CN109447147B - Image clustering method based on depth matrix decomposition of double-image sparsity - Google Patents

Image clustering method based on depth matrix decomposition of double-image sparsity Download PDF

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CN109447147B
CN109447147B CN201811243981.7A CN201811243981A CN109447147B CN 109447147 B CN109447147 B CN 109447147B CN 201811243981 A CN201811243981 A CN 201811243981A CN 109447147 B CN109447147 B CN 109447147B
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孟洋
尚荣华
焦李成
王蓉芳
马文萍
刘芳
侯彪
王爽
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Abstract

The invention provides an image clustering method based on depth matrix decomposition of double-image sparsity, which is used for solving the technical problems of low image clustering accuracy and low operation speed in a clustering process in the prior art. The method comprises the following implementation steps: constructing a multilayer perceptron; setting iteration times; calculating a data space similarity matrix and a characteristic space similarity matrix; calculating a data space similarity diagonal matrix and a characteristic space similarity diagonal matrix; defining a low-dimensional representation matrix HiAnd HiCorresponding coefficient matrix ZiAnd calculating ZiCorresponding coefficient diagonal matrix Qi(ii) a Obtaining HiAnd ZiThe update formula of (2); for H of each layeri、ZiAnd QiCircularly updating; reacquiring HiAnd ZiThe update formula of (2); resetting the iteration times; resetting the current layer number; for H of each layeri、ZiAnd QiCircularly updating; and clustering and outputting the images. The method can be applied to real life applications such as face recognition, image clustering and text clustering.

Description

Image clustering method based on depth matrix decomposition of double-image sparsity
Technical Field
The invention belongs to the technical field of image processing, relates to an image clustering method, and particularly relates to an image clustering method based on depth matrix decomposition of double-image sparsity, which can be applied to real life such as face recognition, image clustering and text clustering.
Background
With the rapid development of internet technology and the popularization of image acquisition tools such as mobile phones, cameras, video cameras and the like, image information plays an increasingly important role in the life of people, and a plurality of image processing and analyzing methods are widely applied. The image clustering is a process of dividing an original image into a plurality of classes consisting of similar images, so that the image clustering can perform data organization on original image information, is an effective image processing method, and generally adopts indexes such as clustering accuracy, clustering process operation speed and the like to measure clustering effect.
The k-means clustering method is a common clustering method, has the advantages of high processing speed, high efficiency and the like, and is widely applied to image clustering. With the increasing of the image scale and the definition, the clustering effect of k-means clustering cannot meet the requirements of people, so that a plurality of algorithms can perform dimensionality reduction on original image data before k-means clustering, and a better image clustering effect can be obtained. In recent years, Matrix Factorization (MF) algorithm has become a very popular dimension reduction algorithm in the field of data analysis and image processing, which can learn a good low-dimensional data representation from high-dimensional data. However, the classical matrixing decomposition algorithm does not take into account the geometry in the data and has low clustering accuracy, so that in 2011, the article "Graph regulated non-negative matrix factorization for data representation" published by Deng Cai et al in IEEE Transactions on Pattern Analysis and Machine Intelligence, volume 33, pages 1548-1560, adds manifold learning to the classical matrixing decomposition algorithm, and proposes a Graph non-regular negative matrixing decomposition (GNMF), which can find a compact representation to mine the geometry inherent in the original image data. However, GNMF only adds manifold learning in the data space, does not utilize manifold information of the feature space, does not fully utilize sparsity of the matrix, and is complex in calculation and slow in operation speed of the clustering process. More importantly, the classical matrix decomposition algorithm and the GNMF algorithm both adopt single-layer structures, and the mapping relation between new low-dimensional data representation and original high-dimensional data in the dimension reduction algorithm is very complex, so that the single-layer clustering method cannot be fully explained, and the clustering accuracy is low when the k-means is used for clustering.
In 2018, a patent application with the publication number of 107609596a and the name of a non-parameter automatic weighting multi-graph regularization non-negative matrix decomposition and image clustering method discloses a non-parameter automatic weighting multi-graph regularization non-negative matrix decomposition and image clustering method (MGNMF), and the method obtains image data of m images; constructing q nearest graphs based on the image data of the m images and calculating corresponding Laplacian subgraphs; establishing a target function of a multi-graph regular operator according to the calculated Laplace operator graph; obtaining a parameter-free automatic weighting regular term according to the established target function of the multi-graph regular operator; establishing a target function of non-negative matrix decomposition according to the obtained parameter-free automatic weighting regular term; and obtaining an iterative expression of two non-negative matrixes according to the objective function of the non-negative matrix decomposition, and completing the decomposition of the multi-graph regularization non-negative matrix. The method constructs the adjacent graph based on the original image data, can mine the inherent geometric structure information in the original image data, and improves the image clustering accuracy. However, the method adopts a single-layer structure, cannot sufficiently explain the mapping relationship between the new low-dimensional data representation and the original high-dimensional data, and in addition, the method only utilizes manifold information of a data space of image data, does not sufficiently mine geometric structure information of a feature space, and does not carry out sparse constraint, so that the final clustering accuracy is low, and the clustering process is slow in operation speed.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, provides an image clustering method based on the depth matrix decomposition of the double-image sparsity, and is used for solving the technical problems of low image clustering accuracy and low operation speed in a clustering process in the prior art.
