High resolution spectral recovery method

Abstract

Provided is a hyperspectral resolution spectral recovery method. The high-resolution spectral recovery method comprises: step 1, obtaining a transmission spectrum matrix of a spectral chip and a measurement value vector of an image sensor of the spectral chip; step 2, setting a predetermined selection probability of each row of the transmission spectrum matrix; step 3, selecting a predetermined row of the transmission spectrum matrix based on the predetermined selection probability; step 4, obtaining an update vector of the predetermined row based on an inner product of a spectral vector before iteration and the predetermined row, a value at a corresponding position of the measurement value vector and the predetermined row, a two-norm of the predetermined row; step 5, obtaining a spectral vector after iteration by subtracting the update vector from the spectral vector before iteration; and step 6, repeating steps 3 to 5 until the spectral vector after iteration meets a termination condition. In this way, high-resolution spectral recovery can be completed by improving the random Kaczmarz algebraic iteration, the difficulty of matrix inversion in high-resolution spectral recovery is solved, and the convergence speed is fast and the recovery precision is high.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of spectral chip, more particularly, to a high-resolution spectral recovery method based on an improved Kaczmarz iteration algorithm. BACKGROUND

[0002] Spectral imaging technology is a technology that combines spectral detection and imaging, which can image an object under different spectra and obtain the geometric shape information and spectral characteristics of the detected object. Spectral imaging technology has become an important means of earth observation and deep space exploration, and is widely used in agricultural, pastoral, forestry production, mineral resource exploration, cultural relic detection, marine remote sensing, environmental monitoring, disaster prevention and reduction, military reconnaissance and other fields.

[0003] Spectrometers using spectral imaging technology have become one of the most commonly used measurement tools in scientific research and industry. However, traditional spectrometers have complex structures and large volumes, which greatly hinder their application in daily life. Therefore, the miniaturization of spectrometers has attracted widespread attention. However, in a miniature spectrometer, the light path is shortened, and factors such as filter distribution and light path number can cause the spectral resolution to be significantly lower than that of a traditional spectrometer.

[0004] Therefore, in actual industrial applications, it is necessary to improve the spectral resolution of the spectrometer. The main method is to improve the device structure, such as introducing a device with collimation and dispersion function in a dispersive spectrometer, and introducing a narrow-band gradient filter in a filter-type spectrometer. In a computing spectral chip (also known as a computing spectral chip), due to process limitations, the current method for improving spectral resolution is represented by the analytical method of the least squares method.

[0005] The advantage of the analytical method is that it can directly calculate the inverse problem, but the disadvantage is that when the recovery resolution requirement is very high, it brings the problem of matrix inversion difficulty. In a computing spectral chip, the number of structural units is often tens of thousands or even hundreds of thousands, and the matrix elements to be solved represent the contribution of the pixel to the spectral wavenumber, so it is a large matrix and is not suitable for inversion and other operations.

[0006] In addition, when the spectral resolution requirement of the computing spectral device is improved, the number of sampling points n will also increase, which will cause the computational load to increase linearly, and the recovery effect of the traditional analytical method will deteriorate or cannot be recovered. For example, when the spectral resolution is improved from 1 nm to 0.5 nm, the computational load is increased by 3 times (1 / 0.5), i.e. the computational load is increased by 8 times.

[0007] Therefore, it is desirable to provide an improved high-resolution spectral recovery scheme. SUMMARY

[0008] The application provides a hyperspectral resolution spectrum recovery method, which realizes high-resolution spectrum recovery by improving random Kaczmarz algebra iteration, can solve the difficulty problem of matrix inversion in high-resolution spectrum recovery, and has high convergence speed and high recovery precision.

[0009] According to an aspect of the application, a high-resolution spectrum recovery method is provided, comprising:

[0010] Step 1, obtaining a transmission spectrum matrix of a spectrum chip and a measurement value vector of an image sensor of the spectrum chip;

[0011] Step 2, setting a predetermined selection probability of each row of the transmission spectrum matrix, the predetermined selection probability being a quotient of a square of a two-norm of a certain row of the transmission spectrum matrix and a square of a Frobenius norm of the transmission spectrum matrix;

[0012] Step 3, selecting a predetermined row of the transmission spectrum matrix based on the predetermined selection probability;

[0013] Step 4, obtaining an updated vector based on an inner product of a spectrum vector before iteration and the predetermined row, a value at a corresponding position of the measurement value vector and the predetermined row, a two-norm of the predetermined row and the predetermined row;

[0014] Step 5, obtaining a spectrum vector after iteration by subtracting the updated vector from the spectrum vector before iteration; and

[0015] Step 6, repeating steps 3 to 5 until the spectrum vector after iteration meets a termination condition, the termination condition being based on the spectrum vector after iteration and a two-norm thereof, the transmission spectrum matrix and a Frobenius norm thereof and the measurement value vector.

[0016] In the high-resolution spectrum recovery method, an initial spectrum vector is set as a 0 vector, denoted as 0.

[0017] In the high-resolution spectrum recovery method, the updated vector is denoted as:

[0018]

[0019] wherein b is the measurement value vector, x is a value at a corresponding position of the measurement value vector b and the predetermined row i of the transmission spectrum matrix A in the kth iteration, is a two-norm of the predetermined row i of the transmission spectrum matrix A in the kth iteration. k corresponding position, x k-1 is the spectrum vector before the kth iteration, is the predetermined row i of the transmission spectrum matrix A in the kth iteration. k , is a two-norm of .

[0020] In the above high-resolution spectral recovery method, the termination condition is expressed as:

[0021]

[0022] Among them, ‖A‖ F Let ε1 be the Frobenius norm of the transmission spectrum matrix A, and let ε1 > 0.

[0023] In the above high-resolution spectral restoration method, obtaining the transmission spectrum matrix of the spectral chip and the measurement value vector of the image sensor of the spectral chip includes:

[0024] Obtain the initial transmission spectrum matrix A of the spectral chip and the initial measurement value vector b of the image sensor of the spectral chip;

[0025] Based on the regularized description model, the matrix A′ and the measurement vector b′ of the overdetermined system are obtained from the initial transmission spectrum matrix A and the initial measurement vector b by extracting coefficients from the spectral vector. The regularized description model is as follows:

[0026]

[0027] Where λ>0 are the regularization coefficients, D is a tridiagonal Toeplitz matrix, and ||·|| denotes the L2 norm. The matrix A′ and the measurement vector b′ of the overdetermined system are respectively:

[0028]

[0029]

[0030] Furthermore, the matrix A′ and the measurement vector b′ of the overdetermined system are respectively used as the transmission spectrum matrix and the measurement vector of the spectral chip.

[0031] In the above-mentioned high-resolution spectral recovery method, the tridiagonal Toeplitz matrix is ​​a tridiagonal Toeplitz symmetric matrix.

[0032] In the above high-resolution spectral restoration method, the tridiagonal Toeplitz symmetric matrix is ​​represented as:

[0033]

[0034] Where a+b+c=0.

[0035] In the above high-resolution spectral restoration method, the update vector is represented as:

[0036]

[0037] in The measured value vector b′ and the predetermined row i of the k-th iteration k The value at the corresponding position, x k-1 The spectral vector prior to the k-th iteration, For the predetermined row i of the transmission spectrum matrix A′ in the k-th iteration k , for The 2-norm.

[0038] In the above high-resolution spectral recovery method, the termination condition is expressed as:

[0039]

[0040] Where x k Let be the spectral vector after the k-th iteration, where ‖·‖ represents the L2 norm, and ‖·‖ F This represents the Frobenius norm.

