. In (Equation 1), x is the kernel norm of the matrix H, and ０ is the ideal restored signal X free of noise, y is the measured signal Y containing noise, x is the restored signal X with reduced noise, and F is the Frobenius norm. (Item 3) The restored signal X, in which noise has been reduced in an optimized state. (Item 4) The noise reduction method according to any one of items 1 to 3, wherein the Bayesian optimization is performed according to the algorithm of a Tree-structured Parzen estimator (TPE). (Item 5) The noise reduction method according to any one of items 1 to 4, wherein the matrix H is a Hankel matrix or a block Hankel matrix. (Item 6) The noise reduction method according to any one of items 1 to 4, wherein the measurement signal Y is expressed in the time domain, frequency domain, spatial axis, and phase axis. (Item 6) A signal acquisition unit that acquires the measurement signal Y, and sample data y of multiple points N from the measurement signal Y. Ｎ A sampling unit that extracts the sample data y from multiple points. ＮNoise reduction device comprising: a pluralization unit that creates a pluralized matrix H from; a singular value decomposition unit that performs singular value decomposition on the matrix H to find the singular values ​​σ of the matrix H; an optimization unit that finds the regularization parameter λ by Bayesian optimization of the regularization parameter λ, in a formula expressed as the sum of a first term relating to the sum of the singular values ​​σ, a second term expressed as the product of a term relating to the difference between the measured signal Y and the restored signal X and a regularization parameter λ, the difference between the standard deviation Z of the absolute value of the difference between the restored signal X and the measured signal Y and the standard deviation δ of the noise contained in the measured signal Y; and a signal restoration unit that generates the restored signal X based on the singular values ​​σ when the regularization parameter λ is optimized. (Item 7) A program for causing a computer to perform each step of the method according to any one of items 1 to 5.

[0009] According to the present invention, it is possible to reduce noise components contained in the measurement data.

[0010] This is a diagram illustrating the usage of a noise reduction device according to one embodiment of the present invention. This is a diagram illustrating the signal processing performed by a noise reduction method according to one embodiment of the present invention. This is a diagram illustrating the signal processing performed by a noise reduction method according to one embodiment of the present invention. This is a block diagram illustrating the function of a noise reduction device according to one embodiment of the present invention. This is a flowchart illustrating the processing procedure performed by a noise reduction device according to one embodiment of the present invention. This is a table illustrating conventional methods for improving sensitivity by signal processing related to solid-state NMR. This is a diagram illustrating the signal processing performed by a noise reduction method according to another embodiment of the present invention. This is a graph showing the relationship between the score of the evaluation function and the regularization parameter λ in Example 1. This is the spectrum of the restored signal obtained from the regularization parameter λ obtained by optimization in Example 1. This is the spectrum before and after noise reduction obtained by applying the noise reduction method of the present invention to the CP / MAS spectrum of PMMA in Example 2. This is the spectrum before and after noise reduction obtained by applying the noise reduction method of the present invention to glycine in Example 3. １３These are spectra before and after noise reduction, obtained by applying the C chemical shift anisotropy (CSA) spectrum to proline. In Example 4, the noise reduction method of the present invention was used for proline. １５ These are spectra before and after noise reduction, obtained by applying the noise reduction method of the present invention to an NDNP-NMR spectrum in Example 5. These are spectra before and after noise reduction, obtained by applying the noise reduction method of the present invention to a two-dimensional magic angle turning (MAT) spectrum of glycine in Example 5. These are spectra before and after noise reduction, obtained by applying the noise reduction method of the present invention to a two-dimensional magic angle turning (MAT) spectrum of glycine in Example 5. These are ESR signals before and after noise reduction, obtained by applying the noise reduction method of the present invention to an ESR signal in Example 6. These are fluorescence lifetime measurement signals before and after noise reduction, obtained by applying the noise reduction method of the present invention to a fluorescence lifetime measurement signal in Example 7. These are fluorescence lifetime measurement signals before and after noise reduction, obtained by applying the noise reduction method of the present invention to a fluorescence lifetime measurement signal in Example 8. These are fluorescence lifetime measurement signals before and after noise reduction, obtained by applying the noise reduction method of the present invention to a fluorescence lifetime measurement signal in Example 9.

[0011] Embodiments of the present invention will be described in detail below with reference to the accompanying drawings. In the following description and drawings, the same reference numerals indicate the same or similar components, and therefore, redundant explanations of the same or similar components will be omitted.

[0012] In this specification, the term "signal" can be interpreted not only as literally meaning a signal, but also as meaning data. Similarly, the term "data" can be interpreted not only as literally meaning data, but also as meaning a signal.

[0013] The term "Fourier transform" refers not only to the literal Fourier transform, but also to the discrete Fourier transform and the fast Fourier transform. The same applies to the inverse transform.

[0014] [Device Overview] <Usage> Figure 1 is a diagram illustrating the usage of a noise reduction device according to one embodiment of the present invention.

[0015] In one embodiment, the noise reduction device 1 acquires a measurement signal from a measuring device 90 and reduces noise in the acquired measurement signal. The measuring device 90 is a device that performs measurements using various measurement methods such as nuclear magnetic resonance (NMR) spectroscopy, time-resolved electron spin resonance (ESR) spectroscopy, characteristic X-ray spectroscopy, time-resolved fluorescence spectroscopy, transient absorption spectroscopy, and magnetic resonance imaging (MRI). In this embodiment, the measuring device 90 is a solid-state NMR device. The noise reduction device 1 acquires the measurement signal acquired by the measuring device 90, for example, via a network 9. The noise reduction device 1 can use a general-purpose computer such as a personal computer.

[0016] The measurement signal measured by the measuring device 90 contains additive white Gaussian noise (hereinafter referred to as Gaussian noise). Gaussian noise is a signal component in which the noise component does not depend on the input signal, its power spectrum (Fourier transform of autocorrelation) is constant at all frequencies, and its amplitude intensity follows a normal distribution. Thus, the measurement signal has random perturbations added to it according to a normal distribution, with its true value as the average. The variance of these perturbations becomes the width of the noise.

