\Z_2^\times \ar[ddr,"\textup{I}"']\ar[r,"\textup{II}"] \ar[dr,"\textup{I}"description] &[5 pt]\{\pm 1\}\times \langle 9\rangle \ar[r,"\textup{V}"]\ar[dr,"\textup{IV}"description]\ar[ddr,"\textup{IV}"description] & [10 pt] \{\pm 1\}\times \langle 17\rangle \ar[r,"\textup{V}"]\ar[dr,"\textup{IV}"description]\ar[ddr,"\textup{IV}"description] &[5 pt]\cdots \ar[r,"\textup{V}"]\ar[dr,"\textup{IV}"description]\ar[ddr,"\textup{IV}"description]& [5 pt]\{\pm 1\}\times \langle 2^{k}+1\rangle  \ar[r,"\textup{V}"]\ar[dr,"\textup{IV}"description]\ar[ddr,"\textup{IV}"description]&[10 pt] \{\pm 1\}\times \langle 2^{k+1}+1\rangle  \ar[r,"\textup{V}"]\ar[dr,"\textup{IV}"description]\ar[ddr,"\textup{IV}"description]& \cdots\rar &\{\pm 1\}\ar[ddr,"\textup{VII}"] & [-10pt]\\
& \langle3\rangle \ar[dr,"\textup{III}"description] & \langle7\rangle \ar[dr,"\textup{VI}"description]& \cdots\ar[dr,"\textup{VI}"description]& \langle 2^{k-1}-1\rangle \ar[dr,"\textup{VI}"description]& \langle 2^{k}-1\rangle \ar[dr,"\textup{VI}"description]&\cdots\ar[drr] && \\
&  \langle5\rangle \ar[r,"\textup{III}"'] &  \langle9\rangle\ar[r,"\textup{VI}"'] & \cdots\ar[r,"\textup{VI}"'] &  \langle 2^{k-1}+1\rangle \ar[r,"\textup{VI}"'] & \langle 2^{k}+1\rangle \ar[r,"\textup{VI}"'] & \cdots\ar[rr] &&\{1\}.
\end{tikzcd}
\end{equation*}

We start with the known Mackey functor $\underline{\Pic}(\KR)$ in Case VII and use the descent spectral sequence in <Ref>. Set $G=\langle 2^k+1\rangle$ or $\langle 2^k-1\rangle$ for $k\ge 2$. Recall that $\{\pm1\}\times \langle2^k-1\rangle= \{\pm1\}\times \langle2^k+1\rangle$ as subgroups of $\Z_2^\times$, we have

\begin{equation*}
\left(KO^\wedge_2\right)^{h\langle2^k-1\rangle} \simeq \left(KU^\wedge_2\right)^{h(\{\pm1\}\times \langle2^k-1\rangle)}\simeq \left(KU^\wedge_2\right)^{h(\{\pm1\}\times \langle2^k+1\rangle)}\simeq \left(KO^\wedge_2\right)^{h\langle2^k+1\rangle}.
\end{equation*}

Notice that $\cd_2(G)=\cd_2(\Z_2)=1$. This implies the descent spectral sequences

\begin{align*}
H_c^s\left(G; \pi_t\pic_{K(1)}\left(KO^\wedge_2\right)\right)&\Longrightarrow \pi_{t-s}\pic_{K(1)}\left(\left(KO^\wedge_2\right)^{hG}\right)\cong\pi_{t-s}\pic_{K(1)}\left(E_1^{h(\{\pm 1\}\times G)}\right)\\
H_c^s\left(G;\pi_t\pic_{K(1)}\left(KU^\wedge_2\right)\right)&\Longrightarrow \pi_{t-s}\pic_{K(1)}\left(\left(KU^\wedge_2\right)^{hG}\right)\cong\pi_{t-s}\pic_{K(1)}\left(E_1^{hG}\right)
\end{align*}

collapse on the $E_2$-pages,  yielding extension problems:

\begin{equation*}
\begin{tikzcd}[row sep =0]
0\rar &H^1_c\left(G;\pi_0\left(KO^\wedge_2\right)^\times\right)\rar& \Pic_{K(1)}\left[\left(KO^\wedge_2\right)^{hG}\right]\rar& \left[\Pic_{K(1)}\left(KO^\wedge_2\right)\right]^{G}\rar&0,\\
0\rar &H^1_c\left(G;\pi_0\left(KU^\wedge_2\right)^\times\right)\rar& \Pic_{K(1)}\left[\left(KU^\wedge_2\right)^{hG}\right]\rar& \left[\Pic_{K(1)}\left(KU^\wedge_2\right)\right]^{G}\rar&0.
\end{tikzcd}
\end{equation*}

