Method and system for spectral unmixing of smoked fresco pigments

Abstract

The present application relates to a method for spectral unmixing of smoked mural pigments, belonging to the field of hyperspectral unmixing. The method establishes a pure pigment spectral library and a smoked spectral library as data basis through S1, constructs a sparse unmixing model based on the linear combination of endmembers of the smoked spectral library through S2, introduces a spatial adaptive mechanism based on spectral similarity measurement through S3, dynamically adjusts the regularization weight to optimize the objective function through abundance gradient analysis, and finally realizes the accurate unmixing of smoked pigment components through S4 by adopting the alternating least squares method combined with spectral smoothing constraint to implement hierarchical iterative optimization. The method effectively overcomes the interference of smoked layers and the nonlinear effect of mixed pixels through dynamic weight adjustment, and integrates spectral similarity analysis and abundance spatial distribution characteristics. It is mainly applied to the quantitative analysis of the original pigment components under the smoked covering layer in ancient murals, and provides accurate material composition data support for cultural relic restoration.

Description

Technical Field

[0001] The present invention relates to the field of hyperspectral unmixing, and more specifically, to a method and system for unmixing spectral pigments of smoked murals. Background Art

[0002] Hyperspectral imaging technology, due to its non-contact and non-destructive nature, has become a research hotspot for identifying pigments in murals. In recent years, the issue of smoke contamination in ancient murals has gained increasing attention. Smoke pollution, primarily a complex mixture of carbon black particles, saturated fatty acid glycerides, or saturated fatty acids, often forms a thin film on the surface of the mural, altering the spectral reflectance properties of the pigments. This results in the measured spectrum being a mixture of smoke pollution and pigments, necessitating spectral unmixing to identify the specific pigments.

[0003] Traditional hyperspectral unmixing methods typically use linear mixture models (LMMs), which can be broadly categorized into three types: geometric, statistical, and sparse. Geometric unmixing primarily exploits the fact that all pixels in spectral data are distributed in a high-dimensional singlet space or a positive convex cone, considering the vertices of the singlet enclosing the dataset as endmembers. This type of method offers an intuitive geometric interpretation. Statistical unmixing aims to utilize parameter estimation techniques to determine the endmember set and abundance matrix, requiring little prior knowledge. Sparse regression-based unmixing primarily utilizes the prior knowledge of a spectral library as a dictionary, transforming the spectral unmixing problem into a sparse regression problem through a semi-supervised approach. These classic models have limitations when addressing the unmixing problem of smoky mural pigments. First, the spectral coupling between smoke pollution and the pigment layer exhibits spatial heterogeneity, making traditional endmember extraction algorithms (determining the basic pigments that make up the mixed pixel) insensitive to sparsely distributed trace components. Second, the nonlinear spectral superposition of the smoke and pigment layers leads to cumulative errors in abundance inversion (calculating the proportion of each basic pigment in the mixed pixel), which is particularly prone to artifact interference in areas with low signal-to-noise ratios. Therefore, unmixing research on smoke murals is urgently needed to fill the academic gap in the field of hyperspectral unmixing of smoke murals. Summary of the Invention

[0004] An object of the present invention is to provide a method and system for spectral unmixing of smoked mural pigments to at least solve the above-mentioned problems.

[0005] In order to achieve the purpose and other advantages of the present invention, a method for spectral unmixing of smoky mural pigments is provided, comprising: S1, data preparation and preprocessing: establishing a pure pigment spectrum library and a smoky spectrum library based on hyperspectral imaging data; S2, sparse unmixing model construction: expressing the spectral signal of each pixel point in the smoky pigment hyperspectral image as a linear combination of end-member spectra in the smoky spectrum library, and constructing a sparse unmixing model; S3, objective function design: dynamically adjusting the regularization weight based on spectral similarity measurement and abundance gradient to construct the objective function of the sparse unmixing model; the dynamic adjustment of the regularization weight adopts a spatial adaptive mechanism based on spectral similarity, and the weight update is achieved by calculating the abundance; S4, hierarchical optimization and unmixing: taking the end-member matrix constructed from the pure pigment spectrum library as input, the alternating least squares method combined with spectral smoothness constraint iterative optimization objective function is adopted to obtain the unmixing result of the smoky pigment.

[0006] Preferably, in S1, the hyperspectral imaging data needs to be radiometrically corrected before use. The radiometric correction formula is:

[0007]

[0008] Among them, R is the corrected target hyperspectral data, R raw is the original hyperspectral data of the target object, R dark is the dark current data, R white This is the standard reflector data.

[0009] Preferably, in S1, the pure pigment comprises at least three mineral pigments selected from ochre, ultramarine, graphite, malachite and clam powder, and the pure pigment spectral library satisfies: spectral resolution ≤ 5 nm, signal-to-noise ratio ≥ 50 dB, and the spectral average of more than 10 pollution-free samples is collected for each pigment; the smoky pigment is obtained by placing a simulated mural sample in a closed smoking environment and smoking it for a preset time, and is divided into three categories: mild, moderate and severe according to the smoking time.

[0010] Preferably, in S2, the sparse unmixing model is: j =D1a j +n j

[0011] Among them, y j is the spectral vector of the jth pixel in the smoke pigment hyperspectral image, D1 is the end member matrix constructed by the smoke spectral library, a j is the sparse coefficient vector of the j-th pixel, n j is the noise term of the j-th pixel.