[0017] The inventors have found that by performing signal processing on a measurement signal containing such Gaussian noise, combining singular value decomposition and Bayesian optimization, it is possible to identify the main components in the measurement data and reduce the noise components contained in the measurement data.

[0018] <Overview of Signal Processing> Figures 2 and 3 are diagrams illustrating the signal processing performed by a noise reduction method according to one embodiment of the present invention.

[0019] In the following, we will describe the signal processing performed by a noise reduction method according to one embodiment, using the case where the measuring device 90 is a solid-state NMR device and the measuring device 90 measures a one-dimensional NMR signal as an example.

[0020] - Acquisition of Measurement Signal First, acquire the measurement signal Y to be subjected to noise reduction. In this embodiment, the measurement signal Y is a signal expressed in the time domain or the frequency domain. The measurement device 90 (solid-state NMR device) measures a free induction decay (FID) signal 91 of the target sample as illustrated in FIG. 2. The FID signal 91 is a signal in the time domain and includes a superposition of waveforms oscillating at various frequencies and noise. In order to clearly show the state of rotation of the nuclear spins in the waveform, usually, the FID signal 91 is Fourier-transformed and converted into an NMR spectrum 92. The NMR spectrum 92 is a signal in the frequency domain. Between these FID signal 91 and NMR spectrum 92, only the method of expressing information regarding the state of rotation of the nuclear spins is different, and these FID signal 91 and NMR spectrum 92 contain the same information regarding the state of rotation of the nuclear spins.

[0021] Therefore, the measurement signal Y to be subjected to noise reduction may be either the FID signal 91 or the NMR spectrum 92, and the measurement signal Y to be subjected to noise reduction may be any signal that can be expressed as a sum of exponential functions. Hereinafter, the case where the FID signal 91 is acquired from the measurement device 90 as the measurement signal Y will be described.

[0022] - Sampling and Diversification As shown in FIG. 3(A), a plurality of N (N is a natural number) sample data y Ｎ are taken out from the acquired measurement signal Y. Preferably, the interval at which the sample data y Ｎ is taken out is a constant interval. Next, as shown in FIG. 3(B), a diversified matrix H is created from the taken-out plurality of N sample data y Ｎ . Preferably, the diversified matrix H is a Hankel matrix.

[0023] - Singular Value Decomposition, Bayesian Optimization, and Restoration of Measurement Signal As shown in FIG. 3(C), the created matrix H is subjected to singular value decomposition to obtain the singular values σ of the matrix H. Among the obtained plurality of singular values σ, those whose values approach 0 (zero) correspond to the noise components of the measurement signal Y, and those that do not correspond to the main components of the measurement signal Y. In the illustrated example, the singular values σ １ , σ ２corresponds to the main components, and the other singular values correspond to noise components. Therefore, when restoring the signal using the components of the measurement signal Y corresponding to the singular values σ １ , σ ２ , a restored signal X with reduced noise components can be obtained.

[0024] In reducing the noise of the measurement signal Y, it is important not only to simply approximate the matrix H with a low rank by singular value decomposition, but also to appropriately distinguish which of the plurality of singular values σ obtained by singular value decomposition correspond to the main components and which correspond to the noise components. If it can be appropriately distinguished, a restored signal X with more ideally reduced noise components than a mere low rank approximation can be obtained. That is, in a method of identifying the main components of the measurement signal Y based on the result of singular value decomposition and restoring the original measurement signal Y from the identified main signal components, the problem is how to minimize the difference between the original measurement signal Y and the restored signal X by any method.

[0025] The inventors of the present invention solve such a minimization problem by combining the method of singular value decomposition with the method of Bayesian optimization. That is, in a mathematical formula represented by the sum of a first term related to the sum of the singular values σ, a second term represented by the product of a term related to the difference between the measurement signal Y and the restored signal X, and a regularization parameter λ, the regularization parameter λ that minimizes (Z - δ), which is the difference between the standard deviation Z of the absolute value of the difference between the restored signal X and the measurement signal Y and the standard deviation δ of the noise included in the measurement signal Y, is obtained by Bayesian optimization of the regularization parameter λ. Preferably, the Bayesian optimization is performed according to the algorithm of Tree-structured Parzen estimator (TPE).

[0026] By Bayesian optimization, the regularization parameter λ that minimizes the value of the above-described first term related to the sum of the singular values σ is obtained. Therefore, based on the singular value σ when the regularization parameter λ is optimized, the restored signal X is generated. As a result, a restored signal X with more ideally reduced noise components than a mere low rank approximation is obtained.

[0027] <Details of Signal Processing> In the following example, a free induction decay (FID) signal obtained by NMR measurement is used as the signal to be processed. The FID signal, which is the signal to be processed, can theoretically be expressed by the following equation (2) as the sum of R exponential functions.

[0028]

[0029] Here, R is the number of signals observed in the frequency domain signal, a ｒ is the signal intensity of the r-th component, and f ｒ is the resonance frequency of the r component, and τ ｒ is the attenuation coefficient. The actually observed measurement signal y includes noise and is as shown in the following equation (3).

[0030]

[0031] Here, x ０ represents an ideal FID signal (ideal restored signal) without noise, and z is Gaussian noise with an average of 0 (zero) and a variance of σ ２ . y, x ０ described in bold in Equation (3) are represented in vector form or matrix form. The bold notations x in the following description are the same. The Hankel matrix Hx ０ can be formed from x ０ as follows.

[0032]

[0033] When the number of signals R is small compared to the size (N + 1) of the Hankel matrix (usually R < 0.1(N + 1)), the Hankel matrix is known to be of low rank. At this time, the signal to be restored (restored signal) from the measurement signal y containing noise can be formulated as a convex optimization problem as shown in the following equation (5).

[0034]

[0035] Here, λ represents the kernel norm (sum of singular values) of the Hankel matrix, and λ is the regularization parameter. y is the measured signal containing noise, x is the reconstructed signal X with reduced noise, and F is the Frobenius norm (L2 norm). The solution to equation (5) can be obtained, for example, by the alternating direction method of multipliers (ADMM). An example of the ADMM algorithm is shown below.

