CN111273351B - Structural guide direction generalized total variation regularization method for seismic data denoising - Google Patents

Structural guide direction generalized total variation regularization method for seismic data denoising Download PDF

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CN111273351B
CN111273351B CN201911145443.9A CN201911145443A CN111273351B CN 111273351 B CN111273351 B CN 111273351B CN 201911145443 A CN201911145443 A CN 201911145443A CN 111273351 B CN111273351 B CN 111273351B
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CN111273351A (en
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王德华
高静怀
丁小丽
张丽丽
周宏安
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Xian Technological University
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    • G01V1/28—Processing seismic data, e.g. for interpretation or for event detection
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Abstract

The invention discloses a structural self-adaptive direction generalized total variation regularization method for seismic data denoising, which comprises the following steps of: (1) based on the tensor of the gradient structure, a novel calculation method of the seismic event space-variant dip angle is provided; (2) will l2The constant angle theta in the DTGV regularization model is generalized to the space-variant angle thetai,jFurther, a new structure self-adaptive direction generalized total variation regularization model (l) is constructed2-SADTGV regularization model); (3) using a chambole-Pock primitive-duality algorithm pair l2-the SADTGV regularization model is solved. The method provided by the invention is suitable for processing seismic data with a complex geological structure. Compared with the prior art, the SADTGV regularization technology has the following advantages: (1) the signal-to-noise ratio of the noise-containing seismic data can be effectively improved; (2) the method has the advantages that the transverse continuity of the seismic event is enhanced, the longitudinal resolution of a seismic section is improved, and geological structure characteristics such as fault information of seismic data are kept; (3) has better amplitude-keeping capability.

Description

Structural guide direction generalized total variation regularization method for seismic data denoising
The technical field is as follows:
the invention belongs to the technical field of seismic exploration, and relates to a Structure-oriented Total Generalized Total Variation (SADTGV) regularization method for denoising seismic data.
Background art:
noise suppression plays a very important role in seismic data processing and is the primary task of improving seismic data interpretation accuracy. Random noise is randomly distributed in a time-space domain of seismic data, covers an effective frequency band of the seismic data, and therefore random noise removal is an important problem in seismic data processing. The seismic data denoising method commonly used in the industry includes a wavelet transform method, median filtering, KL transform, singular value decomposition, and the like.
In recent years, variation regularization methods for solving inverse problems have been applied to seismic data random noise suppression and have achieved certain effects. Among them, the Total Variation (TV) regularization method is an innovative work, and its model is defined as follows:
Figure GDA0003511844580000011
in the formula, BV (omega) represents a bounded variation function space defined on omega, f represents noisy data, u represents denoised data, and lambda > 0 is a regularization parameter. Further, TV (u) is defined as
Figure GDA0003511844580000012
In the formula, v represents a Euclidean closed unit sphere B centered on the origin2(0) A dual variable of (c). Because the function in the BV (omega) function space is 'sliced' smoothly, the data denoised by the method can generate 'step effect'; meanwhile, since tv (u) has rotation invariance, the method does not consider the space-variant characteristic of the seismic data structure feature.
Chinese patent (ZL201710053646.X) is a seismic data joint denoising method, which comprises the following steps: s1, decomposing the seismic section data by using variational mode decomposition to obtain new data; s2, carrying out denoising processing on the new data by using an improved total variation method; s3, combining and reconstructing the de-noised data; and obtaining final seismic section data. The invention adopts a total variation regularization method in the denoising processing link, and has the following defects: (1) TV regularization can cause the de-noised data to generate a 'step effect'; (2) TV regularization does not take into account the space-variant nature of seismic data structure features.
Bredie et al proposed a Generalized Total Variation (TGV) functional that overcomes the disadvantage of "step effect" that would result from TV regularization method (1) by introducing a high order derivative, and Generalized Total Variation functional (2) whose second order situation is defined as:
Figure GDA0003511844580000021
in the formula,
Figure GDA0003511844580000022
the representation is defined in
Figure GDA0003511844580000023
Second order symmetric tensor space above, for matrix valued functions
Figure GDA0003511844580000024
(divW)TAnd div2W is respectively defined as
Figure GDA0003511844580000025
And
Figure GDA0003511844580000026
B2×2(0) is represented by a matrix
Figure GDA0003511844580000027
A unit ball of wherein (v)i1,vi2)T∈B2(0) And (v)1j,v2j)T∈B2(0)，λ1,λ2> 0 is a defined regularization parameter. However, due to the rotational invariance of the TGV functional, TGV regularization, like TV regularization, cannot adaptively process seismic data with directional information.
