CN107145827A - Cross-camera person re-identification method based on adaptive distance metric learning - Google Patents

Abstract

The invention discloses a kind of across video camera pedestrian recognition methods again learnt based on adaptive distance metric, sample pair is constituted first with pedestrian's picture from different cameras in training set, constraint is provided for learning distance metric；Then the ga s safety degree according to training sample in original feature space is different samples to adaptively distributing training weight；Then use the object function for accelerating near-end gradient algorithm to adjust the distance metric learning to be solved, obtain mahalanobis distance metric matrix；The distance matrix metric for learning to obtain finally is substituted into mahalanobis distance metric function, and calculates the mahalanobis distance between test phase pedestrian picture feature vector, similitude ranking results are obtained.The present invention has taken into full account difference of the different training samples during learning distance metric, the distance metric function that study is obtained is had stronger identification, so as to reach higher across video camera pedestrian recognition accuracy again.

Description

Cross-camera person re-identification method based on adaptive distance metric learning

technical field

The invention relates to a method in the technical field of video image processing, in particular to a cross-camera pedestrian re-identification method based on adaptive distance metric learning.

Background technique

It has broad application prospects to match and recognize designated target pedestrians in a camera surveillance network with non-overlapping lenses. This technology is called cross-camera pedestrian re-identification, which is the basis of cross-camera target tracking and behavior analysis in intelligent surveillance systems. and prerequisites. Due to its huge commercial value, cross-camera person re-identification has received extensive attention and research in recent years. The key and difficult points in cross-camera person re-identification research are the differences in lighting and viewing angles between cameras, as well as changes in the pose and occlusion of pedestrians themselves. In addition, low-resolution surveillance video makes information such as faces no longer applicable in most cases.

In order to overcome the above problems, many researchers hope to design features that are robust to cross-camera visual changes. However, due to the strong uncertainty in the differences between cameras in real-world scenes and the many variables in the poses of pedestrians, it is difficult to find a feature that is robust to all these changes and can effectively distinguish different pedestrians. Therefore, distance metric learning is introduced to the cross-camera person re-identification problem. Specifically, the distance metric learning uses labeled pedestrian sample pairs (positive sample pairs indicate that two pictures belong to the same pedestrian, and negative sample pairs indicate that two pictures belong to different pedestrians) as the training set. Optimize the distance, and learn a distance metric matrix, which can project all samples to a new feature space. In this feature space, the mutual distance between positive sample pairs is small, and the mutual distance between negative sample pairs is larger. For example, the paper "Reidentification by relative distance comparison" (cross-camera pedestrian matching based on relative distance comparison) published by Zheng Weishi et al. ) use the training samples to learn the optimal probability relative distance measure, and use this standard to measure the distance of other pictures in the database. By mapping features to a common space, distance metric learning can address the discrepancy between different cameras to a certain extent. However, existing distance metric learning algorithms usually treat all samples equally during training, without considering the differences between different samples. Since different samples have different discriminability on the original feature space, the importance for distance metric learning is substantially different.

Contents of the invention

Aiming at the deficiencies of the above-mentioned prior art, the present invention provides a cross-camera pedestrian re-identification method based on adaptive distance metric learning, which can adaptively classify and weight the samples according to the distribution of the original feature space of the training samples, so that different The samples can play different roles in the process of distance metric learning, so as to train a more discriminative distance metric function.

In order to achieve the above purpose, the present invention first uses pedestrian images from different cameras in the training set to form sample pairs to provide constraints for distance metric learning; then according to the distinguishability of training samples in the original feature space, different sample pairs are adaptively Assign training weights; then use the accelerated proximal gradient algorithm to solve the objective function of distance metric learning to obtain the Mahalanobis distance metric matrix; finally, substitute the learned distance metric matrix into the Mahalanobis distance metric function, and calculate the pedestrian image features in the test phase Mahalanobis distance between vectors to get similarity ranking results.

