An optimization method for broadband filter sets with equal optical efficiency
By optimizing the optical efficiency of the broadband filter set and keeping it consistent on each channel, the problem of inconsistent optical efficiency between broadband filters in the prior art is solved, and a higher quality and robust spectral reconstruction effect is achieved.
Patent Information
- Application Number
- CN202410721535.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-06-05
- Publication Date
- 2025-05-13
- Estimated Expiration
- 2044-06-05
AI Technical Summary
In the existing broadband spectral technology, the optical efficiency between multiple broadband filters is inconsistent, resulting in inconsistent signal-to-noise ratios of each spectral acquisition channel, which is not conducive to high-quality spectral reconstruction.
By establishing a mathematical model of spectral imaging of broadband filter sets, the constraint relationship between the relative error of the solution and the number of conditions is obtained, and the cost function of the equal-efficient broadband filter set is constructed, and the broadband filter set is optimized by using the method of minimizing the number of conditions.
The consistent signal-to-noise ratio of each spectral acquisition channel is achieved, the quality and robustness of spectral reconstruction are improved, and the peak signal-to-noise ratio, structural similarity and spectral angle matching of the reconstructed spectral image are significantly improved.
Smart Images
Figure CN118913440B_ABST
Abstract
Description
Technical Field
[0001] The invention relates to a broadband filter set with equal light efficiency (BFSELE) optimization method, which is used to achieve high-quality and real-time spectrum acquisition, can perform color reproduction and spectrum reconstruction, and belongs to the technical field of spectral imaging. Background Art
[0002] Spectral imaging technology has important research and application potential in various fields such as remote sensing and biomedicine. Traditional spectral imaging techniques, including dispersion, filter-based and interference techniques, are impractical for high-quality and real-time spectral imaging. In recent years, broadband spectral imaging technology has emerged as a new method to break the current bottleneck. In broadband spectral imaging systems, the accuracy of spectral reconstruction is closely related to the spectral sensitivity function (SSF) of broadband filters. Therefore, there have been many studies on the optimization design of spectral sensitivity functions. Current research mainly focuses on selecting filter sets from various commercial filters, appropriately designing the spectral sensitivity function of filters, or using tunable optical devices to modulate the spectral domain. In spectral imaging systems, the optimal broadband filter set can provide better performance than the narrowband filter set. The reason is that broadband filters have higher optical efficiency and can obtain more spectral information than narrowband filters. However, these studies did not consider the impact of the difference in optical efficiency between multiple broadband filters. The difference in optical efficiency makes the signal-to-noise ratio of each spectral acquisition channel inconsistent, which is not conducive to high-quality spectral reconstruction. Therefore, it is necessary to design broadband filter sets with equal optical efficiency. Summary of the invention
[0003] In order to solve the problems of inconsistent light efficiency and weak noise robustness among multiple broadband filters in the existing broadband spectroscopy technology, the present invention proposes an optimization method for a group of broadband filters with equal light efficiency.
[0004] The technical solution of the present invention to solve the technical problem is:
[0005] A method for optimizing a broadband filter set of equal optical efficiency, the method comprising the following steps:
[0006] Step 1, establishing a mathematical model for spectral imaging of broadband filter sets;
[0007] Step 2, obtain the constraint relationship between the relative error of the solution and the condition number;
[0008] Step 3: construct the cost function of broadband filter set with equivalent optical efficiency (BFSELE).
[0009] The step 1 is specifically as follows: during the operation of the spectral imaging system with broadband filters, the incident spectrum of the target scene is modulated multiple times in the spectral dimension through a series of broadband filters; then, a modulation coded image of the incident spectrum of the target scene is measured and captured by a CCD / CMOS sensor; finally, the incident spectrum information of the target scene is obtained by reconstruction calculation; S i is the i-th pixel value of the sensor, which can be expressed as
[0010]
[0011] where λ is the wavelength, λ k is the kth spectral band Λ k The central wavelength of [λ min ,λ max ] is the spectral range of the system spectral response, R(λ) is the distribution of incident spectral intensity with wavelength, θ i (λ) is the spectral sensitivity function of the i-th broadband filter; so far, the mathematical model of the broadband filter group spectral imaging process can be expressed as the following equation group
[0012]
[0013] S is a column vector, where each element represents the i-th measurement value; Θ is a sensor matrix, where each row vector corresponds to the spectral sensitivity function of the broadband filter; R is a column vector, where each element represents the incident spectral intensity to be solved.