To characterize the Directional information of the data, Bayram et al propose Directional Total Variation (DTV) as defined below:
Figure GDA0003511844580000031
in the formula,
Figure GDA0003511844580000032
set of ellipsoids E representing a closeda,θ(0) A dual variable (as shown in fig. 1). Note the book
Figure GDA0003511844580000033
And
Figure GDA0003511844580000034
then for v (x) ε B2(0) The method comprises the following steps of (1) preparing,
Figure GDA0003511844580000035
kongskov et al combine DTV with TGV, introduce Directional information into the TGV, and propose Directional Total Generalized Variation (DTGV), whose second order form is defined as follows:
Figure GDA0003511844580000036
in the formula,
Figure GDA0003511844580000037
is represented by a matrix
Figure GDA0003511844580000038
The space is formed, wherein,
Figure GDA0003511844580000039
and
Figure GDA00035118445800000310
λ1,λ2> 0 is a defined regularization parameter. Note that for V (x) ε B2×2(0) Is provided with
Figure GDA00035118445800000311
However, the DTGV regularization method is applicable if the texture structure of the data has a single directivity, and obviously, for a complex geological structure, the event axis of the seismic data has a spatially varying dip, and therefore, the method cannot be directly used for processing the seismic data.
The invention content is as follows:
the invention aims to provide a Structure-oriented Total Generalized Total Variation (SADTGV) regularization method for removing random noise of seismic data, which aims to overcome the problem that the prior art is not suitable for processing seismic data with space-variant direction information.
In order to achieve the purpose, the invention adopts the technical scheme that:
the structural guide direction generalized total variation regularization method for seismic data denoising is characterized by comprising the following steps of:
(1) estimating dip of the same phase axis of the seismic section at any point
First, Gaussian filtering is performed on noisy data, and then the gradient of the noisy data is calculated and recorded as
Figure GDA0003511844580000041
Where f denotes noisy data, σ denotes the standard deviation of the Gaussian function, and for two-dimensional data, the component of g is written as
Figure GDA0003511844580000042
Secondly, mapping the gradient g into the following tensor by the parallel product operation
Figure GDA0003511844580000043
Then, convolving T with the Gaussian kernel G (x; rho) to obtain the gradient structure tensor of f
Figure GDA0003511844580000044
Thirdly, calculating gradient structure tensor T of the seismic data at a certain pointρAnd the eigenvector corresponding to the largest eigenvalue is the normal vector of the reflection event at that point, the dip of the seismic event is limited to the interval [0, pi ], and the dip at any point is defined as:
Figure GDA0003511844580000045
where e denotes the normal vector reflecting the in-phase axis at point x, e1、e2Respectively represent an edge x1And x2The unit vector of the direction is,<·,·>representing the inner product operation of the vector.
(2) Construction of
Figure GDA00035118445800000410
-SADTGV regularized denoising model
Will be provided with
Figure GDA0003511844580000046
Figure GDA0003511844580000047
The constant angle theta in the medium-direction generalized total variation is generalized to the space-variant angle theta defined by the formula (8)i,jMeasuring the inclination angles of the seismic reflection event at different points by self-adaption; wherein,
Figure GDA0003511844580000048
the representation is defined in
Figure GDA0003511844580000049
The second-order symmetric tensor space above,
Figure GDA0003511844580000051
is represented by a matrix
Figure GDA0003511844580000052
Formed space of λ1,λ2The more than 0 is a determined regularization parameter, and further the direction generalized total variation is popularized to the structure self-adaptive direction generalized total variation, and the discrete form is
Figure GDA0003511844580000053
Where v is a tensor, its elemental form is expressed as:
Figure GDA0003511844580000054
λ1and λ2Is a parameter, | · non conducting phosphorFRepresenting the Frobenius norm. For the
Figure GDA0003511844580000055
Is provided with
Figure GDA0003511844580000056
In the formula,
Figure GDA0003511844580000057
and
Figure GDA0003511844580000058
respectively represent an edge x1And x2A forward difference operator of a direction that satisfies a symmetric boundary condition,
Figure GDA0003511844580000059
and
Figure GDA00035118445800000510
the rotation matrix and the scale-scaling matrix are represented separately,
Figure GDA00035118445800000511
is a directionally symmetric derivative of the tensor v, whose elemental form is expressed as
Figure GDA00035118445800000512
In the formula,
Figure GDA00035118445800000513
and
Figure GDA00035118445800000514
respectively is corresponding to
Figure GDA00035118445800000515
And
Figure GDA00035118445800000516
the backward difference operator of (2);
L2the norm is suitable for characterizing random noise that follows a Gaussian distribution by combining the regularization terms SADTGV (u) and SADTGV (u) defined by equation (9)
Figure GDA00035118445800000519
Data fitting terms constructed as
Figure GDA00035118445800000520
-SADTGV variational regularization model
Figure GDA00035118445800000517
In the formula,
Figure GDA00035118445800000518
representing noisy and de-noised data separately，
Figure GDA00035118445800000521
Is L2Discrete representation of norm.