The group $G$ acts trivially on $\pi_0$ and $\pi_1$ of the Picard spectra $\pic_{K(1)}\left(KO^\wedge_2\right)$ and $\pic_{K(1)}\left(KU^\wedge_2\right)$. It follows that the $H^1_c$ in the short exact sequences above is isomorphic to $\hom_c$, and every element in the Picard groups is fixed. As the group $G\cong \Z_2$ is a $2$-completed pro-cyclic group and $\pi_0\left(KO^\wedge_2\right)^\times\cong \pi_0\left(KU^\wedge_2\right)^\times\cong \Z_2^\times$ is also $2$-completed, we have isomorphisms:

\begin{align*}
H^1_c\left(G;\pi_0\left(KO^\wedge_2\right)^\times\right)&\cong \hom_c\left(G,\pi_0\left(KO^\wedge_2\right)^\times\right)\cong \Z_2^\times,\\ H^1_c\left(G;\pi_0\left(KU^\wedge_2\right)^\times\right)&\cong\hom_c\left(G,\pi_0\left(KU^\wedge_2\right)^\times\right)\cong \Z_2^\times .
\end{align*}

In the end, the extension problems are:

\begin{equation*}
\begin{tikzcd}[row sep =0]
0\rar &\Z_2^\times \rar& \Pic_{K(1)}\left[\left(KO^\wedge_2\right)^{hG}\right]\rar& \Z/8\rar&0,\\
0\rar &\Z_2^\times\rar& \Pic_{K(1)}\left[\left(KU^\wedge_2\right)^{hG}\right]\rar& \Z/2\rar&0.
\end{tikzcd}
\end{equation*}

To solve them, our starting point is the base case computed in <cit.>:

\begin{equation*}
\Pic_{K(1)}\left(E_1^{h\Z_2^\times}\right)\cong \Z_2\langle1,\sigma\rangle/(4(\sigma-1))\oplus \Z/2\cong \Z_2\oplus\Z/4 \oplus\Z/2.
\end{equation*}

We will bootstrap from there using restriction maps in group cohomology. Fix $G=\langle3 \rangle$ or $\langle5\rangle$. To compute $\Pic_{K(1)}\left(E_1^{hG}\right)$, consider the following diagram between extensions:

\begin{equation}\label{eqn:caseI_proof}
\begin{tikzcd}[column sep =small]
0\rar &\hom_c\left(G,\pi_0\left(KO^\wedge_2\right)^\x\right)\rar\dar[symbol=\cong]& \Pic_{K(1)}\left[\left(KO^\wedge_2\right)^{hG}\right]\rar\dar[symbol=\cong]& \left[\Pic_{K(1)}\left(KO^\wedge_2\right)\right]^{G}\rar\dar[symbol=\cong]&0\\ [-2.5 ex]
0\rar &\Z_2^\times \rar["{g\mapsto (1+\sigma,0)}","{-1\mapsto (0,1)}"']\vResAr[1]& \Z_2\langle1,\sigma\rangle/(4(\sigma-1))\oplus \Z/2\rar\vResAr[\res]& \Z/8\rar\vResAr[1] &0\\ [2.5 ex]
0\rar & \Z_2^\times \rar["i"]\dar[symbol=\cong] \vTrAr[(-)^2] & {\color{red}??}
\rar\dar[symbol=\cong] \vTrAr[\tr ]
& \Z/2 \rar\dar[symbol=\cong] \vTrAr[0]&0\\ [-2.5 ex]
0\rar &\hom_c\left(G,\pi_0\left(KU^\wedge_2\right)^\x\right)\rar& \Pic_{K(1)}\left[\left(KU^\wedge_2\right)^{hG}\right]\rar& \left[\Pic_{K(1)}\left(KU^\wedge_2\right)\right]^{G}\rar&0
\end{tikzcd}
\end{equation}

The left column in this diagram is obtained by applying the functor $\hom_c(G,(-)^\times)$ to the Tambara functor:

\begin{equation*}
\begin{tikzcd}[column sep=tiny]
\pi_0(KO^\wedge_2)\rar[symbol=\cong]&\Z_2\vResAr[1]\\
\pi_0(KU^\wedge_2)\rar[symbol=\cong]& \Z_2.\vTrAr[2]
\end{tikzcd}
\end{equation*}

The right column is obtained by applying the $G$-fixed point functor to the Mackey functor $\underline{\Pic}(\KR)$ in Case VII. An $\ext$-group computation shows $\Pic_{K(1)}\left(E_1^{hG}\right)$ is either $\Z_2\oplus \Z/2$ or $\Z_2\oplus \Z/2\oplus \Z/2$. Consider the image of the generator $g\in\Z_2^\times$ under the map $i\colon \Z_2^\times \to \Pic_{K(1)}\left[\left(KU^\wedge_2\right)^{hG}\right]$. For the left square in (<ref>) to commute under restriction maps, the projection $i(g)$ onto the free summand must be a multiple of $2$. This forces an isomorphism:

\begin{equation*}
\Pic_{K(1)}\left(E_1^{hG}\right)\cong \Z_2\oplus\Z/2.
\end{equation*}

The restriction and transfer maps can then be computed via diagram chase. This finishes the computation of Case I.  All other cases can be proved in similar ways.