[0014] The specific step 2 is as follows: there is a discrete error between the spectral imaging process described by equation (2) in step 1 and the actual physical process; Θ and S are obtained through the measurement process, and the measurement error is introduced in the process; when solving the equation, the above error is transmitted, which eventually leads to a decrease in the accuracy of the reconstructed spectrum; assuming that Θ and S have δ Θ and δ S If the error is small, the equation has an exact solution R+δ R ,Right now
[0015] (S+δ S )=(Θ+δ Θ )(R+δ R ). (3)
[0016] According to the matrix principle, when the number of measurements is equal to the number of spectral bands, the relative error of the solution ||δ R || / ||R|| is
[0017]
[0018] where κ(Θ) is the condition number of Θ, defined as κ(Θ) = ||Θ||||Θ + ||. Θ + is the pseudo-inverse of Θ, which is equal to Θ when Θ is non-singular. -1 For the 2-norm, κ(Θ)=σ max / σ min ≥1 is the ratio of the largest and smallest singular values of Θ. Similarly, when the number of measurements is greater than the number of spectral bands, the relative error of the solution ||δ R || / ||R|| is
[0019]
[0020] Equations (12) and (13) show that the relative error of the solution δ R / R is restricted by the condition number κ(Θ). When κ(Θ) is close to 1, the solution has better robustness.
[0021] The step three is specifically as follows: the optical efficiency of each broadband filter is defined as
[0022]
[0023] For a set of broadband filters with equal optical efficiency, all elements of the filter vector η i The sum of should be equal; then, in order to find the equivalent optical efficiency broadband filter set, it can be achieved by solving the following problem, that is,
[0024]
[0025] ρ∈(0,m) is a constant representing the target light efficiency level. To minimize the condition number κ(Θ) in equation (15), a scheme for minimizing the condition number based on soft orthogonal constraints is proposed, which directly takes the angle between paired row vectors as the minimum optimal target. Finally, the cost function of the broadband filter set with equal light efficiency (BFSELE) is expressed as
[0026]
[0027] Among them, Θ T is the transposed matrix of Θ, ΘΘ T The off-diagonal elements in are the dot products of the paired row vectors in Θ, γ represents the paired angle matrix, θ i represents the row vector of the sensor matrix Θ, Represents θ i So far, the three parameters of the number of spectral bands, the number of broadband filters, and the target light efficiency level are set in advance, and the cost function of the broadband filter set with equal light efficiency (BFSELE) is solved to obtain the broadband filter set with equal light efficiency (BFSELE).
[0028] The technical effect of the present invention is that the method of the present invention utilizes the constraint relationship between the relative error of the solution and the condition number and the equal light efficiency constraint regularization term to construct the cost function of the equal light efficiency broadband filter set (BFSELE). This method is very flexible, and only the parameters such as the number of spectral bands, the number of broadband filters and the target light efficiency level need to be simply set to quickly obtain the equal light efficiency broadband filter set that meets the needs of practical applications. The equal light efficiency broadband filter set obtained by the method of the present invention has a consistent signal-to-noise ratio for each spectral acquisition channel, and has better color reproduction and spectral reconstruction capabilities under various noise levels. The peak signal-to-noise ratio (PSNR), structural similarity (SSIM) and spectral angle mapper (SAM) of the reconstructed spectral image have obvious gains compared with the non-equal light efficiency broadband filter set (BFSULE). BRIEF DESCRIPTION OF THE DRAWINGS
[0029] Figure 1 , Workflow diagram of the spectral imaging system with broadband filters.
[0030] Figure 2 , Visualization of spectral reconstruction results (PSNR / SSIM / SAM) on the CAVE dataset. DETAILED DESCRIPTION
[0031] The present invention is further described in detail below with reference to the accompanying drawings.
[0032] A method for optimizing a broadband filter set of equal optical efficiency, the method comprising the following steps:
[0033] Step 1: Establish a mathematical model for spectral imaging of broadband filter sets.
[0034] During the operation of the spectral imaging system with broadband filters, the incident spectrum of the target scene is modulated multiple times in the spectral dimension through a series of broadband filters. Then, the CCD / CMOS sensor is used to measure and capture the modulation coded image of the incident spectrum of the target scene. Finally, the incident spectrum information of the target scene is obtained through reconstruction calculation. Figure 1 In, S i is the i-th pixel value of the sensor, which can be expressed as
[0035]
[0036] where λ is the wavelength, λ k is the kth spectral band Λ k The central wavelength of [λ min ,λ max] is the spectral range of the system spectral response, R(λ) is the distribution of incident spectral intensity with wavelength, θ i (λ) is the spectral sensitivity function of the i-th broadband filter. So far, the mathematical model of the broadband filter set spectral imaging process can be expressed as the following set of equations:
[0037]
[0038] S is a column vector, where each element represents the i-th measurement value. Θ is a sensor matrix, where each row vector corresponds to the spectral sensitivity function of the broadband filter. R is a column vector, where each element represents the incident spectral intensity to be solved.