[0036]

[0037] Here, This represents a vector whose elements are all 1. This is the adjoint operator of H, and by taking the sum of the opposite angles of each term, the Hankel matrix is ​​transformed into a vector. Singular value decomposition Then, the threshold operator S applied to matrix X １／β (X) is, It is given by, however, That is the case.

[0038] In the ADMM algorithm shown as an example, equation i) is an update equation that gives the x that minimizes equation (5) given the variables Z and D. Equation ii) is called soft thresholding and corresponds to the process of reducing singular values ​​increased by noise to an appropriate value. Equation iii) corresponds to the update equation for the Lagrange multiplier.

[0039] The FID signal (reconstruction signal) obtained in the algorithm described above. This strongly depends on the regularization parameter λ. In this embodiment, the Tree-structured Parzen estimator (TPE), a Bayesian optimization method, is used to optimize this regularization parameter λ. Among several Bayesian optimization methods, the TPE algorithm is characterized by obtaining the best results among other hyperparameter search methods reported to date, while also exhibiting low variability in results.

[0040] Bayesian optimization methods explore the parameter space of interest and find the optimal solution that minimizes or maximizes the value of the target function. In the signal processing performed in the noise reduction method according to this embodiment, the optimal solution for the regularization parameter λ is found using Bayesian optimization. Because Bayesian optimization methods can explore both unexplored and promising explored regions in a balanced manner, they can find better parameters with fewer trials compared to other hyperparameter search methods such as grid search or random search.

[0041] Let's outline the Bayesian optimization algorithm using the example of setting the hyperparameter to be searched to the regularization parameter λ described above. In the Bayesian optimization algorithm, assuming that the search for parameter λ is performed n times (where n is a natural number), information is obtained from the results of these n searches that "the noise-reduced result is likely to take values ​​within this range," and the parameter λ is narrowed down (i.e., the search for the next (n+1)th parameter λ) is performed. More specifically, an expected improvement amount is defined within the function (Bayesian optimization algorithm), and the hyperparameter that maximizes this expected improvement amount is searched for. The expected improvement amount is defined as a quantity that represents how much the search result for parameter λ (n+1) is improved compared to the results of the n search trials for parameter λ. Various libraries written in programming languages ​​such as Python can be used for the Bayesian optimization algorithm.

[0042] [Device Configuration] Figure 4 is a block diagram illustrating the function of a noise reduction device according to one embodiment of the present invention.

[0043] One embodiment of the noise reduction device 1 comprises a data processing unit 10, an auxiliary storage device 20, an input unit 31, a display unit 32, and a communication interface unit (communication I / F unit) 33. The noise reduction device 1 can be configured using, for example, a general-purpose computer such as a personal computer, a laptop PC, or a tablet terminal.

[0044] In this embodiment, the noise reduction device 1 comprises, as hardware components, an auxiliary storage device 20, an input unit 31, a display unit 32, and a communication I / F unit 33. Although not shown, the noise reduction device 1 further comprises, as hardware components, a processor such as a CPU that performs data processing, and memory used by the processor as a work area for data processing.

[0045] The auxiliary storage device 20 is a non-volatile storage device that stores the operating system (OS), various control programs, and data generated by the programs, and is composed of, for example, flash memory, eMMC (embedded Multi Media Card), SSD (Solid State Drive), etc. In this embodiment, the auxiliary storage device 20 stores measurement signal data 21, sample data 22, Hankel matrix data 23, singular value data 24, restored signal data 25, and noise reduction program 29.

[0046] The noise reduction program 29 is a computer program for realizing the various parts 11 to 16 within the data processing unit 10, which is a software-based functional block described later. These functional blocks are realized by installing the noise reduction program 29 into the auxiliary storage device 20 or memory of the noise reduction device 1, and by the processor executing the noise reduction program 29. The noise reduction program 29 may also be installed into the noise reduction device 1 via a network 9 such as the Internet, connected by the communication I / F unit 33. Alternatively, the noise reduction program 29 may be installed into the noise reduction device 1 by having the noise reduction device 1 read a computer-readable, non-temporary, tangible recording medium, such as a memory card, on which the noise reduction program 29 is recorded. The noise reduction program 29 can also be an application for, for example, a tablet terminal.

[0047] The input unit 31 can be configured as, for example, a mouse or keyboard. The display unit 32 can be configured as, for example, a liquid crystal display or an organic EL display. The input unit 31 and the display unit 32 can also be integrated as a touch panel.

[0048] The communication interface unit 33 transmits and receives data with external devices such as measuring devices 90 via a wired or wireless network 9. The communication interface unit 33 may be various wired or wireless connections such as Ethernet®, Bluetooth®, and Wi-Fi®.

[0049] In this embodiment, the noise reduction device 1 includes a data processing unit 10 as part of its software configuration. The data processing unit 10 is a functional block realized by the processor executing a noise reduction program 29.

[0050] The signal acquisition unit 11 acquires a measurement signal Y expressed in the time domain or frequency domain. In this embodiment, the signal acquisition unit 11 acquires measurement signal data 21 from the measurement device 90 via the network 9. The acquired measurement signal data 21 is recorded in the auxiliary storage device 20.

[0051] The sampling unit 12 extracts sample data y from multiple points N (where N is a natural number) from the acquired measurement signal Y. Ｎ Extract the sample data y of the multiple points N extracted. Ｎ This is recorded in the auxiliary storage device 20 as sample data 22.

[0052] The diversification unit 13 extracts sample data y from multiple points N. Ｎ From this, a multidimensional matrix H is created. The created matrix H is recorded in the auxiliary storage device 20 as Hankel matrix data 23.

[0053] The singular value decomposition unit 14 performs singular value decomposition on the created matrix H to obtain the singular values ​​σ of matrix H. The obtained singular values ​​σ are recorded as singular value data 24 in the auxiliary storage device 20.