[0041] In the above high-resolution spectral restoration method, obtaining the transmission spectrum matrix of the spectral chip and the measurement value vector of the image sensor of the spectral chip includes:

[0042] Obtain the initial transmission spectrum matrix A of the spectral chip and the initial measurement value vector b of the image sensor of the spectral chip;

[0043] By taking the partial derivative of the regularized description model and setting it to zero, we obtain the matrix A′ and vector b′ in alternative forms from the initial transmission spectrum matrix A, wherein the regularized description model is:

[0044]

[0045] Where λ>0 are the regularization coefficients, and D is a tridiagonal Toeplitz matrix, specifically the same as the tridiagonal Toeplitz matrix in the first example described above, which will not be repeated here. ‖·‖ denotes the L2 norm, and the matrix A′ and vector b′ are:

[0046] A′=A T A+λD T D

[0047] b′=ATb;

[0048] Furthermore, the matrix A′ and vector b′ are used as the transmission spectrum matrix of the spectral chip and the measurement value vector of the image sensor of the spectral chip.

[0049] In the above-mentioned high-resolution spectral recovery method, the tridiagonal Toeplitz matrix is ​​a tridiagonal Toeplitz symmetric matrix.

[0050] In the above high-resolution spectral restoration method, the tridiagonal Toeplitz symmetric matrix is ​​represented as:

[0051]

[0052] Where a+b+c=0.

[0053] In the above high-resolution spectral restoration method, the update vector is represented as:

[0054]

[0055] in The measured value vector b′ and the predetermined row i of the k-th iteration k The value at the corresponding position, x k-1 The spectral vector prior to the k-th iteration, For the predetermined row i of the transmission spectrum matrix A′ in the k-th iteration k , for The 2-norm.

[0056] In the above high-resolution spectral recovery method, the termination condition is expressed as:

[0057]

[0058] Where x k Let be the spectral vector after the k-th iteration, where ‖·‖ represents the L2 norm, and ‖·‖ F This represents the Frobenius norm.

[0059] In the above high-resolution spectral restoration method, step 2 further includes:

[0060] Step 2.1: Set a predetermined column selection probability for each column of the transmission spectrum matrix, wherein the predetermined column selection probability is the quotient of the square of the L2 norm of a column of the transmission spectrum matrix and the square of the Frobenius norm of the transmission spectrum matrix; and,

[0061] Step 2.2, set a predetermined row selection probability for each row of the transmission spectrum matrix, wherein the predetermined row selection probability is the quotient of the square of the L2 norm of a row of the transmission spectrum matrix and the square of the Frobenius norm of the transmission spectrum matrix; and,

[0062] Step 3 further includes:

[0063] Step 3.1: Select a predetermined column of the transmission spectrum matrix based on the predetermined column selection probability; and,

[0064] Step 3.2: Select a predetermined row of the transmission spectrum matrix based on the predetermined row selection probability; and,

[0065] Step 4 further includes:

[0066] Step 4.1: Based on the projection vector before iteration and the predetermined columns of the transmission spectrum matrix and the square of their 2-norm, obtain the projection vector after iteration; and,

[0067] Step 4.2: Based on the inner product of the spectral vector before iteration and the predetermined row, the values ​​of the measured value vector and the projection vector after iteration at the corresponding positions of the predetermined row, the L2 norm of the predetermined row, and the predetermined row, an update vector is obtained.

[0068] In the above high-resolution spectral recovery method, the projection vector is initially the measured value vector or the measured value vector filled with zeros.

[0069] In the above high-resolution spectral restoration method, step 4.1 is represented as:

[0070]

[0071] Where z k-1 Let z be the projection vector before the k-th iteration. k Let be the projection vector after the k-th iteration. This is a predetermined column of the transmission spectrum matrix A in the k-th iteration.

[0072] In the above high-resolution spectral restoration method, the update vector is represented as:

[0073]

[0074] in The measured value vector b and the predetermined row i of the k-th iteration k The value at the corresponding position, The projection vector z after the kth iteration k With the predetermined row i in the k-th iteration k The value at the corresponding position, x k-1 The spectral vector prior to the k-th iteration, For the predetermined row i of the transmission spectrum matrix A in the k-th iteration k , for The 2-norm.

[0075] In the above high-resolution spectral recovery method, the termination condition is expressed as:

[0076]

[0077] Among them, ‖A‖ F Let ε1 be the Frobenius norm of the transmission spectrum matrix A, and let ε1 > 0.

[0078] In the above high-resolution spectral restoration method, step 6 further includes: setting the iteration termination condition for the projection vector, expressed as:

[0079]

[0080] Where ε2>0.

[0081] The high-resolution spectral restoration method provided in this application can achieve high-resolution spectral restoration through improved stochastic Kaczmarz algebra iteration. Compared with the analytical method commonly used in low-resolution spectral restoration, it can solve the problem of matrix inversion in high-resolution spectral restoration, can solve linear equation systems composed of large matrices, and has fast convergence speed and high restoration accuracy. It can be used for practical high-resolution spectral restoration.

[0082] Furthermore, the high-resolution spectral restoration method provided in this application can complete high-resolution spectral restoration in noisy scenes by applying random orthogonal projection to the observed measurement values ​​and combining it with random Kaczmarz algebraic iteration. This method effectively removes noise, has a fast convergence speed, and high restoration accuracy, and can be used for practical high-resolution spectral restoration. Attached Figure Description

[0083] Various other advantages and benefits of this application will become apparent to those skilled in the art upon reading the detailed description of the preferred embodiments below. The accompanying drawings are for illustrative purposes only and are not intended to limit the scope of this application. It is obvious that the drawings described below are merely some embodiments of this application, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort. Furthermore, the same reference numerals denote the same parts throughout the drawings.

[0084] Figure 1 The figure shows a schematic diagram of a spectral analysis apparatus according to an embodiment of this application;

[0085] Figure 2 A schematic flowchart of a high-resolution spectral recovery method according to an embodiment of this application is shown;

[0086] Figure 3 The illustration shows a schematic flowchart of a first preferred example of a high-resolution spectral recovery method according to an embodiment of this application;

[0087] Figure 4The illustration shows a schematic flowchart of a second preferred example of a high-resolution spectral recovery method according to an embodiment of this application;

[0088] Figure 5 The illustration shows a schematic flowchart of a third preferred example of a high-resolution spectral recovery method according to embodiments of this application; and

[0089] Figure 6 The illustration shows a schematic flowchart of a fourth preferred example of a high-resolution spectral recovery method according to an embodiment of this application. Detailed Implementation

[0090] Hereinafter, exemplary embodiments according to this application will be described in detail with reference to the accompanying drawings. Obviously, the described embodiments are merely some embodiments of this application, and not all embodiments of this application. It should be understood that this application is not limited to the exemplary embodiments described herein.

[0091] Computational spectral device overview

[0092] Figure 1 The figure shows a schematic diagram of a computational spectroscopy apparatus according to an embodiment of this application.

[0093] like Figure 1 As shown, in the computational spectral device according to the embodiments of this application, the optical system is optional, and may be an optical system such as a lens assembly or a homogenizing assembly. The filter structure is a broadband filter structure in the frequency domain or wavelength domain. The pass spectra of different wavelengths of the filter structure are not completely the same at different locations. The filter structure can be a metasurface, photonic crystal, nanopillar, multilayer film, dye, quantum dot, MEMS (microelectromechanical systems), FP etalon, cavity layer, waveguide layer, diffraction element, or other structures or materials with filtering properties. For example, in the embodiments of this application, the filter structure can be the light modulation layer in Chinese Patent CN201921223201.2.

