CN112508049B - Clustering method based on group sparse optimization - Google Patents
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Abstract
The invention provides a group sparse optimization-based class aggregation method, which is characterized by firstly processing data and aiming at obtaining a similarity matrix target matrix, an error minimum term and a sparse constraint term among data set samples; secondly, an optimization model based on group sparse constraint is constructed, and the purpose of the optimization model is to restrain noise influence by utilizing more powerful group sparse constraint; then, the invention provides an optimization algorithm based on the alternating direction multiplier (Alternating Direction Method of Multipliers) to quickly solve the constructed optimization model; finally, the invention provides a rapid optimized clustering algorithm, which aims to combine redundant clustering results and further improve performance. The method of the invention restricts each sample to be approximately represented by only one sample, thereby effectively improving the robustness of the algorithm; on the other hand, the obtained target matrix does not need to carry out spectral clustering analysis, so that an end-to-end clustering effect is achieved.
Description
Technical Field
The invention relates to the technical field of digital image evidence obtaining, in particular to a clustering method based on group sparse optimization.
Background
The digital image device evidence obtaining technology refers to the technology of judging the imaging device by only relying on the image content, and has important research value in the fields of image processing, information evidence obtaining and the like. Early methods aimed at judging whether an image to be detected is photographed by a specific camera or not are widely used in scenes such as copyright protection, evidence detection, and the like. However, such methods have a large limitation in practical application because they require acquisition of identification features of the imaging apparatus in advance. At present, given a set of images, it is more feasible to determine which images are taken by the same camera and which are not. The scheme does not depend on any priori knowledge, can be suitable for a digital image equipment traceability scene in an open environment, and has greater application potential. At present, given a group of images with unknown sources, the equipment tracing technology is mainly realized by a blind clustering method, namely, clustering is carried out according to shooting equipment of the images on the premise of not depending on any priori knowledge (such as unknown equipment quantity).
The theoretical basis for digital image device traceability stems from the photo-response non-Uniformity characteristics (Photo Response Non-Uniformity, PRNU). Since there are different degrees of errors in photosensitivity of photosensitive elements in image forming apparatuses, different apparatuses naturally have their own noise patterns. This unique noise pattern to the imaging device is recorded into the digital image during imaging, and is a key feature (i.e., PRNU) to enable device traceability. In theory, by comparing and matching the PRNU features of the images to be detected, it can be determined whether the images are captured by the same camera.
However, PRNU extracted from a single image tends to be noisy, severely degrading the accuracy and reliability of the tracing. Specifically, the PRNU mainly exists in noise residuals of an image, and a large amount of various types of noise introduced in the imaging process are mixed. Due to noise interference, the correlation between noise residuals of some homologous images may be low, such that they are misjudged as non-homologous images. The existence of the outliers makes the tracing result unreliable, so that the performance of the existing clustering method is seriously weakened, and the reliability in practical application is greatly reduced. Therefore, how to accurately and quickly determine which images are from the same device source in the noise interference scenario is an urgent need to solve the problem.
At present, similar researches are carried out by the scientific research team. For example, G.J. Bloy Blind camera fingerprinting and image clustering, "IEEE Trans. On Pattern Analysis & Machine Intelligence," vol.30, no.3, pp.532-534,2008, proposes a hierarchical clustering method, and samples with smaller noise interference are preferentially aggregated together to obtain a better PRNU, so that the reliability of the PRNU is gradually improved, and possible outliers are eliminated. The method has the advantages that good accuracy is obtained, and feasibility of the blind tracing technology is verified on a small-scale data set.
In addition, F.Marra, G.Poggi, C.Sansone and L.Verdoliva 'Blind prnu-based image clustering for source identification' IEEE Trans.on Information Forensics and Security, vol.12, no.9, pp.2197-2211,2017 disclose a Blind clustering device traceability algorithm based on an integration strategy. Specifically, the number of different devices present in the dataset is determined by traversing all the category numbers to obtain a preliminary clustering result. And then the most effective tracing result is obtained through a simple optimizing step.
However, the above prior art still has a certain distance from the actual application. For example, in the method described in document 1, some features in which noise interferes more severely may cause accumulated errors, thereby reducing the reliability of the tracing result. In addition, in the method described in document 2, in the face of a larger-scale dataset, the method thereof requires integration of a large number of sub-cluster models, thereby severely reducing recognition efficiency.
Disclosure of Invention
The embodiment of the invention provides a clustering method based on group sparse optimization, which is used for solving the problems in the prior art.
