CN109859123B - Image denoising method and system based on Primal-dual - Google Patents
Image denoising method and system based on Primal-dual Download PDFInfo
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Abstract
The invention discloses an image denoising method and system based on Primal-dual. The denoising method comprises the following steps: acquiring a training sample set, an original vector space and a dual vector space of the original vector space of a sample image; acquiring a first regularization item corresponding to an original variable in an original vector space and a second regularization item corresponding to a dual variable in a dual vector space; constructing a training matrix according to the training sample set; determining a training sample set constant and an iteration step length according to the training matrix; establishing a non-smooth objective function according to the first regularization term, the second regularization term and the training sample set constant; adopting an individual output form, updating a primary variable and a dual variable according to a non-smooth objective function and an iteration step length, and determining the primary variable of the individual output and the dual variable of the individual output; and denoising the sample image according to the original variable output by the individual and the dual variable output by the individual. The denoising method and the denoising system provided by the invention can improve the image denoising efficiency.
Description
Technical Field
The invention relates to the field of image denoising, in particular to a Primal-dual-based image denoising method and system.
Background
The traditional image denoising method only obtains sparsity based on primary variables or dual variables, so that the sparsity can only be achieved on feature selection or sample number, the denoising effect of processing a large-scale image data set is not excellent enough, and meanwhile, the traditional image denoising method only obtains sparsity under the condition of non-smooth function type
The convergence rate of (c); the Primal-dual method can further improve the convergence rate, but most of solutions of the Primal-dual method for image application are output in an average form, individual output is relatively less, a regularization term is not added, generalization capability is poor, and good sparsity cannot be obtained, so that the method cannot be suitable for the requirement of a high-precision denoising effect, and the image denoising efficiency of the traditional image denoising method based on the Primal-dual is low.
Disclosure of Invention
The invention aims to provide a Primal-dual-based image denoising method and a Primal-dual-based image denoising system, which aim to solve the problem of low image denoising efficiency of the traditional Primal-dual-based image denoising method.
In order to achieve the purpose, the invention provides the following scheme:
an image denoising method based on Primal-dual comprises the following steps:
acquiring a training sample set, an original vector space and a dual vector space of the original vector space of a sample image; the training sample set comprises a feature vector of a sample image and a classification type corresponding to the feature vector;
initializing a first regularization item corresponding to a primary variable in the primary vector space and a second regularization item corresponding to a dual variable in the dual vector space;
constructing a training matrix according to the training sample set;
determining a training sample set constant and an iteration step length according to the training matrix;
establishing a non-smooth objective function according to the first regularization term, the second regularization term and the training sample set constant;
adopting an individual output form, updating the original variable and the dual variable according to the non-smooth objective function and the iteration step length, and determining the original variable of the individual output and the dual variable of the individual output;
and denoising the sample image according to the original variable output by the individual and the dual variable output by the individual.
Optionally, the determining a training sample set constant and an iteration step according to the training matrix specifically includes:
and determining the constant of the training sample set and the iteration step according to the two-norm of the training matrix.
Optionally, the establishing a non-smooth objective function according to the first regularization term, the second regularization term, and the training sample set constant specifically includes:
according to the formula
Establishing a non-smooth objective function; wherein the content of the first and second substances,
is a non-smooth loss function, X is the original vector space, Y is the dual vector space, w ∈ X, α∈ Y, α are dual variables based on w, w is X for a training sample setiThe optimized weight vector of (2); lambda [ alpha ]1R (w) is a first regularization term; lambda [ alpha ]2R (α) is a second regularization term, wTIs a transposed matrix of w αTα, H (S, y), a (S, y), b (S, y), d are constants based on the training sample set S.
Optionally, the determining, in an individual output form, the original variable and the dual variable according to the non-smooth objective function and the iteration step length, and the original variable and the dual variable of the individual output specifically include:
in individual output form, according to formula
Formula (II)
And formulas
Determining a primary variable output by an individual and a dual variable output by the individual; wherein G isα(wt-1，βt-1) Is composed of
Partial derivative at α, Gw(wt-1，αt) Is composed of
Partial derivative at w β is a vector connecting w, α in the construction iteration, wt，αt，βt(T ═ 0, 1, 2,....... times, T) is the output vector of w, α after T cycles;
c＝D2the/m and D are x in the training sample set SiIs the maximum of the two norms.