[0012] Preferably, in S3, the objective function is specifically:

[0013] Where X is the sparse coefficient matrix, D is the end member matrix of pure pigment, y is the hyperspectral data matrix of smoked pigment, ‖‖ F represents the norm, ∑ i,j ω i,j |X i,j | is the adaptive L1 regularization term, |X i,j | is the absolute value of the coefficient value of the corresponding i-th band and j-th pixel in the sparse coefficient matrix. The proportion of non-zero elements in the sparse coefficient vector does not exceed 5%, and the absolute value of the non-zero elements is ≥ 0.1. ω i,j is the adaptive regularization weight, and the calculation formula is:

[0014]

[0015] Among them, S i,j is the spectral similarity measure, S i,j ∈[0,1], β is the smoothness control parameter, is the abundance gradient.

[0016] Preferably, in S3, the spectral similarity measure S i,j Calculated by the following formula:

[0017]

[0018] Where, P i is the spectrum vector of the i-th band in the pure pigment spectrum library, y j is the spectral vector of the j-th pixel in the smoky pigment hyperspectral image, and the calculated results are normalized.

[0019] Preferably, in S4, the hierarchical optimization specifically includes:

[0020] (a) Fixed end member matrix D (k) , update the sparse coefficient matrix X through sparse regression (k+1) :

[0021]

[0022] Among them, X (k+1) is the updated sparse coefficient matrix, D (k) is the end member matrix, the initial end member matrix is ​​constructed from the pure pigment spectrum library, X (k) is a sparse coefficient matrix, and the initial sparse coefficient matrix is ​​preliminarily estimated by least squares or sparse coding;

[0023] (b) Fixed sparse coefficient matrix X (k+1) , update the endmember matrix D by combining the least squares method with the spectral smoothing constraint (k +1) :

[0024]

[0025] wherein D (k+1) is the updated endmember matrix, λ is a regularization parameter, D T is the transpose of the endmember matrix, is a spectral smoothness constraint matrix;

[0026] (c) repeating steps (a) and (b) until the objective function converges or reaches a preset number of iterations, the convergence criterion being that the relative change rate of the objective function value of the last 5 iterations is <0.01% or the F-norm change amount is 1e -4 .

[0027] Preferably, the smoke fresco pigment spectral unmixing method further comprises: comparing the unmixing result with the spectral data in the pure pigment spectral library, and performing precision evaluation using at least one of root mean square error (RMSE) and spectral information divergence (SID), wherein RMSE≤0.1 and / or SID≤0.05 meet the precision requirement.

[0028] The application further provides a smoke fresco pigment spectral unmixing system, comprising: a data preparation and preprocessing module for establishing a pure pigment spectral library and a smoke spectral library based on hyperspectral imaging data; a sparse unmixing model construction module for representing the spectral signal of each pixel point in the smoke pigment hyperspectral image as a linear combination of endmember spectra in the smoke spectral library, and constructing a sparse unmixing model; an objective function design module for dynamically adjusting the regularization weight based on spectral similarity measurement and abundance gradient, constructing the objective function of the sparse unmixing model, and dynamically adjusting the regularization weight using a spatial adaptive mechanism based on spectral similarity, and updating the weight by calculating the abundance; and a hierarchical optimization and unmixing module for taking the endmember matrix constructed by the pure pigment spectral library as input, iteratively optimizing the objective function using the alternating least squares method combined with spectral smoothness constraints, and obtaining the unmixing result of the smoke pigment.

[0029] The application at least has the following beneficial effects:

[0030] First, by constructing a dual-spectral library system and a hierarchical optimization mechanism, the adaptability of pigment unmixing in smoked murals was effectively improved. The pure pigment spectral library provides baseline endmember characteristics, while the smoked spectral library quantifies the spectral distortion patterns of the smoke layer. Their synergistic use can cover the mixing characteristics of pigments at different levels of contamination. A dynamic weighting mechanism based on spectral similarity adjusts the regularization strength according to the degree of local mixing. This reduces the sparsity constraint in areas with thin smoke layers to avoid over-suppression of true endmembers, while strengthening the constraint in areas with dense smoke deposits to suppress noise interference. The hierarchical optimization alternately updates the endmember matrix and the sparse coefficient matrix, combined with spectral smoothing constraints. This method maintains the computational efficiency of the linear model while mitigating the nonlinear effects caused by smoke through iterative approximation, ultimately achieving a balance between computational complexity and accuracy. This method is highly robust to endmember spectral overlap and noise interference in mixed pixels, reducing unmixing error by approximately 30% compared to traditional methods.

[0031] Second, the types of pure pigments and spectral library parameters are limited (resolution ≤ 5nm, signal-to-noise ratio ≥ 50dB) to ensure that the spectral library covers the spectral characteristics of common mineral pigments in murals, while reducing the error of single sample collection through multi-sample averaging. Smoked samples are classified according to the length of smoking (mild / moderate / severe), and spectral variation models with different degrees of pollution can be constructed. When used in conjunction with a pure spectral library, the optical properties of the smoked layer can be more accurately characterized. The dual constraints of spectral resolution and signal-to-noise ratio can effectively distinguish subtle differences between smoked objects and pigments in the near-infrared band, such as the difference between the characteristic absorption peak of malachite at 750nm and the scattering effect of smoked carbon black in this band, thereby improving the distinguishability of end-member spectra.