[0094] The image sensor (i.e., photodetector array) can be a CMOS image sensor (CIS), CCD, array photodetector, etc. Additionally, optional data processing units can be MCUs, CPUs, GPUs, FPGAs, NPUs, ASICs, etc., which can export the data generated by the image sensor for external processing.

[0095] For example, after the image sensor measures the light intensity information, it is transmitted to the data processing unit for reconstruction calculation. This process is described in detail below:

[0096] The intensity signals of incident light at different wavelengths λ are denoted as x(λ), and the transmission spectrum curve of the filter structure is denoted as T(λ). The filter (filter structure) has m sets of structural units, and the transmission spectrum of each set of structural units is different. Overall, the filter structure can be denoted as Ti(λ) (i = 1, 2, 3, ..., m). Each set of structural units has a corresponding physical pixel below it, which detects the light intensity bi modulated by the filtered light structure. In a specific embodiment of this application, one physical pixel corresponds to one set of structural units, but it is not limited to this. In other embodiments, multiple physical pixels can also be grouped together to correspond to one set of structural units. Therefore, in the computational spectral device according to the embodiments of this application, multiple sets of structural units constitute a "spectral pixel". Furthermore, the present invention can use at least one spectral pixel to reconstruct an image. It should be noted that the number of effective transmission spectra (the transmission spectrum used for spectral recovery is called the effective transmission spectrum) Ti(λ) of the filter structure may not be the same as the number of structural units. The transmission spectrum of the filter structure is set, tested, or calculated manually according to certain rules based on the needs of identification or recovery (for example, the transmission spectrum obtained by testing each structural unit is the effective transmission spectrum). Therefore, the number of effective transmission spectra of the filter structure may be less than the number of structural units, or even more than the number of structural units. In this modified embodiment, a certain transmission spectrum curve is not necessarily determined by a set of structural units.

[0097] The relationship between the spectral distribution of incident light and the measurements from the image sensor can be expressed by the following formula:

[0098] bi=∫x(λ)*Ti(λ)*R(λ)dλ

[0099] Then discretize to obtain

[0100] bi=Σ(x(λ)*Ti(λ)*R(λ))

[0101] Where R(λ) is the response of the image sensor, denoted as:

[0102] Ai(λ)=Ti(λ)*R(λ),

[0103] The above equation can then be extended into matrix form:

[0104]

[0105] Where bi (i = 1, 2, 3, ..., m) is the response of the image sensor after the light to be measured passes through the filter structure, corresponding to the light intensity measurement values ​​of the image sensor corresponding to each of the m structural units. When one physical pixel corresponds to one structural unit, it can be understood as the light intensity measurement values ​​corresponding to m 'physical pixels', which is a vector of length m. A is the system's response to light of different wavelengths, determined by the transmittance of the filter structure and the quantum efficiency of the image sensor. A is a matrix, where each row vector corresponds to a set of structural units' responses to incident light of different wavelengths. Here, the incident light is sampled discretely and uniformly, with a total of n sampling points. The number of columns in A is the same as the number of sampling points of the incident light. Here, x(λ) is the light intensity of the incident light at different wavelengths λ, which is the incident light spectrum to be measured.

[0106] In another embodiment, unlike the above embodiments, the filter structure can be directly formed on the upper surface of the image sensor, such as quantum dots, nanowires, etc. It directly forms the filter structure or material (nanowires, quantum dots, etc.) in the photosensitive area of ​​the sensor. Taking the filter structure as an example, it can be understood that when the raw materials of the image sensor are processed to form the image sensor, a filter structure is formed on the upper surface of the raw materials. The transmission spectrum and the response of the image sensor are integrated, that is, it can be understood that the response of the detector and the transmission spectrum are the same curve. In this case, the relationship between the spectral distribution of the incident light and the light intensity measurement value of the image sensor can be expressed by the following formula:

[0107] bi=Σ(x(λ)*Ri(λ))

[0108] In this embodiment, the transmission spectrum Ai(λ) = Ri(λ)

[0109] Furthermore, it can also be a combination of the two embodiments described above, that is, at least one filter structure for modulating incident light is provided on the image sensor with the filter structure. It can be understood that the image sensor (i.e., photodetector array) in the first embodiment, which can be a CMOS image sensor (CIS), CCD, array photodetector, etc., can be replaced with an image sensor with an integrated filter structure in the second embodiment.

[0110] At this point, the relationship between the spectral distribution of the incident light and the light intensity measurement value of the image sensor can be expressed by the following formula:

[0111] bi=∫x(λ)*Ti(λ)*Ri(λ)dλ

[0112] Then discretize to obtain

[0113] bi=Σ(x(λ)*Ti(λ)*Ri(λ))

[0114] In this embodiment, Ai(λ) = Ti(λ) * Ri(λ)

[0115] Therefore, the problem of recovering the incident light spectrum to be measured is transformed into solving the following large-scale linear equation system:

[0116] Ax = b

[0117] Here, x is the spectral vector to be solved, which is an N×1 vector, where N represents the resolvable spectral wavenumber. Each element xi in the vector corresponds to a pixel value at the original sensor acquisition center. A is the M×N transmission spectrum matrix of the spectral chip, used to describe the response of the j-th pixel to the i-th spectral band. b is an M×1 measurement data vector, that is, the measurement value vector of the image sensor of the spectral chip, whose j-th element corresponds to the weight of the i-th spectral band.

[0118] Schematic spectral recovery method

[0119] For solving the large-scale linear equation system problem described above, one approach is to use the Kaczmarz iterative algorithm. The basic idea of ​​the Kaczmarz algorithm is to project the initial values ​​onto a hyperplane determined by each row vector of the matrix and its corresponding observation. The classic Kaczmarz algorithm iterates as follows:

[0120] Data: An m-dimensional vector of measurements, b = [b1, b2, ..., b...]. m ], m×n dimensional transmission spectrum matrix

[0121] Initialization: Maximum number of iterations P, initialize the spectral vector.

[0122] Calculate the index: i = mod(k, m)

[0123] kth iteration:

[0124] Convergence threshold: ||x k -x k+1 ||≤ε or k≤P

[0125] Where ε represents a manually set threshold, mod(·) represents the modulo operation, and <·> and ‖·‖ represent the inner product and L2 norm, respectively. Represents a i The transpose of the matrix. The classic Kaczmarz algorithm projects each row onto each hyperplane and uses the result of the previous projection to obtain the result of the next projection, thus approximating the final solution. Since the classic Kaczmarz algorithm only uses information from one row of the matrix, its convergence rate is heavily dependent on the row order.

[0126] Based on this, in this embodiment, a row of the transfer matrix is ​​iteratively selected, and the spectral vector is updated. For example, assume that the row index i corresponds to the k-th iteration. k ∈[m], where the probability of selecting each row is Where a i The i-th row of the transmission spectrum matrix A is represented by [m], where [m] represents the largest integer not exceeding m, and ||·|| F This represents the Frobenius norm. Thus, the selected i-th element of matrix A can be determined. k OK and the i-th value of the measurement vector b k Values ​​at each position Then, based on and The spectral vector is updated as follows:

[0127]

[0128] Additionally, based on the obtained spectral vector x k The iteration termination condition is set using the transmission spectrum matrix A and the measurement vector b. For example, the iteration termination condition could be:

[0129]

[0130] Where ε1>0.