A Primal-dual based image denoising system, comprising:
the system comprises a sample image parameter acquisition module, a parameter acquisition module and a parameter acquisition module, wherein the sample image parameter acquisition module is used for acquiring a training sample set, an original vector space and a dual vector space of the original vector space of a sample image; the training sample set comprises a feature vector of a sample image and a classification type corresponding to the feature vector;
the regularization item acquisition module is used for initializing a first regularization item corresponding to an original variable in the original vector space and a second regularization item corresponding to a dual variable in the dual vector space;
the training matrix constructing module is used for constructing a training matrix according to the training sample set;
a training sample set constant and iteration step length determining module for determining a training sample set constant and an iteration step length according to the training matrix;
a non-smooth objective function determination module, configured to establish a non-smooth objective function according to the first regularization term, the second regularization term, and the training sample set constant;
the output module is used for updating the original variable and the dual variable according to the non-smooth objective function and the iteration step length by adopting an individual output form, and determining the original variable of the individual output and the dual variable of the individual output;
and the denoising processing module is used for denoising the sample image according to the original variable output by the individual and the dual variable output by the individual.
Optionally, the training sample set constant and iteration step determining module specifically includes:
and the training sample set constant and iteration step length determining unit is used for determining the training sample set constant and the iteration step length according to the two-norm of the training matrix.
Optionally, the non-smooth objective function establishing module specifically includes:
a non-smooth objective function establishing unit for establishing an objective function according to a formula
Establishing a non-smooth objective function; wherein the content of the first and second substances,
is a non-smooth loss function, X is the primary vector space, Y is the dual vector space, w ∈ X, α∈ Y, α are basesA dual variable of w, w being for the training sample set xiThe optimized weight vector of (2); lambda [ alpha ]1R (w) is a first regularization term; lambda [ alpha ]2R (α) is a second regularization term, wTIs a transposed matrix of w αTα, H (S, y), a (S, y), b (S, y), d are constants based on the training sample set S.
Optionally, the output module specifically includes:
an output unit for adopting an individual output form according to a formula
Formula (II)
And formulas
Determining a primary variable output by an individual and a dual variable output by the individual; wherein G isα(wt-1，βt-1) Is composed of
Partial derivative at α, Gw(wt-1，αt) Is composed of
Partial derivative at w β is a vector connecting w, α in the construction iteration, wt，αt，βt(T ═ 0, 1, 2,....... times, T) is the output vector of w, α after T cycles;
c＝D2the/m and D are x in the training sample set SiIs the maximum of the two norms.
According to the specific embodiment provided by the invention, the invention discloses the following technical effects: by adopting the image denoising method and system based on the Primal-dual provided by the invention, regularization terms are added to both the original variable and the dual variable, and the original variable output by the individual and the dual variable output by the individual are determined by adopting an output mode of the individual output, so that double sparsity based on the dual variable and the dual variable is obtained, and the sparsity of understanding is ensured.
The number of non-zero dimensions in the solution vector output by an individual is reduced as much as possible, so that the number of support vectors is greatly reduced, the method has important significance for feature selection of high-dimensional data, sparsity is obtained in feature selection or sample number, excellent effects are achieved in processing of large-scale data and high-dimensional data, prediction time can be greatly reduced for image samples with sparse models, and image denoising efficiency is improved.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without inventive exercise.
FIG. 1 is a flowchart of an image denoising method based on Primal-dual according to the present invention;
FIG. 2 is a flowchart of an image denoising method based on Primal-dual, taking two classifications as examples, according to the present invention;
FIG. 3 is a structural diagram of a Primal-dual-based image denoising system provided by the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention aims to provide a Primal-dual-based image denoising method and system, which can improve the image denoising efficiency.
In order to make the aforementioned objects, features and advantages of the present invention comprehensible, embodiments accompanied with figures are described in further detail below.
Fig. 1 is a flowchart of a Primal-dual-based image denoising method provided by the present invention, and as shown in fig. 1, the Primal-dual-based image denoising method includes:
step 101: acquiring a training sample set, an original vector space and a dual vector space of the original vector space of a sample image; the training sample set comprises feature vectors of sample images and classification types corresponding to the feature vectors.