[0032] Third, through a linear combination model, the smoke spectral library is used as the endmember matrix D1 to explicitly model the coverage effect of the smoke layer on the original pigment. The introduction of the noise term can separate the random scattered noise and system noise in the smoke layer, avoiding misidentification of them as endmember components. While maintaining the simplicity of the linear mixing framework, this model expands the endmembers of the smoke spectral library and treats the smoke layer as an independent endmember for unmixing, thus bypassing the modeling difficulties of nonlinear smoke-pigment mixing in traditional methods. Experiments show that the model's adaptability to changes in smoke layer thickness is improved by 40%, especially in moderately smoked areas, where the error of endmember abundance estimation can be controlled within 8%.

[0033] Fourth, the adaptive L1 regularization term dynamically adjusts the weights by spectral similarity and abundance gradient to achieve spatially adaptive sparse constraints. In areas with low spectral similarity (such as the smoke-pigment mixing boundary), the weights are increased to suppress redundant end members; in areas with gentle gradients (such as uniform smoke coverage), the weights are reduced to retain weak signal end members. The exponential term in the formula This smoothes out sudden abundance changes and avoids weight fluctuations caused by local gradient anomalies. This mechanism allows the model to maintain overall sparsity while preserving the spatial distribution characteristics of fine pigment particles. Testing has shown that it can reduce false endmember misjudgments by over 30% while reducing the missed detection rate of valid endmembers to below 5%.

[0034] Fifth, the inverse cosine function is used to calculate spectral similarity, and the degree of similarity between the pure end member and the mixed spectrum is measured by the vector angle. Compared with the traditional Euclidean distance, it has a better band correlation characterization capability. Normalization processing eliminates the influence of energy differences in different bands on the similarity measurement. For example, cross-band comparability is achieved between the visible light band (400-600nm) with strong absorption by smoke and the relatively stable near-infrared band (800-1000nm). This measurement method is sensitive to spectral band shifts caused by the smoke layer (such as the right shift of the reflection peak of cinnabar at 550nm) and can accurately identify contaminated end members. Experiments show that its mismatch rate is reduced by about 25% compared with spectral angle matching (SAM).

[0035] Sixth, hierarchical optimization gradually approaches the optimal solution in iteration by alternately updating the endmember matrix D and the sparse coefficient matrix X, combined with spectral smoothing constraints. When D is fixed to update X, sparse regression retains the main endmember components; when X is fixed to update D, spectral smoothing constraints suppress the high-frequency noise of the endmember spectrum and maintain its physical rationality (such as the smoothing of the absorption valley of azurite at 680nm). This strategy avoids the local minimum problem caused by optimizing all parameters at once, and the convergence speed is about 50% faster than traditional global optimization. F-norm change threshold (1e -4 ) and the relative rate of change of the objective function (<0.01%) to ensure that the unmixing result is stable within a limited number of iterations. The measured number of iterations usually does not exceed 20.

[0036] Seventh, unmixing accuracy is assessed using the dual metrics of RMSE and SID, quantifying the reliability of the results from the perspectives of numerical error and distribution similarity, respectively. The RMSE constraint ensures that the absolute error between the abundance estimate and the actual pigment ratio is manageable, for example, the error for clam powder abundance is ≤10%. The SID constraint ensures waveform consistency between the unmixed spectrum and the reference spectrum, avoiding misinterpretations of overall reflectance shifts due to absorption from the smoke layer. This evaluation system provides quantitative acceptance criteria for cultural relic restoration. In field measurements, unmixing results that meet these dual-metric constraints have a consistency of over 98% in expert visual inspection, significantly outperforming single-metric evaluation methods.

[0037] Other advantages, objectives and features of the present invention will be reflected in part from the following description and will be understood by those skilled in the art through study and practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1The figure is a flow chart of a method for spectral unmixing of smoked mural pigments according to an embodiment of the present invention. DETAILED DESCRIPTION

[0039] The present invention will be further described in detail below with reference to the embodiments and drawings so that those skilled in the art can implement the invention with reference to the description.

[0040] It should be understood that terms such as “having”, “including” and “comprising” used herein do not preclude the existence or addition of one or more other elements or combinations thereof.

[0041] It should be noted that the experimental methods described in the following embodiments are conventional methods unless otherwise specified, and the reagents and materials can be obtained from commercial channels unless otherwise specified.

[0042] like Figure 1 As shown, in one embodiment of the present invention, a method for spectral unmixing of smoked mural pigments is provided, including: S1, data preparation and preprocessing: establishing a pure pigment spectrum library and a smoked spectrum library based on hyperspectral imaging data; S2, sparse unmixing model construction: expressing the spectral signal of each pixel point in the smoked pigment hyperspectral image as a linear combination of end-member spectra in the smoked spectrum library, and constructing a sparse unmixing model; S3, objective function design: dynamically adjusting the regularization weight based on spectral similarity measurement and abundance gradient to construct the objective function of the sparse unmixing model; dynamically adjusting the regularization weight adopts a spatial adaptive mechanism based on spectral similarity, and updating the weight is achieved by calculating the abundance; S4, hierarchical optimization and unmixing: using the end-member matrix constructed from the pure pigment spectrum library as input, the alternating least squares method combined with spectral smoothing constraint iterative optimization objective function is adopted to obtain the unmixing result of the smoked pigment.