* $G=\langle 3\rangle$ or $\langle 5\rangle\le \Z_2^\x$.

\begin{equation*}
\begin{tikzcd}[column sep =small]
0\rar &\hom_c\left(G,\pi_0\left(KO^\wedge_2\right)^\x\right)\rar\dar[symbol=\cong]& \Pic_{K(1)}\left[\left(KO^\wedge_2\right)^{hG}\right]\rar\dar[symbol=\cong]& \left[\Pic_{K(1)}\left(KO^\wedge_2\right)\right]^{G}\rar\dar[symbol=\cong]&0\\ [-2.5 ex]
0\rar &\Z_2^\times \rar["{g\mapsto (1+\sigma,0)}","{-1\mapsto (0,1)}"']\vResAr[1]& \Z_2\langle1,\sigma\rangle/(4(\sigma-1))\oplus \Z/2\rar\vResAr[\substack{(1,0),(\sigma,0)\mapsto (1,0)\\(0,1)\mapsto (0,1)}]& \Z/8\rar\vResAr[1] &0\\ [2.5 ex]
0\rar & \Z_2^\times \rar["{g\mapsto (2,0)}","{-1\mapsto (0,1)}"'] \dar[symbol=\cong] \vTrAr[(-)^2] & \Z_2\oplus \Z/2\rar\dar[symbol=\cong] \vTrAr[\substack{(1,0)\mapsto (1+\sigma,0) \\(0,1)\mapsto(0,0)} ]& \Z/2 \rar\dar[symbol=\cong] \vTrAr[0]&0\\ [-2.5 ex]
0\rar &\hom_c\left(G,\pi_0\left(KU^\wedge_2\right)^\x\right)\rar& \Pic_{K(1)}\left[\left(KU^\wedge_2\right)^{hG}\right]\rar& \left[\Pic_{K(1)}\left(KU^\wedge_2\right)\right]^{G}\rar&0
\end{tikzcd}
\end{equation*}

* $\{\pm 1\}\times(1+8\Z_2)\le \Z_2^\x$.

\begin{equation*}
\begin{tikzcd}[column sep =small]
0\rar &\hom_c\left(1+4\Z_2,\pi_0\left(KO^\wedge_2\right)^\x\right)\rar\dar[symbol=\cong]& \Pic_{K(1)}\left[\left(KO^\wedge_2\right)^{h(1+4\Z_2)}\right]\rar\dar[symbol=\cong]& \left[\Pic_{K(1)}\left(KO^\wedge_2\right)\right]^{1+4\Z_2}\rar\dar[symbol=\cong]&0\\ [-2.5 ex]
0\rar &\Z_2^\times \rar["{g\mapsto (1+\sigma,0)}","{-1\mapsto (0,1)}"']\vResAr[(-)^2]& \Z_2\langle1,\sigma\rangle/(4(\sigma-1))\oplus \Z/2\rar\vResAr[\substack{(1,0)\mapsto (1,1,0)\\ (\sigma,0)\mapsto (1,-1,0)\\ (0,1)\mapsto (0,0,0)}]& \Z/8\rar\vResAr[1] &0\\ [2.5 ex]
0\rar & \Z_2^\times \rar["{g\mapsto (1,0,0)}", "{-1\mapsto (0,0,1)}"'] \dar[symbol=\cong] \vTrAr[1] & \Z_2\oplus \Z/8\oplus \Z/2\rar\dar[symbol=\cong] \vTrAr[\substack{(1,0,0)\mapsto(1+\sigma,0)\\ (0,1,0)\mapsto (1-\sigma, 0)\\ (0,0,1)\mapsto (0,0,1)}]& \Z/8 \rar\dar[symbol=\cong] \vTrAr[2]&0\\ [-2.5 ex]
0\rar &\hom_c\left(1+8\Z_2,\pi_0\left(KO^\wedge_2\right)^\x\right)\rar& \Pic_{K(1)}\left[\left(KO^\wedge_2\right)^{h(1+8\Z_2)}\right]\rar& \left[\Pic_{K(1)}\left(KO^\wedge_2\right)\right]^{1+8\Z_2}\rar&0
\end{tikzcd}
\end{equation*}

* $\langle 9\rangle\le G=\langle 3\rangle$ or $\langle 5\rangle$.

