CN112991299A - Method for constructing smooth point diffusion function in image processing - Google Patents
Method for constructing smooth point diffusion function in image processing Download PDFInfo
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Abstract
The invention discloses a method for constructing a point spread function based on a Smooth Principal Component Analysis (SPCA) in image processing, which comprises the steps of 1, fitting a PSF by using an expectation-maximization principal component analysis (EMPCA), and acquiring a j-th-order principal component Pj，kAnd coefficient of principal component Ci，j(ii) a Step 2, utilizing Moffatlets basis function Fl，kFitting each order principal component Pj，kAnd determining the coefficients of the basis functions Dj，l(ii) a And 3, updating the j order to be the j +1 order, and repeating the steps 1 and 2 until a satisfactory result is obtained. The invention combines the advantages of two schemes of the basis function and the EMPCA and can realize the aims of smoothness, good orthogonality and no overfitting.
Description
Technical Field
The invention belongs to the technical field of image processing, and particularly relates to a method for constructing a smooth point diffusion function in image processing.
Background
The Point Spread Function (PSF) is a problem that cannot be circumvented in the field of image acquisition and processing, and its essence is due to the diffraction effect of light, which reflects the aperture of the device for acquiring images, the support, and the pattern caused by the turbulence of the air flow on the light path. That is, a point source is not imaged as a point source, but rather as an extended pattern. Typically the aperture and the fourier transform of the stent pattern, which is smoothly continuous in the image two-dimensional space. The skilled person would measure this PSF prior to the study and then deconvolute this PSF by some means to obtain the original point source. How to accurately measure the PSF will then directly affect the deconvolution and the final effect on the measured image.
The methods for constructing PSF in the prior art are basically divided into two types: one is to consider the PSF as an expansion of a series of orthogonal basis functions by means of basis functions, and determine the final PSF by fitting the coefficients of these basis functions. The other is Expectation Maximization Principal Component Analysis (EMPCA), which is a method capable of estimating a measurement variance while acquiring a Principal Component, thereby maximizing Expectation.
The above two methods have advantages and disadvantages: the basis function method has the advantages of smoothness and quick fitting; the disadvantages are that the basis functions are not completely orthogonal due to the limited influence of pixelation and image size, resulting in cross-correlation of coefficients and that they do not make sense for higher order angle-dependent substructures due to their dependence on circularly symmetric basis functions. The EMPCA method has the advantages that the orthogonality is good, so that the interpolation of coefficients is facilitated, the reconstructed principal component is compact, and a high-order angle structure and even an irregular structure can be reconstructed; the disadvantage is the inclusion of noise and the lack of smoothness, i.e. overfitting, of the higher order components.
Disclosure of Invention
The technical problem to be solved by the present invention is to provide a method for constructing a point spread function by combining a smooth basis function with an EMPCA, aiming at the above-mentioned deficiencies of the prior art.
In order to achieve the technical purpose, the technical scheme adopted by the invention is as follows:
a method for constructing a smooth point spread function in image processing, comprising:
step 1, fitting PSF by using Expectation Maximization Principal Component Analysis (EMPCA) and obtaining a j-th order principal component Pj，kAnd coefficient of principal component Ci，j；
i is a positive integer and represents the serial number of the point source;
k is a positive integer and represents the kth pixel of the ith point source, namely the pixel serial number;
j is a non-negative integer representing the order of the principal component;
Pj，kis the corresponding value at the kth pixel in the jth order principal component;
step 2, utilizing Moffatlets basis function Fl，kFitting each order principal component Pj，kAnd determining the coefficients of the basis functions Dj，l；
And 3, updating the j th order to be a j +1 th order, and repeating the steps 1 and 2 until the residual error meets the set threshold value.
In order to optimize the technical scheme, the specific measures adopted further comprise:
in the step 1, the brightness of a series of point light sources is assumed to be Oi，kI.e. the PSF of the observation point source, is a two-dimensional density profile in the focal plane.