[0043] In the above technical solution, the spectral unmixing method for smoked mural pigments includes four core steps. The "pure pigment spectral library" refers to a collection of uncontaminated mineral pigment spectral data collected using hyperspectral imaging technology; the "smoked spectral library" is a collection of spectral data obtained by fumigating mural samples to varying degrees in a simulated smoked environment; the "sparse unmixing model" represents the spectral signal of smoked pigments as a linear combination of endmember spectra in the smoked spectral library; the "alternating least squares method" is an iterative algorithm that optimizes the objective function by alternating fixed variables; and the "spectral smoothness constraint" is a mathematical constraint imposed on the smoothness of the endmember spectra.

[0044] Specifically, in the S1 stage, hyperspectral data of pure mineral pigments (such as ochre and ultramarine) are first collected. Equipment noise and environmental interference are eliminated through radiation correction to construct a standardized pure pigment spectral library. At the same time, simulated mural samples are placed in a closed smoking device and smoked for a preset time (such as 20 minutes for light, 40 minutes for moderate, and 60 minutes for heavy) to obtain spectral data of different pollution levels to form a smoked spectral library. In the S2 stage, each pixel of the smoked hyperspectral image is modeled as a linear combination of the endmembers of the smoked spectral library, and the sparsity assumption is introduced, that is, each pixel is composed of only a small number of endmembers. When designing the objective function in the S3 stage, the regularization weight is dynamically adjusted according to the spectral similarity (calculated by the inverse cosine function) and the abundance gradient (characterizing the change in spatial distribution). Specifically, the weight is reduced in areas with thin smoke layers to retain weak signals, and the weight is increased in areas with dense smoke to suppress noise. The S4 stage adopts an alternating optimization strategy: first, the endmember matrix is ​​fixed and the sparse coefficients are updated through sparse regression; then the sparse coefficient matrix is ​​fixed and the endmember matrix is ​​updated in combination with the spectral smoothing constraint, and the cycle is iterated until convergence.

[0045] By leveraging a dynamic weighting mechanism and a hierarchical optimization strategy, the unmixing accuracy of smoked murals has been significantly improved. Experiments show that compared to traditional methods, the error in endmember abundance estimation is reduced by approximately 30%. In particular, the spatial continuity of the unmixing results is improved in areas where the smoke layer thickness varies. Furthermore, the alternating optimization strategy effectively mitigates the impact of nonlinear mixing effects while maintaining computational efficiency. Convergence is typically achieved within 20 iterations, meeting the real-time requirements of cultural relic restoration.

[0046] In another embodiment of the present invention, in S1, radiometric correction is required before the hyperspectral imaging data is used. The radiometric correction formula is:

[0047]

[0048] Among them, R is the corrected target hyperspectral data, R raw is the original hyperspectral data of the target object, R dark is the dark current data, R white This is the standard reflector data.

[0049] In the above technical solution, "radiation correction" refers to the process of eliminating the inherent noise and environmental interference of hyperspectral imaging equipment through mathematical processing; "dark current data" is the noise signal recorded by the sensor under light-free conditions; "standard reflector data" is the reference data obtained through a standard white plate with known reflectivity, which is used to calibrate the reflectivity of the target object.

[0050] Specifically, in the data preprocessing stage, the original hyperspectral data, dark current data and standard whiteboard data of the target object (mural surface) are first collected. The dark current data is obtained by turning off the light source and recording the sensor output, which represents the background noise of the equipment. The standard whiteboard data is used to calibrate the reflectivity benchmark. During radiation correction, the dark current noise is subtracted from the original data, then divided by the difference between the whiteboard and the dark current, and finally multiplied by 99% to retain the effective signal dynamic range. This step can eliminate the influence of sensor dark current noise and ambient light fluctuations, ensuring the accuracy of the physical meaning of the spectral data.

[0051] After radiometric correction, the signal-to-noise ratio of the spectral data increases to over 50dB, effectively eliminating baseline drift caused by instrument noise. For example, in the visible light band (400-700nm), the reflectance curve of the corrected data more closely resembles the actual pigment characteristics, and the separation between the absorption characteristics of the smoky layer and the pigment reflectance peak is improved by approximately 20%, providing reliable input for subsequent demixing.

[0052] In another embodiment of the present invention, in S1, the pure pigment comprises at least three mineral pigments selected from ochre, ultramarine, graphite, malachite and clam powder, and the pure pigment spectral library satisfies: spectral resolution ≤ 5 nm, signal-to-noise ratio ≥ 50 dB, and the spectral mean of more than 10 pollution-free samples is collected for each pigment; the smoky pigment is obtained by placing a simulated mural sample in a closed smoking environment for a preset time, and is divided into three categories: mild, moderate and severe according to the smoking time.

[0053] In the above technical solution, "spectral resolution ≤ 5nm" means that the width of each band in the spectral library does not exceed 5 nanometers, ensuring the capture of subtle spectral features; "signal-to-noise ratio ≥ 50dB" means that the effective signal strength is 10 times the noise strength. 5 times or more; “uncontaminated sample” means that the pigment sample surface has no physical damage or chemical contamination, and the spectral data reflects the characteristics of pure substances.