\begin{equation*}
\begin{tikzcd}[column sep =small]
0\rar &\hom_c\left(G,\pi_0\left(KU^\wedge_2\right)^\x\right)\rar\dar[symbol=\cong]& \Pic_{K(1)}\left[\left(KU^\wedge_2\right)^{hG}\right]\rar\dar[symbol=\cong]& \left[\Pic_{K(1)}\left(KU^\wedge_2\right)\right]^{G}\rar\dar[symbol=\cong]&0\\ [-2.5 ex]
0\rar &\Z_2^\times \rar["{g\mapsto (2,0)}","{-1\mapsto (0,1)}"']\vResAr[(-)^2]& \Z_2\oplus \Z/2\rar\vResAr[\substack{(1,0)\mapsto (1,0,1)\\ (0,1)\mapsto (0,0,0)}]& \Z/2\rar\vResAr[1] &0\\ [2.5 ex]
0\rar & \Z_2^\times \rar["{g\mapsto (1,0,0)}","{-1\mapsto(0,1,0)}"'] \dar[symbol=\cong] \vTrAr[1] & \Z_2\oplus \Z/2\oplus\Z/2\rar\dar[symbol=\cong] \vTrAr[\substack{(1,0,0) \mapsto (2,0)\\ (0,1,0)\mapsto(0,1)\\ (0,0,1)\mapsto (0,0)}]& \Z/2 \rar\dar[symbol=\cong] \vTrAr[0]&0\\ [-2.5 ex]
0\rar &\hom_c\left(1+8\Z_2,\pi_0\left(KU^\wedge_2\right)^\x\right)\rar& \Pic_{K(1)}\left[\left(KU^\wedge_2\right)^{h(1+8\Z_2)}\right]\rar& \left[\Pic_{K(1)}\left(KU^\wedge_2\right)\right]^{1+8\Z_2}\rar&0
\end{tikzcd}
\end{equation*}

* $G=\langle 2^k-1\rangle$ or $\langle 2^k+1\rangle\le \{\pm 1\}\times (1+2^k\Z_2)$, where $k\ge 3$.

\begin{equation*}
\begin{tikzcd}[column sep =small, ampersand replacement = \&]
0\rar \&\hom_c\left(G,\pi_0\left(KO^\wedge_2\right)^\x\right)\rar\dar[symbol=\cong]\& \Pic_{K(1)}\left[\left(KO^\wedge_2\right)^{hG}\right]\rar\dar[symbol=\cong]\& \left[\Pic_{K(1)}\left(KO^\wedge_2\right)\right]^{G}\rar\dar[symbol=\cong]\&0\\ [-2.5 ex]
0\rar \&\Z_2^\times \rar\vResAr[1]\& \Z_2\oplus \Z/2\oplus \Z/8\rar\vResAr[\tiny{\begin{pmatrix}
\end{pmatrix}}]\& \Z/8\rar\vResAr[1] \&0\\ [2.5 ex]
0\rar \& \Z_2^\times \rar["{g\mapsto (1,0,0)}","{-1\mapsto(0,1,0)}"'] \dar[symbol=\cong] \vTrAr[(-)^2] \& \Z_2\oplus \Z/2\oplus \Z/2\rar\dar[symbol=\cong] \vTrAr[\tiny{\begin{pmatrix}
\end{pmatrix}}]\& \Z/2 \rar\dar[symbol=\cong] \vTrAr[0]\&0\\ [-2.5 ex]
0\rar \&\hom_c\left(G,\pi_0\left(KU^\wedge_2\right)^\x\right)\rar\& \Pic_{K(1)}\left[\left(KU^\wedge_2\right)^{hG}\right]\rar\& \left[\Pic_{K(1)}\left(KU^\wedge_2\right)\right]^{G}\rar\&0
\end{tikzcd}
\end{equation*}

* $\{\pm 1\}\times(1+2^{k+1}\Z_2)\le \{\pm 1\}\times(1+2^k\Z_2)$, where $k\ge 3$.

\begin{equation*}
\begin{tikzcd}[column sep =small,ampersand replacement=\&]
0\rar \&\hom_c\left(1+2^k\Z_2,\pi_0\left(KO^\wedge_2\right)^\x\right)\rar\dar[symbol=\cong]\& \Pic_{K(1)}\left[\left(KO^\wedge_2\right)^{h(1+2^k\Z_2)}\right]\rar\dar[symbol=\cong]\& \left[\Pic_{K(1)}\left(KO^\wedge_2\right)\right]^{1+2^k\Z_2}\rar\dar[symbol=\cong]\&0\\ [-2.5 ex]
0\rar \&\Z_2^\times \rar\vResAr[(-)^2]\& \Z_2\oplus \Z/2\oplus\Z/8\rar\vResAr[\tiny{\begin{pmatrix}
\end{pmatrix}}]\& \Z/8\rar\vResAr[1] \&0\\ [2.5 ex]
0\rar \& \Z_2^\times \rar \dar[symbol=\cong] \vTrAr[1] \& \Z_2\oplus \Z/2\oplus \Z/8\rar\dar[symbol=\cong] \vTrAr[\tiny{\begin{pmatrix}
\end{pmatrix}}]\& \Z/8 \rar\dar[symbol=\cong] \vTrAr[2]\&0\\ [-2.5 ex]
0\rar \&\hom_c\left(1+2^{k+1}\Z_2,\pi_0\left(KO^\wedge_2\right)^\x\right)\rar\& \Pic_{K(1)}\left[\left(KO^\wedge_2\right)^{h(1+2^{k+1}\Z_2)}\right]\rar\& \left[\Pic_{K(1)}\left(KO^\wedge_2\right)\right]^{1+2^{k+1}\Z_2}\rar\&0
\end{tikzcd}
\end{equation*}

