CN108896499A - In conjunction with principal component analysis and the polynomial spectral reflectance recovery method of regularization - Google Patents
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Abstract
The invention discloses a kind of combination principal component analysis and the polynomial spectral reflectance recovery method of regularization, include the following steps:1) using the spectroscopic data of multi-optical spectrum imaging system acquisition training sample;2) dimensionality reduction is carried out using spectroscopic data of the principal component analytical method to training sample, to reduce the spectroscopic data amount of training sample;3) training sample set is constructed by the spectroscopic data for handling obtained training sample through step 2), the extension of polynomial regression channel response is carried out to training sample set again, then construct the spectral reflectivity to reconstruct and between actual value the minimum target of error objective function, and limit entry is added into objective function using Tikhonov regularization method, finally solve the objective function, the spectral reflectivity that must be rebuild, it completes to combine principal component analysis and the polynomial spectral reflectance recovery of regularization, this method can rebuild to obtain accurate spectral reflectivity.
Description
Technical field
The invention belongs to digital image processing field, it is related to a kind of combination principal component analysis and the polynomial spectrum of regularization
Reflectivity method for reconstructing.
Background technique
The method that tradition obtains object spectra reflectivity is using the point-to-point measurement of spectrophotometer, and workload is very
Greatly, due to people life in most of body surfaces spectral reflectivity be it is smooth, in order to be efficiently obtained body surface
Spectral reflectivity can use multispectral colouring information of the multi-optical spectrum imaging system acquisition object under multiple channels, reuse
Algorithm for reconstructing efficiently reappears the continuous spectrum of body surface, and this method is known as the spectral reflectance based on multi-optical spectrum imaging technology
Rate method for reconstructing, this method can obtain the information of target object from spectral Dimensions and Spatial Dimension simultaneously.However, traditional colour
Degree Spatial Dimension only has 1/10th of the dimension of spectral space, when indicating color, needs to handle using spectral reflectivity big
The spectroscopic data of amount, so that treatment effeciency is low, memory space increases, therefore, can be to big under conditions of meeting reconstruction precision
It measures spectroscopic data and carries out data compression, to achieve the effect that spectrum dimensionality reduction.Currently, common spectroscopic data by dimension method have it is main at
Divide the related algorithm of analysis method, independent principal component analytical method and other improvements, however above method cannot reconstruct standard
True spectral reflectivity.
Summary of the invention
It is an object of the invention to overcome the above-mentioned prior art, a kind of combination principal component analysis and canonical are provided
Change polynomial spectral reflectance recovery method, this method can rebuild to obtain accurate spectral reflectivity.
In order to achieve the above objectives, combination principal component analysis of the present invention and the polynomial spectral reflectivity weight of regularization
Construction method includes the following steps:
1) using the spectroscopic data of multi-optical spectrum imaging system acquisition training sample;
2) dimensionality reduction is carried out using spectroscopic data of the principal component analytical method to training sample, to reduce the spectrum of training sample
Data volume;
3) training sample set is constructed by the spectroscopic data for handling obtained training sample through step 2), then to training sample
Set carries out the extension of polynomial regression channel response, then constructs the spectral reflectivity to reconstruct and error is most between actual value
The small objective function for target, and limit entry is added into objective function using Tikhonov regularization method, finally solve institute
Objective function is stated, the spectral reflectivity of reconstruction is obtained, completes to combine principal component analysis and the polynomial spectral reflectivity weight of regularization
It builds.
Multi-optical spectrum imaging system in step 1) is SpectroCam VIS CCD monochrome cameras.