[0039] Step 2: Obtain the constraint relationship between the relative error of the solution and the condition number.
[0040] There is a discrete error between the spectral imaging process described by equation (2) in step 1 and the actual physical process. Both Θ and S are obtained through the measurement process, and the measurement error is introduced in the process. When solving the equation, the above error is transmitted, which eventually leads to a decrease in the accuracy of the reconstructed spectrum. Assume that Θ and S have δ Θ and δ S If the error is small, the equation has an exact solution R+δ R ,Right now
[0041] (S+δ S )=(Θ+δ Θ )(R+δ R ). (11)
[0042] According to the matrix principle, when the number of measurements is equal to the number of spectral bands, the relative error of the solution ||δ R || / ||R|| is
[0043]
[0044] where κ(Θ) is the condition number of Θ, defined as κ(Θ) = ||Θ||||Θ + ||. Θ + is the pseudo-inverse of Θ, which is equal to Θ when Θ is non-singular. -1 For the 2-norm, κ(Θ)=σ max / σ min ≥1 is the ratio of the largest and smallest singular values of Θ. Similarly, when the number of measurements is greater than the number of spectral bands, the relative error of the solution ||δ R || / ||R|| is
[0045]
[0046] Equations (12) and (13) show that the relative error of the solution δR / R is limited by the condition number κ(Θ). When κ(Θ) is close to 1, the solution has better robustness. So far, the influence of broadband filter set error and sensor noise on the accuracy of reconstructed spectrum has been analyzed from a theoretical level, and the broadband filter set can be optimized by minimizing the condition number method.
[0047] Step 3: construct the cost function of broadband filter set with equivalent optical efficiency (BFSELE).
[0048] The optical efficiency of each broadband filter is defined as
[0049]
[0050] For a set of broadband filters with equal optical efficiency, all elements of the filter vector η i The sum of should be equal. Then, in order to find the equivalent broadband filter set, it can be achieved by solving the following problem, that is,
[0051]
[0052] ρ∈(0,m) is a constant representing the target light efficiency level. In order to minimize the condition number κ(Θ) in equation (15), a scheme for minimizing the condition number based on soft orthogonal constraints is proposed, which directly takes the angle between paired row vectors as the minimum optimal target. Finally, the cost function of the broadband filter set with equal light efficiency (BFSELE) can be expressed as
[0053]
[0054] Among them, Θ T is the transposed matrix of Θ, ΘΘ T The off-diagonal elements in are the dot products of the paired row vectors in Θ, γ represents the paired angle matrix, θ i represents the row vector of the sensor matrix Θ, Represents θ i So far, the three parameters of the number of spectral bands, the number of broadband filters, and the target light efficiency level are set in advance, and the cost function of the broadband filter set with equal light efficiency (BFSELE) is solved to obtain the broadband filter set with equal light efficiency (BFSELE).
[0055] Example:
[0056] Firstly, the number of spectral bands is set to 16, the number of broadband filters is set to 16, and the target light efficiency levels are set to 7, 8, and 9 respectively. The adaptive motion estimation algorithm (Adam) based on the penalty function method is used to solve the cost function shown in Formula 8, and three broadband filter groups with equal light efficiency (BFSELE) are obtained. The 2-norm-matrix condition number and light efficiency level are calculated, as shown in Table 1.
[0057] Then, three non-equivalent broadband filter sets (BFSULE) are selected from commercial filters, and their 2-norm-matrix condition numbers are kept consistent with the above-mentioned equi-equivalent broadband filter set (BFSELE), as shown in Table 2.
[0058] Next, the six filter sets in Table 1 were used to simulate spectral imaging, and a spectral reconstruction experiment was performed on the CAVE dataset. The inverse matrix method was used to solve Equation 2 to obtain the reconstructed spectrum.
[0059] Finally, the error evaluation indicators between the target spectrum and the reconstructed spectrum are calculated, including peak signal-to-noise ratio (PSNR), structural similarity (SSIM), and spectral angle matching (SAM), to compare the performance of different broadband filter sets in spectral imaging. The spectral reconstruction results (PSNR / SSIM / SAM) of each set of broadband filter sets on the CAVE dataset are shown in Table 3, and the visualization results are shown in Figure 2 shown.