In step 1, the two-dimensional spatial distribution of any light source in EMPCA is expressed by the sum formula of principal components of each order, and the brightness I of the light source at the ith point isi，kExpressed as:
n is the maximum order of the principal component used to construct the PSF;
Ci，jis a principal component coefficient matrix representing coefficients on the j-th order principal component of the ith point source.
In step 2 above, Moffatlets basis functions are used to fit the principal components P of each orderj，kObtaining smooth PSF;
for a principal component Pj，kExpressed in terms of Moffatlets basis functions:
m is the maximum order of the basis function used to fit the j-th order principal component;
Dj，lis a basis function coefficient matrix which is expressed as a coefficient corresponding to the ith order basis function used for fitting the jth order principal component;
Fl，kis the value of the first order basis function at the kth pixel;
l is a non-negative integer representing the order of the basis function.
In the above step 2, the method for constructing D is carried out by the least square methodj，lLinear system of equations χ2Function, solved to obtain Dj，l；
The x2The function is:
Wi，kis a weight function.
For Wi，kAnd selecting local noise.
The invention has the following beneficial effects:
the invention combines the advantages of two schemes of the basis function and the EMPCA and can realize the aims of smoothness, good orthogonality and no overfitting.
Each item of SPCA (smooth PCA, smooth principal component analysis) is analyzed, so that the calculation is convenient and rapid; overfitting is avoided, so that signals and noise are effectively distinguished; SPCA is smooth and therefore closer to reality.
Drawings
In fig. 1, 4 PSFs simulated at 4 signal-to-noise SNR levels;
fig. 2 shows the results of comparing PSFs constructed using EMPCA and SPCA for four conditions with PSFs of the PhoSim simulation input under 4 SNR levels.
Detailed Description
Embodiments of the present invention are described in further detail below with reference to the accompanying drawings.
In practical observation, the brightness of a series of point light sources is assumed to be Oi，kAlso, it isI.e., PSF, is a two-dimensional density profile at the focal plane.
On the other hand, theoretically, in principal component analysis, the two-dimensional spatial distribution of any one light source can be expressed by a sum expression of principal components of respective orders, and the luminance I of the source at the ith pointi，kCan be expressed as
i is a positive integer and represents the serial number of the point source;
k is a positive integer and represents the kth pixel of the ith point source, namely the pixel serial number;
j is a non-negative integer representing the order of the principal component;
n is the maximum order of the principal component used to construct the PSF;
Ci，jis a principal component coefficient matrix which represents the coefficient on the j-th order principal component of the ith point source;
Pj，kis the corresponding value at the kth pixel in the jth order principal component;
in the past, the PSF constructed by expectation-maximization principal component analysis (EMPCA) directly obtains principal component Pj，kAnd coefficient matrix Ci，jThereby determining the best fit for the PSF.
However, the EMPCA method is prone to overfitting, i.e., fitting together with noise, and the result deviates from the original input PSF, so that the resulting PSF is not smooth and contains noise. Based on the method, the EMPCA method is improved, and Moffatlets basis functions are used for fitting each order of principal component Pj，kThis results in a smooth PSF, avoiding overfitting.
For a principal component Pj，kExpressed in terms of Moffatlets basis functions:
l is a non-negative integer representing the order of the basis function;
k is a positive integer and represents the kth pixel, i.e., the pixel number;
j is a non-negative integer representing the order of the principal component;
m is the maximum order of the basis function used to fit the j-th order principal component;
Dj，lis a basis function coefficient matrix which is expressed as a coefficient corresponding to the ith order basis function used for fitting the jth order principal component;
Fl，kis the value of the first order basis function at the kth pixel;
in step 2, the method of least squares is used to construct the structure of Dj，lLinear system of equations χ2Function, solved to obtain Dj，l；
Then to construct the PSF, the goal is to minimize the following χ2Function(s)
Wi，kIs a weight function and is known, while minimizing χ2The function can obtain Moffat basis function and coefficient D thereofi，lThereby obtaining a smooth PSF.
Wherein Wi，kLocal noise is typically taken as a weight, i.e., minimized.