The specific operation process of step 3) is:
1a) obtain the channel response g of multispectral camera and matched i channel filteri, i.e.,
Wherein, l (λ), fi(λ), o (λ) and s (λ) are function of spectral power distribution, the multi-optical spectrum imaging system of lighting source
The spectrum sensitive function of the transmissivity of i-th of channel filter, the spectral transfer function of camera lens and multispectral camera;
2a) with k orthogonal basis vectors biLinear combination be approximately by spectral reflectance data collection R:
Wherein, B=[b1,b2,Λ,bk] it is characterized vector matrix, a=[a1,a2,Λ,ak]TFor transition matrix;
3a) the corresponding channel response g of each color cardiFor gi=[g1,g2,Λ,gC]T, utilize polynomial regression pair
Original channel response giCarry out higher order polynomial multiplication extension, wherein giResult after extensionFor:
WithThe matrix for indicating m × K dimension, obtains transition matrix Q+For:
Wherein, using polynomial regression to original channel response giThe purpose for carrying out higher order polynomial multiplication extension exists
In keeping the difference between the spectral reflectivity reconstructed and actual value minimum, that is, construct objective function:
E||R-Q+g||2→Minimum (5)
Wherein, E is mean square deviation, | | | |2For L2Norm, Minimum are to minimize;
4a) using Tikhonov regularization method in formula (5) | | R-Q+G | | middle addition limit entry then adds limitation
Objective function is after:
Wherein, λ is regularization parameter;
5a) according to the limit entry of addition to matrixConditional number Cond changed accordingly, and by matrixChange
ForSo that matrixCharacteristic value can fall inIn range, wherein matrixItem
Number of packages meets:
6a) withAnd lg | | Q+| | it is respectively the abscissa and ordinate of L-curve, when λ makesAnd lg | | Q+| | while when obtaining sufficiently small, then correspond to the angle point of the L-curve, wherein the angle point is L bent
The maximum curvature point of line, ifη (λ)=| | Q+| |, then L-curve curvature K (λ) calculation formula is:
As d [K (λ)]/d λ=0, curvature K (λ) is maximized, and obtains optimal regularization parameter λ;
7a) the optimal regularization parameter λ for calculating step 6a) is substituted into formula (6), is then solved objective function, is obtained
The spectral reflectivity of reconstruction.
The invention has the advantages that:
Combination principal component analysis of the present invention is specifically being grasped with the polynomial spectral reflectance recovery method of regularization
When making, using the spectroscopic data of multi-optical spectrum imaging system acquisition training sample, recycle principal component analytical method to training sample
Spectroscopic data carry out dimensionality reduction, to reduce the data volume of training sample, then to training sample set carry out polynomial regression
Channel response extension, then objective function is constructed, and limit entry is added into objective function using Tikhonov regularization method, lead to
Crossing limit entry, data are unstable and random noise bring ill-conditioning problem limiting, to reduce in multispectral data treatment process
Data volume achievees the purpose that the precision for improving the spectral reflectivity rebuild.
Detailed description of the invention
Fig. 1 is mural painting reference color block label figure in emulation experiment;
Fig. 2 is the multispectral image figure of mural painting in emulation experiment;
Fig. 3 a is the reconstruction result map of 1 position in Fig. 1;
Fig. 3 b is the reconstruction result map of 2 positions in Fig. 1;
Fig. 3 c is the reconstruction result map of 3 positions in Fig. 1;
Fig. 3 d is the reconstruction result map of 4 positions in Fig. 1;
Fig. 3 e is the reconstruction result map of 5 positions in Fig. 1;
Fig. 3 f is the reconstruction result map of 6 positions in Fig. 1.
Specific embodiment
The invention will be described in further detail with reference to the accompanying drawing:
Combination principal component analysis of the present invention includes following with the polynomial spectral reflectance recovery method of regularization
Step:
1) using the spectroscopic data of multi-optical spectrum imaging system acquisition training sample, specifically, passing through multi-optical spectrum imaging system
The image information of picture to be processed is acquired, and using collected information as the spectroscopic data of training sample;
2) dimensionality reduction is carried out using spectroscopic data of the principal component analytical method to training sample, to reduce the spectrum of training sample
Data volume;
3) training sample set is constructed by the spectroscopic data for handling obtained training sample through step 2), then to training sample
Set carries out the extension of polynomial regression channel response, then constructs the spectral reflectivity to reconstruct and error is most between actual value
The small objective function for target, and limit entry is added into objective function using Tikhonov regularization method, finally solve institute
Objective function is stated, the spectral reflectivity of reconstruction is obtained, completes to combine principal component analysis and the polynomial spectral reflectivity weight of regularization
It builds.