[0054] The optimization unit 15 determines the regularization parameter λ that minimizes (Z - δ), which is the difference between the standard deviation Z of the absolute value of the difference between the restored signal X and the measured signal Y and the standard deviation δ of the noise contained in the measured signal Y, in a formula expressed as the sum of a first term relating to the sum of singular values ​​σ and a second term expressed as the product of a term relating to the difference between the measured signal Y and the restored signal X and the regularization parameter λ, by Bayesian optimization of the regularization parameter λ.

[0055] The optimization unit 15 performs the following processes. First, for a suitable initial value of the regularization parameter λ, equation 5 is satisfied. The process is then explored. Next, the multi-element unit 13 and the singular value decomposition unit 14 repeatedly perform the processes of creating a multi-element matrix X, finding the singular values ​​σ of matrix H, and performing soft thresholding, so that the value of the restored signal X changes repeatedly until it finally satisfies equation 5. The result obtained from the above process is the regularization parameter λ of a certain value. Whether it is the best is evaluated using an evaluation function expressed by the following formula: Evaluation function (Score) = Standard deviation of (measured signal Y - reconstructed signal X) - Standard deviation of the noise region. The regularization parameter λ that makes this evaluation function the smallest value is found by Bayesian optimization to obtain the final reconstructed signal X.

[0056] The signal restoration unit 16 generates a restored signal X based on the singular value σ when the regularization parameter λ is optimized. The generated restored signal X is recorded in the auxiliary storage device 20 as restored signal data 25. The signal restoration unit 16 displays the generated restored signal X on, for example, the display unit 32.

[0057] [Processing Procedure] Figure 5 is a flowchart illustrating the processing procedure performed by a noise reduction device according to one embodiment of the present invention.

[0058] In step S1 (signal acquisition step), a measurement signal Y, expressed in the time domain or frequency domain, is acquired.

[0059] In step S2 (sampling step), sample data y of multiple points N (where N is a natural number) is obtained from the acquired measurement signal Y. Ｎ Take it out.

[0060] In step S3 (diversification step), the sample data y of multiple points N that were extracted Ｎ From this, we create a multi-element matrix H.

[0061] In step S4 (singular value decomposition step), the created matrix H is subjected to singular value decomposition to find the singular values ​​σ of matrix H.

[0062] In step S5 (optimization step), in a formula expressed as the sum of a first term relating to the sum of singular values ​​σ, a second term expressed as the product of a term relating to the difference between the measured signal Y and the reconstructed signal X and the regularization parameter λ, the regularization parameter λ that minimizes (Z - δ), which is the difference between the standard deviation Z of the absolute value of the difference between the reconstructed signal X and the measured signal Y and the standard deviation δ of the noise contained in the measured signal Y, is determined by Bayesian optimization of the regularization parameter λ.

[0063] In step S6 (signal restoration step), the restored signal X is generated based on the singular value σ when the regularization parameter λ is optimized.

[0064] [Effects] As described above, the noise reduction device 1 and method according to one embodiment of the present invention can reduce noise components contained in the measurement data.

[0065] The measurement signal measured by the measuring device 90 contains additive white Gaussian noise, and in measurement methods with low measurement sensitivity, the presence of such noise makes post-measurement analysis difficult. In contrast, the noise reduction device 1 and method according to one embodiment perform signal processing on the measurement signal containing Gaussian noise by combining a singular value decomposition method and a Bayesian optimization method. This makes it possible to identify the main components in the measurement data and reduce the noise components contained in the measurement data. When it becomes possible to reduce the noise components in the measurement signal after measurement by the measuring device 90, it becomes possible to shorten the measurement time by the measuring device 90 and to measure trace components by improving the measurement sensitivity.

[0066] Furthermore, efforts to improve the sensitivity of measurement methods with low sensitivity have included the development of new equipment and pulse programs. Methods for improving sensitivity in low-sensitivity methods, such as solid-state NMR, include the development of equipment such as high-field superconducting magnets, cryoprobes, magic-angle spinning (MAS), and dynamic nuclear polarization (DNP), as well as the development of pulse programs such as cross-polarization (CP) and proton detection. However, equipment development presents the problem of high initial costs. Pulse program development has the problem of being limited to specific measurement methods and target nuclides.

[0067] In contrast, the signal processing applied to the measurement signal in the noise reduction device 1 and method according to one embodiment, which combines singular value decomposition and Bayesian optimization, is a mathematical and statistical method. Therefore, the noise reduction device 1 and method according to one embodiment can be used in combination with the sensitivity improvement method by device development described above, or the sensitivity improvement method by pulse program development described above, enabling synergistic sensitivity improvement. Since it is a signal processing method using a computer, the introduction cost is also low.

[0068] The type of measuring device 90 used to measure the measurement signal to be subjected to noise reduction is not limited to the solid-state NMR apparatus exemplified. The measuring device 90 can be various NMR apparatuses that target not only solid-phase but also liquid-phase, gas-phase, and supercritical samples. Not limited to the NMR apparatus exemplified, the measuring device 90 can be an apparatus that performs measurements using various measurement methods such as time-resolved electron spin resonance (ESR), characteristic X-ray spectroscopy, time-resolved fluorescence spectroscopy, transient absorption spectroscopy, and magnetic resonance imaging (MRI). Therefore, the noise reduction apparatus 1 and method according to one embodiment can reduce noise components in measurement signals from various measuring devices using various measurement methods.

[0069] This embodiment will describe in detail a case in which noise components of a measurement signal measured by a solid-state NMR device are reduced.