The inventive method is realized through the following specific steps:

A cross-camera pedestrian re-identification method based on adaptive distance metric learning, including the following steps:

Step 1: Input the training data for distance metric learning, and represent the pedestrian pictures under different cameras as query sets and the candidate set in is the feature vector of the i-th query image and the j-th candidate image, and with is the identity label of the corresponding pedestrian, n is the total number of pictures in the query set in the training phase, and m is the total number of pictures in the candidate set in the training phase;

Step 2: Select the query image x _i in the query set and the candidate image y _j in the candidate set to form a sample pair ( _xi , y _j ), and assign a binary label z _ij to the sample pair, where when When z _ij = 1, ( _xi , y _j ) is called a positive sample pair, and when When z _ij = -1, ( _xi , y _j ) is called a negative sample pair; define the Mahalanobis distance measurement function between any sample pair ( _xi , y _j ) as:

Where M is the distance metric matrix in the Mahalanobis distance;

Step 3: Use the Logistic loss function to establish a loss function for distance metric learning for each sample pair ( _xi ,y _j ) in the training set:

Where ξ is the mean Euclidean distance between all sample pairs in the training set. When ( _xi ,y _j ) is a positive sample pair, the loss function constrains the Mahalanobis distance d _M ( _xi ,y _j ) to be less than And when ( _xi ,y _j ) is a negative sample pair, the loss function constrains the Mahalanobis distance d _M ( _xi ,y _j ) to be greater than

Step 4: Considering the loss function constraints between all sample pairs in the training set at the same time, define the overall objective function of adaptive distance metric learning as:

Step 5: According to the difference of training samples, assign different training weights w _ij to the loss function ψ( _xi ,y _j ) of different sample pairs, so that the objective function of adaptive distance metric learning becomes:

Step 6: According to the objective function, the adaptive distance metric learning is defined as the following optimization problem:

Step 7: Use the accelerated proximal gradient algorithm to solve the optimization problem in step 6, and obtain the corresponding distance metric matrix M;

Step 8: In the test phase, for a given query template picture feature vector x ^p and other camera suspicious pedestrian target constituted by the feature vector group of candidate pictures N is the total number of pictures in the candidate set in the test phase, and the distance metric matrix M is substituted into the Mahalanobis distance metric function in step 2 to calculate x ^p and the feature vector of each candidate picture Mahalanobis distance between And sort the candidate pictures according to the Mahalanobis distance, so that the candidate pictures with smaller Mahalanobis distances to x ^p are at the front of the queue.

Further: according to the difference of training samples described in step 5, assign different training weights w _ij to the loss function ψ( _xi , y _j ) of different sample pairs, the specific implementation process is:

Step 5.1: For the feature vector x _i of the query image in the training set, calculate the candidate set The characteristic distances between all candidate picture feature vectors and x _i in the Euclidean space, and then in order of distance from small to large Sort the pictures in;

Step 5.2: According to the ranking results in step 5.1, the candidate set Divide into the hard set for x _i medium set and simple collections

difficult set is defined as:

In the formula, r _i (y _j ) represents the position of the candidate picture y _j in the sorting queue obtained in step 5.1, and It means that _xi is in the candidate set The correct match in y ⁺ position in the sorted queue;

medium set is defined as:

where m is the candidate set The total number of pictures of pedestrians in the middle, that is, the total length of the sorting queue in step 5.1;

simple set is defined as:

Step 5.3: According to the candidate set in step 5.2 The division result of the sample pair ( _xi ,y _j ) adaptively assigns the training weight w _ij to the loss function of the sample pair (xi ,y j ); when ( _xi ,y _j ) is a positive sample pair, the training weight of ψ( _xi ,y _j ) The weight w _ij is set to 1/N ⁺ , where N ⁺ is the total number of positive sample pairs in the training set; and when ( _xi , y _j ) is a negative sample pair, the loss function ψ( _xi , y j ) is defined by the following formula _j ) training weight w _ij :

Among them, N ^- indicates the total number of negative sample pairs whose weight is not zero in the training set, and β ₁ and β ₂ are balance parameters to adjust the range of training weight variation.