Step 102: initializing a first regularization item corresponding to a primary variable in the primary vector space and a second regularization item corresponding to a dual variable in the dual vector space.
Step 103: and constructing a training matrix according to the training sample set.
Constructing a matrix P based on a picture training sample set S, and solving a parameter c (c ═ D)2The/m and D are x in the training sample set SiThe maximum of the two norms), determining the step length gamma, taking the value range, constructing a vector β connecting w and α in iteration, and initializing α0＝0，β0＝0。
Step 104: and determining a constant of a training sample set and an iteration step according to the training matrix.
Step 105: and establishing a non-smooth objective function according to the first regularization term, the second regularization term and the training sample set constant.
As shown in FIG. 2, taking two classes as an example, the sample set is trained for independent and identically distributed pictures
xiFor feature vectors based on picture samples, yiTo be directed to sample xiType (+1, -1), R of classificationnIs the hilbert space. The optimized non-smooth objective function based on the image denoising application problem can be expressed as follows:
wherein the content of the first and second substances,
is a non-smooth loss function, X is the original vector space, Y is the dual vector space, w ∈ X, α∈ Y, α are dual variables based on w, w is X for a training sample setiThe optimized weight vector of (2); lambda [ alpha ]1R (w) is a first regularization term; lambda [ alpha ]2R (α) is a second regularization term, both the first regularization term and the second regularization term are L1 regularization terms, wTIs a transposed matrix of w αTα, H (S, y), a (S, y), b (S, y), d are constants based on the training sample set S, and the non-smooth Hinge loss function.
Step 106: and adopting an individual output form, updating the original variable and the dual variable according to the non-smooth objective function and the iteration step length, and determining the original variable of the individual output and the dual variable of the individual output.
Initializing a regularization parameter λ1、λ2And step length gammatThe loop is executed T times in the following manner:
i)
ii)
iii)
wherein G isα(wt-1，βt-1) Is composed of
Partial derivative at α, Gw(wt-1，αt) Is composed of
Partial derivative at w β is a vector connecting w, α in the construction iteration, wt，αt，βt(T ═ 0, 1, 2,....... times, T) is the output vector of w, α after T cycles;
c＝D2the/m and D are x in the training sample set SiIs the maximum of the two norms.
Output α after T cyclesT，wTThe dual variables and the original variables are respectively output by individuals, and if the image denoising effect or the classification effect (namely, the function error value) reaches an expected value, the operation is terminated; otherwise, adjusting the modification step size gammatRegularization parameter λ1And λ2。
By updating the solution vector w each time, the solution vector w approaches to a theoretical optimal solution, so that the non-smooth loss function gradually becomes smaller through each cycle, and the expected effect is achieved.
Step 107: and denoising the sample image according to the original variable output by the individual and the dual variable output by the individual.
According to the method, L1 regularization terms are added to both the original variable and the dual variable of the image data to be denoised, and the form of individual output is adopted, so that the sparsity of understanding is ensured; the number of non-zero dimensions in the solution vector output by an individual is as small as possible, so that the number of support vectors is greatly reduced, the method has important significance for feature selection of high-dimensional data, can greatly reduce prediction time for an image sample with a sparse model, and has extremely important application value for application occasions with strong timeliness.
Fig. 3 is a structural diagram of a Primal-dual-based image denoising system provided by the present invention, and as shown in fig. 3, an image denoising system based on Primal-dual includes:
a sample image parameter obtaining module 301, configured to obtain a training sample set, an original vector space, and a dual vector space of the original vector space of a sample image; the training sample set comprises feature vectors of sample images and classification types corresponding to the feature vectors.
A regularization term obtaining module 302, configured to initialize a first regularization term corresponding to a primitive variable in the primitive vector space and a second regularization term corresponding to a dual variable in the dual vector space.
A training matrix constructing module 303, configured to construct a training matrix according to the training sample set.
A training sample set constant and iteration step determining module 304, configured to determine a training sample set constant and an iteration step according to the training matrix.