* $ \langle 2^{k+1}+1 \rangle \le G= \langle 2^k+1 \rangle$ or $ \langle 2^k-1 \rangle $, where $k\ge 3$.

\begin{equation*}
\begin{tikzcd}[column sep =small,ampersand replacement=\&]
0\rar \&\hom_c\left(G,\pi_0\left(KU^\wedge_2\right)^\x\right)\rar\dar[symbol=\cong]\& \Pic_{K(1)}\left[\left(KU^\wedge_2\right)^{hG}\right]\rar\dar[symbol=\cong]\& \left[\Pic_{K(1)}\left(KU^\wedge_2\right)\right]^{G}\rar\dar[symbol=\cong]\&0\\ [-2.5 ex]
0\rar \&\Z_2^\times \rar\vResAr[(-)^2]\& \Z_2\oplus \Z/2\oplus\Z/2\rar\vResAr[\tiny{\begin{pmatrix}
\end{pmatrix}}]\& \Z/2\rar\vResAr[1] \&0\\ [2.5 ex]
0\rar \& \Z_2^\times \rar \dar[symbol=\cong] \vTrAr[1] \& \Z_2\oplus \Z/2\oplus \Z/2\rar\dar[symbol=\cong] \vTrAr[\tiny{\begin{pmatrix}
\end{pmatrix}}]\& \Z/2 \rar\dar[symbol=\cong] \vTrAr[0]\&0\\ [-2.5 ex]
0\rar \&\hom_c\left(1+2^{k+1}\Z_2,\pi_0\left(KU^\wedge_2\right)^\x\right)\rar\& \Pic_{K(1)}\left[\left(KU^\wedge_2\right)^{h(1+2^{k+1}\Z_2)}\right]\rar\& \left[\Pic_{K(1)}\left(KU^\wedge_2\right)\right]^{1+2^{k+1}\Z_2}\rar\&0
\end{tikzcd}
\end{equation*}

When $p=2$, the group $\Pic_{K(1)}\cong \mathrm{RO}(C_2)^\wedge_2/(4(1-\sigma))\oplus\Z/2$ is not topologically cyclic. The element $S^1_{K(1)}$ corresponds to $(1,0)$ under the identification. From the formulas in <Ref>, we observe that:

\begin{equation*}
\left(\Sigma E_1^{hG} \in \Pic_{K(1)}\left(E_1^{hG}\right)\right) =\begin{cases}
(1,0)\in \mathrm{RO}(C_2)^\wedge_2/(4(1-\sigma))\oplus \Z/2, & G=\Z_2^\times;\\
(1,0)\in \Z_2\oplus \Z/2, & G=\langle 3\rangle \text{ or }\langle 5\rangle;\\
(2^{k-3},1,0)\in \Z_2\oplus\Z/8\oplus \Z/2, &  G=\{\pm 1\}\times \langle 2^k+1\rangle, k\ge 3;\\
(2^{k-3},1,0)\in \Z_2\oplus\Z/2\oplus \Z/2, & G=\langle 2^k-1\rangle \text{ or }\langle 2^k+1\rangle, k\ge 3;\\
1\in \Z/8, &G=\{\pm 1\};\\
1\in \Z/2, & G=\{1\}.
\end{cases}
\end{equation*}

As a result, the element $\Sigma E_1^{hG}$ is not divisible by $2$ (or its powers) in the Picard group $\Pic_{K(1)}\left(E_1^{hG}\right)$ at prime $2$ for any closed subgroup $G$.  This is very different from the odd prime case in  <Ref>.