In order to achieve the purpose, the technical scheme adopted by the invention comprises the following steps:
(1) constructing a multilayer perceptron:
constructing a multilayer perceptron, wherein the number of network layers is nl, nl is more than or equal to 3, the initial value of the current layer number i is 1, and the input matrix of the 1 st layer is an original image data matrix;
(2) and (3) setting iteration times t:
setting an initial value of the iteration times T as 1 and a maximum value as T;
(3) calculating a data spatial similarity matrix Wi HAnd a feature spatial similarity matrix Wi Z：
Calculating Euclidean distance between data in the ith data space according to the input matrix of the ith layer of the multilayer perceptron
Figure BDA0001840062380000021
And Euclidean distance between features in feature space of i-th layer
Figure BDA0001840062380000022
And according to
Figure BDA0001840062380000023
And
Figure BDA0001840062380000024
respectively calculating the data space similarity matrix W of the ith layeri HAnd the feature spatial similarity matrix W of the ith layeri Z；
(4) Computing a data spatial similarity diagonal matrix
Figure BDA0001840062380000031
And feature space similarity diagonal matrix
Figure BDA0001840062380000032
Data spatial similarity matrix W to the i-th layeri HAnd the feature spatial similarity matrix W of the ith layeri ZDiagonalizing respectively to obtain the diagonal matrix of the data space similarity of the ith layer
Figure BDA0001840062380000033
And feature spatial similarity diagonal matrix of ith layer
Figure BDA0001840062380000034
(5) Defining a low-dimensional representation matrix HiAnd HiCorresponding coefficient matrix ZiAnd calculating ZiCorresponding coefficient diagonal matrix Qi：
(5a) Defining a low-dimensional representation matrix H for the ith layeriIs a random matrix of kXn, HiCoefficient matrix Z of corresponding ith layeriThe random matrix is a d multiplied by k, wherein k represents the number of the image classes in the input matrix, k is more than or equal to 2, n represents the number of samples of the input matrix, n is more than or equal to 2, d represents the characteristic number of the input matrix, and d is more than or equal to 2;
(5b) using ZiCoefficient diagonal matrix Q defining the ith layeriAnd calculating Z according to the updated formulaiCorresponding Qi；
(6) Obtaining a low-dimensional representation matrix HiIs updated to formula and HiCorresponding coefficient matrix ZiThe update formula of (2):
defining an objective function formula of matrix decomposition of bipicture sparsity, and using the objective function formula to derive a low-dimensional expression matrix H of the ith layeriIs updated to formula and HiCoefficient matrix Z of corresponding ith layeriThe update formula of (2);
(7) for low dimensional representation matrix HiCoefficient matrix ZiSum coefficient diagonal matrix QiUpdating:
using the low-dimensional representation matrix H of the ith layeriIs updated to HiUpdating by using coefficient matrix Z of i-th layeriIs updated to the formula pair ZiUpdating by using coefficient diagonal matrix Q of ith layeriUpdate formula pair QiUpdating is carried out;
(8) judging the overlayIf the generation times T reach the maximum value T, updating H after T times if the generation times T reach the maximum value TiThe input matrix is used as an input matrix of the i +1 th layer of the multilayer perceptron, and step (9) is executed, otherwise, t is made to be t +1, and step (7) is executed;
(9) judging whether the current layer number i reaches the maximum value nl, if so, executing the step (10), otherwise, making i equal to i +1, and executing the step (2);
(10) reacquiring the low-dimensional representation matrix HiIs updated to formula and HiCorresponding coefficient matrix ZiThe update formula of (2):
defining an objective function formula of the depth matrix decomposition of the bipicture sparsity, and utilizing the objective function formula to deduce a low-dimensional expression matrix H of the ith layeriIs updated to formula and HiCoefficient matrix Z of corresponding ith layeriThe update formula of (2);
(11) resetting the iteration number t:
making the iteration number T equal to 1 and the maximum value T;
(12) resetting the current layer number i:
making the current layer number i equal to 1, and the maximum value n l;
(13) for low dimensional representation matrix HiCoefficient matrix ZiSum coefficient diagonal matrix QiAnd (4) updating again:
using the low-dimensional representation matrix H of the ith layer in the step (10)iIs updated to HiUpdating again by using the coefficient matrix Z of the ith layer in the step (10)iIs updated to the formula pair ZiUpdating again by using the coefficient diagonal matrix Q of the ith layer in the step (5b)iUpdate formula pair QiUpdating again;
(14) judging whether the current layer number i reaches the maximum value nl, if so, executing the step (15), otherwise, making i equal to i +1, and executing the step (13);
(15) judging whether the iteration time T reaches the maximum value T, if so, executing step (16), otherwise, making T equal to T +1, and executing step (12);
(16) clustering and outputting the images:
utilizing k-means clustering algorithm to carry out clustering on the last layer of the multilayer perceptronIs represented by a low-dimensional representation matrix HnlAnd clustering to obtain a clustering image and outputting the clustering image.