Multi-optical spectrum imaging system in step 1) is SpectroCam VIS CCD monochrome cameras.
The concrete operations of step 3) are:
The spectral reflectivity of object 1a) is rebuild using rebuilding spectrum algorithm based on multi-optical spectrum imaging technology, is obtained first more
The digital response g of spectrum camera and matched i channel filteri, wherein
Wherein, l (λ), fi(λ), o (λ) and s (λ) are respectively the function of spectral power distribution of lighting source, multispectral imaging
The spectrum sensitive function of the transmissivity of i-th of channel filter, the spectral transfer function of camera lens and multispectral camera in system.
Construct single spectral function qi(λ)=l (λ) fi(λ) o (λ) s (λ), biAnd niThe dark current respectively represented under i-th of optical filter is rung
Answer and system generate noise.After being sampled according to N point into row interval to the spectral reflectivity in visible-range 380-780nm,
The vector R that will obtain a N × 1 dimension indicates that the camera channel of the dimensional vector of C × 1 responds with g, will using vector matrix form
Formula (1) is rewritten as:
G=QR+b+n (2)
Wherein, Q is C × N-dimensional spectral response matrix, and formula (2) is rewritten as G=QR+n using G=g-b, passes through mark
Quasi- blank correction removal dark current b, wherein the updating formula for removing dark current is:
Wherein, GwhiteFor the response of standard white plate, GdataFor the real response of system, GdarkTo use lens cap to block phase
System response after machine camera lens;The influence of noise n is not considered after removing system dark current, then the basic conversion shape of formula (2)
Formula is:
G=QR (4)
2a) with k orthogonal basis vectors biLinear combination spectral reflectance data collection R is approximately indicated
For:
Wherein, B=[b1,b2,Λ,bk] it is characterized vector matrix, a=[a1,a2,Λ,ak]TFor transition matrix, base vector
Principal component analytical method can be used to be calculated, it is assumed that N*M ties up matrix R=(R1,R2,Λ,Rm)TBy M unequal spectrum
Reflectivity composition carries out singular value decomposition (SVD) to matrix R, then has:
R=USVT (6)
Wherein, S is diagonal matrix, is made of a singular value arranged from big to small of r=rank (R), i.e.,
S=diag (σ1,σ2,Λ,σi) (7)
By RRTFeature vector composition N*N tie up unit orthogonal matrix U, RTThe unit matrix of the feature vector composition M*M dimension of R
The linear combination of V, former spectral reflectance data are indicated by preceding k principal component, wherein the number of principal component is contributed by principal component
Rate determines that the principal component accumulation contribution rate of preceding k feature vector is:
Wherein, when principal component number meets the preceding k accumulation contribution rate ρ of acquirement spectral reflectivityk>When 99.95%, then will
Formula (5) is substituted into formula (4) and is obtained:
g0=QBa0 (9)
Wherein, g0For the channel response of training sample, Ba0For the spectral reflectivity collection R of training sample0Feature vector square
Battle array and transition matrix, QB are the response values of preceding k feature vector principal component, obtain coefficient transition matrix a by formula (9)0
For:
a0=[(QB)T(QB)]-1(QB)Tg0 (10)
The value of QB can be found out according to training sample in practice, wherein
Meanwhile the value of QB changes with system response, if using new test sample, coefficient transition matrix can be with
It is expressed as:
Wherein, g is the channel response of test sample, the then spectral reflectivity rebuild using principal component analytical methodFor:
3a) for each color card, corresponding channel response contains C element, i.e. gi=[g1,g2,Λ,gC]T, lead to
In normal situation, number of active lanes is more, and the linearity of imaging system is better, but the filter disc of multi-optical spectrum imaging system can not be without limitation
Increase.Under the premise of not increasing the number of optical filter, for the linearity for improving system, polynomial regression pair can use
Original channel response carries out higher order polynomial multiplication extension, i.e. result after gi extension is
WithIndicate that (m is a m × KThe number of middle element) dimension matrix, obtain new transition matrix Q+For:
Use the difference that the purpose that polynomial regression is extended is between the spectral reflectivity for making to reconstruct and actual value
Minimum, i.e.,:
E||R-Q+g||2→Minimum (16)
Wherein, | | | |2Represent L2Norm.
