CN113205462B - Photon reflectivity image denoising method based on neural network learning prior - Google Patents

Photon reflectivity image denoising method based on neural network learning prior Download PDF

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CN113205462B
CN113205462B CN202110367190.0A CN202110367190A CN113205462B CN 113205462 B CN113205462 B CN 113205462B CN 202110367190 A CN202110367190 A CN 202110367190A CN 113205462 B CN113205462 B CN 113205462B
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田昕
何访
陈葳
李松
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Abstract

The invention provides a photon reflectivity image denoising method based on neural network learning prior, which mainly comprises the construction of an imaging model and the networking of the model. Firstly, a photon counting imaging model is constructed, the model is solved to obtain corresponding subproblems, then the model of the subproblems is networked to construct a convolutional neural network, a training set is given, corresponding parameters of a priori and the model are learned, and finally the denoising result of the photon reflectivity image is obtained. Experiments prove that the method provided by the invention networks the model through learning prior, and adopts the convolutional neural network to denoise the image, so that the indexes of the denoised image are close to ideal values.

Description

Photon reflectivity image denoising method based on neural network learning prior
Technical Field
The invention belongs to the field of photon counting imaging, and relates to a photon reflectivity image denoising method based on neural network learning prior, which is suitable for various photon counting imaging application scenes.
Background
Photon counting imaging can be used to detect targets in extremely low light conditions, while other imaging techniques have difficulty acquiring valid data in extremely low light conditions. Therefore, the method has attracted extensive research interest in a variety of fields, such as spatial monitoring, biological imaging, fluorescence, and microscopy. Unlike most imaging techniques that measure the intensity of reflected light from a target, photon counting imaging computes each photon collected from reflected light from a different spatial point. Under the condition of extremely low illumination, the background noise is relatively large.
Considering the effect of noise under low light conditions, a poisson distribution is often used to measure the counting process of reflected photons. To remove poisson noise, the poisson distribution may be approximated with a gaussian distribution using a variance-stabilized transform (VST), and denoised using conventional denoising algorithms such as block matching and three-dimensional filtering (BM 3D). However, in most single photon counting imaging applications, this approximation is inaccurate because the number of photons received is very low, resulting in high frequency artifacts. In recent work on single photon counting imaging, reconstruction from the count data has always been done by minimizing the negative poisson log-likelihood term. In 2014, the Ahmed Kirmani team published a First-photon imaging (FPI) system on Science, which proposed to recover 3D structure and reflectivity information from the First detected photon of each pixel. Dongeek Shin proposed a reliable method to estimate reflectivity based on FPI that could better yield a reflectivity image using a fixed dwell time for each pixel. The deep learning method is also applied to single photon counting imaging, but the existing methods are all based on a black box model and cannot combine a real physical model with a network. How to effectively remove the noise in the photon counting image has become a focus of attention and research of many researchers at home and abroad.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a photon reflectivity image denoising method based on neural network learning prior.
The technical scheme adopted by the invention is as follows: a photon reflectivity image denoising method based on neural network learning prior mainly comprises the construction of an imaging model and the networking of the model. Firstly, constructing a photon counting imaging model, solving the model to obtain a corresponding subproblem, then networking the model of the subproblem, constructing a convolutional neural network, giving a training set, learning corresponding parameters of a prior and the model, and finally obtaining a denoising result of a photon reflectivity image. The method comprises the following steps:
step 1: constructing a photon counting imaging model, which can be expressed as:
Figure BDA0003007631500000021
where α represents the photon reflectance image, J (α) is the data fidelity term, and K (α) is the a priori constraint term.
And 2, step: and solving the imaging model to obtain corresponding subproblems.
And step 3: and (4) networking the physical model of the subproblem, and constructing a convolutional neural network and a loss function.
And 4, step 4: and training the convolutional neural network, and obtaining a denoising result of the reflectivity image according to the trained network model.
Preferably, the specific implementation of step 1 comprises the following sub-steps:
step 1.1: a data fidelity term J (α) is constructed. In general, a given photon count value ki,jReflectance of alphai,jThe poisson negative log-likelihood function of (a) may be expressed as:
Figure BDA0003007631500000022
wherein N is the number of emitted pulses, η is the detection efficiency of the detector, B is the background count of the pulse repetition period, S is the signal count within the pulse repetition period, and i, j are the spatial point coordinates. The above equation is taken as a data fidelity term.