From the computations in <Ref>, we verify <Ref> in an example below:

The colimits of $\Pic_{K(1)}\left(E_1^{h-}\right)$ as pro-abelian groups under transfinite compositions of restriction maps are:

\begin{align*}
\colim_{\res} \Pic_{K(1)}\left[\left(KO^\wedge_2\right)^{h(1+2^k\Z_2)}\right]&\cong \Z/8\cong \Pic_{K(1)}\left(KO^\wedge_2\right),\\
\colim_{\res} \Pic_{K(1)}\left[\left(KU^\wedge_2\right)^{h(1+2^k\Z_2)}\right]&\cong \Z/2\cong \Pic_{K(1)}\left(KU^\wedge_2\right).
\end{align*}

By <Ref>,  the restriction maps

\begin{align*}
\Pic_{K(1)}\left[\left(KO^\wedge_2\right)^{h(1+2^k\Z_2)}\right]\longrightarrow \Pic_{K(1)}\left(KO^\wedge_2\right)&&\Pic_{K(1)}\left[\left(KU^\wedge_2\right)^{h(1+2^k\Z_2)}\right]\longrightarrow \Pic_{K(1)}\left(KU^\wedge_2\right)
\end{align*}

are projections onto summands when $k\ge 3$. The claim now follows by reading off the restriction maps in Cases II, V and Cases I, III, respectively.

Similar to <Ref>, the claim above fails when we take colimits in $\Ab$. There will be an extra $\Q_2$ summand in the colimit then. This gives another example of <Ref>.

One important class in $\Pic_{K(1)}\left(S^{0}_{K(1)}\right)$ at $p=2$ is the exotic element $\Ecal_{K(1)}$.  Under the isomorphism $\Pic_{K(1)}\left(S^{0}_{K(1)}\right)\cong \mathrm{RO}(C_2)^\wedge_2/(4(1-\sigma))\oplus \Z/2$, the exotic element corresponds to $2-2\sigma$. Our computation above implies:

The exotic element in $\Pic_{K(1)}\left(S^{0}_{K(1)}\right)$ is detected by a closed subgroup $G\le \Z_2^\times$ iff $\{\pm 1\}\le G$. Equivalently, $E_1^{hG}\hsmash \Ecal_{K(1)}\simeq E_1^{hG}$ iff $-1\notin G$.

We will compute the images of $\Ecal_{K(1)}$ under the restriction maps in the $K(1)$-local Picard Mackey functor. By <Ref>, the index $2$ subgroups of $\Z_2^\times$ are $\{\pm 1\}\times 1+8\Z_2$, $\langle 3\rangle$, and $\langle5\rangle$. Reading off the restriction maps in <Ref>, we can see:

* Case I  implies  images of $\Ecal_{K(1)}$ in $\Pic_{K(1)}\left(E_1^{h\langle3\rangle}\right)\cong \Pic_{K(1)}\left(E_1^{h\langle5\rangle}\right)\cong \Z_2\oplus \Z/2$ are both zero.

* Case II implies its image n $\Pic_{K(1)}\left(E_1^{h(\{\pm 1\}\times 1+8\Z_2)}\right)\cong \Z_2\oplus\Z/8\oplus \Z/2$ is $(0,4,0)$.

* Subgroups of $\Z_2^\times$ not contained in $\langle 3\rangle$, and $\langle5\rangle$ are of the form $\{\pm 1\}\times 1+2^k\Z_2$. Case V states that the restriction map  $\Pic_{K(1)}\left(E_1^{h(\{\pm 1\}\times 1+2^k\Z_2)}\right)\to \Pic_{K(1)}\left(E_1^{h(\{\pm 1\}\times 1+2^{k+1}\Z_2)}\right)$ restricts to identity on the $\Z/8$-summands. Hence images of $\Ecal_{K(1)}$ in them are all no trivial.

* It is well-known that $KO^\wedge_2\hsmash \Ecal_{K(1)}\simeq \Sigma^4 KO^\wedge_2$ is nontrivial. This can also be obtained from the transfinite restriction computation in <Ref>.

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journaltitleCompos. Math.
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Institut des Hautes Études Scientifiques

journaltitlePublications Mathématiques de l'IHÉS
titleÉléments de géométrie algébrique : III. Étude cohomologique des faisceaux cohérents, Première partie

noteTo appear in Algebr. Geom. Topol.
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Amer. Math. Soc., Providence, RI

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titleOn Hopkins' Picard group ${\rm Pic}_2$ at the prime 3

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titleUne résolution projective pour le second groupe de Morava pour $p \ge 5$ et applications
Morava stabilizer group ; formal group law deformation ; profinite group cohomology ; Picard group ; Groupe stabilisateur de Morava ; déformations des lois de groupe formel ; cohomologie des groupes profinis ; groupe de Picard

titleDerived Algebraic Geometry III: Commutative Algebra

Princeton University Press, Princeton, NJ

isbn978-0-691-14049-0; 0-691-14049-9
seriesAnnals of Mathematics Studies
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titleHigher algebra
.ias.edu/ lurie/papers/HA.pdf

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.ias.edu/ lurie/papers/Elliptic-II.pdf

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Princeton University Press, Princeton, NJ

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Cambridge Univ. Press, Cambridge

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# Multi-target tracking for video surveillance using deep affinity network: A
brief review