Further: use the accelerated proximal gradient algorithm described in step 7 to solve the optimization problem in step 6, and obtain the corresponding distance metric matrix M, which specifically includes the following steps:

Step 7.1: Initialize M ₀ and M _-1 as the identity matrix, and update it iteratively;

Step 7.2: Specifically, for the tth iteration, first calculate the aggregation forward matrix S _t :

where coefficient is set to 0 on the first iteration;

Step 7.3: Define the matrix in is the gradient of the overall objective function Ψ(M) of distance metric learning at S _t , and η _t is the iteration step size;

Step 7.4: Perform an eigenvalue decomposition on the matrix P _t and express it as Then update the distance metric matrix M of the t-th iteration to obtain the iteration result M _t :

in Used to ensure that the distance metric matrix M _t is positive semi-definite;

Step 7.5: Repeat steps 7.2 to 7.4 until the convergence condition is reached:

Step 7.6: Return M _t at convergence as the training result M of adaptive distance metric learning.

Compared with the prior art, the beneficial effects of the present invention are as follows:

1) The present invention performs adaptive classification and weighting on the sample pairs according to the distinguishability of the training samples in the original feature space. Compared with the existing distance metric learning technology that treats all training samples equally during the training process, the present invention Can make better use of the differences between training samples;

2) In the present invention, by setting the training weights of some sample pairs to zero, the negative samples that are easily distinguished in part of the original feature space in the training set are discarded, which not only effectively alleviates the problem of too many negative sample pairs in the pedestrian re-identification problem, but also Reduced computation during training;

3) The present invention can use the labeling information of the training samples to learn a specific Mahalanobis distance measurement function, so that the feature distances of pictures of the same pedestrian are relatively close, and the feature distances of pictures of different pedestrians are relatively far, effectively overcoming the difference between different cameras. The effect of variance on cross-camera person re-identification.

Description of drawings

FIG. 1 is a schematic diagram of the overall flow of the cross-camera pedestrian re-identification method based on adaptive distance metric learning in the present invention.

detailed description

In order to make the purpose, technical solution and advantages of the present invention clearer, the technical solution of the present invention will be further described in detail below in conjunction with the drawings and specific embodiments.

The following examples are carried out on the premise of the technical solutions of the present invention, and detailed implementation methods and specific operation processes are provided, but the protection scope of the present invention is not limited to the following examples.

Example

In this embodiment, the cross-camera pedestrian re-identification is completed by taking the pedestrian who has appeared in camera A as the target, and looking for the suspicious pedestrian object most similar to the pedestrian target in camera B which does not overlap with camera A. In an embodiment of the invention, the method includes the following steps:

Step 1: Input the training data for distance metric learning, and define the collection of pedestrian pictures that have appeared in camera A as the query set And the candidate set of pedestrian pictures that have appeared in camera B in is the feature vector of the i-th query image and the j-th candidate image, and with is the identity label of the corresponding pedestrian. n is the total number of pictures in the query set in the training phase, and m is the total number of pictures in the candidate set in the training phase;

Step 2: Select the query image x _i in the query set and the candidate image y _j in the candidate set to form a sample pair ( _xi , y _j ), and assign a binary label z _ij to the sample pair, where when When z _ij = 1, ( _xi , y _j ) is called a positive sample pair, and when When z _ij =-1, ( _xi ,y _j ) is called a negative sample pair. Define the Mahalanobis distance metric function between any sample pair ( _xi , y _j ) as:

where M is the distance metric matrix in Mahalanobis distance.