In order to achieve the above purpose, the present invention adopts the following technical scheme.
A clustering method based on group sparse optimization, comprising:
s1, obtaining a similarity matrix between images based on an image set, and defining a target matrix, an error minimum term and a sparse constraint term;
s2, constructing an optimization model based on group sparse constraint based on a similarity matrix, a target matrix, an error minimum term and a sparse constraint term;
s3, solving the optimization model based on an alternate direction multiplier method to obtain a clustering result;
s4, merging redundant clustering results through an optimized clustering algorithm.
Preferably, step S1 specifically includes:
s100, obtaining a noise residual error of each image through a wavelet denoising algorithm, and carrying out normalization processing on the noise residual error;
s101, measuring the similarity between adjacent noise residuals through a cosine similarity formula, sequentially arranging the similarity to obtain a similarity matrix S, and carrying out normalization processing on the similarity matrix S;
s102 defining a target matrixThe target matrix->The method meets the following conditions: all elements consist of 0 or 1; any column has only one element of 1; wherein N is the number of images given in the image set;
s103 passing through typeA linear form of a sample representing an image set, where C ji Representing sample j is involved in representing the coefficient of sample i, r i Representing a certain sample;
s104 through typeRepresenting error e i Equivalent to r i A weighted dissimilarity sum with other samples; wherein N is the number of images given in the image set, S ji Elements representing the j-th row and i-th column of the similarity matrix S;
s105, obtaining an error minimum term tr (C (1-S)) (4);
s106, calculating a target matrixThe sum measure of all row maxima in (1) gets the sparsity penalty and gets +.>Wherein C is i: Representing all elements of row i of matrix C.
Preferably, constructing an optimization model based on the set of sparse constraints based on the similarity matrix, the target matrix, the error min terms, and the sparse constraint terms comprises:
s201, the objective function of the optimization model based on the group sparse constraint is that
Preferably, obtaining the clustering result by solving the optimization model based on an alternate direction multiplier method comprises:
s300, loosening elements in the target matrix C to be 0 or 1 to be between 0 and 1, introducing a new variable W, and restraining the new variable W to be equal to the target matrix C to obtain deformation of an objective function of an optimization model based on group sparse constraint
S301, transforming an objective function of an optimization model based on group sparsity constraint into a Lagrange form to obtain
u is an augmented lagrangian super parameter, and delta is a diagonal matrix;
s302 defines C (0) ,W (0) ,Δ (0) For a diagonal matrix with dimension N, iterating errorThe upper limit of the difference is epsilon=0.001, the upper limit of the iteration times is T=100, and the weight factor lambda=0.0003;
s303 passing through type
Updating W (k) ；
S304 through type
Update C (k+1) ；
S305 through type
Δ (k+1) ＝Δ (k) +u(C (k+1) -W (k+1) ) (11) update delta (k) The method comprises the steps of carrying out a first treatment on the surface of the Wherein W is a new variable, (k) is an initial value of the kth iteration, and (k+1) is a result after the kth iteration;
s306 calculates the iteration error C respectively (k+1) -W (k+1) || ∞ And C (k+1) -C (k) || ∞ If the iteration error C (k +1) -W (k+1) || ∞ And C (k+1) -C (k) || ∞ If at least one of the above is greater than the preset upper limit epsilon of iteration error or the iteration number k is less than the set upper limit T of iteration number, returning to the execution sub-step S303, otherwise, outputting the target matrix C obtained by solving (k+1) ；
S307 for the target matrix C (k+1) And carrying out normalization processing, and aggregating samples corresponding to the same column vector into the same class, so that the image set is divided into a plurality of clusters.
Preferably, merging redundant clustering results by optimizing a clustering algorithm includes:
s401 is based on a plurality of clusters, and if the number of contained samples is greater than 1, the number is recorded asAnd step S402 is performed, if the number of the included samples is equal to 1, the number is marked as +.>And performs step S406;
s402 for arbitrary clustersBy->Calculate the cluster +.>Average value of similarity of all samples in the cluster, obtaining the cluster +.>Similarity alpha i ；
S403 for arbitrary clustersAnd arbitrary cluster->By->Calculating the arbitrary cluster +.>And arbitrary cluster->Average value beta of similarity between samples of (a) ij The method comprises the steps of carrying out a first treatment on the surface of the Wherein S is lk Elements representing the kth column of the first row, l representing the elements belonging to the cluster +.>The number of samples in (K) represents the number belonging to the cluster +.>Sample number in (a);
s404 through typeObtain arbitrary cluster->And arbitrary cluster->Is a combination coefficient gamma of (2) ij ；
S405 if any clusterAnd arbitrary cluster->Is a combination coefficient gamma of (2) ij Greater than 0.5, the arbitrary cluster +.>And arbitrary cluster->
S406 calculating arbitrary clustersAverage similarity to other clusters, and +.>Cluster +.>Merging, the cluster->Satisfy->In (1) the->Representing non-single sample clusters, G i Representing a single sample cluster.
