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Person re-identification via semi-supervised adaptive graph embedding

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Abstract

Video surveillance is an indispensable part of the smart city for public safety and security. Person Re-Identification (Re-ID), as one of elementary learning tasks for video surveillance, is to track and identify a given pedestrian in a multi-camera scene. In general, most existing methods has firstly adopted a CNN based detector to obtain the cropped pedestrian image, it then aims to learn a specific distance metric for retrieval. However, unlabeled gallery images are generally overlooked and not utilized in the training. On the other hands, Manifold Embedding (ME) has well been applied to Person Re-ID as it is good to characterize the geometry of database associated with the query data. However, ME has its limitation to be scalable to large-scale data due to the huge computational complexity for graph construction and ranking. To handle this problem, we in this paper propose a novel scalable manifold embedding approach for Person Re-ID task. The new method is to incorporate both graph weight construction and manifold regularized term in the same framework. The graph we developed is discriminative and doubly-stochastic so that the side information has been considered so that it can enhance the clustering performances. The doubly-stochastic property can also guarantee the graph is highly robust and less sensitive to the parameters. Meriting from such a graph, we then incorporate the graph construction, the subspace learning method in the unified loss term. Therefore, the subspace results can be utilized into the graph construction, and the updated graph can in turn incorporate discriminative information for graph embedding. Extensive simulations is conducted based on three benchmark Person Re-ID datasets and the results verify that the proposed method can achieve better ranking performance compared with other state-of-the-art graph-based methods.

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Acknowledgements

This work is partially supported by National Natural Science Foundation of China (61971121, 72004174), partially supported by the Science and Technology Research Project of Chongqing Education Commission under Grant KJQN202000612.

Author information

Authors and Affiliations

  1. Shanghai University of Engineering Science, Shanghai, China

    Jiao Liu

  2. City University of Hong Kong, Hong Kong, Hong Kong

    Jiao Liu, Mingquan Lin, Mingbo Zhao & Choujun Zhan

  3. Nanfang College Guangzhou, Guangzhou, China

    Mingbo Zhao & Choujun Zhan

  4. Wuhan University of Technology, Wuhan, China

    Bing Li

  5. Hong Kong Metropolitan University, Hong Kong, Hong Kong

    John Kwok Tai Chui

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  1. Jiao Liu
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  2. Mingquan Lin
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  3. Mingbo Zhao
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Correspondence to Mingbo Zhao or Choujun Zhan.

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Appendix

Appendix

We in this A derive (16) from (15). We first rewrite (16) as follows:

$$ \left\{\begin{aligned} n =& 1{W_{t}^{0}}{1^{T}} + \left| T \right|t + n\mu {1^{T}} + n{\text{1}}{\mu^{T}}\\ \left| T \right| =& 1{W_{t}^{0}}T{1^{T}} + 1{T^{2}}{1^{T}}t + 1{\mu^{T}}\left| T \right| + n\mu T{1^{T}}\\ {1^{T}} =& {W_{t}^{0}}{1^{T}} + t T{1^{T}} + n{\mu^{T}} + {1^{T}}\mu {1^{T}} \end{aligned} \right. $$
(18)

Noted that in (14), \(t = {{\left ({ - Tr\left ({{W_{t}^{0}}T} \right ) - 2\mu T{1^{T}}} \right )} \mathord {\left / {\vphantom {{\left ({ - Tr\left ({{W_{t}^{0}}T} \right ) - 2\mu T{1^{T}}} \right )} {\left | T \right |}}} \right . \kern -\nulldelimiterspace } {\left | T \right |}}\). By replacing it in (15), we have

$$ \left\{\begin{aligned} n =& 1{W_{t}^{0}}{1^{T}} - Tr\left( {{W_{t}^{0}}T} \right) - 2\mu T{1^{T}} + 2n\mu {1^{T}}\\ \left| T \right| =& 1{W_{t}^{0}}T{1^{T}} - \frac{{Tr\left( {{W_{t}^{0}}T} \right)\left( {1{T^{2}}{1^{T}}} \right)}}{{\left| T \right|}} \\ &- \frac{{2\mu T{1^{T}}\left( {1{T^{2}}{1^{T}}} \right)}}{{\left| T \right|}}+ 1{\mu^{T}}\left| T \right| + n\mu T{1^{T}}\\ {1^{T}} =& {W_{t}^{0}}{1^{T}} - \frac{{Tr\left( {{W_{t}^{0}}T} \right)T{1^{T}}}}{{\left| T \right|}} - \frac{{2T{1^{T}}\mu T{1^{T}}}}{{\left| T \right|}}\\ &+ n{\mu^{T}} + {1^{T}}\mu {1^{T}} \end{aligned} \right. $$
(19)