[0060] Comparing the error evaluation index results of the broadband filter groups with equal optical efficiency of filter groups No. 1, No. 2, and No. 3 and the broadband filter groups with non-equivalent optical efficiency of filter groups No. 4, No. 5, and No. 6 in Table 3, it can be seen that the PSNR, SSIM, and SAM of broadband filter groups No. 1, No. 2, and No. 3 are better than those of broadband filter groups No. 4, No. 5, and No. 6. This significant gain can be seen more intuitively from Table 3. The index curves of the broadband filter groups with equal optical efficiency of filter groups No. 1, No. 2, and No. 3 are better than those of the broadband filter groups with non-equivalent optical efficiency of filter groups No. 4, No. 5, and No. 6. This shows that the broadband filter groups with equal optical efficiency represented by broadband filter groups No. 1, No. 2, and No. 3 have better color reproduction and spectral reconstruction capabilities, proving the effectiveness of the optimization method of the broadband filter groups with equal optical efficiency of the present invention.
[0061] Table 1BFSELE (No. 1, No. 2 and No. 3) condition number and light efficiency
[0062]
[0063]
[0064] Table 2 BFSULE (No. 4, No. 5 and No. 6) condition number and light efficiency
[0065]
[0066]
[0067] Table 3 Spectral reconstruction results (PSNR / SSIM / SAM) on the CAVE dataset. The best results are highlighted in bold.
[0068]
Claims
1. A method for optimizing a broadband filter set with equal optical efficiency, characterized in that: The method comprises the following steps: Step 1, establishing a mathematical model for spectral imaging of broadband filter sets; Step 2, obtain the constraint relationship between the relative error of the solution and the condition number; Step 3, constructing the cost function of the broadband filter set with equal optical efficiency; During the operation of the spectral imaging system with broadband filters, the incident spectrum of the target scene is modulated multiple times in the spectral dimension through a series of broadband filters; then, the modulation coded image of the incident spectrum of the target scene is measured and captured by the CCD / CMOS sensor; finally, the incident spectrum information of the target scene is obtained by reconstruction calculation; S i is the i-th pixel value of the sensor, which can be expressed as where λ is the wavelength, k is the kth spectral band Λ k The central wavelength of [λ min ,λ max ] is the spectral range of the system spectral response, R(λ) is the distribution of incident spectral intensity with wavelength, θ i (λ) is the spectral sensitivity function of the i-th broadband filter; so far, the mathematical model of the broadband filter group spectral imaging process can be expressed as the following equation group S is a column vector, where each element represents the i-th measurement value; Θ is a sensor matrix, where each row vector corresponds to the spectral sensitivity function of the broadband filter; R is a column vector, where each element represents the incident spectral intensity to be solved.
2. The method for optimizing a broadband filter set of equal optical efficiency according to claim 1, characterized in that: The specific step 2 is as follows: there is a discrete error between the spectral imaging process described by equation (2) in step 1 and the actual physical process; Θ and S are obtained through the measurement process, and the measurement error is introduced in the process; when solving the equation, the above error is transmitted, which eventually leads to a decrease in the accuracy of the reconstructed spectrum; assuming that Θ and S have δ Θ and δ S If the error is small, the equation has an exact solution R+δ R ,Right now (S+δ S )=(Θ+δ Θ )(R+δ R ) (3) According to the matrix principle, when the number of measurements is equal to the number of spectral bands, the relative error of the solution ||δ R || / ||R|| is where κ(Θ) is the condition number of Θ, defined as κ(Θ) = ||Θ||Θ + ||, Θ + is the pseudo-inverse of Θ, which is equal to Θ when Θ is non-singular. -1 , for the 2-norm, κ(Θ)=σ max / σ min ≥1 is the ratio of the largest and smallest singular values of Θ; similarly, when the number of measurements is greater than the number of spectral bands, the relative error of the solution ||δ R || / ||R|| is Equations (4) and (5) show that the relative error of the solution δ R / R is restricted by the condition number κ(Θ). When κ(Θ) is close to 1, the solution has better robustness.
3. The method for optimizing a broadband filter set of equal optical efficiency according to claim 2, characterized in that: The step three is specifically as follows: the optical efficiency of each broadband filter is defined as For a set of broadband filters with equal optical efficiency, all elements of the filter vector η i The sum of should be equal; then, to find an equal optical efficiency broadband filter set, this can be achieved by solving the following problem, namely ρ∈(0,m) is a constant representing the target light efficiency level. To minimize the condition number κ(Θ) in equation (7), a scheme for minimizing the condition number based on soft orthogonal constraints is proposed, which directly takes the angle between paired row vectors as the minimum optimal target. Finally, the cost function of the equal light efficiency broadband filter set is expressed as Among them, Θ T is the transposed matrix of Θ, ΘΘ T The off-diagonal elements in are the dot products of the paired row vectors in Θ, γ represents the paired angle matrix, θ i represents the row vector of the sensor matrix Θ, Represents θ i The normalized vector of ; So far, the three parameters of the number of spectral bands, the number of broadband filters, and the target light efficiency level are set in advance, and the cost function of the equal light efficiency broadband filter group is solved to obtain the equal light efficiency broadband filter group.