[0070] Solid-state NMR is an indispensable tool for elucidating the structure and dynamics of materials at the molecular and atomic levels, and is applied to a wide range of materials, including organic compounds, inorganic compounds, polymer materials, and biomaterials. One of the drawbacks of solid-state NMR is its lower sensitivity compared to other spectroscopic methods. This stems from the small difference in the Boltzmann distribution between the ground state and excited states of nuclear spins. Therefore, in solid-state NMR, signal integration is usually performed multiple times to suppress noise, but this results in significantly longer measurement times. This problem is particularly pronounced in multidimensional NMR measurements. Multidimensional NMR measurements can provide useful information about magnetic interactions and dynamics between nuclear spins, which are difficult to evaluate with one-dimensional NMR measurements. However, the measurement time is several to tens of times longer than that of one-dimensional NMR measurements. Furthermore, if the sample is in an amorphous state, the signal becomes broader due to the distribution of bond lengths and bond angles, reducing the signal-to-noise ratio (S / N ratio) and requiring longer signal integration times. To improve the sensitivity of such solid-state NMR methods, research has been conducted from the perspectives of equipment development, pulse program development, and signal processing.

[0071] A noise reduction device 1 and method according to one embodiment of the present invention reduce the noise component of a measurement signal by a mathematical statistical signal processing method that combines a singular value decomposition method and a Bayesian optimization method.

[0072] Figure 6 is a table illustrating conventional signal processing techniques for improving sensitivity in solid-state NMR. The phase correlation method, principal component analysis method, Cadzow method, and CHORD method in the table are sensitivity improvement techniques for one-dimensional NMR measurements. The method described in Patent Document 1 corresponds to the phase correlation method in the table. The compressed sensing method, multivariate curve resolution (MCR), and wavelet noise reduction method in the table are sensitivity improvement techniques for two-dimensional NMR measurements.

[0073] The noise reduction methods for one-dimensional NMR measurements listed in the table are explained below. Phase correlation and principal component analysis methods require a data set for noise reduction, and there is a problem that noise reduction processing cannot be applied to signals that have already undergone numerous integrations. For the phase correlation method, the data set for noise reduction is a data set measured while modulating the phase of the excitation pulse, and for the principal component analysis method, it is a data set when the measurement conditions are changed. The Cadzow method is a method based on singular value decomposition, but because this method is a noise reduction method that truncates singular values, there is a problem that distortion occurs in the spectrum after noise reduction, or false signals called artifacts are mixed into the spectrum. The CHORD method is also a method based on singular value decomposition, but its application is limited to one-dimensional NMR measurements.

[0074] The noise reduction methods for two-dimensional NMR measurements listed in the table are explained below. If the original measurement signal itself has a low signal-to-noise ratio (S / N ratio) before noise reduction processing, the S / N ratio after noise reduction processing does not improve significantly regardless of which method is used in the table.

[0075] In contrast, the noise reduction device 1 and method according to one embodiment of the present invention achieve a more efficient (approximately 30 times or more) improvement in the signal-to-noise ratio (S / N ratio) than any of the conventional methods listed in the table, for both one-dimensional NMR measurement and two-dimensional NMR measurement (details will be described later). In particular, even when the S / N ratio of the original measurement signal itself before noise reduction processing is low, the noise component can be efficiently reduced and the S / N ratio can be improved. For the signal after noise reduction processing, if the S / N ratio is 10 or higher, there is almost no spectral distortion. The regularization parameter λ is also automatically adjusted. Generally, the S / N ratio of an NMR spectrum improves in proportion to the square root of the number of integration steps n. The noise reduction device 1 and method according to one embodiment of the present invention can obtain an equivalent S / N ratio without increasing the number of integration steps n.

[0076] [Other Embodiments] Although the present invention has been described above with reference to specific embodiments, the present invention is not limited to the embodiments described above.

[0077] Figure 7 is a diagram illustrating the signal processing performed by a noise reduction method according to another embodiment of the present invention. In the above embodiment, the noise reduction device 1 reduces noise in a one-dimensional NMR signal, but the NMR signal that the noise reduction device 1 processes is not limited to a one-dimensional NMR signal. In other embodiments, the noise reduction device 1 can reduce noise in a two-dimensional NMR signal. Similarly, the noise reduction device 1 can reduce noise in a multi-dimensional NMR signal.

[0078] The signal processing procedure will be explained. First, similar to a one-dimensional NMR signal, sample data y of multiple points N (where N is a natural number) is obtained from the acquired measurement signal Y. Ｎ Next, extract the sample data y of the multiple points N extracted, as shown in Figure 7(A). Ｎ From this, a multi-element matrix H is created. Here, when dealing with a one-dimensional NMR signal, the Hankel matrix is ​​used to create the multi-element matrix H, but when dealing with a two-dimensional NMR signal, a block Hankel matrix is ​​used instead of the Hankel matrix to create the multi-element matrix H. The subsequent signal processing procedure is the same as for a one-dimensional NMR signal.

[0079] In the above embodiment, the measurement signal to be subjected to noise reduction is a signal that can be expressed as a sum of exponential functions, but the functional type of the measurement signal to be subjected to noise reduction is not limited to exponential functions. For example, as explained in the above embodiment, the FID signal 91 and the NMR spectrum 92 differ only in how they express information about the rotation of nuclear spins, and these FID signal 91 and NMR spectrum 92 contain the same information about the rotation of nuclear spins. Therefore, the NMR spectrum 92 may be converted to an FID signal 91 by, for example, an inverse Fourier transform, and then noise reduction processing may be performed. In this case, for example, a step of performing an inverse Fourier transform on the acquired measurement signal Y may be further included between step S1 (signal acquisition step) and step S2 (sampling step). In this case, the noise reduction device 1 may further include, for example, an inverse Fourier transform unit in the data processing unit 10 as a software functional block that performs an inverse Fourier transform on the acquired measurement signal Y.

[0080] In the above embodiment, the measurement signal to be subjected to noise reduction is a signal expressed in the time domain or the frequency domain. However, the expression of the measurement signal to be subjected to noise reduction is not limited to the time domain or the frequency domain. In addition to the time domain or the frequency domain, the measurement signal to be subjected to noise reduction may also be a signal expressed in, for example, the spatial axis (spatial coordinates) or the phase axis (phase angle). That is, the measurement signal to be subjected to noise reduction may be a measurement signal expressed in one-dimensional data, as exemplified by one-dimensional NMR in the above embodiment; it may be a measurement signal expressed in two-dimensional data, as exemplified by two-dimensional NMR in the other embodiment described above; or it may be a measurement signal expressed in multi-dimensional data. The type of measurement signal is also not limited to NMR signals. Various signals representing physical quantities, such as electron spin resonance (ESR) signals and PL signals described later, can be subjected to noise reduction.