(3)
Figure GDA00035118445800000522
-solution of the SADTGV regularization model
In view of the simplicity of the algorithm implementation,
Figure GDA00035118445800000523
solving of the SADTGV regularization model adopts a chambole-Pock original-dual algorithm to improve the calculation efficiency.
Compared with the prior art, the invention has the following advantages and effects:
1. the main innovation of the invention is that: (1) based on the tensor of the gradient structure, a novel calculation method of the seismic event space-variant dip angle is provided; (2) will be provided with
Figure GDA0003511844580000061
The constant angle θ in the DTGV regularization model is generalized to the space-variant angle θ defined by equation (8)i,jFurther, a new structural adaptive direction generalized total variation regularization model is constructed (
Figure GDA0003511844580000062
-SADTGV regularization model).
2. Based on
Figure GDA0003511844580000063
The structure self-adaptive characteristic of the SADTGV regularization model has the following beneficial effects compared with the prior art: (1) the signal-to-noise ratio can be improved to a greater extent; (2) the transverse continuity and the longitudinal resolution of the seismic event can be better maintained; (3) geological structural characteristics of the seismic data, such as fault information and the like, can be better maintained; (4) amplitude information of the seismic data may be better preserved.
Description of the drawings:
FIG. 1 is a schematic view ofA closed ellipsoid set E introduced by describing direction information of the same phase axis of the seismic dataa,θ(0)；
FIG. 2 is a schematic diagram of seismic event local dip constructed using gradient result tensors;
FIG. 3 is a comparison graph of the denoising results of model data by different methods;
(a) reflection coefficient model
(b) Synthesizing seismic data
(c) Noisy seismic data
(d) Median filtering result (SNR 17.70dB)
(e) KL transform result (SNR 13.98dB)
(f) SVD transform result (SNR 14.41dB)
(g) TGV De-noising result (SNR 16.81dB)
(h) DTGV denoise result (SNR 17.20dB)
(i) SADTGV denoising result (SNR 19.51dB)
FIG. 4 is a comparison graph of results of data difference profiles before and after model data are denoised by different methods;
(a) median filtered difference profile
(b) KL transform difference profile
(c) SVD transform difference profile
(d) TGV differential Profile
(e) DTGV differential profile
(f) SADTGV differential profile
FIG. 5 is actual post-stack seismic data for an oil field;
FIG. 6 is a comparison graph of the de-noising results of actual seismic data using different methods;
(a) median filtering denoising result
(b) KL transform denoising result
(c) SVD transform denoising result
(d) TGV denoise results
(e) DTGV denoising result
(f) SADTGV denoising result
FIG. 7 is a comparison graph of results of data difference profiles before and after de-noising actual data by different methods;
(a) median filtered difference profile
(b) KL transform difference profile
(c) SVD differential profile
(d) TGV differential Profile
(e) DTGV differential profile
(f) SADTGV differential profile
FIG. 8 is a single trace spectral contrast before and after de-noising actual seismic data using different methods.