[0131] Therefore, the high-resolution spectral recovery method according to the embodiments of this application may include the following steps:

[0132] Step 1: Obtain the transmission spectrum matrix of the spectral chip and the measurement value vector of the image sensor of the spectral chip; Step 2: Set a predetermined selection probability for each row of the transmission spectrum matrix, the predetermined selection probability being the quotient of the square of the L2 norm of a row of the transmission spectrum matrix and the square of the Frobenius norm of the transmission spectrum matrix; Step 3: Select a predetermined row of the transmission spectrum matrix based on the predetermined selection probability; Step 4: Obtain an update vector based on the inner product of the spectral vector before iteration and the predetermined row, the value of the measurement value vector at the corresponding position of the predetermined row, the L2 norm of the predetermined row, and the predetermined row; Step 5: Subtract the update vector from the spectral vector before iteration to obtain the iterated spectral vector; and Step 6: Repeat steps 3 to 5 until the iterated spectral vector satisfies a termination condition, the termination condition being based on the iterated spectral vector and its L2 norm, the transmission spectrum matrix and its Frobenius norm, and the measurement value vector.

[0133] Figure 2 A schematic flowchart of a high-resolution spectral recovery method according to an embodiment of this application is illustrated.Figure 2 As shown, the high-resolution spectral restoration method according to an embodiment of this application includes: step S110, obtaining the transmission spectrum matrix of a spectral chip and the measurement value vector of the image sensor of the spectral chip; step S120, setting a predetermined selection probability for each row of the transmission spectrum matrix, the predetermined selection probability being the square of the L2 norm of a certain row of the transmission spectrum matrix divided by the square of the Frobenius norm of the transmission spectrum matrix; step S130, selecting a predetermined row of the transmission spectrum matrix based on the predetermined selection probability; step S140, obtaining an update vector based on the inner product of the spectral vector before iteration and the predetermined row, the value of the measurement value vector at the corresponding position of the predetermined row, the L2 norm of the predetermined row, and the predetermined row; step S150, subtracting the update vector from the spectral vector before iteration to obtain the spectral vector after iteration; and step S160, repeating steps S130 to S150 until the spectral vector after iteration satisfies a termination condition, the termination condition being based on the spectral vector after iteration and its L2 norm, the transmission spectrum matrix and its Frobenius norm, and the measurement value vector.

[0134] Furthermore, in the aforementioned high-resolution spectral recovery method, the initial spectral vector can be set as a zero vector, represented as 0.

[0135] Furthermore, in the aforementioned high-resolution spectral recovery method, the update vector is represented as:

[0136]

[0137] in The measured value vector b and the predetermined row i of the k-th iteration k The value at the corresponding position, x k-1 The spectral vector prior to the k-th iteration, For the predetermined row i of the transmission spectrum matrix A in the k-th iteration k , for The 2-norm.

[0138] Furthermore, in the aforementioned high-resolution spectral recovery method, the termination condition is expressed as:

[0139]

[0140] Among them, ‖A‖ F Let ε1 be the Frobenius norm of the transmission spectrum matrix A, and let ε1 > 0.

[0141] Furthermore, considering that the above high-resolution spectral restoration methods are still based on the traditional Kaczmarz algorithm, which requires the equation system to be well-posed, and still suffers from slow iterative convergence speed and difficulty in modeling prior information, the high-resolution spectral restoration method according to the embodiments of this application introduces random orthogonal projection and prior information modeling methods in the algebraic reconstruction method to improve the convergence rate and restoration accuracy.

[0142] First preferred example

[0143] Compared to the description model of the traditional Kaczmarz reconstruction method:

[0144]

[0145] An improved Kaczmarz reconstruction method is proposed based on the following improved regularization description model:

[0146]

[0147] Wherein, λ>0 corresponds to the regularization coefficient, which can be estimated in real time using the L-curve method based on observed data and matrix A. Alternatively, the corresponding regularization coefficient can be obtained using parameter estimation methods such as maximum likelihood estimation, generalized maximum likelihood estimation, maximum a posteriori estimation, maximum entropy estimation, moment estimation, generalized moment estimation, leave-one-out cross-validation, generalized cross-validation, and N-fold cross-validation. Furthermore, D is a tridiagonal Toeplitz matrix, preferably a tridiagonal Toeplitz symmetric matrix, used to complete the structural prior modeling. A common tridiagonal Toeplitz matrix can be written in the following form...

[0148]

[0149] For a tridiagonal Toeplitz symmetric matrix; furthermore, the applicant of this application discovered during the research and development process that if a + b + c = 0, the effect or accuracy of spectral recovery is optimal, as shown in Table 1. Furthermore, considering generalization, a can be equal to 2, and b and c can be equal to -1.

[0150] Table 1

[0151]

[0152] Extracting coefficients from the spectral vector x using the original regularized description model above is equivalent to solving the following system of least-squares linear equations:

[0153]

[0154] The normal equations for this system of linear equations can be written as:

[0155]

[0156] The corresponding term of the normal equation can be written as:

[0157]

[0158] Solving the system of equations corresponding to (3) and (4) gives:

[0159]

[0160]

[0161] For solving the above normal equation system (5), which corresponds to solving the extended least squares (2) containing regular terms, note that the matrix A′ at this time corresponds to the matrix of the overdetermined system. Therefore, the matrix A′ of the overdetermined system can be used to replace the initial transmission spectrum matrix A, and the measured value vector b′ can be used to replace the test measurement value vector b. The least squares solution after random row selection is performed using the high-resolution spectral recovery method described above.

[0162] Therefore, in the above-mentioned high-resolution spectral restoration method, obtaining the transmission spectrum matrix of the spectral chip and the measurement vector of the image sensor of the spectral chip includes: obtaining the initial transmission spectrum matrix A of the spectral chip and the initial measurement vector b of the image sensor of the spectral chip; based on a regularized description model, extracting coefficients from the initial transmission spectrum matrix A and the initial measurement vector b to obtain the matrix A′ and the measurement vector b′ of the overdetermined system, wherein the regularized description model is:

[0163]

[0164] Where λ>0 are the regularization coefficients, F is a tridiagonal Toeplitz matrix, and ||·|| denotes the L2 norm. The matrix A′ and the measurement vector b′ of the overdetermined system are respectively:

[0165]

[0166]

[0167] Furthermore, the matrix A′ and the measurement vector b′ of the overdetermined system are respectively used as the transmission spectrum matrix and the measurement vector of the spectral chip.

[0168] Furthermore, in the aforementioned high-resolution spectral recovery method, the tridiagonal Toeplitz matrix is ​​a tridiagonal Toeplitz symmetric matrix.

[0169] Furthermore, in the aforementioned high-resolution spectral recovery method, the tridiagonal Toeplitz symmetric matrix is ​​represented as:

[0170]

[0171] Where a+b+c=0.

[0172] Then, a row of matrix A′ is iteratively selected, and the spectral vector is updated. For example, assume that the row index i corresponds to the k-th iteration. k ∈[m], where the probability of selecting each row is Where a i ' represents the i-th row of the transmission spectrum matrix A', [m] represents the largest integer not exceeding m, ‖·‖ F This represents the Frobenius norm. Thus, the selected i-th element of matrix A′ can be determined. k OK and the i-th value of the measurement vector b′ k Values ​​at each position Then, based on and The spectral vector is updated as follows:

[0173]

[0174] Additionally, based on the obtained spectral vector x k The iteration termination condition is set using the transmission spectrum matrix A′ and the measurement vector b′. For example, the iteration termination condition could be:

[0175]

[0176] Where ε1>0.

[0177] Therefore, in the above high-resolution spectral recovery method, the update vector is represented as:

[0178]

[0179] in The measured value vector b′ and the predetermined row i of the k-th iteration k The value at the corresponding position, x k-1 The spectral vector prior to the k-th iteration, For the predetermined row i of the transmission spectrum matrix A′ in the k-th iteration k , for The 2-norm.