As can be seen from the technical scheme provided by the embodiment of the invention, the group sparse optimization-based class aggregation method provided by the invention is characterized by firstly processing data, and aims to obtain a similarity matrix target matrix, an error minimum term and a sparse constraint term among data set samples; secondly, an optimization model based on group sparse constraint is constructed, and the purpose of the optimization model is to restrain noise influence by utilizing more powerful group sparse constraint; then, the invention provides an optimization algorithm based on the alternating direction multiplier (Alternating Direction Method of Multipliers) to quickly solve the constructed optimization model; finally, the invention provides a rapid optimized clustering algorithm, which aims to combine redundant clustering results and further improve performance. The method of the invention restricts each sample to be approximately represented by only one sample, thereby effectively improving the robustness of the algorithm; on the other hand, the obtained target matrix does not need to carry out spectral clustering analysis, so that an end-to-end clustering effect is achieved.
Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for the description of 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 other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a process flow diagram of a clustering method based on group sparse optimization provided by the invention;
FIG. 2 is a flowchart of a preferred embodiment of a cluster method based on group sparse optimization provided by the present invention;
FIG. 3 is a flow chart of data preprocessing in a clustering method based on group sparse optimization, which is provided by the invention;
FIG. 4 is a flowchart of another part of data preprocessing in a clustering method based on group sparse optimization provided by the invention;
FIG. 5 is a flow chart of solving an optimization model in a clustering method based on group sparse optimization provided by the invention;
FIG. 6 is a flowchart of an optimized clustering algorithm in a clustering method based on group sparse optimization provided by the invention;
FIG. 7 is a schematic diagram of a target matrix of three exemplary cluster optimization algorithms that implement a cluster method based on group sparse optimization provided by the present invention;
fig. 8 is a schematic diagram of results of different stages of a clustering method based on group sparse optimization provided by the invention.
Detailed Description
Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals refer to the same or similar elements or elements having the same or similar functions throughout. The embodiments described below by referring to the drawings are exemplary only for explaining the present invention and are not to be construed as limiting the present invention.
As used herein, the singular forms "a", "an", "the" and "the" are intended to include the plural forms as well, unless expressly stated otherwise, as understood by those skilled in the art. It will be further understood that the terms "comprises" and/or "comprising," when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. It will be understood that when an element is referred to as being "connected" or "coupled" to another element, it can be directly connected or coupled to the other element or intervening elements may also be present. Further, "connected" or "coupled" as used herein may include wirelessly connected or coupled. The term "and/or" as used herein includes any and all combinations of one or more of the associated listed items.
It will be understood by those skilled in the art that, unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. It will be further understood that terms, such as those defined in commonly used dictionaries, should be interpreted as having a meaning that is consistent with their meaning in the context of the prior art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein.
For the purpose of facilitating an understanding of the embodiments of the invention, reference will now be made to the drawings of several specific embodiments illustrated in the drawings and in no way should be taken to limit the embodiments of the invention.
The invention aims to perform equipment tracing evidence obtaining on a group of images with unknown sources, and the method is to aggregate the images with the same equipment source together in a blind clustering mode. According to the invention, the similarity matrix between the images is obtained through the data preprocessing module, and the similarity matrix is subjected to iterative optimization by using an optimization method based on group sparse constraint, so that the image set is subjected to cluster analysis. Finally, according to the obtained clustering result, the method uses the calculated intra-class/inter-class similarity to further optimize the clustering result, and finally obtains the clustering result. The invention can effectively process the influence caused by noise, thereby obtaining better evidence obtaining performance of the image equipment.
Referring to fig. 1, the clustering method based on group sparse optimization provided by the invention comprises the following steps:
s1, obtaining a similarity matrix among images based on an image set containing N unknown sources, and defining a target matrix, an error minimum term and a sparse constraint term;
s2, constructing an optimization model based on group sparse constraint based on a similarity matrix, a target matrix, an error minimum term and a sparse constraint term;
s3, solving the optimization model based on an alternate direction multiplier method to obtain a preliminary clustering result;
s4, merging redundant image clusters in the preliminary clustering result through an optimized clustering algorithm.