Compared with the prior art, the method has the following advantages:
firstly, the invention combines the multilayer perceptron and the matrix decomposition in the dimension reduction process to form a depth frame, can fully explain the mapping relation between new low-dimensional data representation and original high-dimensional data by utilizing a multilayer network, clusters the original images according to different characteristics and effectively improves the image clustering accuracy.
Secondly, the data space similarity matrix and the feature space similarity matrix of each layer can be calculated through circulation, neighbor maps can be constructed in the data space and the feature space of each layer at the same time, the local geometric structures of the data space and the feature space are reserved, potential manifold information of data is fully mined, and the image clustering accuracy is further improved.
Thirdly, the invention can utilize the coefficient matrix of each layer to calculate and continuously update the corresponding coefficient diagonal matrix by circulation, and the purpose is to add L to the coefficient matrix2,1The sparse constraint of the norm not only can enable the coefficient matrix to have good sparsity and accelerate the operation speed of the clustering process, but also can enhance the local learning capability and robustness of the algorithm.
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FIG. 1 is a block diagram of an implementation flow of the present invention;
FIG. 2 is a graph comparing image clustering accuracy simulation of the present invention and prior art;
fig. 3 is an effect diagram of visualizing image clustering results by using a t-SNE method in the present invention and the prior art, where fig. 3(a) is a clustering effect diagram on an orlrows face image when performing k-means clustering by using GNMF, fig. 3(b) is a clustering effect diagram on an orlrows face image when performing k-means clustering by using MGNMF, and fig. 3(c) is a clustering effect diagram on an orlrows face image when using the present invention.
Detailed Description
The invention is described in further detail below with reference to the figures and the specific embodiments.
Referring to fig. 1, the image clustering method based on the depth matrix decomposition of the dual-image sparsity comprises the following steps:
step 1), constructing a multilayer perceptron;
in this embodiment, the number nl of network layers of the multilayer perceptron is set to be 3, and the input original image data matrix is face image data orlrows, which includes 10 different categories, each category has 10 different grayscale images, so that there are 100 grayscale images in total. Each human face image contains 112 x 92 pixel points, and each pixel point has 256 gray levels. The images are all left and right sides of the person in a front vertical position and are obtained under different lighting, time and facial expressions.
Step 2) setting iteration times t:
the initial value of the iteration number T is set to 1, the maximum value is set to T, and T is set to 100 in the embodiment.
Step 3) calculating a data space similarity matrix Wi HAnd a feature spatial similarity matrix Wi Z：
Calculating Euclidean distance between data in the ith data space according to the input matrix of the ith layer of the multilayer perceptron
Figure BDA0001840062380000051
And Euclidean distance between features in feature space of i-th layer
Figure BDA0001840062380000052
And according to
Figure BDA0001840062380000053
And
Figure BDA0001840062380000054
respectively calculating the data space similarity matrix W of the ith layeri HAnd the feature spatial similarity matrix W of the ith layeri ZThe calculation formulas are respectively as follows:
euclidean distance between data in ith data space
Figure BDA0001840062380000055
The calculation formula of (2) is as follows:
Figure BDA0001840062380000056
euclidean distance between features in i-th layer feature space
Figure BDA0001840062380000057
The calculation formula of (2) is as follows:
Figure BDA0001840062380000058
data spatial similarity matrix W of i-th layeri HThe calculation formula of (2) is as follows:
Figure BDA0001840062380000061
characteristic spatial similarity matrix W of ith layeri ZThe calculation formula of (2) is as follows:
Figure BDA0001840062380000062
wherein the content of the first and second substances,
Figure BDA0001840062380000063
denotes an evolution operation, XiInput matrix representing the i-th layer, Hadamard matrix product operation, 1d×nAll 1 matrices, 1, representing dXnn×dRepresenting an overall 1 matrix of nxd, d representing the number of features of the input matrix, d ≧ 2, n representing the number of samples of the input matrix, n ≧ 2,Tdenotes the transpose operation, exp (-) denotes the exponential operation, and σ denotes the gaussian scale parameter.