The training sample set constant and iteration step determining module 304 specifically includes: and the training sample set constant and iteration step length determining unit is used for determining the training sample set constant and the iteration step length according to the two-norm of the training matrix.
A non-smooth objective function determination module 305, configured to establish a non-smooth objective function according to the first regularization term, the second regularization term, and the training sample set constant.
The non-smooth objective function establishing module 305 specifically includes: a non-smooth objective function establishing unit for establishing an objective function according to a formula
Establishing a non-smooth objective function; wherein the content of the first and second substances,
is a non-smooth loss function, X is the primary vector space, Y is the dual vector space, w ∈ X, α∈ Y, α are dual variables based on w, which is for the training sample set xiThe optimized weight vector of (2); lambda [ alpha ]1R (w) is a first regularization term; lambda [ alpha ]2R (α) is a second regularization term, wTIs a transposed matrix of w αTα, H (S, y), a (S, y), b (S, y), d are constants based on the training sample set S.
An output module 306, configured to update the original variable and the dual variable according to the non-smooth objective function and the iteration step length in an individual output form, and determine the original variable of the individual output and the dual variable of the individual output.
The output module 306 specifically includes: an output unit for adopting an individual output form according to a formula
Formula (II)
And formulas
Determining a primary variable output by an individual and a dual variable output by the individual; wherein G isα(wt-1，βt-1) Is composed of
Partial derivative at α, Gw(wt-1，αt) Is composed of
Partial derivative at w β is a vector connecting w, α in the construction iteration, wt，αt，βt(T ═ 0, 1, 2,....... times, T) is the output vector of w, α after T cycles;
c＝D2the/m and D are x in the training sample set SiIs the maximum of the two norms.
And a denoising module 307, configured to denoise the sample image according to the primitive variable of the individual output and the dual variable of the individual output.
The traditional method only obtains sparsity on original variables or dual variables, and can only achieve sparsity on feature selection or sample number; according to the method, the L1 regularization items are added to both the original variable and the dual variable to obtain dual sparsity based on the dual variable, the sparsity can be obtained in feature selection or sample number at the same time, and the method has an excellent effect on processing large-scale data and high-dimensional data.
The invention has the advantages that the individual solution vector is input, the individual convergence rate can be obtained through the conversion, scaling, recursion and other skills, and the invention can be better popularized and applied to various machine learning applications.
The invention can obtain the average convergence rate (T is expressed as iteration times) of O (1/T) aiming at the special properties (such as bilinear function) of some special functions in the non-smoothness problem, and is one order of magnitude faster than the traditional first-order gradient algorithm
The embodiments in the present description are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are referred to each other. For the system disclosed by the embodiment, the description is relatively simple because the system corresponds to the method disclosed by the embodiment, and the relevant points can be referred to the method part for description.
The principles and embodiments of the present invention have been described herein using specific examples, which are provided only to help understand the method and the core concept of the present invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, the specific embodiments and the application range may be changed. In view of the above, the present disclosure should not be construed as limiting the invention.
Claims (6)
1. An image denoising method based on Primal-dual is characterized by comprising the following steps:
acquiring a training sample set, an original vector space and a dual vector space of the original vector space of a sample image; the training sample set comprises a feature vector of a sample image and a classification type corresponding to the feature vector;
initializing a first regularization item corresponding to a primary variable in the primary vector space and a second regularization item corresponding to a dual variable in the dual vector space;
constructing a training matrix according to the training sample set;
determining a training sample set constant and an iteration step length according to the training matrix;
establishing a non-smooth objective function according to the first regularization term, the second regularization term and the training sample set constant;
adopting an individual output form, updating the original variable and the dual variable according to the non-smooth objective function and the iteration step length, and determining the original variable of the individual output and the dual variable of the individual output; the determining, in an individual output form, the original variable and the dual variable according to the non-smooth objective function and the iteration step length, and the original variable and the dual variable of the individual output specifically include:
in individual output form, according to formula
Formula (II)
And formulas
Determining a primary variable output by an individual and a dual variable output by the individual; wherein G isα(wt-1，βt-1) Is composed of
Partial derivative at α, Gw(wt-1，αt) Is composed of
Partial derivative at w β is a vector connecting w, α in the construction iteration, wt，αt，βt(T ═ 0, 1, 2,....... times, T) is the output vector of w, α after T cycles;
c＝D2the/m and D are x in the training sample set SiThe maximum of the two norms of (a);
is a non-smooth loss function, α is a dual variable based on w, which is a feature vector x for picture-based samplesiThe optimized weight vector of (2); lambda [ alpha ]1、λ2Are all regularization parameters;
and denoising the sample image according to the original variable output by the individual and the dual variable output by the individual.