4a) to solve ill ill-posed problem, Tikhonov regularization method can be used, limit is added in objective function
Item processed allows the minimization problem of the objective function after limit entry is added suitable fixed, wherein addition limitation from the angle of optimum theory
Objective function is changed into after condition:
Previous item in formula (17) guarantees the authenticity of spectroscopic data, the flatness of latter control solution, and λ is regularization ginseng
Number adjusts the weight relationship between data validity and the flatness of solution.
By formula (4) it is found that the precision of spectral reflectance recovery depends on transition matrix Q+, Q+It is the reflection by training sample
What rate matrix R and response matrix g was obtained, therefore, from the aspect of spectroscopic data and channel response two, using dimensionality reduction and regularization
Method solve ill-condition equation simultaneously, it is necessary first to the spectral signature vector B of spectroscopic data is extracted using principal component analytical method,
Channel response and in this, as the input variable of training sample, after extension is then adjusted using regularization Controlling object functionAnd in this, as the output variable of training sample, wherein be added to Tikhonov restrictive condition and principal component analysis dimensionality reduction
Transition matrix can change:
Wherein, I is unit matrix, and formula (18) are carried out singular value decomposition, are obtained:
The stability of system can be generally measured by the conditional number of matrix, the conditional number of matrix is defined as:
Wherein, σmaxAnd σminFor matrixMaximum singular value and minimum singular value.
After restrictive condition 5a) is added using Tikhonov method, matrixConditional number Cond needs become accordingly
Change, because of matrixFor a positive semidefinite symmetrical matrix, matrixIt is changed toSo that matrixSpy
Value indicative can be fallen inIn range, thus restrictive condition coefficient, it is ensured that conditional number can obtain minimum value, i.e., its
Number of packages meets:
Formula (21) shows that conditional number can be controlled by regularization parameter λ, the disease bigger for those conditional numbers
State matrix, so that it may best λ be chosen by L-curve method to reduce its ill-conditioning problem, wherein the finding process of best λ is:
With | | Q+| | it is the function of regularization parameter λ, withAnd lg | | Q+| | it is respectively
The abscissa and ordinate of L-curve, when λ makesAnd lg | | Q+| | while when obtaining sufficiently small, then correspond to L song
Line angle point, and the angle point is L-curve point of maximum curvature, usually best λ is determined by the maximum point, if settingη
(λ)=| | Q+| |, then L-curve curvature K (λ) calculation formula is:
As d [K (λ)]/d λ=0, curvature K (λ) is maximized, and this value is optimal regularization parameter λ.
Emulation experiment
In order to verify the application effect of (PTP method) of the invention in a practical situation, ratio is chosen for the true mural painting of a width
More typical six color reference regions are studied, and mural painting color lump chosen area is obtained by multispectral system acquisition mural painting
Multispectral image under 11 channels, analysis PTP method carry out the precision of rebuilding spectrum.It is with reference to color lump with six in mural painting
Test sample carries out rebuilding spectrums using tri- kinds of methods of PCA, RPR and PTP, six color lump spectral accuracies that PTP method is rebuild with
PCA with RPR method is compared, and average RMSE value, average GFC value, average ISSD value have apparent reduction, using PTP method into
Row spectral reflectance recovery can obtain preferable spectral accuracy.In order to accurately show the reference of PCA, RPR and PTP method reconstruction
The coloration precision of color lump, by mural painting with reference to the L in color lump coloration*, a*, b*Component value calculates, L*For brightness, a*To be red green, b*For
Champac.It can be concluded that original chrominance of the color lump chromatic value of PTP method reconstruction closer to color lump, the present invention can be compared
The better coloration precision of PCA and RPR method, wherein 1 comparing result of table.