[0081] In the above embodiment, the noise reduction device 1 is connected to the measuring device 90 via a network 9, but the connection configuration between the noise reduction device 1 and the measuring device 90 is not limited thereto. The noise reduction device 1 may also be integrated with a control console that controls the measurement operation of the measuring device 90.

[0082] In the above embodiment, the noise reduction device 1 is implemented as an integrated device, but the noise reduction device 1 does not need to be an integrated device; the processor, memory, auxiliary storage device 20, etc. may be located separately and connected to each other via a network. The input unit 31 and the display unit 32 do not necessarily need to be located in the same place; they may be located separately and connected to each other via a network 9 for communication.

[0083] In the above embodiment, each functional block of the noise reduction device 1 is executed by a single processor. However, these functional blocks do not necessarily need to be executed by a single processor; they may be distributed and processed by multiple processors. Alternatively, an FPGA (Field Programmable Gate Array) may perform the processing instead of a processor, or a GPU (Graphics Processing Unit) may be used as an accelerator to assist the parallel processing performed by the processor. In other words, processing performed by a processor includes processing performed by a processor or FPGA using an accelerator such as a GPU.

[0084] In the above embodiment, each functional block 11 to 16 constituting the data processing unit 10 is implemented by software, but each functional block 11 to 16 may be partially or entirely implemented as hardware. The processing of each functional block 11 to 16 constituting the data processing unit 10 does not need to be processed by a single processor, but may be distributed and processed by multiple processors. Each functional block 11 to 16 constituting the data processing unit 10 and the data items 21 to 25 in the auxiliary storage device 20 may be partially or entirely cloudified on another server device (not shown) connected via the communication I / F unit 33.

[0085] The following examples illustrate the features of the present invention. Unless otherwise specified, the noise reduction method of the present invention refers to the noise reduction method described in the above examples.

[0086] In Example 1, the optimal number of Bayesian updates was investigated by varying the number of Bayesian optimization cycles. For this investigation, an evaluation function expressed by the following equation was introduced: Evaluation Function (Score) = Standard deviation of (Measured signal Y - Reconstructed signal X) - Standard deviation of the noise region

[0087] Figure 8 is a graph showing the relationship between the score of the evaluation function and the regularization parameter λ in Example 1. The graph is shown for each iteration of Bayesian optimization. Figure 9 is the spectrum of the reconstructed signal obtained from the regularization parameter λ obtained by optimization in Example 1. The spectrum of the reconstructed signal is shown for each iteration of Bayesian optimization.

[0088] In this Example 1, as shown in Figure 8, we were able to find the regularization parameter λ that produced the smallest score in the evaluation function with a small number of updates. Furthermore, as shown in Figure 9, no difference in the spectrum was observed due to differences in the number of updates. It was confirmed that 15 to 30 repetitions of Bayesian optimization are sufficient to find the regularization parameter λ that produces the smallest score in the evaluation function.

[0089] In Example 2, the noise reduction method of the present invention was applied to a one-dimensional CP / MAS spectrum of polymethyl methacrylate (PMMA). Three measurement signals with different signal integration counts were prepared to apply the noise reduction method (hereinafter simply referred to as "original signals").

[0090] Figure 10 shows the spectra before and after noise reduction, obtained by applying the noise reduction method of the present invention to the CP / MAS spectrum of PMMA in Example 2. In each of (A) to (C), the upper panel shows the spectra before and after noise reduction, and the lower panel shows a graph showing the relationship between the evaluation function score and the regularization parameter λ. The spectrum after noise reduction is shown by a dashed line in the upper panel.

[0091] The number of times the original signal was integrated was 2 in (A), 16 in (B), and 512 in (C). The signal-to-noise ratio of the original signal was approximately 5.6 in (A), approximately 9.3 in (B), and approximately 84 in (C). The signal-to-noise ratio after noise reduction was approximately 517 in (B) and approximately 506 in (C).

[0092] As shown in the lower part of Figure 10(A), when the original signal-to-noise ratio (S / N) was low at approximately 5.6, although the search for a regularization parameter λ near the minimum value was successful, baseline distortion and artifacts were suspected to be present in the spectrum after noise reduction, as shown by the dashed line in the upper part of (A). In contrast, as shown in Figures 10(B) and (C), when the original signal-to-noise ratio was approximately 9.3 and approximately 84, no baseline distortion or artifacts were found in the spectrum after noise reduction. The S / N ratio after noise reduction was approximately 517 and approximately 506, respectively, and it was confirmed that the improvement rate of the S / N ratio was approximately 56 times and approximately 6 times, respectively.

[0093] In Example 3, the noise reduction method of the present invention is used with glycine １３ The method was applied to 13C chemical shift anisotropy (CSA) spectra. Similar to Example 2, three original signals with different signal integration counts were prepared.

[0094] Figure 11 shows the noise reduction method of the present invention in Example 3, using glycine １３ These are spectra before and after noise reduction, obtained by applying the function to C chemical shift anisotropy (CSA) spectra. In each of (A) to (C), the upper panel shows the spectra before and after noise reduction, and the lower panel shows a graph illustrating the relationship between the evaluation function score and the regularization parameter λ. The spectrum after noise reduction is shown as a dashed line in the upper panel.

[0095] The number of times the original signal was integrated was 128 times for (A), 2048 times for (B), and 32768 times for (C). The signal-to-noise ratio of the original signal was approximately 2.6 for (A), approximately 4.9 for (B), and approximately 23 for (C). The signal-to-noise ratio after noise reduction was approximately 122 for (B) and approximately 238 for (C).