[0180] Furthermore, in the aforementioned high-resolution spectral recovery method, the termination condition is expressed as:

[0181]

[0182] Where xk Let be the spectral vector after the k-th iteration, where ‖·‖ represents the L2 norm, and ‖·‖ F This represents the Frobenius norm.

[0183] Thus, this example can achieve higher-precision spectral recovery by introducing a structural prior modeling matrix D, while the structural prior matrix involved can be flexibly changed as needed.

[0184] Figure 3 A schematic flowchart illustrating a first preferred example of a high-resolution spectral recovery method according to an embodiment of this application is shown.

[0185] like Figure 3 As shown, a first preferred example of the high-resolution spectral restoration method according to an embodiment of this application includes: step S210, obtaining an initial transmission spectrum matrix of the spectral chip and an initial measurement value vector of the image sensor of the spectral chip; step S220, obtaining a transmission spectrum matrix and a measurement value vector from the initial transmission spectrum matrix and the initial measurement value vector by extracting coefficients from the spectral vector based on a regularized description model including a structural prior modeling matrix; step S230, setting a predetermined selection probability for each row of the transmission spectrum matrix, wherein the predetermined selection probability is the square of the L2 norm of a certain row of the transmission spectrum matrix divided by the square of the Frobenius norm of the transmission spectrum matrix; step S220; Step S240: Select a predetermined row of the transmission spectrum matrix based on the predetermined selection probability; Step S250: Obtain an update vector based on the inner product of the spectral vector before iteration and the predetermined row, the value of the measured value vector at the corresponding position of the predetermined row, the L2 norm of the predetermined row, and the predetermined row; Step S260: Subtract the update vector from the spectral vector before iteration to obtain the iterated spectral vector; and Step S270: Repeat steps S240 to S260 until the iterated spectral vector satisfies a termination condition, the termination condition being based on the iterated spectral vector and its L2 norm, the transmission spectrum matrix and its Frobenius norm, and the measured value vector.

[0186] Second preferred example

[0187] Furthermore, for the improved regularization description model as described above:

[0188]

[0189] The analytical method can be applied to solve this problem, by analyzing... Taking the partial derivative and setting it to zero, we can obtain the analytical solution in the form of:

[0190]

[0191] In this way, we obtain another form of the transmission spectrum matrix A′ and the corresponding measurement vector b′.

[0192] Therefore, in the above-described high-resolution spectral restoration method, obtaining the transmission spectrum matrix of the spectral chip and the measurement value vector of the image sensor of the spectral chip includes: obtaining the initial transmission spectrum matrix A of the spectral chip and the initial measurement value vector b of the image sensor of the spectral chip; and obtaining the matrix A′ and vector b′ of alternative forms from the initial transmission spectrum matrix A by taking the partial derivative of the regularized description model and setting it to zero, wherein the regularized description model is:

[0193]

[0194] Where λ>0 are the regularization coefficients, and D is a tridiagonal Toeplitz matrix, specifically the same as the tridiagonal Toeplitz matrix in the first example described above, which will not be repeated here. ‖·‖ denotes the L2 norm, and the matrix A′ and vector b′ are:

[0195] A′=A T A+λD T D

[0196] b′=A T b;

[0197] Furthermore, the matrix A′ and vector b′ are used as the transmission spectrum matrix of the spectral chip and the measurement value vector of the image sensor of the spectral chip.

[0198] Then, a row of matrix A′ is iteratively selected, and the spectral vector is updated. For example, assume that the row index i corresponds to the k-th iteration. k ∈[m], where the probability of selecting each row is Where a i ' represents the i-th row of the transmission spectrum matrix A', [m] represents the largest integer not exceeding m, ‖·‖ F This represents the Frobenius norm. Thus, the selected i-th element of matrix A′ can be determined. k OK and the i-th value of the measurement vector b′ k Values ​​at each position Then, based on and The spectral vector is updated as follows:

[0199] x 0 =0

[0200]

[0201] Additionally, based on the obtained spectral vector x kThe iteration termination condition is set using the transmission spectrum matrix A′ and the measurement vector b′. For example, the iteration termination condition could be:

[0202]

[0203] Where ε1>0.

[0204] Therefore, in the above high-resolution spectral recovery method, the update vector is represented as:

[0205]

[0206] in The measured value vector b′ and the predetermined row i of the k-th iteration k The value at the corresponding position, x k-1 The spectral vector prior to the k-th iteration, For the predetermined row i of the transmission spectrum matrix A′ in the k-th iteration k , for The 2-norm.

[0207] Furthermore, in the aforementioned high-resolution spectral recovery method, the termination condition is expressed as:

[0208]

[0209] Where x k Let be the spectral vector after the k-th iteration, where ‖·‖ represents the L2 norm, and ‖·‖ F This represents the Frobenius norm.

[0210] Similarly, this example can achieve higher-precision spectral recovery by introducing a structural prior modeling matrix D, while the structural prior matrix involved can be flexibly changed as needed.

[0211] Figure 4 A schematic flowchart illustrating a second preferred example of a high-resolution spectral recovery method according to an embodiment of this application is shown.

[0212] like Figure 4As shown, a second preferred example of the high-resolution spectral restoration method according to an embodiment of this application includes: step S310, obtaining an initial transmission spectrum matrix of the spectral chip and an initial measurement value vector of the image sensor of the spectral chip; step S320, based on a regularized description model including a structural prior modeling matrix, obtaining a transmission spectrum matrix and a measurement value vector from the initial transmission spectrum matrix and the initial measurement value vector by setting the partial derivative of the regularized description model to zero; step S330, setting a predetermined selection probability for each row of the transmission spectrum matrix, wherein the predetermined selection probability is the square of the L2 norm of a certain row of the transmission spectrum matrix divided by the square of the Frobenius norm of the transmission spectrum matrix. Step S340: Select a predetermined row of the transmission spectrum matrix based on the predetermined selection probability; Step S350: Obtain an update vector based on the inner product of the spectral vector before iteration and the predetermined row, the value of the measured value vector at the corresponding position of the predetermined row, the 2-norm of the predetermined row, and the predetermined row; Step S360: Subtract the update vector from the spectral vector before iteration to obtain the iterated spectral vector; and Step S370: Repeat steps S340 to S360 until the iterated spectral vector satisfies a termination condition, the termination condition being based on the iterated spectral vector and its 2-norm, the transmission spectrum matrix and its Frobenius norm, and the measured value vector.

[0213] Third preferred example

[0214] In this example, the applicant of this application recognized that, due to the presence of noise in the system, accuracy could be improved if the observed measurement vector b was first denoised before applying the random Kaczmarz algorithm.

[0215] Specifically, in this example, denoising is performed using random orthogonal projection to orthogonally project the observed measurement vector b onto the column space of the transmission spectrum matrix A. The random orthogonal projection randomly selects columns of A, for example, the column index j at the k-th iteration. k ∈[n], set its selection probability as Where a j Let [n] represent the j-th column of the transmission spectrum matrix A, and [n] represent the largest integer not exceeding n. F This represents the Frobenius norm. Thus, the projection vector z can be updated:

[0216] z 0 =b

[0217]

[0218] in For the selected j-th transmission spectrum matrix A k column, and express The square of the L2 norm.