Step 1.2: an a priori constraint term K (α) is constructed. In the traditional method, the wavelet domain is sparse, the gradient is sparse and not necessarily accurate, and in order to obtain better effect, the invention adopts a general nonlinear function to carry out sparsification on the reflectivity image, and the function is recorded as
Figure BDA0003007631500000023
The parameters of which are learnable. Inspired by the powerful nonlinear feature representation capability of the convolutional neural network,will be provided with
Figure BDA0003007631500000024
Designed as a combination of two linear convolution operators separated by one linear rectifying unit (ReLU), as shown in fig. 2. From this, the expression of the a priori constraint term is obtained:
Figure BDA0003007631500000025
step 1.3: and obtaining a final reflectivity image reconstruction model. Since η S α < 1, the following approximation can be made:
1-exp[-(ηSα+B)]≈ηSα+B
the final reconstruction model can thus be obtained:
Figure BDA0003007631500000026
where λ is the regularization parameter and k is the matrix of photon count values for all spatial points.
Preferably, the specific implementation of step 2 comprises the following sub-steps:
step 2.1: solving formula (1) based on an optimization strategy (A fast iterative learning algorithm for linear inverse schemes) of FISTA to obtain the following two sub-problems:
Figure BDA0003007631500000031
Figure BDA0003007631500000032
where ρ is a constant, rxIs the result of the x-th iteration calculation of the intermediate variable r, which is alphaxAnd directly reconstructing a result in the x iteration, wherein the initial value can be obtained according to the maximum likelihood estimation.
Preferably, the specific implementation of step 3 comprises the following sub-steps:
step 3.1: r isxThe model is networked, and in order to maintain the network structure and increase the flexibility of the network, the value of the parameter ρ is not fixed and is made to vary as a network parameter with the learning of the network, and then r is calculated according to equation (2)xThe model becomes:
Figure BDA0003007631500000033
step 3.2: alpha (alpha) ("alpha")xAnd networking the model. To network the model, let r bexAnd
Figure BDA0003007631500000034
are respectively alpha and
Figure BDA0003007631500000035
the following approximation can be obtained for the mean value of:
Figure BDA0003007631500000036
wherein beta is a single bond
Figure BDA0003007631500000037
The associated scalar, applying the above approximation term to equation (3), yields:
Figure BDA0003007631500000038
where θ is β λ, then, we obtain
Figure BDA0003007631500000039
Closed loop form of (1):
Figure BDA00030076315000000310
introduction of
Figure BDA00030076315000000311
Is inverse operation of
Figure BDA00030076315000000312
Definition of
Figure BDA00030076315000000313
Finally, alpha is obtainedxThe networking model of (1):
Figure BDA00030076315000000314
in order to maintain the flexibility of the network and increase the capacity of the network, the network is controlled by the control unit
Figure BDA00030076315000000315
Theta varies with the iteration of the network, and thus, alpha can be obtainedxNetworked final model of (2):
Figure BDA00030076315000000316
FIG. 2 shows rxAnd alphaxHow the closed loop form of (A) is mapped to a deep network, first using maximum likelihood estimation from alphax-1To obtain rx，rxPassing through the network layer
Figure BDA00030076315000000317
Become into
Figure BDA00030076315000000318
In a soft threshold operation
Figure BDA00030076315000000319
To obtain
Figure BDA00030076315000000320
In passing through the network layer
Figure BDA00030076315000000321
Output alphaxAnd completing a closed loop operation.
Step 3.3: a loss function is constructed. Employing data sets
Figure BDA00030076315000000322
Training is carried out, and the network counts the photons to form an image kiAs input, generating a reconstruction result
Figure BDA0003007631500000041
We are satisfying the symmetry condition
Figure BDA0003007631500000042
Is reduced in the case of
Figure BDA0003007631500000043
And group Truth yiFrom this, the following loss function can be designed:
Figure BDA0003007631500000044
wherein:
Figure BDA0003007631500000045
Figure BDA0003007631500000046
wherein N isbIs the total number of training blocks, NpNumber of network layers, NsFor each training block size, γ is a constant.