In the embodiment provided by the present invention, step S1 is a data preprocessing process, which may be performed in a data preprocessing module, for obtaining a similarity matrix between samples of a data set, and the specific process includes the following sub-steps:
s100, acquiring a noise residual error of each image by adopting a wavelet denoising algorithm, and carrying out normalization processing on the noise residual error; specifically, any noise residual error can be subtracted by the mean value and divided by the mode;
s101, measuring the similarity between adjacent noise residuals through a cosine similarity formula, sequentially arranging the similarity to obtain a similarity matrix S, and carrying out normalization processing on the similarity matrix S; wherein S is ij A correlation coefficient representing an ith noise residual at a jth noise residual; the purpose of the normalization process is to ensure that the elements of the similarity matrix are between 0 and 1;
in this embodiment, the process of defining the target matrix is:
s102 defining a target matrixThe target matrix->The method meets the following conditions: all elements consist of 0 or 1; any column has only one element of 1; specifically, each list of C represents the self-representative coefficients of each sample in the image set; where N means the number of images given in the image set.
In this embodiment, the error min term is defined according to the sparse representation theory, any one sample can be represented by a linear weighting of a plurality of samples, the representation error of which is equivalent to the weighting of the similarity value between the sample and the sample involved in the representation, in this embodiment, the cosine similarity is used to measure the representation error, and the specific process includes:
s103 passing through typeA linear form of a sample representing an image set, where C ji Representing sample j is involved in representing the coefficient of sample i, r i Representing a certain sample;
s104 through typeRepresenting error e i Equivalent to r i Sum of weighted dissimilarities with other samples, S ji Elements representing the j-th row and i-th column of the similarity matrix S;
s105, obtaining an error minimum term tr (C (1-S)) (4); where N means the number of images given in the image set, tr is the trace of the matrix calculated as the sum of the diagonals of the matrix in brackets, and this formula (4) is in the form of a matrix of formula (2) (in elemental form).
In this embodiment, 1, ++norm is used to measure sparsity loss of the target matrix, specifically, by constraining the sum of all row maxima to be minimum, the number of non-zero rows in the target matrix is constrained to be as small as possible, which can be expressed in mathematical form as |C | 1,∞ The method comprises the steps of carrying out a first treatment on the surface of the The process specifically comprises the following steps:
s106, calculating a target matrixThe sum measure of all row maxima in (1) gets the sparsity penalty and gets +.>Wherein C is i: All elements of row i of matrix C, i.e. row i. />Meaning the maximum value (max j C i: ) A kind of electronic device.
Further, the objective function of the optimization model based on the group sparse constraint is that
In a preferred embodiment provided by the present invention, a model solving algorithm of an ADMM framework is provided for solving an optimization model based on a group sparse constraint, and specifically includes the following sub-steps:
s300, firstly relaxing constraint conditions of a model, specifically relaxing elements belonging to 0 or 1 in a target matrix C to be between 0 and 1, introducing a new variable W, and constraining the new variable W to be equal to the target matrix C to obtain deformation of an objective function of an optimization model based on group sparse constraint
S301, transforming an objective function of an optimization model based on group sparsity constraint into a Lagrange form, and introducing a Lagrange multiplier delta to obtain
Wherein u is an augmented Lagrangian super parameter, and delta is a diagonal matrix;
s302 defines C (0) ,W (0) ,Δ (0) For a diagonal matrix with a dimension of N, the upper limit of iteration error is epsilon=0.001, the upper limit of iteration times is t=100, and the weight factor lambda=0.0003; performing k iterations according to the following substeps;
specifically, in the kth iteration:
s303 passing through type
Updating W (k) ；
S304W (k+1) As a constant, pass-through type
Update C (k+1) 。
The function of the new variable W is to derive two variables when solving the above formula, respectively, and convert a complex problem into two relatively easy problems, thereby saving the calculation force.
Preferably, the above equation is solved by a stochastic algorithm as in the prior art, for example, J.Duchi, S.Shalev-Shewartz, Y.Singer, and T.Chandra, efficient projections onto the l1-ball for learning in high dimensions in ICML,2008, pp.272-279;
s305 to W (k+1) And C (k+1) As a constant, pass-through type
Δ (k+1) ＝Δ (k) +u(C (k+1) -W (k+1) ) (11) update delta (k) ；
S306 calculates the iteration error C respectively (k+1) -W (k+1) || ∞ And C (k+1) -C (k) || ∞ If the iteration error C (k +1) -W (k+1) || ∞ And C (k+1) -C (k) || ∞ If at least one of the above is greater than the preset upper limit epsilon of iteration error or the iteration number k is less than the set upper limit T of iteration number, returning to the execution sub-step S303, otherwise, outputting the target matrix C obtained by solving (k+1) ；
S307 for the target matrix C (k+1) And carrying out normalization processing, and aggregating samples corresponding to the same column vector into the same class, so that the image set is divided into a plurality of clusters.