[0054] Specifically, the pure pigment spectral library must contain at least three mineral pigments (such as ochre, ultramarine, and graphite), and more than 10 pollution-free samples of each pigment must be selected. Sample data is collected using a hyperspectral imaging device (such as Themis Vision System's VNIR / 400H hyperspectral imager), with a spectral range covering 400-1000nm, and a halogen lamp is used as the light source. During collection, it is necessary to ensure uniform lighting to avoid interference from shadows or reflections. When preparing the smoked samples, the simulated murals are placed in a closed combustion chamber, and spectral data of different pollution levels are generated by controlling the burning time (such as 20 minutes for light, 40 minutes for moderate, and 60 minutes for heavy), and classified and stored in the smoked spectral library.

[0055] Furthermore, the simulated mural samples were produced based on the materials and techniques used in ancient mural production. The samples (24cm x 20cm x 1cm) consisted of a 2 / 3 coarse mud layer, a 1 / 3 fine mud layer, and a base color layer approximately 0.5mm thick. The coarse mud layer was composed of clay and sand in a 2:1 ratio, with 3% by weight of wheat straw approximately 1cm long added. The fine mud layer was composed of clay and sand in a 2:1 ratio, with 3% by weight of hemp silk added. After the specimen dried, a mixture of calcite powder and 5% gelatin was applied to the fine mud layer as a color layer. A variety of mineral pigments that are resistant to chemical reactions with air at high temperatures (such as ochre, ultramarine, graphite, malachite green, and clam powder) were applied to the color layer. These mineral pigments were all produced in Jiang Sixu Hall. Hyperspectral data of the simulated mural samples was then collected to obtain the original mural spectral data P1. The samples were then smoked using a homemade fumigation device. The homemade apparatus is 73 cm tall and 34 cm wide. Combustible material is placed at the bottom, and water is added to the sides as a coolant. A mesh plate is placed in the upper center, on which the sample is placed. Smoke is generated by burning wood chips, charcoal, candle pellets, and incense. The temperature of the fumigation apparatus is controlled below 200°C. First, a sample without a pigment layer (24 cm x 20 cm x 1 cm) was continuously exposed to smoke. Results showed that a smoke exposure of 60 minutes was sufficient to achieve complete coverage with smoke particles. Therefore, the smoke intensity of the murals was categorized into three levels based on the duration of the smoke exposure: light smoke (20 minutes), moderate smoke (40 minutes), and heavy smoke (60 minutes). Data was collected for the smoked murals during these time periods, resulting in three sets of simulated smoked mural data (S1, S2, and S3). After radiometric correction, the original mural spectral data (P1) and the simulated smoked mural data (S1, S2, and S3) were analyzed for the spectral characteristics of the different pigments in each region of P1 and compared with a laboratory spectral library to ensure measurement accuracy. The obtained pure pigment spectral data (ochre, ultramarine, graphite, malachite, and clam powder) constitute the pure pigment spectral library of this experiment, and the obtained S1, S2, and S3 are used to establish the smoked spectrum library D1 according to the degree of smoke.

[0056] The spectral library clearly distinguishes characteristic absorption peaks of pigments. For example, the absorption peak of malachite at 750nm differs significantly from the scattering spectrum of smoked carbon black. Multi-sample averaging improves the stability of the spectral library, reducing the error in single sample acquisition to less than 3%. Spectral data categorized by smoke duration accurately models the effects of varying pollution levels, reducing matching errors by 15% during unmixing.

[0057] In another embodiment of the present invention, in S2, the sparse unmixing model is specifically: j =D1a j +n j

[0058] Among them, y jis the spectral vector of the jth pixel in the smoke pigment hyperspectral image, D1 is the end member matrix constructed by the smoke spectral library, a j is the sparse coefficient vector of the j-th pixel, n j is the noise term of the j-th pixel.

[0059] In the above technical solution, the "endmember matrix" is composed of spectral vectors in the smoked spectrum library arranged in columns, representing possible mixed components; the "sparse coefficient vector" represents the mixing ratio of each endmember in each pixel point; and the "noise term" includes random errors such as sensor noise and environmental interference.

[0060] Specifically, the sparse unmixing model converts the spectrum y of each pixel of the smoked hyperspectral image into j Represented as the end member matrix D1 and the sparse coefficient vector a j A linear combination of , and superimposed noise term n j The column vectors of the endmember matrix are the spectral curves of each endmember in the smoked spectrum library. The sparsity constraint requires that the proportion of non-zero elements in each coefficient vector does not exceed 5%, and the absolute value is ≥ 0.1, ensuring that only the main components are retained in the mixture model.

[0061] By explicitly introducing smoke endmembers, this model treats the smoke layer as an independent component for unmixing, avoiding the modeling challenges of nonlinear smoke-pigment mixing in traditional methods. Experiments show that in moderately smoked areas, the error in endmember abundance estimation is kept within 8%, a 40% improvement in accuracy compared to models without smoke endmembers.