Step 3: Use the Logistic loss function to establish a loss function for distance metric learning for each sample pair ( _xi ,y _j ) in the training set:

where ξ is the mean Euclidean distance between all sample pairs in the training set. When ( _xi ,y _j ) is a positive sample pair, the loss function constrains the Mahalanobis distance d _M ( _xi ,y _j ) to be less than And when ( _xi ,y _j ) is a negative sample pair, the loss function constrains the Mahalanobis distance d _M ( _xi ,y _j ) to be greater than

Step 4: Considering the loss function constraints between all sample pairs in the training set at the same time, define the overall objective function of adaptive distance metric learning as:

Step 5: According to the difference of training samples, assign different training weights w _ij to the loss function ψ( _xi ,y _j ) of different sample pairs, so that the objective function of adaptive distance metric learning becomes:

In this embodiment, according to the difference of training samples described in step 5, different training weights w _ij are assigned to the loss function ψ( _xi ,y _j ) of different sample pairs, and the specific implementation process is as follows:

Step 5.1: For the feature vector x _i of the query image in the training set, calculate the candidate set The characteristic distances between all candidate picture feature vectors and x _i in the Euclidean space, and then in order of distance from small to large Sort the pictures in;

Step 5.2: According to the ranking results in step 5.1, the candidate set Divide into the hard set for x _i medium set and simple collections

difficult set is defined as:

In the formula, r _i (y _j ) represents the position of the candidate picture y _j in the sorting queue obtained in step 5.1, and It means that _xi is in the candidate set The correct match in y ⁺ position in the sorted queue.

medium set is defined as:

where m is the candidate set The total number of pictures of pedestrians in the middle, that is, the total length of the sorting queue in step 5.1.

simple set is defined as:

Step 5.3: According to the candidate set in step 5.2 The division result of the sample pair ( _xi , y _j ) adaptively assigns the training weight w _ij to the loss function. When ( _xi ,y _j ) is a positive sample pair, set the training weight w _ij of ψ( _xi ,y _j ) to 1/N ⁺ , where N ⁺ is the total number of positive sample pairs in the training set. And when ( _xi ,y _j ) is a negative sample pair, the training weight w _ij of its loss function ψ( _xi ,y _j ) is defined by the following formula:

Wherein, N ^- represents the total number of negative sample pairs whose weight is not zero in the training set, β ₁ and β ₂ are balance parameters for adjusting the variation range of the training weight, and in this embodiment, β ₁ =0.75 and β ₂ =0.25.

Step 6: According to the objective function, the adaptive distance metric learning is defined as the following optimization problem:

Step 7: Use the accelerated proximal gradient algorithm to solve the optimization problem in step 6, and obtain the corresponding distance metric matrix M.

In this embodiment, the accelerated proximal gradient algorithm described in step 7 is used to solve the optimization problem in step 6, and the corresponding distance metric matrix M is obtained. The specific implementation steps are as follows:

Step 7.1: Initialize M ₀ and M _-1 as the identity matrix, and update it iteratively;

Step 7.2: Specifically, for the tth iteration, first calculate the aggregation forward matrix S _t :

where coefficient is set to 0 on the first iteration;

Step 7.3: Define the Matrix in is the gradient of the objective function Ψ(M) of distance metric learning at S _t , and η _t is the iteration step size; in this embodiment, η _t is initialized to 2 ⁸ at each iteration, and checks whether the following conditions are satisfied :

Where <·,·> represents the inner product of the matrix, ||·|| _F represents the Frobenius norm of the matrix, when the conditions in the above formula are met, use η _t as the iteration step; otherwise replace η with η _t /2 _t , until the condition is met;

Step 7.4: Perform an eigenvalue decomposition on the matrix P _t and express it as Then update the distance metric matrix M of the t-th iteration to obtain the iteration result M _t :

in Used to ensure that the distance metric matrix M _t is positive semi-definite;

Step 7.5: Repeat steps 7.2 to 7.4 until the convergence condition is reached:

Step 7.6: Return M _t at convergence as the training result M of adaptive distance metric learning.

Step 8: After obtaining the distance metric matrix M, for any pedestrian target template picture that appeared in camera A, extract its feature vector x ^p , and at the same time define the set of unknown pedestrian pictures that appeared in camera B as a candidate set, Extract the feature vector of each candidate picture to form a feature vector group N is the total number of pictures in the candidate set in the test phase. Substitute the distance metric matrix M obtained in step 7 into the Mahalanobis distance metric function in step 2, and calculate x ^p and the feature vector of each candidate picture Mahalanobis distance between And sort the candidate pictures according to the Mahalanobis distance, so that the candidate pictures with smaller Mahalanobis distances to x ^p are at the front of the queue. Ultimately, the higher-ranked candidate images are more likely to be the correct match of the pedestrian object in camera B.