After the above iteration is terminated, the obtained C is further processed. Specifically, the maximum value in each column is set to 1 and the remainder to 0, the purpose of which is to deal primarily with defects due to the relaxed condition.
Where (k) is the initial value of the kth iteration and (k+1) is the result after the kth iteration.
Still further, in a preferred embodiment of the present invention, an optimization algorithm for clustering is provided, i.e. step S4 described above. The aim is to solve the following problems: in steps S1-S3, the images shot by the same device are clustered into one class, and different classes are clustered into different classes, so as to obtain a rough clustering result, where samples clustered into one class are all shot by the same device, but samples of the same class may be divided into several clusters (i.e. redundant clusters). Step S4 consists mainly in merging redundant clusters and further clustering samples that have failed to aggregate. The method specifically comprises the following substeps:
s401 classifies the obtained clusters into classes: one is that the number of samples in the cluster exceeds 1, and the other is that the number of samples in the cluster is equal to 1; respectively marked asAnd->Specifically, a plurality of clusters may be used, and if the number of included samples is greater than 1, the number is marked as +.>And step S402 is performed, if the number of the included samples is equal to 1, the number is marked as +.>And performs step S406;
then toThe cluster of the cluster is combined, and the following process is carried out:
s402, calculating intra-class similarity among clusters, aiming at any clusterBy->Calculate the cluster +.>Average value of similarity of all samples in the cluster, obtaining the cluster +.>Similarity alpha i ；
S403, calculating the similarity among the clusters, aiming at any clusterAnd arbitrary cluster->By->Calculating the arbitrary cluster +.>And arbitrary cluster->Average value beta of similarity between samples of (a) ij The method comprises the steps of carrying out a first treatment on the surface of the Wherein S is lk Elements representing the first row and the k column; l represents belonging to cluster->The number of samples in (K) represents the number belonging to the cluster +.>Sample number in (a);
s404, calculating the merging similarity of clusters, and passing throughObtain arbitrary cluster->And arbitrary cluster->Is a combination coefficient gamma of (2) ij ；
S405 if any clusterAnd arbitrary cluster->Is a combination coefficient gamma of (2) ij Greater than 0.5, the arbitrary cluster +.>And arbitrary cluster->
S406 calculating arbitrary clustersAverage similarity to other clusters, and +.>Cluster +.>Merging, the cluster->Satisfy->In (1) the->Representing non-single sample clusters, G i Representing a single sample cluster.
Through the above steps, pictures taken by the same device are aggregated into the same cluster in a given set of images. The number of clusters obtained is the predicted number of devices.
The present invention also provides an embodiment for exemplarily displaying an implementation of the method of the present invention.
As shown in fig. 2, given an image of unknown device source, comprising the steps of:
step S101: obtaining a similarity matrix S between images;
therein, as shown in fig. 3, for a given image set, a wavelet denoising method is preferably employed to obtain noise residuals for each image. The normalization of the noise residuals obtained is preferably done by subtracting the mean and dividing by the modulus. In addition, for the obtained noise residual signal set, a similarity value between every two signals is calculated. Preferably, a cosine similarity calculation formula is used to measure the similarity value between the two signals. The obtained similarity values are sequentially arranged into a similarity matrix S, and normalization processing is carried out. Preferably, each element of the obtained similarity matrix subtracts the minimum value of the similarity matrix and takes the maximum value to ensure that the value range of all elements in S is between 0 and 1;
step S102: constructing an optimization model based on group sparse constraint;
wherein, as shown in fig. 4, step S102 further includes defining a target matrix, an error min term and a sparse constraint term.
Step S201: a target matrix C is defined. The target matrix C should meet a two-point constraint, i.e. 1) all elements take values of 0 or 1; 2) Only one element in any one column is 1 and the remainder are 0. Each column in C may be considered a label vector for the corresponding image, while samples with the same label vector may be considered aggregated into a class. Unlike typical sparse term constraints, the present invention employs a set of sparse constraints, i.e., each sample can be represented by only one sample. The purpose of the constraint term is to have a stronger constraint effect and to sufficiently suppress interference caused by noise.