[0062] In another embodiment of the present invention, in S3, the objective function is specifically:

[0063]

[0064] Where X is the sparse coefficient matrix, D is the end member matrix of pure pigment, y is the hyperspectral data matrix of smoked pigment, ‖‖ F represents the norm, ∑ i,j ω i,j |X i,j | is the adaptive L1 regularization term, |X i,j | is the absolute value of the coefficient value of the corresponding i-th band and j-th pixel in the sparse coefficient matrix. The proportion of non-zero elements in the sparse coefficient vector does not exceed 5%, and the absolute value of the non-zero elements is ≥ 0.1. ω i,j is the adaptive regularization weight, and the calculation formula is:

[0065]

[0066] Among them, S i,j is the spectral similarity measure, S i,j ∈[0,1], β is the smoothness control parameter, is the abundance gradient.

[0067] In the above technical solution, the "adaptive L1 regularization term" is a regularization method that dynamically adjusts the strength of the sparsity constraint based on the local mixing characteristics; the "spectral similarity measure" quantifies the degree of similarity between the pure endmember and the mixed spectrum; and the "abundance gradient" reflects the rate of change of the endmember ratio in space and is calculated using the spatial difference method.

[0068] Specifically, the objective function includes a data fitting term and an adaptive regularization term. The regularization weight ω is determined by the spectral similarity S and the abundance gradient Jointly determined: the lower the similarity (such as the smoke-pigment mixing boundary), the larger the weight to suppress redundant end members; the larger the gradient (such as the area of ​​pigment distribution mutation), the smaller the weight to avoid over-smoothing. Used to smooth spatial mutations, the β parameter controls the smoothing strength.

[0069] The dynamic weighting mechanism enables the model to retain weak signal endmembers in thin smoke layer areas (high similarity and small gradients) and suppress noise in thick areas (low similarity and large gradients), reducing false endmember misclassification by 30%. The smoothing term prevents weight fluctuations caused by local gradient anomalies and improves spatial continuity by 25%.

[0070] In another embodiment of the present invention, in S3, the spectral similarity measure S i,j Calculated by the following formula:

[0071]

[0072] Where, P i is the spectrum vector of the i-th band in the pure pigment spectrum library, y j is the spectral vector of the j-th pixel in the smoky pigment hyperspectral image, and the calculated results are normalized.

[0073] In the above technical solution, the "inverse cosine function" measures the spectral similarity by calculating the vector angle. The smaller the angle, the higher the similarity. The "normalization processing" scales the similarity value to the range of 0-1 to eliminate the influence of energy differences between bands.

[0074] Specifically, the calculation of spectral similarity S is divided into three steps: 1) Calculate the pure end member spectrum vector X i With mixed spectrum y j The cosine similarity of the two samples is calculated; 2) the angle is converted to radians using the inverse cosine function; and 3) the S value is normalized to a value between 0 (perfect similarity) and 1 (complete dissimilarity). This metric is sensitive to spectral band shifts caused by smoke. For example, when the 550nm reflection peak of cinnabar shifts to the right, the S value increases significantly.

[0075] Compared to the traditional Euclidean distance, the arccosine metric is more sensitive to spectral waveform variations, reducing the mismatch rate by 25%. Normalization ensures cross-band comparability, for example, improving the consistency of similarity assessments between the visible light band, where smoke strongly absorbs, and the stable near-infrared band.

[0076] In another embodiment of the present invention, in S4, the hierarchical optimization specifically includes:

[0077] (a) Fixed end member matrix D (k) , update the sparse coefficient matrix X through sparse regression (k+1) :

[0078]

[0079] Among them, X (k+1) is the updated sparse coefficient matrix, D (k) is the end member matrix, the initial end member matrix is ​​constructed from the pure pigment spectrum library, X (k) is a sparse coefficient matrix, and the initial sparse coefficient matrix is ​​preliminarily estimated by least squares or sparse coding;

[0080] (b) Fixed sparse coefficient matrix X (k+1) , update the endmember matrix D by combining the least squares method with the spectral smoothing constraint (k +1) :

[0081]

[0082] Among them, D (k+1) is the updated end member matrix, λ is the regularization parameter, D T is the transpose of the endmember matrix, is the spectral smoothing constraint matrix;

[0083] (c) Repeat steps (a) and (b) until the objective function converges or the preset number of iterations is reached. The convergence criterion is that the relative change rate of the objective function value for 5 consecutive iterations is less than 0.01% or the F norm changes by 1e -4 .

[0084] In the above technical solution, "sparse regression" is an optimization process for solving sparse coefficients under the condition of a fixed endmember matrix; the "spectral smoothing constraint matrix" is constructed through the Laplace operator to suppress the high-frequency noise of the endmember spectrum.

[0085] Specifically, the hierarchical optimization is divided into two steps of iteration: 1) fix the end member matrix D, update the sparse coefficient X through L1 regularized regression, and retain the main end member components; 2) fix X, combine the least squares method The constraint update D ensures the smoothness of the endmember spectral curve. The convergence criterion adopts a double standard: the change rate of the objective function is less than 0.01% for 5 consecutive iterations, or the F-norm change is less than 1e -4 .

[0086] The alternating optimization strategy improves the calculation efficiency by 50%, and the convergence can be achieved within 20 iterations. The spectral smoothing constraint effectively removes the high-frequency noise of the endmember spectrum, such as the absorption valley shape of azurite at 680 nm being closer to the true physical characteristics, and the endmember reconstruction error is reduced to below 5%.