###### Abstract

Deep learning models are known to function like the human brain. Due to their
functional mechanism, they are frequently utilized to accomplish tasks that
require human intelligence. Multi-target tracking (MTT) for video surveillance
is one of the important and challenging tasks, which has attracted the
researchers’ attention due to its potential applications in various domains.
Multi-target tracking tasks require locating the objects individually in each
frame, which remains a huge challenge as there are immediate changes in
appearances and extreme occlusions of objects. In addition to that, the Multi-
target tracking framework requires multiple tasks to perform i.e. target
detection, estimating trajectory, associations between frame, and re-
identification. Various methods have been suggested, and some assumptions are
made to constrain the problem in the context of a particular problem. In this
paper, the state-of-the-art MTT models, which leverage from deep learning
representational power are reviewed.

Index Terms—  Multi-Target Tracking, Deep Learning, Conventional Neural
Networks, Video Tracking.

##  1 Introduction:

Multi-target tracking has found applications in diverse domains including
surveillance systems [1, 2], anomaly detection [3, 4], self-driving vehicles
[5, 6], sports [7, 8], robotics [9], segmentation [10][11], behaviour analysis
[12][13], virtual reality [14], and pose estimation [15][16] etc. Although
Multi-target tracking has been part of research studies for so many decades
now, it’s not fully optimized yet. The key issues that lead to complicated MTT
tasks are frequent occlusions, path estimation, initialization and termination
of a track, objects having similar appearances, and interaction between them.
The multi-target tracking is divided into two main tasks i.e. object detection
and tracking. MTT algorithms, to differentiate intra-group objects, associate
a unique id to each detected object that remains specific to the object for a
certain time. These Ids are then utilized for generating motion trajectories
for the tracked objects. Detection is usually provided by a pre-trained
detector and affinity model provides the estimation between the detections and
an approach; optimizing association. The precision of the object detecting
system determines how effective a target tracking system is. The precision of
the MTT models is highly influenced by factors like scale changes, frequent id
switches, rotation, illumination variation, etc. Figure 1 illustrates the
output of the MTT algorithm. Furthermore, there are the issues of background
clutter, post shifting, track initialization and termination are complex tasks
in multi-target tracking systems. To overcome these issues, researchers have
taken advantage of deep neural networks, have proposed a variety of
strategies.

Fig. 1: Illustrates the MTT algorithm’s basic workflow

Since deep neural networks (DNNs) have the capability to retrieve abstract and
complex features from input, by learning representations, they have been
utilized in state-of-the-art methodologies to improve performance. In addition
to extracting features, deep neural networks based architectures are designed
for motion detection [17] aiding to resolve the object affiliation task in
tracking problems. This study will briefly review state-of-the-art models that
are proposed.

The organization of this analysis will be structured as,section 2 briefly
review the previous work that has been done in terms of surveys or reviews,
the MOT algorithms are briefly over viewed in section 3, followed by
characterization of algorithms. Discussion of deep learning in multi-goal
tracking is done in section 4. Section 5 covers the evaluation aspects metrics
for assessment, datasets and discussion of benchmarks results, followed by
conclusion.

##  2 Related Work:

In past studies, the problem of multi-target tracking has been formulated
differently and from multiple perspectives. Several surveys and literature
studies have been undertaken on those studies. Some of those reviews are
dedicated to specific fields, such as crowd analysis [18][19][16], intelligent
visual surveillance[20][21],behaviour analysis[22][23] etc. Some are dedicated
to techniques e.g appearance models in visual tracking, Xi Li et al. in
[24][25] reviewed recent developments in 2D appearance models for visual
object tracking. Alreshidi [26] proposed hybrid features for facial emotion
recognition but it could be used for multi object tracking as shown by Jiarui
et al. [27]. Nikolajs at el. [28] identified the recent trends of Multi target
tracking and determined the necessary computer vision building blocks for
application in road traffic scenarios, including surveillance and planning.
Ullah at el. [25] evaluated different hand-crafted and learned features for
the appearance model of the pedestrians. Wenhan Luo at el. [29] investigated
different important components involved in a MOT system as well as uncovered
existing MOT research difficulties to present the first comprehensive review,
deep learning-based strategies, on the other hand, were only briefly discussed
because DNN is used by just a few algorithms. Patrick at el. [30] reviewed
learning based approaches however, they scoped their review around frameworks
tackling data association in MTT tasks. Ullah at el. [31][32] proposed a
Bayesian framework and used hand-crafted and sparse coded deep features for
modeling the appearance of the targets. The framework shows promising results
when detectors gives accurate detection. Yingkun Xu at el. [33] evaluated
existing deep learning-based MOT approaches and discussed the performance
improvement in deep learning methods. They also analyzed the mechanism of deep
feature transferring, and rules to define a new tracking framework. Similarly,
following the batch approach for multi-target tracking, Ullah and Faouzi
[34][35] proposed a deep learning based transportation network where the edge
cost of the graph model through deep features and dynamic programming is used
to optimize the trajectories of the individual objects. Gioele at el. [36]
provided the first extensive study of Deep Learning’s usage in Multiple Object
Tracking, concentrating on 2D data retrieved from single-camera films. This
survey also gathered experimental findings from the widely used MOT datasets
and provided the numerical comparison between them in addition to making
recommendations for possible future research paths.