Therefore, the steps of the present invention for reconstructing the image point spread function PSF are as follows:
step 1, fitting PSF by using Expectation Maximization Principal Component Analysis (EMPCA), and acquiring a j-th order principal component Pj，kAnd coefficient of principal component Ci，j；
Step 2, utilizing Moffatlets basis function Fl，kFitting each order principal component Pj，kAnd determining the coefficients of the basis functions Dj，l；
And 3, updating the j th order to be a j +1 th order, and repeating the steps 1 and 2 until the residual error meets the set threshold value. Results and comparison:
in the example, 2 CCDs in the LSST focal plane were simulated with simulation software, phosimm (version 3.4): R42S21 and R22S11, respectively, in the simulation software for four cases of diffraction effect (diffraction) in on-off mode: r42_ S21_ differentiation _ off, R42_ S21_ differentiation _ on, R22_ S11_ differentiation _ off, and R22_ S11 differentiation _ on.
In fig. 1, 4 PSFs were simulated at 4 signal-to-noise ratio SNR levels ( SNR 300, 200, 100, and 100) and (a) the top row SNR 300; (b) second row SNR 200; (c) third row SNR is 100; (d) the lowest row SNR is random (100-. It is clear that at higher orders SPCA is smoother than EMPCA.
In fig. 2, four conditions of R42_ S21_ differentiation _ off, R42_ S21_ differentiation _ on, R22_ S11_ differentiation _ off, and R22_ S11_ differentiation _ on were compared with PSFs of the phosimm analog input under 4 SNR levels ( SNR 300, 200, 100, and 100) with SNR ratios, wherein χ is a ratio of 300 to SPCA, and PSFs constructed by EMPCA and SPCA, respectively, were compared with PSFs of the phosimm analog input2The closer to 1 the better, the closer to noise the residual is. It is clear that the difference between EMPCA and SPCA is not large when the diffraction effect is off, but that SPCA is significantly better than EMPCA when the diffraction effect is on.
Example results demonstrate that SPCA avoids overfitting of EMPCA.
The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.
Claims (6)
1. A method for constructing a smooth point spread function in image processing is characterized by comprising the following steps:
step 1, fitting PSF by using an expectation maximization principal component analysis method, and acquiring a j-th order principal component Pj，kAnd coefficient of principal component Ci，j；
i is a positive integer and represents the serial number of the point source;
k is a positive integer and represents the kth pixel of the ith point source, namely the pixel serial number;
j is a non-negative integer representing the order of the principal component;
Pj，kis the corresponding value at the kth pixel in the jth order principal component;
step 2, utilizing Moffatlets basis function Fl，kFitting each order principal component Pj，kAnd determining the coefficients of the basis functions Dj，l；
And 3, updating the j th order to be a j +1 th order, and repeating the steps 1 and 2 until the residual error meets the set threshold value.
2. The method of claim 1, wherein in step 1, the brightness of a series of point light sources is assumed to be Oi，kI.e. the PSF of the observation point source, is a two-dimensional density profile in the focal plane.
3. The method of claim 2, wherein in step 1, the two-dimensional spatial distribution of any light source in EMPCA is expressed by a sum of principal components of each order, and the luminance I of the ith point source is represented byi，kExpressed as:
n is the maximum order of the principal component used to construct the PSF;
Ci，jis a principal component coefficient matrix representing coefficients on the j-th order principal component of the ith point source.
4. The method of claim 3, wherein in step 2, Moffatlets basis functions are used to fit principal components P of each orderj，kObtaining smooth PSF;
for oneA main component Pj，kExpressed in terms of Moffatlets basis functions:
m is the maximum order of the basis function used to fit the j-th order principal component;
Dj，lis a coefficient of a basis function, which is expressed as a coefficient corresponding to a basic function of the ith order used for fitting the principal component of the jth order;
Fl，kis the value of the first order basis function at the kth pixel;
l is a non-negative integer representing the order of the basis function.
5. The method of claim 4, wherein in step 2, the smooth point spread function is constructed by least square method with respect to Dj，lLinear system of equations χ2Function, solved to obtain Dj，l；
The x2The function is:
Wi，kis a weight function.
6. The method of claim 5, wherein for W, the method comprisesi，kAnd selecting local noise.
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