[0087] In another embodiment of the present application, the spectral unmixing method of the smoked fresco pigment also includes: comparing the unmixing result with the spectral data in the pure pigment spectral library, and using at least one of the root mean square error (RMSE) and the spectral information divergence (SID) to evaluate the accuracy, wherein RMSE≤0.1 and / or SID≤0.05 meet the accuracy requirement.

[0088] In the above technical solution, the "root mean square error (RMSE)" measures the absolute deviation of the abundance estimate value from the true value; and the "spectral information divergence (SID)" evaluates the spectral waveform consistency through the difference of probability distribution.

[0089] Specifically, after unmixing, the result is compared with the pure spectral library, and RMSE and SID are calculated. RMSE is required to be ≤0.1 to ensure that the abundance error is below 10%; and SID is required to be ≤0.05 to ensure that the unmixing spectral waveform is highly consistent with the reference spectrum. For example, the abundance RMSE of shell powder needs to be ≤0.08, and the SID needs to be ≤0.03. Among them,

[0090] The calculation formula of the root mean square error (RMSE) is:

[0091]

[0092] M is the total number of pixel points in the hyperspectral image of the smoked pigment, j is the jth pixel point, N is the total number of bands in the hyperspectral image of the smoked pigment, P i is the spectral vector of the ith band in the pure pigment spectral library, is the spectral vector of the ith band after unmixing.

[0093] The calculation formula of the spectral information divergence (SID) is:

[0094]

[0095] Among them, R i and P i are the values of the ith band in the two spectral signals, R i is the spectral signal value of the ith band of the unmixing pigment, and P iThe spectrum vector of the i-th band in the pure pigment spectrum library.

[0096] The dual-indicator evaluation system comprehensively quantifies the demixing accuracy. In the actual measurement, more than 95% of the areas meet the constraints, and the expert visual acceptance pass rate exceeds 98%, which is significantly better than the single-indicator evaluation method.

[0097] In another embodiment of the present invention, a spectral unmixing system for smoked mural pigments is provided, comprising: a data preparation and preprocessing module for establishing a pure pigment spectral library and a smoked spectral library based on hyperspectral imaging data; a sparse unmixing model construction module for representing the spectral signal of each pixel point in the smoked pigment hyperspectral image as a linear combination of end-member spectra in the smoked spectral library to construct a sparse unmixing model; an objective function design module for dynamically adjusting the regularization weight based on spectral similarity measurement and abundance gradient to construct an objective function of the sparse unmixing model, dynamically adjusting the regularization weight using a spatial adaptive mechanism based on spectral similarity, and updating the weight by calculating the abundance; a hierarchical optimization and unmixing module for using the end-member matrix constructed from the pure pigment spectral library as input, and iteratively optimizing the objective function using the alternating least squares method combined with spectral smoothing constraints to obtain the unmixing result of the smoked pigment.

[0098] In the above technical solution, the "data preparation and preprocessing module" is responsible for spectral library construction and data correction; the "sparse unmixing model construction module" realizes linear mixture modeling; and the "hierarchical optimization module" performs alternating iterative optimization.

[0099] Specifically, the system consists of four modules connected in series: 1) a preprocessing module that performs radiometric correction on the raw hyperspectral data and constructs a dual-spectral library; 2) a model-building module that models each pixel spectrum as a linear combination of smoked endmembers; 3) an objective function module that dynamically calculates regularization weights; and 4) an optimization module that iteratively solves using alternating least squares and outputs the unmixing results. Parameters are transferred between modules via a data interface, ensuring process automation.

[0100] After system integration, processing efficiency increased by 40%, and the demixing time for a single hyperspectral image (1000×1000 pixels) was reduced to less than 10 minutes. The modular design supports flexible expansion. For example, adding new smoked endmember types requires only updating the spectral library, without modifying the core algorithm.

[0101] The number of devices and processing scales described herein are intended to simplify the description of the present invention. Applications, modifications, and variations of the smoked mural pigment spectrum unmixing method and system of the present invention will be readily apparent to those skilled in the art.

[0102] Although the embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the description and implementation methods. They can be fully applied to various fields suitable for the present invention. For those familiar with the art, additional modifications can be easily implemented. Therefore, without departing from the general concept defined by the claims and the scope of equivalents, the present invention is not limited to specific details.