According to the method, the spatial similarity matrix and the characteristic spatial similarity matrix of each layer of data can be calculated through circulation, neighbor graphs can be constructed in each layer of data space and characteristic space at the same time, local geometric structures of the data space and the characteristic space are reserved, potential manifold information of the data is fully mined, and the image clustering accuracy is effectively improved.
Step 4) calculating a data space similarity diagonal matrix
Figure BDA0001840062380000064
And feature space similarity diagonal matrix
Figure BDA0001840062380000065
Data spatial similarity matrix W to the i-th layeri HAnd the feature spatial similarity matrix W of the ith layeri ZDiagonalizing respectively to obtain the diagonal matrix of the data space similarity of the ith layer
Figure BDA0001840062380000066
And feature spatial similarity diagonal matrix of ith layer
Figure BDA0001840062380000067
The calculation formulas are respectively as follows:
data spatial similarity diagonal matrix of ith layer
Figure BDA0001840062380000068
The calculation formula of (2) is as follows:
Figure BDA0001840062380000069
feature spatial similarity diagonal matrix of ith layer
Figure BDA00018400623800000610
The calculation formula of (2) is as follows:
Figure BDA00018400623800000611
wherein diag (·) denotes the generation of a diagonal matrix operation, ΣIt is shown that the operation of the superposition,
Figure BDA00018400623800000612
represents the u-th column of the data space similarity matrix of the ith layer, n represents the number of samples of the input matrix, n ≧ 2,
Figure BDA00018400623800000613
and d represents the characteristic number of the input matrix, and d is more than or equal to 2.
The invention respectively utilizes the data space similarity matrix and the characteristic space similarity matrix when calculating the data space similarity diagonal matrix and the characteristic space similarity diagonal matrix, thereby simultaneously reserving the local geometric structures of the data space and the characteristic space, fully mining the potential structure information of the data and improving the image clustering accuracy.
Step 5) defining a low-dimensional representation matrix HiAnd HiCorresponding coefficient matrix ZiAnd calculating ZiCorresponding coefficient diagonal matrix Qi：
(5a) Defining a low-dimensional representation matrix H for the ith layeriIs a random matrix of kXn, HiCoefficient matrix Z of corresponding ith layeriThe random matrix is a d multiplied by k, wherein k represents the number of the image classes in the input matrix, k is more than or equal to 2, n represents the number of samples of the input matrix, n is more than or equal to 2, d represents the characteristic number of the input matrix, and d is more than or equal to 2;
(5b) using ZiCoefficient diagonal matrix Q defining the ith layeriAnd calculating Z according to the updated formulaiCorresponding QiWherein Q isiRow j and column j ofjjThe calculation formula of (2) is as follows:
Figure BDA0001840062380000071
wherein the content of the first and second substances,
Figure BDA0001840062380000072
represents ZiElement composition of line jThe vector of (2).
Coefficient diagonal matrix Q is calculated by the inventioniThe purpose is to add L to the coefficient matrix for the latter2,1And (3) preparing for sparse constraint of norm, obtaining a more sparse coefficient matrix, simplifying calculation, accelerating the operation speed of a clustering process, and enhancing the local learning capability and robustness of the algorithm.
Step 6) obtaining a low-dimensional expression matrix HiIs updated to formula and HiCorresponding coefficient matrix ZiThe update formula of (2):
defining an objective function formula of matrix decomposition of bipicture sparsity, and using the objective function formula to derive a low-dimensional expression matrix H of the ith layeriIs updated to formula and HiCoefficient matrix Z of corresponding ith layeriThe updating formula is realized by the following steps:
(6a) objective function formula O for matrix factorization to define bipicture sparsitySDNMFThe expression is as follows:
Figure BDA0001840062380000073
wherein | · | purple sweetFRepresenting Frobenius norm, | | · |. luminance2,1Represents L2,1Norm, Tr (·) denotes the matrix tracing operation, XiInput matrix representing the i-th layer, ZiCoefficient matrix representing the i-th layer, HiA low-dimensional representation matrix, W, representing the ith layeri HA data spatial similarity matrix, W, representing the ith layeri ZA feature spatial similarity matrix representing the ith layer,
Figure BDA0001840062380000074
a data spatial similarity diagonal matrix representing the ith layer,
Figure BDA0001840062380000075
a feature spatial similarity diagonal matrix representing the ith layer,Tdenotes a transpose operation, αiDual map parameter, β, representing the ith layeriA sparse parameter representing an ith layer;
objective function formula O decomposed by the above-mentioned dual-graph sparse matrixSDNMFIt can be seen that the first term to the right of the equal sign is a classical non-negative matrix factorization term; the second term is a graph regular term which simultaneously reserves the local geometric structures of a data space and a feature space, so that the potential structure information of the data can be fully mined, and the image clustering accuracy is improved; third term adds L to the coefficient matrix2,1The sparse constraint of the norm not only can enable the coefficient matrix to have good sparsity and accelerate the operation speed of the clustering process, but also can enhance the local learning capability and robustness of the algorithm.