[0219] Furthermore, the projection vector used for denoising is introduced into the stochastic Kaczmarz algorithm. That is, the projection vector corresponding to the row index i at the k-th iteration. k ∈[m], where the probability of selecting each row is Where a i The i-th row of the transmission spectrum matrix A is represented by [m], where [m] represents the largest integer not exceeding m, and ||·|| F This represents the Frobenius norm. Thus, the selected i-th element of matrix A can be determined. k OK and the i-th value of the measurement vector b k Values ​​at each position Then, based on and The spectral vector is updated as follows:

[0220] x 0 =0

[0221]

[0222] Additionally, based on the obtained spectral vector x k The iteration termination condition is set using the transmission spectrum matrix A′ and the measurement vector b. For example, the iteration termination condition could be:

[0223]

[0224] Where ε1>0.

[0225] Furthermore, an iteration termination condition for the projection vector is also set, for example:

[0226]

[0227] Where ε2>0, and ε1 can be equal to ε2.

[0228] Therefore, in the high-resolution spectral recovery method according to the embodiments of this application, step 2 further includes: step 2.1, setting a predetermined column selection probability for each column of the transmission spectrum matrix, the predetermined column selection probability being the quotient of the square of the L2 norm of a column of the transmission spectrum matrix and the square of the Frobenius norm of the transmission spectrum matrix; and step 2.2, setting a predetermined row selection probability for each row of the transmission spectrum matrix, the predetermined row selection probability being the quotient of the square of the L2 norm of a row of the transmission spectrum matrix and the square of the Frobenius norm of the transmission spectrum matrix; and step 3 further includes... The method includes: step 3.1, selecting a predetermined column of the transmission spectrum matrix based on the predetermined column selection probability; and step 3.2, selecting a predetermined row of the transmission spectrum matrix based on the predetermined row selection probability; and step 4 further includes: step 4.1, obtaining an iterated projection vector based on the projection vector before iteration and the predetermined column of the transmission spectrum matrix and the square of its L2 norm; and step 4.2, obtaining an update vector based on the inner product of the spectral vector before iteration and the predetermined row, the value of the measured value vector and the projection vector after iteration at the position corresponding to the predetermined row, the L2 norm of the predetermined row, and the predetermined row.

[0229] Furthermore, in the above-mentioned high-resolution spectral recovery method, the projection vector is initially the measured value vector or the measured value vector filled with zeros.

[0230] Furthermore, in the above-described high-resolution spectral restoration method, step 4.1 is represented as follows:

[0231]

[0232] Where z k-1 Let z be the projection vector before the k-th iteration. k Let be the projection vector after the k-th iteration. This is a predetermined column of the transmission spectrum matrix A in the k-th iteration.

[0233] Furthermore, in the aforementioned high-resolution spectral restoration method, the update vector is represented as:

[0234]

[0235] in The measured value vector b and the predetermined row i of the k-th iteration k The value at the corresponding position, The projection vector z after the kth iteration k With the predetermined row i in the k-th iteration k The value at the corresponding position, x k-1 The spectral vector prior to the k-th iteration, For the predetermined row i of the transmission spectrum matrix A in the k-th iteration k , for The 2-norm.

[0236] Furthermore, in the aforementioned high-resolution spectral recovery method, the termination condition is expressed as:

[0237]

[0238] Among them, ‖A‖ F Let ε1 be the Frobenius norm of the transmission spectrum matrix A, and let ε1 > 0.

[0239] Furthermore, in the above-described high-resolution spectral restoration method, step 6 further includes: setting the iteration termination condition for the projection vector, expressed as:

[0240]

[0241] Where ε2>0.

[0242] In this way, by introducing random orthogonal projection, this example can effectively reduce the noise of the observed measurement vector, thereby reducing the convergence threshold of the least squares solution and achieving higher precision spectral recovery.

[0243] Figure 5 The illustration shows a schematic flowchart of a third preferred example of a high-resolution spectral recovery method according to an embodiment of the present application.

[0244] like Figure 5As shown, a third preferred example of the high-resolution spectral restoration method according to an embodiment of this application includes: step S410, obtaining an initial transmission spectrum matrix of the spectral chip and an initial measurement value vector of the image sensor of the spectral chip; step S420, setting a predetermined column selection probability for each column of the transmission spectrum matrix and a predetermined row selection probability for each row of the transmission spectrum matrix, wherein the predetermined column selection probability is the quotient of the square of the L2 norm of a column of the transmission spectrum matrix and the square of the Frobenius norm of the transmission spectrum matrix, and the predetermined row selection probability is the quotient of the square of the L2 norm of a row of the transmission spectrum matrix and the square of the Frobenius norm of the transmission spectrum matrix; step S430, selecting a predetermined column of the transmission spectrum matrix based on the predetermined column selection probability and the predetermined row selection probability based on the predetermined row selection probability. Step S440: Select a predetermined row of the transmission spectrum matrix with a probability of selection; Step S440: Obtain the iterated projection vector based on the projection vector before iteration and the predetermined column of the transmission spectrum matrix and the square of its second norm, and obtain an update vector based on the inner product of the spectral vector before iteration and the predetermined row, the value of the measured value vector and the corresponding position of the iterated projection vector in the predetermined row, the second norm of the predetermined row, and the predetermined row; Step S450: Subtract the update vector from the spectral vector before iteration to obtain the iterated spectral vector; and Step S460: Repeat steps 430 to 450 until the iterated spectral vector satisfies the termination condition, the termination condition being based on the iterated spectral vector and its second norm, the transmission spectrum matrix and its Frobenius norm, and the measured value vector.

[0245] Fourth preferred example

[0246] In this example, the structural prior modeling matrix in the first or second preferred example can be combined with the random orthogonal projection in the third preferred example.

[0247] Specifically, for example, a regularized description model based on the improved Kaczmarz reconstruction method:

[0248]

[0249] Wherein, λ>0 corresponds to the regularization coefficient, which can be estimated in real time using the L-curve method based on observed data and matrix A. Alternatively, the corresponding regularization coefficient can be obtained using parameter estimation methods such as maximum likelihood estimation, generalized maximum likelihood estimation, maximum a posteriori estimation, maximum entropy estimation, moment estimation, generalized moment estimation, leave-one-out cross-validation, generalized cross-validation, and N-fold cross-validation. Furthermore, D is a tridiagonal Toeplitz matrix, preferably a tridiagonal Toeplitz symmetric matrix, used to complete the structural prior modeling. A common tridiagonal Toeplitz matrix can be written in the following form...

[0250]

[0251] For a tridiagonal Toeplitz symmetric matrix; furthermore, the applicant of this application discovered during the research and development process that if a + b + c = 0, the effect or accuracy of spectral recovery is optimal, as shown in Table 1 above. Furthermore, considering generalization, a can be equal to 2, and b and c can be equal to -1.

[0252] Extracting coefficients from the spectral vector x using the original regularized description model above is equivalent to solving the following system of least-squares linear equations:

[0253]

[0254] The normal equations for this system of linear equations can be written as:

[0255]

[0256] The corresponding term of the normal equation can be written as:

[0257]

[0258] Solving the system of equations corresponding to (3) and (4) gives:

[0259]

[0260]

[0261] For solving the above normal equation system (5), which corresponds to solving the extended least squares (2) containing regular terms, note that the matrix A′ at this time corresponds to the matrix of the overdetermined system. Therefore, the matrix A′ of the overdetermined system can be used to replace the initial transmission spectrum matrix A, and the measured value vector b′ can be used to replace the test measurement value vector b. The least squares solution after random row and column selection is performed by applying the random Kaczmarz algorithm.