Step S202: the definition represents an error term. For arbitrary samples r i Can be expressed as a linear approximation by several samples within the datasetWherein C is ji Representing sample j participates in the coefficients representing sample i. Thus, it represents the error e i Equivalent to r i Weighted dissimilarity sum with all other samples, i.e. +.>The representation error is the sum of the representation errors of all samples for the whole data set, i.e +.>The expression can be rewritten in matrix form, i.e., tr (C (1-S));
step S203: is fixed to the sparse constraint term. To prevent samples from being self-represented, the present invention employs a set of sparse term constraints to limit the sparsity of the target matrix. Unlike traditional sparsity constraint methods, the present invention measures sparsity loss by computing the sum of all row maxima in the target matrix. The mathematical expression can be expressed as
In combination with the above, the optimization model based on the group sparsity constraint can be written as follows:
step S103: solving an optimization model;
as shown in fig. 5, step S103 uses the ADMM framework to solve, and the steps include:
step S301: the objective function is transformed into a lagrangian form. Due to c ij The existence of E {0,1} and the objective function is a NP-difficult problem. Therefore, the constraint condition is required to be relaxed to be in the range of 0 to 1 of the element value in C, namely C is more than or equal to 0. At the same time, a new variable W is introduced, making it equal to C. Thus, the model can be rewritten as:
the objective function may then be introduced into the Lagrangian multiplier Δ and rewritten as
Step S302: initializing parameters;
wherein, define C (0) ,W (0) ,Δ (0) For a diagonal matrix with a dimension of N, the upper limit of iteration error is epsilon=0.001, the upper limit of iteration times is t=100, and the weight factor lambda=0.0003;
step S303: updating W (k) ；
Wherein, fix C (k) ,Δ (k) Update C by (k) ；
Step S304, update C (k+1) ；
Wherein W is fixed (k+1) ,Δ (k) Update C by (k+1) ；
Preferably, update C (k+1) The method can be quickly solved by adopting a randomization algorithm described in document 3;
step S305: updating delta (k)
Wherein W is fixed (k+1) ,C (k+1) Delta is updated by (k) ：
Δ (k+1) ＝Δ (k) +u(C (k+1) -W (k+1) )
Step S306: and judging an iteration stop condition.
Wherein, respectively calculating the iteration error C (k+1) -W (k+1) || ∞ And C (k+1) -C (k) || ∞ The method comprises the steps of carrying out a first treatment on the surface of the If at least one of the two is greater than the preset upper limit epsilon of the iteration error or the iteration number k is less than the set upper limit T of the iteration number, the iteration is continued from step S303. Otherwise, outputting the obtained target matrix C (k+1) ；
Step S307: normalizing the obtained C (k+1) ；
Wherein, due to relaxation of the constraint, C (k+1) There are cases where the value is between 0 and 1. For this, C is (k+1) The maximum value of each column is set to 1 and the remaining elements are set to 0. Then, aggregating samples corresponding to the same column vector into the same class, thereby dividing the data set into a plurality of clusters;
step S104: and (3) clustering and optimizing. The method aims at further combining samples which are not aggregated, so as to obtain a better clustering effect;
as shown in fig. 6, step S104 further includes:
step S401: for all clusters obtained, if the number of samples contained therein is greater than 1, jumping to step S402; otherwise, jump to step S406;
step S402: calculating the similarity of each cluster;
for the ith cluster, calculating the average value of the pairwise similarity of all samples in the clusters to be used as the intra-cluster similarity alpha of the class i The mathematical expression should be
Step S403: calculating the similarity between the classes of every two clusters;
wherein, for the ith and jth clusters, an average value beta of two-by-two similarities between samples among the clusters is calculated ij The mathematical expression should be
Step S404: calculating the merging coefficients of every two clusters;
wherein, for the ith and jth clusters, if the number of samples of the jth cluster is greater than the number of samples of the ith cluster, the coefficients γ are combined ij Should be the inter-cluster similarity beta ij Dividing the similarity of the jth cluster by the sum of the similarity between clusters; otherwise, the coefficient of merger gamma ij Should be the inter-cluster similarity beta ij Divided by the sum of the similarity of the ith cluster and the similarity between clusters. The mathematical expression should be
Step S405: clusters with a number of samples greater than 1 are merged.
Wherein if the merging coefficient gamma between two clusters ij Greater than 0.5, then the ith cluster and the jth cluster should be merged;
step S406: merging clusters of only one sample;
and calculating the average value of the similarity between the single sample cluster and all other clusters, and combining the single sample cluster with the obtained cluster with the maximum similarity. For the ith single sample cluster, it should be merged at the jth non-single sample cluster, where j satisfies the following equation:
wherein,representing non-single sample clusters, G i Representing a single sample cluster.