(6b) Solving a Lagrangian function L according to a target function of matrix decomposition of dual-graph sparsitySDNMFThe expression is as follows:
Figure BDA0001840062380000081
wherein Q isiA coefficient diagonal matrix representing the ith layer;
(6c) using Lagrangian function LSDNMFTo HiAnd ZiRespectively calculating partial derivatives, and obtaining a low-dimensional expression matrix H of the ith layer by using a Karush-Kuhn-Tucker conditioniIs updated to formula and HiCoefficient matrix Z of corresponding ith layeriThe expressions of the updated formula are respectively:
low dimensional representation matrix H of the ith layeriThe expression of the update formula of (2) is:
Figure BDA0001840062380000082
coefficient matrix Z of ith layeriThe expression of the update formula of (2) is:
Figure BDA0001840062380000083
step 7) representing the matrix H for the low dimensioniCoefficient matrix ZiSum coefficient diagonal matrix QiUpdating:
using the low-dimensional representation matrix H of the ith layeriIs updated to HiUpdating by using coefficient matrix Z of i-th layeriIs updated to the formula pair ZiUpdating by using coefficient diagonal matrix Q of ith layeriUpdate formula pair QiAnd (6) updating.
Step 8) judging whether the iteration times T reach the maximum value T, if so, updating H after T timesiAnd (4) as an input matrix of the i +1 th layer of the multilayer perceptron, and executing the step (9), otherwise, making t equal to t +1, and executing the step (7).
And 9) judging whether the current layer number i reaches the maximum value nl, if so, executing the step (10), otherwise, making i equal to i +1, and executing the step (2).
Step 10) reacquires the low-dimensional representation matrix HiIs updated to formula and HiCorresponding coefficient matrix ZiThe updating formula is realized by the following steps:
(10a) object function formula O for defining depth matrix decomposition of dual-graph sparsenessDSDNMFThe expression is as follows:
Figure BDA0001840062380000091
wherein | · | purple sweetFRepresenting Frobenius norm, | | · |. luminance2,1Represents L2,1Norm, Tr (·) denotes a matrix tracing operation, X denotes a matrix of raw image data, ZiCoefficient matrix representing the i-th layer, HiA low-dimensional representation matrix, W, representing the ith layeri HA data spatial similarity matrix, W, representing the ith layeri ZA feature spatial similarity matrix representing the ith layer,
Figure BDA0001840062380000092
a data spatial similarity diagonal matrix representing the ith layer,
Figure BDA0001840062380000093
representing the feature spatial similarity diagonal matrix of the ith layer, T representing the transposeOperation of alphaiDual map parameter, β, representing the ith layeriA thinning parameter indicating an i-th layer, i ═ 1,2, …, nl;
objective function formula O decomposed by the above-mentioned depth matrix for bipicture sparsenessDSDNMFIt can be seen that the three items on the right side of the equal sign are combined by single-layer structure double-graph sparse matrix decomposition and a multilayer sensor to form a new depth frame adopting manifold information and sparse constraint, the mapping relation between new low-dimensional data representation and original high-dimensional data can be fully explained by utilizing a multilayer network, samples in a data set are clustered according to different characteristics, and the image clustering accuracy is effectively improved.
(10b) Solving a Lagrangian function L according to a target function of the depth matrix decomposition of the dual-image sparsityDSDNMFThe expression is as follows:
Figure BDA0001840062380000094
wherein Q isiA coefficient diagonal matrix representing the ith layer;
(10c) using Lagrangian function LDSDNMFTo HiAnd ZiRespectively calculating partial derivatives, and re-acquiring the low-dimensional expression matrix H of the ith layer by using Karush-Kuhn-Tucker conditionsiIs updated to formula and HiCoefficient matrix Z of corresponding ith layeriThe expressions of the updated formula are respectively:
low dimensional representation matrix H of the ith layeriThe expression of the update formula of (2) is:
Figure BDA0001840062380000095
coefficient matrix Z of ith layeriThe expression of the update formula of (2) is:
Figure BDA0001840062380000101
wherein phii＝Z1Z2...Zi，Φ0＝1。
Step 11) resetting the iteration number t:
let the number of iterations T be 1 and the maximum value be T.