Claims

1. A method for spectral unmixing of smoked mural pigments, characterized in that: include: S1. Data preparation and preprocessing: Building pure pigment spectral libraries and smoke spectral libraries based on hyperspectral imaging data; S2. Sparse unmixing model construction: The spectral signal of each pixel in the smoke pigment hyperspectral image is represented as a linear combination of the endmember spectra in the smoke spectral library to construct a sparse unmixing model; S3. Objective function design: Dynamically adjust the regularization weights based on spectral similarity metrics and abundance gradients to construct the objective function of the sparse unmixing model. Dynamic adjustment of the regularization weights uses a spatial adaptive mechanism based on spectral similarity and updates the weights by calculating abundance. S4. Hierarchical optimization and unmixing: Using the endmember matrix constructed from the pure pigment spectral library as input, the alternating least squares method combined with spectral smoothing constraint iterative optimization of the objective function is used to obtain the unmixing results of the smoky pigment; In S1, the pure pigments include at least three mineral pigments selected from ochre, ultramarine, graphite, malachite, and clam powder. The pure pigment spectral library meets the following requirements: spectral resolution ≤ 5 nm, signal-to-noise ratio ≥ 50 dB, and the spectral average of at least 10 uncontaminated samples is collected for each pigment. The smoky pigments are obtained by placing simulated mural samples in a closed smoking environment for a preset time, and are classified into three categories: mild, moderate, and severe according to the smoking time. In S2, the sparse unmixing model is as follows: in, is the spectral vector of the j-th pixel in the smoky pigment hyperspectral image, is the endmember matrix constructed from the smoked spectral library, It is The sparse coefficient vector of pixels, It is The noise term of pixels; In S3, the objective function is: Among them, X is the sparse coefficient matrix, D is the end member matrix of pure pigment, and y is the hyperspectral data matrix of smoked pigment. represents the norm, is the adaptive L1 regularization term, is the absolute value of the coefficient value of the corresponding i-th band and j-th pixel in the sparse coefficient matrix. The proportion of non-zero elements in the sparse coefficient vector does not exceed 5%, and the absolute value of the non-zero element , is the adaptive regularization weight, and the calculation formula is: in, is the spectral similarity measure, , is the smoothness control parameter, is the abundance gradient.

2. The spectral unmixing method for smoked mural pigments according to claim 1, wherein: In S1, hyperspectral imaging data needs to be radiometrically corrected before use. The radiometric correction formula is: in, R is the corrected target hyperspectral data, The original hyperspectral data of the target object, is the dark current data, This is the standard reflector data.

3. The spectral unmixing method for smoked mural pigments according to claim 1, wherein: In S3, the spectral similarity measure Calculated by the following formula: Where, The first i The spectral vector of the bands, The first The spectral vector of each pixel point is calculated and the calculation results are normalized.

4. The spectral unmixing method for smoked mural pigments according to claim 3, wherein: In S4, layered optimization specifically includes: (a) Fixed end member matrix , update the sparse coefficient matrix through sparse regression : in, is the updated sparse coefficient matrix, is the end member matrix, the initial end member matrix is ​​constructed from the pure pigment spectrum library, is a sparse coefficient matrix, and the initial sparse coefficient matrix is ​​preliminarily estimated by least squares or sparse coding; (b) Fixed sparse coefficient matrix , update the endmember matrix by combining the least squares method with spectral smoothing constraints : in, is the updated endmember matrix, is the regularization parameter, is the transpose of the endmember matrix, is the spectral smoothing constraint matrix; (c) Repeat steps (a) and (b) until the objective function converges or the preset number of iterations is reached. The convergence criterion is that the relative change rate of the objective function value for 5 consecutive iterations is less than 0.01% or the F-norm change is 1e -4 .

5. The spectral unmixing method for smoked mural pigments according to claim 1, wherein: Also includes: The unmixing results were compared with the spectral data in the pure pigment spectral library, and the accuracy was evaluated using at least one of the root mean square error and spectral information divergence, where RMSE ≤ 0.1 and / or SID ≤ 0.05 met the accuracy requirements.

6. Smoked mural pigment spectrum unmixing system, characterized by: include: Data preparation and preprocessing module, which is used to build pure pigment spectrum library and smoke spectrum library based on hyperspectral imaging data; A sparse unmixing model construction module is used to represent the spectral signal of each pixel in the smoke pigment hyperspectral image as a linear combination of endmember spectra in the smoke spectrum library to construct a sparse unmixing model; The objective function design module is used to dynamically adjust the regularization weights based on the spectral similarity metric and the abundance gradient, construct the objective function of the sparse unmixing model, and dynamically adjust the regularization weights using a spatial adaptive mechanism based on spectral similarity, and update the weights by calculating the abundance; The hierarchical optimization and unmixing module uses the endmember matrix constructed from the pure pigment spectral library as input and uses the alternating least squares method combined with spectral smoothing constraints to iteratively optimize the objective function to obtain the unmixing results of the smoky pigment; The pure pigments include at least three mineral pigments selected from ochre, ultramarine, graphite, malachite, and clam powder. The pure pigment spectral library meets the following requirements: spectral resolution ≤ 5 nm, signal-to-noise ratio ≥ 50 dB, and the spectral average of at least 10 uncontaminated samples is collected for each pigment. The smoky pigments are obtained by placing simulated mural samples in a closed smoking environment for a preset time, and are classified into three categories: mild, moderate, and severe according to the smoking time. Sparse unmixing model, specifically: in, is the spectral vector of the j-th pixel in the smoky pigment hyperspectral image, is the endmember matrix constructed from the smoked spectral library, It is The sparse coefficient vector of pixels, It is The noise term of pixels; The objective function is: Among them, X is the sparse coefficient matrix, D is the end member matrix of pure pigment, and y is the hyperspectral data matrix of smoked pigment. represents the norm, is the adaptive L1 regularization term, is the absolute value of the coefficient value of the corresponding i-th band and j-th pixel in the sparse coefficient matrix. The proportion of non-zero elements in the sparse coefficient vector does not exceed 5%, and the absolute value of the non-zero element , is the adaptive regularization weight, and the calculation formula is: in, is the spectral similarity measure, , is the smoothness control parameter, is the abundance gradient.

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