The invention mainly aims at the novel computational imaging technology of single photon counting imaging, and provides a denoising method based on convolutional neural network learning prior, which combines the advantages of a model and a network by networking a physical model.
Drawings
FIG. 1 is a network framework diagram of the present invention.
FIG. 2 is a non-linear function
Figure BDA0003007631500000048
And its inverse function
Figure BDA0003007631500000049
The structure of (1) is formed into a diagram.
Fig. 3 is a comparison of visual quality of a photon reflectance image reconstructed by using simulation data and the method of the present invention and other comparison methods, where (a) is a group Truth, (b) is a photon counting simulation image added with poisson noise and impulse noise, (c) is a denoising result of SPIRALONB, (d) is a denoising result of binomial SPIRAL TV, (e) is a denoising result of NLSPCA, (f) is a denoising result of VST + BM3D, and (g) is a reconstruction result of the method of the present invention.
Detailed Description
In order to facilitate the understanding and implementation of the present invention for those of ordinary skill in the art, the present invention will be described in further detail with reference to the accompanying drawings and examples, it is to be understood that the examples described herein are only for the purpose of illustrating the present invention and are not to be construed as limiting the present invention.
As shown in fig. 1, an embodiment of the present invention provides a photon reflectivity image denoising method based on convolutional neural network learning prior. The method specifically comprises the following steps:
step 1: constructing a photon counting imaging model, which can be expressed as:
Figure BDA0003007631500000047
where α represents the photon reflectance image, J (α) is the data fidelity term, and K (α) is the a priori constraint term. The specific implementation comprises the following substeps:
step 1.1: and constructing a data fidelity item J (alpha). In general, a given photon count value ki,jReflectivity of alphai,jThe poisson negative log-likelihood function of (a) may be expressed as:
Figure BDA0003007631500000051
the above equation is taken as a data fidelity term. Wherein N is the number of emitted pulses, η is the detection efficiency of the detector, B is the background count of the pulse repetition period, S is the signal count within the pulse repetition period, and i, j are the spatial point coordinates. In this embodiment, N is 100, η is 0.35, B is 0, and S is 1.
Step 1.2: an a priori constraint term K (α) is constructed. In the traditional method, the wavelet domain is sparse, the gradient is sparse and not accurate, and in order to obtain a better effect, the invention adopts a general nonlinear function to carry out sparsification on the reflectivity image, and the function is recorded as
Figure BDA0003007631500000052
The parameters of which are learnable. Inspired by the powerful representation capability of the convolutional neural network and the general approximate property thereof, the method is to
Figure BDA0003007631500000053
Designed as a combination of two linear convolution operators separated by one linear rectifying unit (ReLU), as shown in fig. 2. From this, the expression of the a priori constraint term is obtained:
Figure BDA0003007631500000054
step 1.3: and obtaining a final reflectivity image reconstruction model. Since η S α < 1, the following approximation can be made:
1-exp[-(ηSα+B)]≈ηSα+B
the final reconstruction model can thus be obtained:
Figure BDA0003007631500000055
where λ is the regularization parameter and k is the matrix of photon count values for all spatial points.
Step 2: and solving the imaging model to obtain corresponding subproblems. The method specifically comprises the following steps:
step 2.1: solving formula (1) based on an optimization strategy (A fast iterative learning algorithm for linear inverse schemes) of FISTA to obtain the following two sub-problems:
Figure BDA0003007631500000056
Figure BDA0003007631500000057
where ρ is a constant, rxIs the result of the x-th iteration calculation of the intermediate variable r, which is alphaxThe direct reconstruction result at the x-th iteration, the initial value of which can be obtained from the maximum likelihood estimation, i.e.