(a) Median filtered single pass spectral contrast
(b) KL transform single pass spectral contrast
(c) SVD transform single pass spectral contrast
(d) TGV Single pass spectral comparison
(e) DTGV single pass spectral contrast
(f) Single pass spectral comparison of SADTGV
The specific implementation mode is as follows:
aiming at the problem that the seismic data with space-variant direction information are not suitable to be processed in the prior art, the invention provides a novel structure self-adaptive variation regularization method capable of reconstructing the seismic data more accurately. The new method provided by the invention is to popularize the DTGV regularization method suitable for processing the characteristic data with a single direction to the SADTGV regularization method suitable for processing the characteristic data with any direction, and mainly comprises the following steps:
(1) giving out a calculation formula of the same phase axis inclination angle of the seismic section at any point
For two-dimensional seismic data, because the local event has a linear texture structure, the azimuth of the event remains unchanged after the event is rotated by 180 degrees, and therefore, the dip angle of the seismic event can be limited to the interval [0, pi ]. First, Gaussian filtering is performed on noisy data, and then the gradient of the noisy data is calculated and recorded as
Figure GDA0003511844580000081
Where f denotes noisy data, σ denotes the standard deviation of the Gaussian function, and for two-dimensional data, the component of g is written as
Figure GDA0003511844580000091
Secondly, mapping the gradient g into the following tensor by the parallel product operation
Figure GDA0003511844580000092
Then, convolving T with the Gaussian kernel G (x; rho) to obtain the gradient structure tensor of f
Figure GDA0003511844580000093
Thirdly, calculating gradient structure tensor T of the seismic data at a certain pointρAnd the eigenvector corresponding to the largest eigenvalue is the normal vector of the reflection in-phase axis at that point. Thus, the dip of the seismic event at any point (as shown in FIG. 2) can be defined as the dip of the seismic event at any point
Figure GDA0003511844580000094
Where e denotes the normal vector reflecting the in-phase axis at point x, e1、e2Respectively represent an edge x1And x2The unit vector of the direction is,<·,·>representing the inner product operation of the vector. The inclination angle calculation formula (8) has the advantages that: (1) the calculation is simple; (2) the ambiguity of the reflection in-phase axis normal vector is avoided (as in fig. 2, which represents normal vector e with solid and dashed lines).
(2) Construction of
Figure GDA0003511844580000098
-SADTGV regularized denoising model
The invention extends the constant angle theta in the formula (5) DTGV to the space-variant angle theta defined by the formula (8)i,jIs used to adaptivelyThe dip of the seismic reflection event at different points is measured. Further, generalizing DTGV to SADTGV, its discrete form can be written as
Figure GDA0003511844580000095
Where v is a tensor, whose elemental form is expressed as
Figure GDA0003511844580000096
λ1And λ2Is a parameter, | · non conducting phosphorFRepresenting the Frobenius norm. For the
Figure GDA0003511844580000097
Is provided with
Figure GDA0003511844580000101
In the formula,
Figure GDA0003511844580000102
and
Figure GDA0003511844580000103
respectively represent an edge x1And x2A forward difference operator of a direction that satisfies a symmetric boundary condition,
Figure GDA0003511844580000104
and
Figure GDA0003511844580000105
the rotation matrix and the scale-scaling matrix are represented separately,
Figure GDA0003511844580000106
is a directionally symmetric derivative of the tensor v, whose elemental form can be expressed as
Figure GDA0003511844580000107
In the formula,
Figure GDA0003511844580000108
and
Figure GDA0003511844580000109
respectively is corresponding to
Figure GDA00035118445800001010
And
Figure GDA00035118445800001011
the backward difference operator of (1). Due to L2The norm is suitable for characterizing random noise that follows a Gaussian distribution, so to be able to accurately reconstruct noisy seismic data with a space variant reflection in-phase axis, we can combine the regularization terms SADTGV (u) and SADTGV (u) defined by equation (9)
Figure GDA00035118445800001015
(L2Discrete representation of norm) data fit term constructed as follows
Figure GDA00035118445800001016
-SADTGV variational regularization model
Figure GDA00035118445800001012
In the formula,
Figure GDA00035118445800001013
respectively representing noisy data and denoised data. Theoretically, the model can make the variation of the seismic data along the direction of the same phase axis obtain a larger punishment, thereby effectively keeping the geological structure characteristics of the seismic data while denoising.
Example (b):
the following describes in detail the implementation algorithm and experimental effect of the SADTGV regularization technique proposed by the present invention with reference to the accompanying drawings.
Implementation of SADTGV regularization technique
Due to the functional in minimization of the problem (10)
Figure GDA00035118445800001014
Is a convex functional, therefore, the problem can be solved using a convex optimization algorithm. In consideration of the simplicity of algorithm implementation, the invention adopts the existing chambole-Pock original-dual algorithm to solve the minimization problem (10), and the specific solving steps are as follows:
(1) write the primitive-dual format of the minimization problem (10):
Figure GDA0003511844580000111
in the formula,
Figure GDA0003511844580000112
Figure GDA0003511844580000113
Figure GDA0003511844580000114
representing a set of 2 nd order real symmetric matrices.