Then, we some math derivation, we have:

$$ \left\{ {\begin{aligned} \mu {1^{T}} =& \frac{{n - 1{W_{t}^{0}}{1^{T}} + Tr\left( {{W_{t}^{0}}T} \right) }}{{2n}} + \frac{{\mu T{1^{T}}}}{n}\\ \mu T{1^{T}} =& \frac{{n\left| T \right| - 2n\left( {1{W_{t}^{0}}T{1^{T}}} \right) + \left| T \right|\left( {1{W_{t}^{0}}{1^{T}}} \right) }}{{2{n^{2}} + 2\left| T \right| - {{4n1{T^{2}}{1^{T}}} \mathord{\left/ {\vphantom {{4n1{T^{2}}{1^{T}}} {\left| T \right|}}} \right. \kern-\nulldelimiterspace} {\left| T \right|}}}}\\ &+ \frac{{ \left( {\frac{{2n1{T^{2}}{1^{T}}}}{{\left| T \right|}} - \left| T \right|} \right)Tr\left( {{W_{t}^{0}}T} \right)}}{{2{n^{2}} + 2\left| T \right| - {{4n1{T^{2}}{1^{T}}} \mathord{\left/ {\vphantom {{4n1{T^{2}}{1^{T}}} {\left| T \right|}}} \right. \kern-\nulldelimiterspace} {\left| T \right|}}}}\\ {\mu^{T}} =& \frac{I}{n}\left( {{1^{T}} - {W_{t}^{0}}{{\text{1}}^{T}}{\text{ + }}\frac{{Tr\left( {{W_{t}^{0}}T} \right)}}{{\left| T \right|}}T{1^{T}}} \right) \\ & + \frac{{2T{1^{T}}\mu T{1^{T}}}}{{\left| T \right|n}} - \frac{{{1^{T}}\mu {1^{T}}}}{n} \end{aligned}} \right. $$
(20)

By multiply 1T into both sides of the first and second equations, and replacing the derivation results 1Tμ1T and 1TμT1T into the third equation, we have:

$$ \left\{ {\begin{aligned} {1^{T}}\mu {1^{T}} =& \frac{{{1^{T}}1}}{{2n}}\left( {{1^{T}} - {W_{t}^{0}}{{\text{1}}^{T}}{\text{ + }}\frac{{Tr\left( {{W_{t}^{0}}T} \right)}}{{\left| T \right|}}T{1^{T}}} \right) \\ &+ \frac{{{1^{T}}\mu T{1^{T}}}}{n}\\ {1^{T}}\mu T{1^{T}} =& \frac{{{1^{T}}1\left( {2nT - \left| T \right|I} \right)}}{{2{n^{2}} + 2\left| T \right| - {{4n1{T^{2}}{1^{T}}} \mathord{\left/ {\vphantom {{4n1{T^{2}}{1^{T}}} {\left| T \right|}}} \right. \kern-\nulldelimiterspace} {\left| T \right|}}}}\\ &\left( {1 - 1{W_{t}^{0}}{\text{ + 1}}T\frac{{Tr\left( {{W_{t}^{0}}T} \right)}}{{\left| T \right|}}} \right)\\ \mu =&\left( {1 - 1{W_{t}^{0}}{\text{ + }}1T\frac{{Tr\left( {{W_{t}^{0}}T} \right)}}{{\left| T \right|}}} \right) \left( {\frac{I}{n} - \frac{{{1^{T}}1}}{{2{n^{2}}}}} \right)\\ &+ 1T\mu 1\frac{{\text{2}}}{{n\left| T \right|}}\left( {T - \frac{{\left| T \right|}}{{2n}}I} \right) \end{aligned}} \right. $$
(21)

Finally, we prove:

$$ \begin{aligned} \mu =& \left( {1 - {\text{1}}{W_{t}^{0}}{\text{ + 1}}T\frac{{Tr\left( {{W_{t}^{0}}T} \right)}}{{\left| T \right|}}} \right)\\ &\left( {\frac{I}{n} - \frac{{{1^{T}}1}}{{2{n^{2}}}} - \frac{{\left( {T - \frac{{\left| T \right|}}{{2n}}I} \right){1^{T}}1\left( {T - \frac{{\left| T \right|}}{{2n}}I} \right)}}{{{{n1{T^{2}}{1^{T}} - {n^{2}}\left| T \right|} \mathord{\left/ {\vphantom {{n1{T^{2}}{1^{T}} - {n^{2}}\left| T \right|} 2}} \right. \kern-\nulldelimiterspace} 2} - {{{{\left| T \right|}^{2}}} \mathord{\left/ {\vphantom {{{{\left| T \right|}^{2}}} 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right) \end{aligned} $$
(22)

where the final equations hold as in (23) and (24):