Figure BDA0003007631500000058
And 3, step 3: and (4) networking the physical model of the subproblem, and constructing a convolutional neural network and a loss function. The specific implementation comprises the following substeps:
step 3.1: r isxAnd networking the model. In order to maintain the structure of the network and increase the flexibility of the network, the value of the parameter ρ is not fixed, and is made to vary as a network parameter with the learning of the network. Then, r is according to the formula (1)xThe model becomes:
Figure BDA0003007631500000061
in the present embodiment, it is preferred that,
Figure BDA0003007631500000062
step 3.2: alpha (alpha) ("alpha")xAnd networking the model. To network the model, let rxAnd
Figure BDA0003007631500000063
are respectively alpha and
Figure BDA0003007631500000064
the following approximation can be obtained for the mean value of:
Figure BDA0003007631500000065
wherein beta is a single radical of
Figure BDA0003007631500000066
Scalar quantity concerned, applying the above approximation term to the equation (2) and adopting
Figure BDA0003007631500000067
The image is thinned, so that the following results can be obtained:
Figure BDA0003007631500000068
where θ is β λ, then, obtaining
Figure BDA0003007631500000069
Closed-loop form of (c):
Figure BDA00030076315000000610
introduction of
Figure BDA00030076315000000611
Is inverse operation of
Figure BDA00030076315000000612
Definition of
Figure BDA00030076315000000613
Finally obtaining alphaxThe networking model of (1):
Figure BDA00030076315000000614
in order to maintain the flexibility of the network and increase the capacity of the network, the network is controlled by a control unit
Figure BDA00030076315000000615
Theta varies with the iteration of the network, and thus, alpha can be obtainedxNetworked final model of (2):
Figure BDA00030076315000000616
FIG. 2 shows rxAnd alphaxHow to map to a deep network. First using maximum likelihood estimation, from alphax-1To obtain rx，rxPassing through the network layer
Figure BDA00030076315000000617
Become into
Figure BDA00030076315000000618
In a soft threshold operation
Figure BDA00030076315000000619
To obtain
Figure BDA00030076315000000620
In passing through the network layer
Figure BDA00030076315000000621
Output alphaxAnd completing a closed loop operation. In the present embodiment, θ0＝0.0005。
Step 3.3: a loss function is constructed. Using data sets
Figure BDA00030076315000000622
Training is carried out, and the network counts the photons to form an image kiGenerating a reconstruction result as an input
Figure BDA00030076315000000623
We satisfy the symmetry condition
Figure BDA00030076315000000624
Is reduced in the case of
Figure BDA00030076315000000625
And group Truth yiFrom this, the following loss function can be designed:
Figure BDA0003007631500000071
wherein:
Figure BDA0003007631500000072
Figure BDA0003007631500000073
wherein N isbIs the total number of training blocks, NpIs the number of network layers, NsFor each training block size, γ is a constant. In this embodiment, Nb＝192000，Ns＝1089，Np＝9，γ＝0.01。
And 4, step 4: and training the network by using the simulation data, and obtaining a denoising result of the reflectivity image according to the trained network model. In this embodiment, the training speed lr is 0.0001, the batch size is 64, the convolution kernel size is 3 × 3, and the number of channels is 32.
And obtaining a photon reflectivity image based on the operation, selecting a 'Male' image with the resolution of 1024 multiplied by 1024 in an image database of a signal and image processing research institute from southern California university as a true value image in order to quantitatively evaluate the reconstructed image, simulating the true value image to obtain a photon counting image, and selecting a peak signal-to-noise ratio (PSNR) as an evaluation index. For comparison with other methods, we used SPIRALONB method, binomial SPIRAL TV method, NLSPCA method, and VST + BM3D method to compare with our method, we selected 100 pictures in BSD300 data set as training set, and the results are shown in table 1. For a comparison of the visual quality of the reconstructed images of the algorithms, see fig. 3.
TABLE 1 PSNR (dB) comparison of different reconstruction methods (ideal: +∞)
Figure BDA0003007631500000074
It can be seen that the method provided by the inventor networks the model through learning prior, and adopts the convolutional neural network to denoise the image, so that the indexes of the denoised image are close to ideal values.
The method is mainly used for meeting the application requirement of the photon counting imaging image denoising. In consideration of the defects of the traditional method, the photon counting image denoising method based on the learning prior is provided, an optimal sparse representation mode is obtained through learning, and the parameter design process is further simplified through a model networking mode, so that an optimal result is obtained.
It should be understood that parts of the specification not set forth in detail are of the prior art.
It should be understood that the above-mentioned embodiments are described in some detail, and not intended to limit the scope of the invention, and those skilled in the art will be able to make alterations and modifications without departing from the scope of the invention as defined by the appended claims.