[0096] Similar to Example 2, when the original signal-to-noise ratio (S / N) was low at approximately 2.6, distortion of the baseline and spectral shape was observed in the spectrum after noise reduction, as shown by the dashed line in the upper part of Figure 11(A). In contrast, as shown in Figures 11(B) and (C), when the original signal-to-noise ratio was approximately 4.9 and approximately 23, the spectrum after noise reduction showed generally good results. The S / N ratio after noise reduction was approximately 122 and approximately 238, respectively, and it was confirmed that the improvement rate of the S / N ratio was approximately 25 times and approximately 10 times, respectively.

[0097] In Example 4, the dynamic nuclear polarization (DNP) method and the noise reduction method of the present invention are combined to obtain low-sensitivity nuclei. １５ Solid-state NMR measurements were performed on N. １５ Proline was used for the N sample.

[0098] Figure 12 shows the noise reduction method of the present invention in Example 4, using proline １５ These are spectra obtained by applying the DNP-NMR spectrum to the N-DNP spectrum, before and after noise reduction. In the figure, "Mw off" is the spectrum when the DNP method itself is not used, "Mw on" is the spectrum when the DNP method is used, and "Mw on + Denoise" is the spectrum when both the DNP method and the noise reduction method of the present invention are used.

[0099] Without the DNP method, the signal-to-noise ratio (S / N ratio) was approximately 2.43. In contrast, using the DNP method improved the S / N ratio to approximately 473. Furthermore, by using both the DNP method and the noise reduction method of the present invention, the S / N ratio improved to approximately 45,547, which was confirmed to be an improvement of approximately 18,743 times compared to when the DNP method was not used.

[0100] In Example 5, the noise reduction method of the present invention was applied to a two-dimensional NMR spectrum. When applying the noise reduction method of the present invention to a two-dimensional NMR spectrum, a block Hankel matrix, as explained with reference to Figure 7, was used instead of the Hankel matrix.

[0101] Figures 13 and 14 show the spectra before and after noise reduction, obtained in Example 5 by applying the noise reduction method of the present invention to the two-dimensional magic angle turning (MAT) spectrum of glycine. (A) is the two-dimensional spectrum before noise reduction, (B) is the two-dimensional spectrum after noise reduction, and (C) is a cross-section of the two-dimensional spectrum, which is the slice data when the vertical axis is 115 ppm in spectra (A) and (B). In (C), the waveform shown by the white solid line within the black peak of the spectrum is the waveform of the spectrum after noise reduction. The number of integrations of the original signal was 64 in Figure 13 and 16 in Figure 14. In the spectrum shown in Figure 13, the S / N ratio of the original signal was approximately 7.9, and the S / N ratio after noise reduction was approximately 429, resulting in an improvement rate of approximately 54 in the S / N ratio.

[0102] As shown in Figures 13 and 14, it was confirmed that the noise reduction method of the present invention can reduce noise in both two-dimensional spectra, which have different numbers of integration steps for the original signal. It was confirmed that the noise reduction method of the present invention is applicable not only to one-dimensional spectra but also to two-dimensional spectra, and it was suggested that it may also be applicable to higher-dimensional spectra.

[0103] In Example 6, the noise reduction method of the present invention was applied to an electron spin resonance (ESR) signal.

[0104] Figure 15 shows the ESR signals before and after noise reduction, obtained by applying the noise reduction method of the present invention to the ESR signal in Example 6. (A) to (C) show the ESR signal or FID signal according to the procedure when the noise reduction method is applied.

[0105] (A) shows the original ESR signal measured by the measuring device before applying the noise reduction method. In Example 6, the original ESR signal shown in (A) was converted to an FID signal by inverse Fourier transform, and then the noise reduction method of the present invention was applied. (B) shows the FID signals before and after application. In (B), the waveform shown by the gray solid line within the black signal peak is the FID signal after noise reduction.

[0106] Next, the noise-reduced FID signal was converted back to its original signal format, the ESR signal, by performing a Fourier transform. Hereafter, the ESR signal obtained after converting the noise-reduced FID signal back to its original signal format will be referred to as the noise-reduced ESR signal. The ESR signals before and after noise reduction are shown in (C). In (C), the waveform shown by the gray solid line within the black signal peaks is the noise-reduced ESR signal. The signal-to-noise ratio (S / N ratio) of the original ESR signal was approximately 138, and the S / N ratio of the noise-reduced ESR signal was approximately 848. It was confirmed that the improvement in the S / N ratio was approximately 6 times.

[0107] In Examples 7 to 8, the noise reduction method of the present invention was applied to the photoluminescence lifetime measurement signal (hereinafter simply referred to as the PL signal). Figure 16 shows the photoluminescence lifetime measurement signal before and after noise reduction, obtained in Example 7 by applying the noise reduction method of the present invention to the photoluminescence lifetime measurement signal. (A) is the original PL signal before applying the noise reduction method. (B) is a graph showing the relationship between the evaluation function score and the regularization parameter λ. (C) is the PL signal before and after noise reduction. (D) is a graph showing the results of singular value decomposition. In (D), the gray solid line is the result of singular value decomposition of the signal after noise reduction.

[0108] In Example 7, the data enclosed by the dashed line in Figure 16(A) was used as sample data (noise level) to separate the noise component from the main component of the measurement signal to be subjected to noise reduction. The noise level of the PL signal in the area enclosed by the dashed line was approximately 4.96. Note that the PL signal in the area enclosed by the dashed line corresponds to the noise before irradiation with excitation light.

[0109] In (C), the waveform shown by the gray solid line within the black signal peak is the PL signal after noise reduction. In Example 7, the improvement rate of the S / N ratio by the noise reduction method of the present invention was approximately 2.6 times. In Example 7, where the noise before irradiation with excitation light was used as sample data, although the S / N ratio was improved, the expected level of improvement in the S / N ratio was not obtained.

[0110] In Example 8, as in Example 7, the noise reduction method of the present invention was applied to the PL signal. Figure 17 shows the fluorescence lifetime measurement signals before and after noise reduction, obtained by applying the noise reduction method of the present invention to the fluorescence lifetime measurement signal in Example 8.