Table 1
Claims (3)
1. a kind of combination principal component analysis and the polynomial spectral reflectance recovery method of regularization, which is characterized in that including with
Lower step:
1) using the spectroscopic data of multi-optical spectrum imaging system acquisition training sample;
2) dimensionality reduction is carried out using spectroscopic data of the principal component analytical method to training sample, to reduce the spectroscopic data of training sample
Amount;
3) training sample set is constructed by the spectroscopic data for handling obtained training sample through step 2), then to training sample set
The extension of polynomial regression channel response is carried out, the spectral reflectivity to reconstruct then is constructed and error is minimum between actual value
The objective function of target, and limit entry is added into objective function using Tikhonov regularization method, finally solve the mesh
Scalar functions obtain the spectral reflectivity of reconstruction, complete to combine principal component analysis and the polynomial spectral reflectance recovery of regularization.
2. combination principal component analysis according to claim 1 and the polynomial spectral reflectance recovery method of regularization,
It is characterized in that, the multi-optical spectrum imaging system in step 1) is SpectroCam VIS CCD monochrome cameras.
3. combination principal component analysis according to claim 1 and the polynomial spectral reflectance recovery method of regularization,
It is characterized in that, the specific operation process of step 3) is:
1a) obtain the channel response g of multispectral camera and matched i channel filteri, i.e.,
Wherein, l (λ), fi(λ), o (λ) and s (λ) are i-th of function of spectral power distribution, multi-optical spectrum imaging system of lighting source
The spectrum sensitive function of the transmissivity of channel filter, the spectral transfer function of camera lens and multispectral camera;
2a) with k orthogonal basis vectors biLinear combination be approximately by spectral reflectance data collection R:
Wherein, B=[b1,b2,Λ,bk] it is characterized vector matrix, a=[a1,a2,Λ,ak]TFor transition matrix;
3a) the corresponding channel response g of each color cardiFor gi=[g1,g2,Λ,gC]T, using polynomial regression to original
Channel response giCarry out higher order polynomial multiplication extension, wherein giResult after extensionFor:
WithThe matrix for indicating m × K dimension, obtains transition matrix Q+For:
Wherein, using polynomial regression to original channel response giThe purpose for carrying out higher order polynomial multiplication extension is to make
Difference between the spectral reflectivity reconstructed and actual value is minimum, i.e. building objective function:
E||R-Q+g||2→Minimum (5)
Wherein, E is mean square deviation, | | | |2For L2Norm, Minimum are to minimize;
4a) using Tikhonov regularization method in formula (5) | | R-Q+G | | middle addition limit entry then adds mesh after limit entry
Scalar functions are:
Wherein, λ is regularization parameter;
5a) according to the limit entry of addition to matrixConditional number Cond changed accordingly, and by matrixIt is changed toSo that matrixCharacteristic value can fall inIn range, wherein matrixCondition
Number meets:
6a) withAndThe respectively abscissa and ordinate of L-curve, when λ makesAnd
lg||Q+| | while when obtaining sufficiently small, then correspond to the angle point of the L-curve, wherein the angle point is the maximum curvature of L-curve
Point, ifη (λ)=| | Q+| |, then L-curve curvature K (λ) calculation formula is:
As d [K (λ)]/d λ=0, curvature K (λ) is maximized, and obtains optimal regularization parameter λ;
7a) the optimal regularization parameter λ for calculating step 6a) is substituted into formula (6), is then solved objective function, must be rebuild
Spectral reflectivity.
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