Claims (2)

1. A photon reflectivity image denoising method based on neural network learning prior is characterized by comprising the following steps:
step 1, constructing a photon counting imaging model, wherein the photon counting imaging model is expressed as follows:
Figure FDA0003671846170000011
wherein alpha represents a photon reflectivity image, J (alpha) is a data fidelity term, and K (alpha) is a priori constraint term;
the specific implementation of the step 1 comprises the following substeps:
step 1.1, constructing a data fidelity item J (alpha), and giving a photon counting value ki,jReflectivity of alphai,jThe poisson negative log-likelihood function of (a) may be expressed as:
Figure FDA0003671846170000017
wherein N is the number of emitted pulses, eta is the detection efficiency of the detector, B is the background count of the pulse repetition period, S is the signal count within the pulse repetition period, i, j are the coordinates of spatial points, and the above formula is used as a data fidelity term;
step 1.2, constructing a prior constraint term K (alpha),
Figure FDA0003671846170000012
wherein the content of the first and second substances,
Figure FDA0003671846170000013
is a combination of two linear convolution operators separated by one linear rectifying unit;
step 1.3, obtaining a final reflectivity image reconstruction model, wherein because eta S alpha is less than 1, the following approximation can be carried out:
1-exp[-(ηSα+B)]≈ηSα+B
the final reconstruction model can thus be obtained:
Figure FDA0003671846170000014
wherein lambda is a regularization parameter, and k is a matrix formed by photon counting values of all space points;
step 2, solving the imaging model to obtain corresponding subproblems;
the specific implementation manner of the step 2 is as follows:
solving the formula (1) based on the optimization strategy of FISTA to obtain the following two subproblems:
Figure FDA0003671846170000015
Figure FDA0003671846170000016
where ρ is a constant, rxIs the result of the x-th iteration calculation of the intermediate variable r, which is alphaxDirectly reconstructing a result in the x iteration, wherein an initial value can be obtained according to maximum likelihood estimation;
step 3, networking the physical model of the subproblem, and constructing a convolutional neural network and a loss function;
the specific implementation of the step 3 comprises the following substeps:
step 3.1, rxNetworking of the model, in which the value of the parameter p varies as the network is learned, r according to equation (2)xThe model becomes:
Figure FDA0003671846170000021
step 3.2, alphaxModeling, assuming r for modelingxAnd
Figure FDA00036718461700000220
are respectively alpha and
Figure FDA0003671846170000022
the following approximation can be obtained for the mean value of:
Figure FDA0003671846170000023
wherein beta is a single radical of
Figure FDA00036718461700000221
Relative scalar, applying the above approximation term to equation (3) yields:
Figure FDA0003671846170000024
where θ is β λ, then, we obtain
Figure FDA00036718461700000222
Closed loop form of (1):
Figure FDA0003671846170000025
introduction of
Figure FDA0003671846170000026
Is inverse operation of
Figure FDA0003671846170000027
Definition of
Figure FDA0003671846170000028
Finally, alpha is obtainedxThe networked model of (2):
Figure FDA0003671846170000029
in order to maintain the flexibility of the network and increase the capacity of the network, the network is controlled by the control unit
Figure FDA00036718461700000210
Varies with iteration of the network, and thus, α can be obtainedxNetworked final model of (2):
Figure FDA00036718461700000211
step 3.3, constructing a loss function and adopting a data set
Figure FDA00036718461700000212
Training is carried out, and the network counts the photons to form an image kiAs input, generating a reconstruction result
Figure FDA00036718461700000213
When the symmetric condition is satisfied
Figure FDA00036718461700000214
Is reduced in the case of
Figure FDA00036718461700000215
And group Truth yiFrom which the following loss function is designed:
Figure FDA00036718461700000216
wherein:
Figure FDA00036718461700000217
Figure FDA00036718461700000218
wherein N isbFor training blocksTotal number of (2), NpIs the number of network layers, NsFor each training block size, γ is a constant;
and 4, training the convolutional neural network, and obtaining a denoising result of the reflectivity image according to the trained network model.
2. The method of claim 1, wherein the method comprises: r isxCan be obtained from a maximum likelihood estimation, i.e.
Figure FDA00036718461700000219
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