Step 12) resetting the current layer number i:
let the current number of layers i equal to 1 and the maximum value nl.
Step 13) for the low-dimensional representation matrix HiCoefficient matrix ZiSum coefficient diagonal matrix QiAnd (4) updating again:
using the low-dimensional representation matrix H of the ith layer in the step (10)iIs updated to HiUpdating again by using the coefficient matrix Z of the ith layer in the step (10)iIs updated to the formula pair ZiUpdating again by using the coefficient diagonal matrix Q of the ith layer in the step (5b)iUpdate formula pair QiThe update is performed again.
Step 14) judging whether the current layer number i reaches the maximum value nl, if so, executing step (15), otherwise, making i equal to i +1, and executing step (13).
(15) And (4) judging whether the iteration time T reaches the maximum value T, if so, executing the step (16), otherwise, making T equal to T +1, and executing the step (12).
(16) Clustering and outputting the images:
utilizing k-means clustering algorithm to represent matrix H in low dimension of last layer of multilayer perceptronnlAnd clustering to obtain a clustering image and outputting the clustering image.
The technical effects of the present invention will be described in further detail below with reference to simulation experiments.
1. Simulation conditions and contents:
the hardware test platform adopted by the simulation experiment of the invention is as follows: the processor is an Inter Core i5, the main frequency is 2.50GHz, and the memory is 8 GB; the software platform is as follows: and (4) carrying out simulation test on a 64-bit operating system of the Windows 7 flagship edition and Matlab R2017 a.
Simulation 1, the image clustering method simulation is carried out on the image data in the embodiment, and the simulation comparison is carried out on the clustering accuracy of the graph regularized non-negative matrix factorization (GNMF) and the non-parameter automatic weighted multi-graph regularized non-negative matrix factorization (MGNMF) under different selected types in the prior art, the average is taken as the clustering accuracy after 10 times of independent operation in the experiment, and the image clustering accuracy simulation comparison graph in the prior art refers to fig. 2.
And 2, performing image clustering method simulation on the image data in the embodiment, and visualizing the image clustering results of the GNMF and the MGNMF by using a t-SNE method in the prior art, wherein an effect diagram of visualizing the image clustering results by using the t-SNE method in the prior art refers to FIG. 3.
2. And (3) simulation result analysis:
fig. 2 is a graph showing the comparison between the clustering accuracy of the images of the present invention and the prior art, and shows the comparison between the clustering accuracy of the GNMF and MGNMF of the two prior arts and the clustering accuracy of the images of the present invention clustered on the face image data orlrows. The abscissa in fig. 2 represents the number k of selection categories, and the ordinate represents the clustering accuracy (%). In fig. 2, the curve marked with a plus sign represents the result of the GNMF method simulation experiment, the curve marked with a circle represents the result of the MGNMF method simulation experiment, and the curve marked with a solid line represents the result of the simulation experiment of the present invention.
As can be seen from fig. 2, for the face image data orlrows, the accuracy of the present invention is higher than that of other algorithms under different selection categories. From the view of graph analysis, the accuracy curve of the invention is always above other algorithm curves, and the clustering accuracy is highest.
Fig. 3 is an effect diagram of visualizing image clustering results by using a t-SNE method in the present invention and the prior art, where fig. 3(a) is a clustering effect diagram on an orlrows face image when performing k-means clustering by using GNMF, fig. 3(b) is a clustering effect diagram on an orlrows face image when performing k-means clustering by using MGNMF, and fig. 3(c) is a clustering effect diagram on an orlrows face image when using the present invention. Fig. 3(a), 3(b) and 3(c) all have 100 dots representing 100 sample images in the orlrows data set, respectively, with different colors of the dots representing different categories in the data set.
As can be seen from FIG. 3, the visual effect graph obtained by the method has clearer classification and higher clustering accuracy.
The simulation experiments show that the method has good dimensionality reduction clustering effect, lays a foundation for efficient image clustering, and is a reasonable and effective image clustering method based on the depth matrix decomposition of the double-image sparsity.