$$ \begin{aligned} &{1^{T}}\mu {1^{T}}\\ &= {1^{T}}\frac{{n - 1{W_{t}^{0}}{1^{T}} + Tr\left( {{W_{t}^{0}}} \right) }}{{2n}} + \frac{{{1^{T}}\mu T{1^{T}}}}{n}\\ &= \frac{{{1^{T}}{{11}^{T}}}}{{2n}} - \frac{{{1^{T}}1}}{{2n}}{W_{t}^{0}}{1^{T}} + \frac{{{1^{T}}1T{1^{T}}}}{{2n\left| T \right|}}Tr\left( {{W_{t}^{0}}} \right) + \frac{{{1^{T}}\mu T{1^{T}}}}{n}\\ &= \frac{{{1^{T}}1}}{{2n}}\left( {{1^{T}} - {W_{t}^{0}}{1^{T}} + \frac{{Tr\left( {{W_{t}^{0}}} \right)T{1^{T}}}}{{\left| T \right|}}} \right) + \frac{{{1^{T}}\mu T{1^{T}}}}{n} \end{aligned} $$
(23)
$$ \begin{aligned} &{1^{T}}\mu T{1^{T}}\\ &= \frac{{{1^{T}}\left( {2n\left| T \right| - 2n\left( {1{W_{t}^{0}}T{1^{T}}} \right) + \frac{{2n1{T^{2}}{1^{T}}}}{{\left| T \right|}}Tr\left( {{W_{t}^{0}}T} \right)} \right)}}{{2{n^{2}} + 2\left| T \right| - {{4n1{T^{2}}{1^{T}}} \mathord{\left/ {\vphantom {{4n1{T^{2}}{1^{T}}} {\left| T \right|}}} \right. \kern-\nulldelimiterspace} {\left| T \right|}}}}\\ &\qquad{\text{ - }} \frac{{{1^{T}}\left( {n\left| T \right| - \left| T \right|\left( {1{W_{t}^{0}}{1^{T}}} \right) + \left| T \right|Tr\left( {{W_{t}^{0}}T} \right)} \right)}}{{2{n^{2}} + 2\left| T \right| - {{4n1{T^{2}}{1^{T}}} \mathord{\left/ {\vphantom {{4n1{T^{2}}{1^{T}}} {\left| T \right|}}} \right. \kern-\nulldelimiterspace} {\left| T \right|}}}}\\ &= \frac{{2n\left( {{1^{T}}1T{1^{T}} - {1^{T}}1T{W_{t}^{0}}{1^{T}} + {1^{T}}1T\frac{{Tr\left( {{W_{t}^{0}}T} \right)T{1^{T}}}}{{\left| T \right|}}} \right)}}{{2{n^{2}} + 2\left| T \right| - {{4n1{T^{2}}{1^{T}}} \mathord{\left/ {\vphantom {{4n1{T^{2}}{1^{T}}} {\left| T \right|}}} \right. \kern-\nulldelimiterspace} {\left| T \right|}}}}\\ &\qquad{\text{ - }}\frac{{\left| T \right|\left( {{1^{T}}{{11}^{T}} - {1^{T}}1{W_{t}^{0}}{1^{T}} + {1^{T}}1\frac{{Tr\left( {{W_{t}^{0}}T} \right)T{1^{T}}}}{{\left| T \right|}}} \right)}}{{2{n^{2}} + 2\left| T \right| - {{4n1{T^{2}}{1^{T}}} \mathord{\left/ {\vphantom {{4n1{T^{2}}{1^{T}}} {\left| T \right|}}} \right. \kern-\nulldelimiterspace} {\left| T \right|}}}} \end{aligned} $$
$$ \begin{aligned} &=\frac{{2n{1^{T}}1T\left( {{1^{T}} - {W_{t}^{0}}{{\text{1}}^{T}}{\text{ + }}\frac{{Tr\left( {{W_{t}^{0}}T} \right)}}{{\left| T \right|}}T{1^{T}}} \right)}}{{2{n^{2}} + 2\left| T \right| - {{4n1{T^{2}}{1^{T}}} \mathord{\left/ {\vphantom {{4n1{T^{2}}{1^{T}}} {\left| T \right|}}} \right. \kern-\nulldelimiterspace} {\left| T \right|}}}}\\ &\qquad{\text{ - }}\frac{{{1^{T}}1\left| T \right|\left( {{1^{T}} - {W_{t}^{0}}{{\text{1}}^{T}}{\text{ + }}\frac{{Tr\left( {{W_{t}^{0}}T} \right)}}{{\left| T \right|}}T{1^{T}}} \right)}}{{2{n^{2}} + 2\left| T \right| - {{4n1{T^{2}}{1^{T}}} \mathord{\left/ {\vphantom {{4n1{T^{2}}{1^{T}}} {\left| T \right|}}} \right. \kern-\nulldelimiterspace} {\left| T \right|}}}}\\ &= \frac{{2n{1^{T}}1\left( {T - \frac{{\left| T \right|}}{{2n}}I} \right)\left( {{1^{T}} - {W_{t}^{0}}{{\text{1}}^{T}}{\text{ + }}\frac{{Tr\left( {{W_{t}^{0}}T} \right)}}{{\left| T \right|}}T{1^{T}}} \right)}}{{2{n^{2}} + 2\left| T \right| - {{4n1{T^{2}}{1^{T}}} \mathord{\left/ {\vphantom {{4n1{T^{2}}{1^{T}}} {\left| T \right|}}} \right. \kern-\nulldelimiterspace} {\left| T \right|}}}} \end{aligned} $$
(24)

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Liu, J., Lin, M., Zhao, M. et al. Person re-identification via semi-supervised adaptive graph embedding. Appl Intell 53, 2656–2672 (2023). https://doi.org/10.1007/s10489-022-03570-9

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