[0111] Unlike Example 7, Example 8 uses sample data (noise level) to separate the noise component from the main component of the measurement signal to be reduced for noise reduction. In Example 8, the noise level of the PL signal was optimized, resulting in a value of approximately 9.0, which is about 1.8 times that of Example 7.

[0112] In (C), the waveform shown by the gray solid line within the black signal peak is the PL signal after noise reduction. In Example 8, the improvement in the signal-to-noise ratio by the noise reduction method of the present invention was approximately 58 times. Compared with Example 7, which applied the noise reduction method of the present invention to the same PL signal, the signal-to-noise ratio was significantly improved in Example 8. From the comparison between Example 7 and Example 8, it was shown that when applying the noise reduction method of the present invention to a PL signal, significant noise reduction is possible by optimizing the noise level rather than using the noise before irradiation with excitation light as sample data.

[0113] In Example 9, as in Example 7, the noise reduction method of the present invention was applied to a PL signal. In Example 9, the noise reduction method of the present invention was applied to a PL signal from an organic EL light-emitting material (DACT-II), which is a different fluorescent material from that used in Examples 7 and 8.

[0114] Figure 18 shows the fluorescence lifetime measurement signals before and after noise reduction, obtained by applying the noise reduction method of the present invention to the fluorescence lifetime measurement signal in Example 9. (A) and (C) are the PL signals before and after noise reduction, respectively. In (A), the noise level is given in the first way, and in (C), the noise level is given in the second way. (B) and (D) are graphs showing the relationship between the score of the evaluation function and the regularization parameter λ, respectively, corresponding to (A) and (C).

[0115] In Example 9, based on the results of Examples 7 and 8, two methods of providing sample data (noise level) to separate the noise component from the main component of the measurement signal to be subjected to noise reduction were tried, as described below. In the first method, as in Example 7, a portion of the PL signal data was used as sample data. Specifically, the last 300 data points on the time axis of the PL signal, which is time-series data, were used as sample data to calculate the noise level. In the first method, the noise level of the PL signal was approximately 5.64. In the second method, as in Example 8, the noise level of the PL signal was manually set to approximately 9.0.

[0116] In (A) and (C), the waveform shown by the gray solid line within the black signal peak is the PL signal after noise reduction. In Example 9, when the noise level was given in the first way, the improvement rate of the S / N ratio by the noise reduction method of the present invention was approximately 15 times, and when the noise level was given in the second way, the improvement rate of the S / N ratio by the noise reduction method of the present invention was approximately 68 times. In Example 9, the expected level of improvement in the S / N ratio was obtained in both ways of giving the noise level.

[0117] 1 Noise Reduction Device 9 Network 10 Data Processing Unit 11 Signal Acquisition Unit 12 Sampling Unit 13 Diversification Unit 14 Singular Value Decomposition Unit 15 Optimization Unit 16 Signal Restoration Unit 20 Auxiliary Storage Device 21 Measurement Signal Data 22 Sample Data 23 Hankel Matrix Data 24 Singular Value Data 25 Restoration Signal Data 29 Noise Reduction Program 31 Input Unit 32 Display Unit 33 Communication Interface Unit (Communication I / F Unit) 90 Measurement Device 91 Free Induction Attenuation (FID) Signal 92 NMR Spectrum

Claims

1. A signal acquisition step to acquire a measurement signal Y, and a sample data y of multiple points N from the measurement signal Y. Ｎ A sampling step to extract the sample data y of multiple points N, and Ｎ A noise reduction method comprising: a pluripotency step of creating a pluripotency matrix H from; a singular value decomposition step of performing singular value decomposition on the matrix H to obtain the singular values ​​σ of the matrix H; an optimization step of finding the regularization parameter λ that minimizes (Z-δ), which is the difference between the standard deviation Z of the absolute value of the difference between the restored signal X and the measured signal Y and the standard deviation δ of the noise contained in the measured signal Y, in a formula expressed as the sum of a first term relating to the sum of the singular values ​​σ, a second term expressed as the product of a term relating to the difference between the measured signal Y and the restored signal X and a regularization parameter λ, by Bayesian optimization of the regularization parameter λ; and a signal restoration step of generating the restored signal X based on the singular values ​​σ when the regularization parameter λ is optimized.

2. The noise reduction method according to claim 1, wherein the mathematical formula is represented by the following (Formula 1). In (Formula 1), is the nuclear norm of the matrix H, x ０ is the ideal restoration signal X without noise, y is the measurement signal Y with noise, x is the restoration signal X with reduced noise, F is the Frobenius norm, is the restoration signal X with reduced noise in an optimized state.

3. The noise reduction method according to claim 1, wherein the Bayesian optimization is performed according to the algorithm of a Tree-structured Parzen estimator (TPE).

4. The noise reduction method according to claim 1, wherein the matrix H is a Hankel matrix or a block Hankel matrix.

5. The noise reduction method according to claim 1, wherein the measurement signal Y is expressed in any of the time domain, frequency domain, spatial axis, and phase axis.

6. A signal acquisition unit that acquires a measurement signal Y, and a sample data y of multiple points N from the measurement signal Y. Ｎ A sampling unit that extracts the sample data y from multiple points. Ｎ A noise reduction device comprising: a pluralization unit that creates a pluralized matrix H from; a singular value decomposition unit that performs singular value decomposition on the matrix H to find the singular values ​​σ of the matrix H; an optimization unit that finds the regularization parameter λ, which minimizes (Z-δ), the difference between the standard deviation Z of the absolute value of the difference between the restored signal X and the measured signal Y and the standard deviation δ of the noise contained in the measured signal Y, in a formula expressed as the sum of a first term relating to the sum of the singular values ​​σ, a second term expressed as the product of a term relating to the difference between the measured signal Y and the restored signal X and a regularization parameter λ, by Bayesian optimization of the regularization parameter λ; and a signal restoration unit that generates the restored signal X based on the singular values ​​σ when the regularization parameter λ is optimized.

7. A program for causing a computer to perform each step of the method according to any one of claims 1 to 5.

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