(2) The chambole-Pock primitive-dual algorithm for solving the minimization problem (10):
the first step, initialization:
Figure GDA0003511844580000115
q0＝0,p0＝0,W0＝0,e0＝1；
secondly, inputting parameters: f, lambda1,λ2L, τ, η, a and tol;
thirdly, circularly iterating:
while ek＞tol do
Figure GDA0003511844580000116
Figure GDA0003511844580000117
Figure GDA0003511844580000118
Figure GDA0003511844580000119
Figure GDA00035118445800001110
Figure GDA00035118445800001111
Figure GDA00035118445800001112
Figure GDA00035118445800001113
k＝k+1；
end while
the fourth step, output uk+1。
(II) analysis of Experimental Effect
The SADTGV regularization technology provided by the invention is used for denoising synthetic and actual seismic data, and is compared with denoising methods (such as a median filtering method, a KL filtering method and an SVD transformation method) commonly used in the industry and related advanced regularization technologies (such as TGV regularization and DTGV regularization methods) so as to verify the effectiveness of the new method provided by the invention. The parameters involved in the experiment were chosen as follows:
Figure GDA0003511844580000121
(1) experiment of synthetic data
The synthetic data used in this experiment (see fig. 3(b)) was obtained by convolving a reflection coefficient model with a space-variant seismic event axis, as shown in fig. 3(a), with a minimum-phase wavelet with a dominant frequency of 10 Hz. Then, 5dB of white gaussian noise was added to the synthetic seismic data to obtain noisy seismic data as shown in fig. 3 (c).
From the denoising result (such as fig. 3(d) -3(i)), compared with other methods, the new method provided by the invention not only improves the signal-to-noise ratio to a greater extent, but also better maintains the transverse continuity of the seismic event and the longitudinal strong and weak contrast; from the view of the difference profile of the seismic data before and after denoising (as shown in fig. 4), the new method proposed by the invention has little useful structural information of the seismic data compared with other methods, and other methods have damage to the structural information of the seismic data to different degrees. In a word, the new method provided by the invention can effectively improve the signal-to-noise ratio of the noise-containing seismic data and can well keep the geological structure characteristics of the seismic data.
(2) Experiment of actual data
The practical effect of the new method provided by the invention is verified by adopting the actual seismic data after the two-dimensional stacking of the oil field as shown in figure 5, and the actual data is seriously polluted by noise, so that great difficulty is caused for further accurate geological interpretation.
From the denoising result shown in fig. 6, the analysis can be performed from the following three aspects: firstly, as seen from a rectangular area A in a denoising result, compared with other methods, the novel method provided by the invention can effectively enhance the transverse continuity of the seismic event, so that the transverse continuity of the denoised seismic section is better; secondly, the median filtering, KL transformation and SVD transformation methods are analyzed from the rectangular region B in the denoising result, compared with the variation regularization methods TGV, DTGV and SADTGV, the longitudinal resolution of the denoised seismic section is worse, compared with the TGV and DTGV regularization method, the novel method can more clearly present the intensity contrast of the longitudinal homophase axis of the seismic section, the reason is mainly that the TGV is used for constraining the seismic data, the direction characteristic that the seismic event axis has space variation is not considered, the DTGV is suitable for describing the data with the single direction characteristic, obviously, the two methods are not suitable for processing the seismic data with the space variation characteristic, the new method SADTGV regularization technology provided by the invention considers the space variation characteristic of the seismic event, filtering can be carried out along the direction of the same-phase axis, so that the transverse continuity and longitudinal strength contrast of the same-phase axis in the seismic section can be enhanced; thirdly, as can be seen from the area indicated by the black arrow in the denoising result, the new method can more accurately depict the boundary information of the fault layer in the seismic data compared with other methods, and this point is mainly benefited by the fact that the local dip of the seismic event is constructed by utilizing the gradient structure tensor. From the difference profile of the seismic data before and after denoising shown in fig. 7, the SADTGV regularization technique provided by the invention can better maintain the structural information of the seismic event while denoising efficiently, and has less damage to the effective information of the original seismic data. In a word, the SADTGV regularization technology provided by the invention has obvious effects on the aspects of improving the signal-to-noise ratio, enhancing the transverse continuity of the seismic event, improving the longitudinal resolution of a seismic section and maintaining fault information of seismic data.