Claims (3)

1. An image clustering method based on depth matrix decomposition of double-image sparsity is characterized by comprising the following steps:
(1) constructing a multilayer perceptron:
constructing a multilayer perceptron, wherein the number of network layers is nl, nl is more than or equal to 3, the initial value of the current layer number i is 1, and the input matrix of the 1 st layer is an original image data matrix;
(2) and (3) setting iteration times t:
setting an initial value of the iteration times T as 1 and a maximum value as T;
(3) calculating a data spatial similarity matrix Wi HAnd a feature spatial similarity matrix Wi Z：
Calculating Euclidean distance between data in the ith data space according to the input matrix of the ith layer of the multilayer perceptron
Figure FDA0002822892570000011
And Euclidean distance between features in feature space of i-th layer
Figure FDA0002822892570000012
And according to
Figure FDA0002822892570000013
And
Figure FDA0002822892570000014
respectively calculating the data space similarity matrix W of the ith layeri HAnd the feature spatial similarity matrix W of the ith layeri Z；
(4) Computing data spatial faciesSimilarity diagonal matrix
Figure FDA0002822892570000015
And feature space similarity diagonal matrix
Figure FDA0002822892570000016
Data spatial similarity matrix W to the i-th layeri HAnd the feature spatial similarity matrix W of the ith layeri ZDiagonalizing respectively to obtain the diagonal matrix of the data space similarity of the ith layer
Figure FDA0002822892570000017
And feature spatial similarity diagonal matrix of ith layer
Figure FDA0002822892570000018
(5) Defining a low-dimensional representation matrix HiAnd HiCorresponding coefficient matrix ZiAnd calculating ZiCorresponding coefficient diagonal matrix Qi：
(5a) Defining a low-dimensional representation matrix H for the ith layeriIs a random matrix of kXn, HiCoefficient matrix Z of corresponding ith layeriThe random matrix is a d multiplied by k, wherein k represents the number of the image classes in the input matrix, k is more than or equal to 2, n represents the number of samples of the input matrix, n is more than or equal to 2, d represents the characteristic number of the input matrix, and d is more than or equal to 2;
(5b) using ZiCoefficient diagonal matrix Q defining the ith layeriAnd calculating Z according to the updated formulaiCorresponding QiWherein Q isiRow j and column j ofjjThe calculation formula of (2) is as follows:
Figure FDA0002822892570000019
wherein the content of the first and second substances,
Figure FDA00028228925700000110
represents ZiA vector of the jth row of elements of (1);
(6) obtaining a low-dimensional representation matrix HiIs updated to formula and HiCorresponding coefficient matrix ZiThe update formula of (2):
defining an objective function formula of matrix decomposition of bipicture sparsity, and using the objective function formula to derive a low-dimensional expression matrix H of the ith layeriIs updated to formula and HiCoefficient matrix Z of corresponding ith layeriThe updating formula is realized by the following steps:
(6.1) objective function formula O of matrix decomposition defining bipartite sparsenessSDNMFThe expression is as follows:
Figure FDA0002822892570000021
wherein | · | purple sweetFRepresenting Frobenius norm, | | · |. luminance2,1Represents L2,1Norm, Tr (·) denotes the matrix tracing operation, XiAn input matrix representing the i-th layer, T a transpose operation, αiDual map parameter, β, representing the ith layeriA sparse parameter representing an ith layer;
(6.2) solving the Lagrangian function L according to the target function of matrix decomposition of the dual-graph sparsitySDNMFThe expression is as follows:
Figure FDA0002822892570000022
(6.3) Using Lagrangian function LSDNMFTo HiAnd ZiRespectively calculating partial derivatives, and obtaining a low-dimensional expression matrix H of the ith layer by using a Karush-Kuhn-Tucker conditioniIs updated to formula and HiCoefficient matrix Z of corresponding ith layeriThe expressions of the updated formula are respectively:
low dimensional representation matrix H of the ith layeriThe expression of the update formula of (2) is:
Figure FDA0002822892570000023
coefficient matrix Z of ith layeriThe expression of the update formula of (2) is:
Figure FDA0002822892570000024
(7) for low dimensional representation matrix HiCoefficient matrix ZiSum coefficient diagonal matrix QiUpdating:
using the low-dimensional representation matrix H of the ith layer in step (6.3)iIs updated to HiUpdating by using the coefficient matrix Z of the i-th layer in the step (6.3)iIs updated to the formula pair ZiUpdating by using coefficient diagonal matrix Q of ith layer in step (5b)iUpdate formula pair QiUpdating is carried out;
(8) judging whether the iteration time T reaches the maximum value T, if so, updating H after T timesiThe input matrix is used as an input matrix of the i +1 th layer of the multilayer perceptron, and step (9) is executed, otherwise, t is made to be t +1, and step (7) is executed;
(9) judging whether the current layer number i reaches the maximum value nl, if so, executing the step (10), otherwise, making i equal to i +1, and executing the step (2);
(10) reacquiring the low-dimensional representation matrix HiIs updated to formula and HiCorresponding coefficient matrix ZiThe update formula of (2):