The above descriptions are only preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements and improvements made within the spirit and principles of the present invention should be included in the protection of the present invention. within range.

Claims (3)

1. A cross-camera pedestrian re-identification method based on adaptive distance metric learning, is characterized in that comprising the following steps: Step 1: Input the training data for distance metric learning, and represent the pedestrian pictures under different cameras as query sets and the candidate set in is the feature vector of the i-th query image and the j-th candidate image, and with is the identity label of the corresponding pedestrian, n is the total number of pictures in the query set in the training phase, and m is the total number of pictures in the candidate set in the training phase; Step 2: Select the query image x _i in the query set and the candidate image y _j in the candidate set to form a sample pair ( _xi , y _j ), and assign a binary label z _ij to the sample pair, where when When z _ij = 1, ( _xi , y _j ) is called a positive sample pair, and when When z _ij = -1, ( _xi , y _j ) is called a negative sample pair; define the Mahalanobis distance measurement function between any sample pair ( _xi , y _j ) as: <mrow> <msub> <mi>d</mi> <mi>M</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>y</mi> <mi>j</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>M</mi> <mn>2</mn> </msubsup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>y</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mi>M</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>y</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> </mrow> Where M is the distance metric matrix in the Mahalanobis distance; Step 3: Use the Logistic loss function to establish a loss function for distance metric learning for each sample pair ( _xi ,y _j ) in the training set: <mrow> <mi>&amp;psi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>d</mi> <mi>M</mi> </msub> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>j</mi> </msub> </mrow> <mo>)</mo> <mo>-</mo> <mi>&amp;xi;</mi> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> </mrow> </mrow> Where ξ is the mean Euclidean distance between all sample pairs in the training set, when ( _xi ,y _j ) is a positive sample pair, the loss function constrains the Mahalanobis distance d _M ( _xi ,y _j ) to be less than ξ; and When ( _xi ,y _j ) is a negative sample pair, the loss function constrains the Mahalanobis distance d _M ( _xi ,y _j ) to be greater than ξ; Step 4: Considering the loss function constraints between all sample pairs in the training set at the same time, define the overall objective function of adaptive distance metric learning as: <mrow> <mi>&amp;Psi;</mi> <mrow> <mo>(</mo> <mi>M</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mi>&amp;psi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> </mrow> Step 5: According to the difference of training samples, assign different training weights w _ij to the loss function ψ( _xi ,y _j ) of different sample pairs, so that the objective function of adaptive distance metric learning becomes: <mrow> <mi>&amp;Psi;</mi> <mrow> <mo>(</mo> <mi>M</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <msub> <mi>w</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mi>&amp;psi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> </mrow> Step 6: According to the objective function, the adaptive distance metric learning is defined as the following optimization problem: <mrow> <munder> <mi>min</mi> <mi>M</mi> </munder> <mi>&amp;Psi;</mi> <mrow> <mo>(</mo> <mi>M</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> <mi>M</mi> <mo>&amp;GreaterEqual;</mo> <mn>0</mn> </mrow> Step 7: Use the accelerated proximal gradient algorithm to solve the optimization problem in step 6, and obtain the corresponding distance metric matrix M; Step 8: In the test phase, for a given query template picture feature vector x ^p and other camera suspicious pedestrian target constituted by the feature vector group of candidate pictures N is the total number of pictures in the candidate set in the test phase, and the distance metric matrix M is substituted into the Mahalanobis distance metric function in step 2 to calculate x ^p and the feature vector of each candidate picture Mahalanobis distance between And sort the candidate pictures according to the Mahalanobis distance, so that the candidate pictures with smaller Mahalanobis distances to x ^p are at the front of the queue. 