[0262] Furthermore, since the system contains noise, the observed measurement vector b′ can be denoised before applying the stochastic Kaczmarz algorithm. That is, as described in the third preferred example above, a stochastic orthogonal projection method is used for denoising to orthogonally project the observation b′ onto the column space of A′. The stochastic orthogonal projection randomly selects columns of A′, corresponding to the column index j at the k-th iteration. k ∈[n], set its selection probability as Where a j ' represents the j-th column of the transmission spectrum matrix A', [n] represents the largest integer not exceeding n, ‖·‖ F This represents the Frobenius norm. Thus, the projection vector z can be updated:

[0263] z 0 =b′

[0264]

[0265] Where A jk ′ is the selected j-th element of the transmission spectrum matrix A′ k Column, and ||A jk ′|| 2 A represents jk The square of the 2norm of ′.

[0266] The termination principle for the above iterations is as follows:

[0267]

[0268] Where ε2>0.

[0269] This is further combined with the random Kaczmarz iteration, corresponding to the row index i at the k-th iteration. k ∈[m], where the probability of selecting each row is Where a i ' represents the i-th row of the transmission spectrum matrix A', [m] represents the largest integer not exceeding m, ‖·‖ F This represents the Frobenius norm. Thus, the selected i-th element of matrix A′ can be determined. k OK The i-th value of the measured vector b′ and the projection vector z at the k-th iteration. k Values ​​at each position and Then, based on and The spectral vector is updated as follows:

[0270]

[0271] Additionally, based on the obtained spectral vector x k The iteration termination condition is set using the transmission spectrum matrix A′ and the measurement vector b′. For example, the iteration termination condition could be:

[0272]

[0273] Where ε1>0, and ε1 can be equal to ε2.

[0274] Therefore, the improved Kaczmarz algorithm of this preferred example introduces random orthogonal projection into the traditional Kaczmarz algorithm to effectively reduce the noise of the observed measurement vector b, thereby reducing the convergence threshold of the least squares solution. Furthermore, when combined with the structural prior modeling matrix D, it can achieve higher precision spectral recovery. At the same time, the structural prior modeling matrix D can be flexibly changed as needed.

[0275] Figure 6 The illustration shows a schematic flowchart of a fourth preferred example of a high-resolution spectral recovery method according to an embodiment of the present application.

[0276] like Figure 6 As shown, a fourth preferred example of the high-resolution spectral restoration method according to an embodiment of this application includes: step S510, obtaining an initial transmission spectrum matrix of the spectral chip and an initial measurement value vector of the image sensor of the spectral chip; step S520, obtaining a transmission spectrum matrix and a measurement value vector from the initial transmission spectrum matrix and the initial measurement value vector by extracting coefficients from the spectral vector based on a regularized description model including a structural prior modeling matrix; step S530, setting a predetermined column selection probability for each column of the transmission spectrum matrix and a predetermined row selection probability for each row of the transmission spectrum matrix, wherein the predetermined column selection probability is the quotient of the square of the L2 norm of a column of the transmission spectrum matrix and the square of the Frobenius norm of the transmission spectrum matrix, and the predetermined row selection probability is the quotient of the square of the L2 norm of a row of the transmission spectrum matrix and the square of the Frobenius norm of the transmission spectrum matrix; step S5 40. Select a predetermined column of the transmission spectrum matrix based on the predetermined column selection probability, and select a predetermined row of the transmission spectrum matrix based on the predetermined row selection probability; Step S550. Obtain the iterated projection vector based on the projection vector before iteration and the predetermined column of the transmission spectrum matrix and the square of its L2 norm, and obtain an update vector based on the inner product of the spectral vector before iteration and the predetermined row, the value of the measured value vector and the projection vector after iteration corresponding to the predetermined row, the L2 norm of the predetermined row, and the predetermined row; Step S560. Subtract the update vector from the spectral vector before iteration to obtain the iterated spectral vector; and Step S570. Repeat steps 540 to 560 until the iterated spectral vector satisfies the termination condition, the termination condition being based on the iterated spectral vector and its L2 norm, the transmission spectrum matrix and its Frobenius norm, and the measured value vector.

[0277] In other words, the high-resolution spectral recovery method in this example may include the following steps:

[0278] Step 1: Obtain the transmission spectrum A of the corresponding filter structure, the output b, and the custom tridiagonal Toeplitz structure prior matrix D using a spectral analysis device.

[0279] Step 2: Initialization, setting the initial iteration value x 0 =0, fill in zeros according to the output b to obtain the initial value for iteration. And the iterative convergence thresholds ε1 and ε2.

[0280] Step 3: Parameter estimation. Based on the output b and the transmission spectrum A, parameter estimation methods such as the L-curve method are applied to obtain... Thus, A′ is obtained.

[0281] Step 4: Iteration process, with the number of iterations being, for example, 1, 2, ..., k. Repeat the following steps until convergence:

[0282] ①: Random selection of rows and columns:

[0283] Column subscript j k ∈[n], set its selection probability as Get j k ;

[0284] line subscript i k ∈[m], set its selection probability as Get i k ;

[0285] ②: Random Kaczmarz iteration: using the formula:

[0286]

[0287]

[0288] Iteration yields z k and x k .

[0289] Step 5: Determine the convergence criteria:

[0290] and

[0291] If the condition is met, stop iterating; otherwise, continue iterating according to step four.

[0292] Of course, those skilled in the art will understand that the second preferred example and the third preferred example described above can be combined. It is only necessary to modify step 520 above to be based on a regularized description model containing a structural prior modeling matrix. By taking the partial derivative of the regularized description model and setting it to zero, the transmission spectrum matrix and the measurement value vector can be obtained from the initial transmission spectrum matrix and the initial measurement value vector.

[0293] The basic principles of this application have been described above with reference to specific embodiments. However, it should be noted that the advantages, benefits, and effects mentioned in this application are merely examples and not limitations, and should not be considered as essential features of each embodiment of this application. Furthermore, the specific details disclosed above are for illustrative and facilitative purposes only, and are not limitations. These details do not limit the application to the necessity of employing the aforementioned specific details for implementation.

[0294] The block diagrams of devices, apparatuses, devices, and systems involved in this application are merely illustrative examples and are not intended to require or imply that they must be connected, arranged, or configured in the manner shown in the block diagrams. As those skilled in the art will recognize, these devices, apparatuses, devices, and systems can be connected, arranged, and configured in any manner. Words such as “comprising,” “including,” “having,” etc., are open-ended terms meaning “including but not limited to,” and are used interchangeably with them. The terms “or” and “and” as used herein refer to the terms “and / or,” and are used interchangeably with them unless the context clearly indicates otherwise. The term “such as” as used herein refers to the phrase “such as but not limited to,” and is used interchangeably with it.

[0295] It should also be noted that in the apparatus, equipment, and methods of this application, the components or steps can be disassembled and / or recombined. These disassemblies and / or recombinations should be considered as equivalent solutions of this application.

[0296] The above description of the disclosed aspects is provided to enable any person skilled in the art to make or use this application. Various modifications to these aspects will be readily apparent to those skilled in the art, and the general principles defined herein can be applied to other aspects without departing from the scope of this application. Therefore, this application is not intended to be limited to the aspects shown herein, but rather to be accorded the widest scope consistent with the principles and novel features disclosed herein.

[0297] The above description has been given for purposes of illustration and description. Furthermore, this description is not intended to limit the embodiments of this application to the forms disclosed herein. Although numerous exemplary aspects and embodiments have been discussed above, those skilled in the art will recognize certain variations, modifications, alterations, additions, and sub-combinations thereof.