The target matrices for three typical cluster optimization algorithms are shown from left to right in fig. 7, respectively. Fig. 7 (a) is a similarity matrix, typically as an input signal for spectral clustering. Because the method is not robust to noise, the sparse subspace clustering (Sparse Subset Clustering, SSC) method obtains a sparse representation coefficient matrix through sparse constraint, and the sparse representation coefficient matrix is used as a purer input signal for spectral clustering analysis. Fig. 7 (c) shows the target matrix of the optimization algorithm based on the group sparsity constraint of the present invention. On one hand, the method of the invention restricts each sample to be approximately represented by only one sample, thereby effectively improving the robustness of the algorithm; on the other hand, the obtained target matrix does not need to carry out spectral clustering analysis, so that an end-to-end clustering effect is achieved.
For a better understanding of the calculation scheme of the present invention, fig. 7 presents a schematic view of the results of the different phases. FIG. 8 (a) shows the input similarity matrix, red, green, and blue representing different classes of samples, respectively; fig. 8 (b) is obtained through the proposed optimization algorithm based on the group sparsity constraint. Wherein, any one column has only one sample of 1, and the rest is 0. While samples with the same column vector are considered to be aggregated together. Although the optimization algorithm can accurately distinguish between different classes of samples, samples of the same class are often divided into classes. In this regard, the present invention further optimizes the results in fig. 8 (b), and combines different clusters of the same class, thereby obtaining the final clustering result, as shown in fig. 8 (c).
In summary, the group sparse optimization-based class gathering method provided by the invention is firstly used for processing data and aims to obtain a similarity matrix target matrix, an error minimum term and a sparse constraint term among data set samples; secondly, an optimization model based on group sparse constraint is constructed, and the purpose of the optimization model is to restrain noise influence by utilizing more powerful group sparse constraint; then, the invention provides an optimization algorithm based on the alternating direction multiplier (Alternating Direction Method of Multipliers) to quickly solve the constructed optimization model; finally, the invention provides a rapid optimized clustering algorithm, which aims to combine redundant clustering results and further improve performance. The method of the invention restricts each sample to be approximately represented by only one sample, thereby effectively improving the robustness of the algorithm; on the other hand, the obtained target matrix does not need to carry out spectral clustering analysis, so that an end-to-end clustering effect is achieved.
Those of ordinary skill in the art will appreciate that: the drawing is a schematic diagram of one embodiment and the modules or flows in the drawing are not necessarily required to practice the invention.
From the above description of embodiments, it will be apparent to those skilled in the art that the present invention may be implemented in software plus a necessary general hardware platform. Based on such understanding, the technical solution of the present invention may be embodied essentially or in a part contributing to the prior art in the form of a software product, which may be stored in a storage medium, such as a ROM/RAM, a magnetic disk, an optical disk, etc., including several instructions for causing a computer device (which may be a personal computer, a server, or a network device, etc.) to execute the method described in the embodiments or some parts of the embodiments of the present invention.
In this specification, each embodiment is described in a progressive manner, and identical and similar parts of each embodiment are all referred to each other, and each embodiment mainly describes differences from other embodiments. In particular, for apparatus or system embodiments, since they are substantially similar to method embodiments, the description is relatively simple, with reference to the description of method embodiments in part. The apparatus and system embodiments described above are merely illustrative, wherein the elements illustrated as separate elements may or may not be physically separate, and the elements shown as elements may or may not be physical elements, may be located in one place, or may be distributed over a plurality of network elements. Some or all of the modules may be selected according to actual needs to achieve the purpose of the solution of this embodiment. Those of ordinary skill in the art will understand and implement the present invention without undue burden.
The present invention is not limited to the above-mentioned embodiments, and any changes or substitutions that can be easily understood by those skilled in the art within the technical scope of the present invention are intended to be included in the scope of the present invention. Therefore, the protection scope of the present invention should be subject to the protection scope of the claims.