2. A cross-camera pedestrian re-identification method based on adaptive distance metric learning according to claim 1, characterized in that: according to the difference of training samples described in step 5, it is the loss function ψ of different sample pairs ( _xi , y _j ) assign different training weights w _ij , the specific implementation process is: Step 5.1: For the feature vector x _i of the query image in the training set, calculate the candidate set The characteristic distances between all candidate picture feature vectors and x _i in the Euclidean space, and then in order of distance from small to large Sort the pictures in; Step 5.2: According to the ranking results in step 5.1, the candidate set Divide into the hard set for x _i medium set and simple collections difficult set is defined as: In the formula, r _i (y _j ) represents the position of the candidate picture y _j in the sorting queue obtained in step 5.1, and It means that _xi is in the candidate set The correct match in y ⁺ position in the sorted queue; medium set is defined as: where m is the candidate set The total number of pictures of pedestrians in the middle, that is, the total length of the sorting queue in step 5.1; simple set is defined as: Step 5.3: According to the candidate set in step 5.2 The division result of the sample pair ( _xi ,y _j ) adaptively assigns the training weight w _ij to the loss function of the sample pair (xi ,y j ); when ( _xi ,y _j ) is a positive sample pair, the training weight of ψ( _xi ,y _j ) The weight w _ij is set to 1/N ⁺ , where N ⁺ is the total number of positive sample pairs in the training set; and when ( _xi , y _j ) is a negative sample pair, the loss function ψ( _xi , y j ) is defined by the following formula _j ) training weight w _ij : Among them, N ^- indicates the total number of negative sample pairs whose weight is not zero in the training set, and β ₁ and β ₂ are balance parameters to adjust the range of training weight variation. 3. a kind of cross-camera pedestrian re-identification method based on adaptive distance metric learning according to claim 1, is characterized in that: using the acceleration proximal gradient algorithm described in step 7 to solve the optimization problem in step 6, obtain The corresponding distance metric matrix M, which specifically includes the following steps: Step 7.1: Initialize M ₀ and M _-1 as the identity matrix, and update it iteratively; Step 7.2: Specifically, for the t-th iteration, first calculate the aggregation forward matrix s _t : <mrow> <msub> <mi>S</mi> <mi>t</mi> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mi>t</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>t</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mn>1</mn> </mrow> <msub> <mi>&amp;alpha;</mi> <mi>t</mi> </msub> </mfrac> <mrow> <mo>(</mo> <msub> <mi>M</mi> <mrow> <mi>t</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>M</mi> <mrow> <mi>t</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> 2 where coefficient is set to 0 on the first iteration; Step 7.3: Define the matrix in is the gradient of the overall objective function Ψ(M) for distance metric learning at s _t , and η _t is the iteration step size; Step 7.4: Perform an eigenvalue decomposition on the matrix P _t and express it as Then update the distance metric matrix M of the t-th iteration to obtain the iteration result M _t : <mrow> <msub> <mi>M</mi> <mi>t</mi> </msub> <mo>=</mo> <msub> <mi>U</mi> <mi>t</mi> </msub> <msubsup> <mi>&amp;Lambda;</mi> <mi>t</mi> <mo>+</mo> </msubsup> <msubsup> <mi>U</mi> <mi>t</mi> <mi>T</mi> </msubsup> </mrow> in Used to ensure that the distance metric matrix M _t is positive semi-definite; Step 7.5: Repeat steps 7.2 to 7.4 until the convergence condition is reached: <mrow> <mfrac> <mrow> <mo>|</mo> <mi>&amp;Psi;</mi> <mrow> <mo>(</mo> <msub> <mi>M</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;Psi;</mi> <mrow> <mo>(</mo> <msub> <mi>M</mi> <mrow> <mi>t</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mi>&amp;Psi;</mi> <mrow> <mo>(</mo> <msub> <mi>M</mi> <mrow> <mi>t</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>|</mo> </mrow> </mfrac> <mo>&amp;le;</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> </mrow> Step 7.6: Return M _t at convergence as the training result M of adaptive distance metric learning.