2. The Primal-dual based image denoising method of claim 1, wherein the determining a training sample set constant and an iteration step size according to the training matrix specifically comprises:
and determining the constant of the training sample set and the iteration step according to the two-norm of the training matrix.
3. The Primal-dual based image denoising method according to claim 1, wherein the establishing a non-smooth objective function according to the first regularization term, the second regularization term, and the training sample set constant specifically comprises:
according to the formula
Establishing a non-smooth objective function; wherein the content of the first and second substances,
is a non-smooth loss function, X is the primitive vector space, Y is the dual vector space, w ∈ X, α∈ Y, α are w-based dual variables, w is a feature vector X for picture-based samplesiThe optimized weight vector of (2); lambda [ alpha ]1R (w) is a first regularization term; lambda [ alpha ]2R (α) is a second regularization term, wTIs a transposed matrix of w αTα, H (S, y), a (S, y), b (S, y), d are constants based on the training sample set S.
4. An image denoising system based on Primal-dual, comprising:
the system comprises a sample image parameter acquisition module, a parameter acquisition module and a parameter acquisition module, wherein the sample image parameter acquisition module is used for acquiring a training sample set, an original vector space and a dual vector space of the original vector space of a sample image; the training sample set comprises a feature vector of a sample image and a classification type corresponding to the feature vector;
the regularization item acquisition module is used for initializing a first regularization item corresponding to an original variable in the original vector space and a second regularization item corresponding to a dual variable in the dual vector space;
the training matrix constructing module is used for constructing a training matrix according to the training sample set;
a training sample set constant and iteration step length determining module for determining a training sample set constant and an iteration step length according to the training matrix;
a non-smooth objective function determination module, configured to establish a non-smooth objective function according to the first regularization term, the second regularization term, and the training sample set constant;
the output module is used for updating the original variable and the dual variable according to the non-smooth objective function and the iteration step length by adopting an individual output form, and determining the original variable of the individual output and the dual variable of the individual output; the output module specifically includes:
an output unit for adopting an individual output form according to a formula
Formula (II)
And formulas
Determining a primary variable output by an individual and a dual variable output by the individual; wherein G isα(wt-1，βt-1) Is composed of
Partial derivative at α, Gw(wt-1，αt) Is composed of
Partial derivative at w β is a vector connecting w, α in the construction iteration, wt，αt，βt(T ═ 0, 1, 2,....... times, T) is the output vector of w, α after T cycles;
c＝D2the/m and D are x in the training sample set SiThe maximum of the two norms of (a);
is a non-smooth loss function, α is a dual variable based on w, which is a feature vector x for picture-based samplesiThe optimized weight vector of (2); lambda [ alpha ]1、λ2Are all regularization parameters;
and the denoising processing module is used for denoising the sample image according to the original variable output by the individual and the dual variable output by the individual.
5. The Primal-dual based image denoising system of claim 4, wherein the training sample set constant and iteration step determining module specifically comprises:
and the training sample set constant and iteration step length determining unit is used for determining the training sample set constant and the iteration step length according to the two-norm of the training matrix.
6. The Primal-dual based image denoising system of claim 4, wherein the non-smooth objective function establishing module specifically comprises:
a non-smooth objective function establishing unit for establishing an objective function according to a formula
Establishing a non-smooth objective function; wherein the content of the first and second substances,
is a non-smooth loss function, X is the primitive vector space, Y is the dual vector space, w ∈ X, α∈ Y, α are w-based dual variables, w is a feature vector X for picture-based samplesiThe optimized weight vector of (2); lambda [ alpha ]1R (w) is a first regularization term; lambda [ alpha ]2R (α) is a second regularization term, wTIs a transposed matrix of w αTα, H (S, y), a (S, y), b (S, y), d are constants based on the training sample set S.
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