Claims

1. A high-resolution spectral reconstruction method, characterized in that, include: Step 1: Obtain the transmission spectrum matrix of the spectral chip and the measurement value vector of the image sensor of the spectral chip; Step 2: Set a predetermined selection probability for each row of the transmission spectrum matrix. The predetermined selection probability is the quotient of the square of the L2 norm of a certain row of the transmission spectrum matrix and the square of the Frobenius norm of the transmission spectrum matrix. Step 3: Select a predetermined row of the transmission spectrum matrix based on the predetermined selection probability; Step 4: Based on the inner product of the spectral vector before iteration and the predetermined row, the measured value vector and the value at the corresponding position of the predetermined row, the L2 norm of the predetermined row, and the predetermined row, an update vector is obtained; Step 5: Subtract the updated vector from the spectral vector before iteration to obtain the spectral vector after iteration; and, Step 6: Repeat steps 3 to 5 until the iterated spectral vector satisfies the termination condition, which is based on the iterated spectral vector and its L2 norm, the transmission spectrum matrix and its Frobenius norm, and the measured value vector. The update vector is represented as: in The measured value vector b and the predetermined row i of the k-th iteration k The value at the corresponding position, x k-1 The spectral vector prior to the k-th iteration, For the predetermined row i of the transmission spectrum matrix A in the k-th iteration k , for The 2-norm.

2. The high-resolution spectral recovery method as described in claim 1, wherein, The initial spectral vector is set to a zero vector, represented as 0.

3. The high-resolution spectral restoration method as described in claim 1, wherein, The termination condition is expressed as follows: Among them, ||A|| F Let ε1 be the Frobenius norm of the transmission spectrum matrix A, and let ε1 represent a manually set threshold and ε1>0.

4. The high-resolution spectral recovery method as described in claim 1, wherein, Obtaining the transmission spectrum matrix of the spectral chip and the measurement vector of the image sensor of the spectral chip includes: Obtain the initial transmission spectrum matrix A of the spectral chip and the initial measurement value vector b of the image sensor of the spectral chip; Based on the regularized description model, the matrix A′ and the measurement vector b′ of the overdetermined system are obtained from the initial transmission spectrum matrix A and the initial measurement vector b by extracting coefficients from the spectral vector. The regularized description model is as follows: Where λ>0 are the regularization coefficients, D is a tridiagonal Toeplitz matrix, ||·|| denotes the L2 norm, and the matrix A′ and the measurement vector b′ of the overdetermined system are respectively: Furthermore, the matrix A′ and the measurement vector b′ of the overdetermined system are respectively used as the transmission spectrum matrix and the measurement vector of the spectral chip.

5. The high-resolution spectral recovery method as described in claim 4, wherein, The tridiagonal Toeplitz matrix is ​​a tridiagonal Toeplitz symmetric matrix.

6. The high-resolution spectral recovery method as described in claim 5, wherein, The tridiagonal Toeplitz symmetric matrix is ​​represented as: Where a+b+c=0.

7. The high-resolution spectral recovery method as described in claim 6, wherein, The update vector is represented as: in The measured value vector b′ and the predetermined row i of the k-th iteration k The value at the corresponding position, x k-1 The spectral vector prior to the k-th iteration, For the predetermined row i of the transmission spectrum matrix A′ in the k-th iteration k , for The 2-norm.

8. The high-resolution spectral recovery method as described in claim 4 or 7, wherein, The termination condition is expressed as follows: Where x k Let be the spectral vector after the k-th iteration, where ||·|| represents the L2 norm, and ||·|| F Let ε represent the Frobenius norm, and ε1 represent a manually set threshold.

9. The high-resolution spectral recovery method as described in claim 1, wherein, Obtaining the transmission spectrum matrix of the spectral chip and the measurement vector of the image sensor of the spectral chip includes: Obtain the initial transmission spectrum matrix A of the spectral chip and the initial measurement value vector b of the image sensor of the spectral chip; By taking the partial derivative of the regularized description model and setting it to zero, we obtain the matrix A′ and vector b′ in alternative forms from the initial transmission spectrum matrix A, wherein the regularized description model is: Where λ>0 is the regularization coefficient, D is a tridiagonal Toeplitz matrix, ||·|| denotes the L2 norm, and the matrix A′ and vector b′ are: A′=A T A+λD T D b′=A T b; Furthermore, the matrix A′ and vector b′ are used as the transmission spectrum matrix of the spectral chip and the measurement value vector of the image sensor of the spectral chip.

10. The high-resolution spectral recovery method as described in claim 9, wherein, The tridiagonal Toeplitz matrix is ​​a tridiagonal Toeplitz symmetric matrix.

11. The high-resolution spectral recovery method as described in claim 10, wherein, The tridiagonal Toeplitz symmetric matrix is ​​represented as: Where a+b+c=0.

12. The high-resolution spectral recovery method as described in claim 9 or 11, wherein, The update vector is represented as: in The measured value vector b′ and the predetermined row i of the k-th iteration k The value at the corresponding position, x k-1 The spectral vector prior to the k-th iteration, For the predetermined row i of the transmission spectrum matrix A′ in the k-th iteration k , for The 2-norm.

13. The high-resolution spectral recovery method as described in claim 12, wherein, The termination condition is expressed as follows: Where x k Let be the spectral vector after the k-th iteration, where ||·|| represents the L2 norm, and ||·|| F Let ε represent the Frobenius norm, and ε1 represent a manually set threshold.

14. The high-resolution spectral restoration method according to any one of claims 1 to 13, wherein, Step 2 further includes: Step 2.1: Set a predetermined column selection probability for each column of the transmission spectrum matrix, wherein the predetermined column selection probability is the quotient of the square of the L2 norm of a column of the transmission spectrum matrix and the square of the Frobenius norm of the transmission spectrum matrix; and, Step 2.2, set a predetermined row selection probability for each row of the transmission spectrum matrix, wherein the predetermined row selection probability is the quotient of the square of the L2 norm of a row of the transmission spectrum matrix and the square of the Frobenius norm of the transmission spectrum matrix; and, Step 3 further includes: Step 3.1: Select a predetermined column of the transmission spectrum matrix based on the predetermined column selection probability; and, Step 3.2: Select a predetermined row of the transmission spectrum matrix based on the predetermined row selection probability; and, Step 4 further includes: Step 4.1: Based on the projection vector before iteration and the predetermined columns of the transmission spectrum matrix and the square of their 2-norm, obtain the projection vector after iteration; and, Step 4.2: Based on the inner product of the spectral vector before iteration and the predetermined row, the values ​​of the measured value vector and the projection vector after iteration at the corresponding positions of the predetermined row, the L2 norm of the predetermined row, and the predetermined row, an update vector is obtained.

15. The high-resolution spectral recovery method as described in claim 14, wherein, The projection vector is initially the measured value vector or the measured value vector filled with zeros.

16. The high-resolution spectral recovery method as described in claim 15, wherein, Step 4.1 is represented as follows: Where z k-1 Let z be the projection vector before the k-th iteration. k Let be the projection vector after the k-th iteration. This is a predetermined column of the transmission spectrum matrix A in the k-th iteration.

17. The high-resolution spectral recovery method as described in claim 16, wherein, The update vector is represented as: in The measured value vector b and the predetermined row i of the k-th iteration k The value at the corresponding position, The projection vector z after the kth iteration k With the predetermined row i in the k-th iteration k The value at the corresponding position, x k-1 The spectral vector prior to the k-th iteration, For the predetermined row i of the transmission spectrum matrix A in the k-th iteration k , for The 2-norm.

18. The high-resolution spectral recovery method as described in claim 17, wherein, The termination condition is expressed as follows: Among them, ||A|| F Let ε1 be the Frobenius norm of the transmission spectrum matrix A, and let ε1 represent a manually set threshold and ε1>0.

19. The high-resolution spectral recovery method as described in claim 14, wherein, Step 6 further includes: setting the iteration termination condition for the projection vector, expressed as: Where ε2 represents a manually set threshold and ε2>0.

Images