Finally, the invention also arbitrarily extracts a piece of data from the denoising results of different methods, compares the Fourier spectrums of the data, and compares the amplitude preservation capability of the seismic data before and after denoising by various methods. Firstly, as can be seen from the whole of the black curve (original single-channel data spectrum) and the red curve (denoised single-channel data spectrum) in fig. 8, the SADTGV regularization technique provided by the present invention can better maintain the amplitude information of the original seismic data; secondly, as can be seen from the results shown in the "blue rectangular box", the new method can better maintain the detailed information of the spectrum; thirdly, as can be seen from the area indicated by the 'blue arrow', the novel method provided by the invention can better maintain the high-frequency information of the seismic data. In a word, the comparison result of the Fourier spectrum shows that the SADTGV regularization technology provided by the invention can effectively improve the signal-to-noise ratio and better keep the amplitude information of the seismic data.

Claims (1)

1. The structural guide direction generalized total variation regularization method for seismic data denoising is characterized by comprising the following steps of:
(1) estimating the dip angle of the same phase axis of the seismic section at any point:
first, Gaussian filtering is performed on noisy data, and then the gradient of the noisy data is calculated and recorded as
Figure FDA0003511844570000015
Where f denotes noisy data, σ denotes the standard deviation of the Gaussian function, and for two-dimensional data, the component of g is written as
Figure FDA0003511844570000011
Secondly, mapping the gradient g into the following tensor by the parallel product operation
Figure FDA0003511844570000012
Then, convolving T with the Gaussian kernel G (x; rho) to obtain the gradient structure tensor of f
Figure FDA0003511844570000013
Thirdly, calculating gradient structure tensor T of the seismic data at a certain pointρAnd the eigenvector corresponding to the largest eigenvalue is the normal vector of the reflection event at that point, the dip of the seismic event is limited to the interval [0, pi ], and the dip at any point is defined as:
Figure FDA0003511844570000014
where e denotes the normal vector reflecting the in-phase axis at point x, e1、e2Respectively represent an edge x1And x2The unit vector of the direction is,<·,·>representing an inner product operation of the vectors;
(2) construction of
Figure FDA0003511844570000016
Regularization denoising model:
will be provided with
Figure FDA0003511844570000021
The constant angle theta in the medium-direction generalized total variation is generalized to the space-variant angle theta defined by the formula (8)i,jMeasuring the inclination angles of the seismic reflection event at different points by self-adaption; wherein,
Figure FDA0003511844570000022
the representation is defined in
Figure FDA0003511844570000023
The second-order symmetric tensor space above,
Figure FDA0003511844570000024
is represented by a matrix
Figure FDA0003511844570000025
Formed space of λ1,λ2> 0 is a defined regularization parameter, and hence squareGeneralized total variation in the direction of generalized total variation to structure adaptation, in discrete form
Figure FDA0003511844570000026
Where v is a tensor, its elemental form is expressed as:
Figure FDA0003511844570000027
λ1and λ2Is a parameter, | · non conducting phosphorFRepresents the Frobenius norm for
Figure FDA0003511844570000028
Is provided with
Figure FDA0003511844570000029
In the formula,
Figure FDA00035118445700000210
and
Figure FDA00035118445700000211
respectively represent an edge x1And x2A forward difference operator of a direction that satisfies a symmetric boundary condition,
Figure FDA00035118445700000212
and
Figure FDA00035118445700000213
the rotation matrix and the scale-scaling matrix are represented separately,
Figure FDA00035118445700000214
is a directionally symmetric derivative of the tensor v, whose elemental form representsIs composed of
Figure FDA00035118445700000215
In the formula,
Figure FDA00035118445700000216
and
Figure FDA00035118445700000217
respectively is corresponding to
Figure FDA00035118445700000218
And
Figure FDA00035118445700000219
the backward difference operator of (2); l is2The norm is suitable for characterizing random noise that follows a Gaussian distribution by combining the regularization terms SADTGV (u) and SADTGV (u) defined by equation (9)
Figure FDA00035118445700000220
Data fitting terms constructed as
Figure FDA00035118445700000221
Variational regularization model
Figure FDA0003511844570000031
In the formula, the ratio of f,
Figure FDA0003511844570000032
respectively representing noisy data and de-noised data,
Figure FDA0003511844570000033
is L2A discrete representation of the norm;
(3)
Figure FDA0003511844570000034
solving the regularization model:
Figure FDA0003511844570000035
the regularization model is solved by using a chambole-Pock primitive-dual algorithm.
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