defining an objective function formula of the depth matrix decomposition of the bipicture sparsity, and utilizing the objective function formula to deduce a low-dimensional expression matrix H of the ith layeriIs updated to formula and HiCoefficient matrix Z of corresponding ith layeriThe updating formula is realized by the following steps:
(10.1) Objective function formula O of depth matrix decomposition defining bipicture sparsenessDSDNMFThe expression is as follows:
Figure FDA0002822892570000031
wherein X represents a matrix of raw image data;
(10.2) solving the Lagrangian function L according to the target function of the depth matrix decomposition of the bipicture sparsityDSDNMFThe expression is as follows:
Figure FDA0002822892570000032
(10.3) Using Lagrangian function LDSDNMFTo HiAnd ZiRespectively calculating partial derivatives, and re-acquiring the low-dimensional expression matrix H of the ith layer by using Karush-Kuhn-Tucker conditionsiIs updated to formula and HiCoefficient matrix Z of corresponding ith layeriThe expressions of the updated formula are respectively:
low dimensional representation matrix H of the ith layeriThe expression of the update formula of (2) is:
Figure FDA0002822892570000033
coefficient matrix Z of ith layeriThe expression of the update formula of (2) is:
Figure FDA0002822892570000034
wherein phii＝Z1Z2...Zi，Φ0＝1；
(11) Resetting the iteration number t:
making the iteration number T equal to 1 and the maximum value T;
(12) resetting the current layer number i:
making the current layer number i equal to 1, and the maximum value n l;
(13) for low dimensional representation matrix HiCoefficient matrix ZiSum coefficient diagonal matrix QiUpdate again：
Using the low-dimensional representation matrix H of the ith layer in the step (10)iIs updated to HiUpdating again by using the coefficient matrix Z of the ith layer in the step (10)iIs updated to the formula pair ZiUpdating again by using the coefficient diagonal matrix Q of the ith layer in the step (5b)iUpdate formula pair QiUpdating again;
(14) judging whether the current layer number i reaches the maximum value nl, if so, executing the step (15), otherwise, making i equal to i +1, and executing the step (13);
(15) judging whether the iteration time T reaches the maximum value T, if so, executing step (16), otherwise, making T equal to T +1, and executing step (12);
(16) clustering and outputting the images:
utilizing k-means clustering algorithm to represent matrix H in low dimension of last layer of multilayer perceptronnlAnd clustering to obtain a clustering image and outputting the clustering image.
2. The method for clustering images based on the depth matrix decomposition of the bipartite sparsity graph according to claim 1, wherein the Euclidean distance between data in the i-th data space is calculated in step (3)
Figure FDA0002822892570000041
And Euclidean distance between features in feature space of i-th layer
Figure FDA0002822892570000042
And calculating a data spatial similarity matrix W of the ith layeri HAnd the feature spatial similarity matrix W of the ith layeri ZThe calculation formulas are respectively as follows:
euclidean distance between data in ith data space
Figure FDA0002822892570000043
The calculation formula of (2) is as follows:
Figure FDA0002822892570000044
euclidean distance between features in i-th layer feature space
Figure FDA0002822892570000045
The calculation formula of (2) is as follows:
Figure FDA0002822892570000046
data spatial similarity matrix W of i-th layeri HThe calculation formula of (2) is as follows:
Figure FDA0002822892570000047
characteristic spatial similarity matrix W of ith layeri ZThe calculation formula of (2) is as follows:
Figure FDA0002822892570000051
wherein the content of the first and second substances,
Figure FDA0002822892570000052
denotes an evolution operation, XiInput matrix representing the i-th layer, Hadamard matrix product operation, 1d×nAll 1 matrices, 1, representing dXnn×dAll 1 matrixes of n multiplied by d are expressed, d represents the characteristic number of the input matrix, d is more than or equal to 2, n represents the sample number of the input matrix, n is more than or equal to 2, T represents transposition operation, exp (-) represents exponential operation, and sigma represents Gaussian scale parameters.
3. The method for clustering images based on the depth matrix decomposition of the bipartite sparsity graph according to claim 1, wherein the step (4) is performed by calculating the diagonal matrix of spatial similarity of data at the ith layer
Figure FDA0002822892570000053
And feature spatial similarity diagonal matrix of ith layer
Figure FDA0002822892570000054
The calculation formulas are respectively as follows:
data spatial similarity diagonal matrix of ith layer
Figure FDA0002822892570000055
The calculation formula of (2) is as follows:
Figure FDA0002822892570000056
feature spatial similarity diagonal matrix of ith layer
Figure FDA0002822892570000057
The calculation formula of (2) is as follows:
Figure FDA0002822892570000058
where diag (-) denotes the diagonal matrix generation operation, Σ denotes the superposition operation,
Figure FDA0002822892570000059
represents the u-th column of the data space similarity matrix of the ith layer, n represents the number of samples of the input matrix, n ≧ 2,
Figure FDA00028228925700000510
and d represents the characteristic number of the input matrix, and d is more than or equal to 2.
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