Claims (1)
1. A clustering method based on group sparse optimization, comprising:
s1, obtaining a similarity matrix between images based on an image set, and defining a target matrix, an error minimum term and a sparse constraint term; the method specifically comprises the following steps:
s100, obtaining a noise residual error of each image through a wavelet denoising algorithm, and carrying out normalization processing on the noise residual error;
s101, measuring the similarity between adjacent noise residuals through a cosine similarity formula, sequentially arranging the similarity to obtain a similarity matrix S, and carrying out normalization processing on the similarity matrix S;
s102 defining a target matrixThe target matrix->The method meets the following conditions: all elements consist of 0 or 1; any column has only one element of 1; wherein N is the number of images given in the image set;
s103 passing through typeA linear form of a sample representing an image set, where C ji Representing sample j is involved in representing the coefficient of sample i, r i Representing a certain sample;
s104 through typeRepresenting error e i Equivalent to r i A weighted dissimilarity sum with other samples; wherein N is the number of images given in the image set, S ji Elements representing the j-th row and i-th column of the similarity matrix S;
s105, obtaining the error minimum term tr (C (1-S)) (4);
s106, calculating the target matrixThe sum measure of all row maxima in (1) gets the sparsity penalty and gets +.>Wherein C is i: All elements representing the ith row of matrix C;
s2, constructing an optimization model based on group sparse constraint based on a similarity matrix, a target matrix, an error minimum term and a sparse constraint term; the method specifically comprises the following steps:
s201, the objective function of the optimization model based on the group sparse constraint is thatWherein λ is a weight factor;
s3, solving the optimization model based on an alternate direction multiplier method to obtain a clustering result; the method specifically comprises the following steps:
s300, loosening elements in the target matrix C to be 0 or 1 to be between 0 and 1, introducing a new variable W, and restraining the new variable W to be equal to the target matrix C to obtain deformation of an objective function of an optimization model based on group sparse constraint
Wherein λ is a weight factor;
s301, transforming an objective function of the optimization model based on the group sparse constraint into a Lagrange form to obtain
u is an augmented lagrangian super parameter, delta is a diagonal matrix, and lambda is a weight factor;
s302 defines C (0) ,w (0) ,Δ (0) Is diagonal with dimension NThe upper limit of the iteration error is epsilon=0.001, the upper limit of the iteration times is T=100, and the weight factor lambda=0.0003;
s303 passing through type
Updating W (k) ；
S304 through type
Update C (k+1) ；
S305 through type
Δ (k+1) ＝Δ (k) +u(C (k+1) -W (k+1) ) (11) update delta (k) The method comprises the steps of carrying out a first treatment on the surface of the Wherein W is a new variable, (k) is an initial value of the kth iteration, and (k+1) is a result after the kth iteration;
s306 calculates the iteration error C respectively (k+1) -W (k+1) || ∞ And C (k+1) -C (k) || ∞ If the iteration error C (k+1) -W (k +1) || ∞ And C (k+1) -C (k) || ∞ If at least one of the above is greater than the preset upper limit epsilon of iteration error or the iteration number k is less than the set upper limit T of iteration number, returning to the execution sub-step S303, otherwise, outputting the target matrix C obtained by solving (k+1) ；
S307 for the target matrix C (k+1) Carrying out normalization processing, and aggregating samples corresponding to the same column vector into the same class to divide an image set into a plurality of clusters;
s4, merging redundant clustering results through an optimized clustering algorithm; the method specifically comprises the following steps:
s401, based on the clusters, if the number of contained samples is greater than 1, marking asAnd step S402 is performed, if the number of the included samples is equal to 1, the number is marked as +.>And performs step S406;
s402 for arbitrary clustersBy->Calculate the cluster +.>Average value of similarity of all samples in the cluster, obtaining the cluster +.>Similarity alpha i The method comprises the steps of carrying out a first treatment on the surface of the Wherein S is lk Elements representing the first row and the k column;
s403 for arbitrary clustersAnd arbitrary cluster->By->Calculating the arbitrary clusterAnd arbitrary cluster->Average value beta of similarity between samples of (a) ij The method comprises the steps of carrying out a first treatment on the surface of the Wherein S is lk Elements representing the kth column of the first row, l representing the elements belonging to the cluster +.>The number of samples in (K) represents the number belonging to the cluster +.>Sample number in (a);
s404 through typeObtain arbitrary cluster->And arbitrary cluster->Is a combination coefficient gamma of (2) ij ；
S405 if any clusterAnd arbitrary cluster->Is a combination coefficient gamma of (2) ij Greater than 0.5, the arbitrary cluster +.>And arbitrary cluster->
S406 calculating arbitrary clustersAverage similarity to other clusters, and +.>And the institute are connected withCluster of maximum similarity obtained +.>Merging, the cluster->Satisfying the requirements
In (1) the->Representing non-single sample clusters, G i Representing a single sample cluster, S lk Representing the elements of row i and column k.
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