##  3 Multi-target tracking: An overview:

The challenge of estimating the number of targets while maintaining their
states or trajectories is defined as multi-target tracking (MTT). Various
researchers have approached this topic in various ways over the years, and
they have proposed algorithms accordingly. MTT models can be classified as
following:

###  3.1 Classification of MTT algorithms:

Depending on how objects are initialized, MOT implementations can be
classified as Detection based (DBT) or Detection free tracking (DFT). However,
MTT models are standardized around Detection based training, where detections
(identification of objects in a frame) are retrieved as a pre-tracking step.
Because an object-detector is required to identify the objects in DBT,
performance is largely dependent on the detector’s quality, hence choosing a
detection framework is crucial. Some of the DFT and DBF algorithms are
reviewed in the following section:

####  3.1.1 Detection free tracking (DFT):

The output of the detector is often utilized as input to the tracker, and the
tracker’s output is provided to a motion forecasting algorithm that
anticipates where objects will move in the following several seconds. However,
that’s not the case in detection free tracking. DFT based models require that
a fixed number of objects must be manually initialized in the first frame and
these objects must then be localized in following frames. DFT is a difficult
task since there is limited information about the object to be tracked, and
that information is unclear. As a result, the initial bounding box merely
approximates the object of interest from the background, and the object’s
appearance may change drastically over time [37].

Luo at el. [38] proposed a completely convolutional technique to develop a
one-stage detector that uses spatio-temporal information acquired by a 3D
sensor to conduct simultaneous 3D detection, tracking, and prediction. The
model was said to function effectively in complex settings, tracking tiny and
large moving vehicles accurately. Better feature representations were
envisioned as a component of future effort. Katerina at el. [39] proposed a
method for video segmentation based on spectral clustering of trajectories,
with the goal of segmenting and tracking closely interacting and defoaming
agents in a video. This method first computes a video figure-ground
segmentation, then assigns repulsive forces between foreground trajectories
belonging to distinct connected components in any frame. Lu Zhang at el. [37]
developed a model-free tracker that tracks many objects at the same time by
merging multiple single-object trackers with constraints on the objects’
spatial organization. When spatial limitations between objects are taken into
account, the performance improves. They also suggested that integrating
structural limitations in DBT can help them perform better. Ming Yang at el.
[40] from a game-theoretic standpoint, gave a new viewpoint to MTT. They
presented a decentralized MTT model in which a collection of independent
kernel-based trackers are deployed, each of which contributes towards bridging
the joint motion estimation and the Nash Equilibrium of a game. In both
synthetic and actual video sequence, the proposed method achieved promising
results. They include the idea of the incorporation of motion dynamic models
in the trackers as a future work. Detection based tracking (DBT) Object
detection is used in DBT-based techniques to produce classification hypotheses
for each object per frame, which are subsequently linked into trajectories.
For these types of training models, the object detector is trained in advance,
making the performance of such models highly dependent on the performance of
the employed object detector. In addition to that, it makes DBT focus on the
specific kind of targets such as pedestrians, vehicles, etc. Bastian Leibe at
el. [41] suggested a model for dynamic scene interpretation from a moving
vehicle that incorporates multi-view point and multi-category object detection
for fully automatic scene geometry estimate, 2D object identification, 3D
localization, trajectory estimation, and tracking. This study also proposed a
novel tracking approach that uses a moving camera to locate and track a large
number of objects. B. Bose at el. [42] proposed a framework for the spatial
proximity of objects, ideal tracks are often fragmented into components, or
are merged with other tracks into common components. In this proposed method,
the goal was to monitor a wide range of items in an open-world, unrestricted
environment without requiring scene-specific training. This technique unifies
fragments and merged track segments to build individual and full tracks for
each object, employing foreground blobs obtained by background subtraction
from stationary camera measurements. The tracking algorithm currently works in
batch, but can be made online. Bi Song at el. [43] studied the problem of long
term tracking & analyzes the statistical features and statistics of the
targets along a proposed long-term track for each target, and proposes a
stochastic graph evaluation framework to understand the link between tracklets
in order to generate long term inter dependencies between the targets where
the context information isn’t readily available. Michael at el. [44] proposed
a unique approach for multi-person tracking by-detection in a particle
filtering framework, employing an online trained classifier for data
association, they investigated how unreliable information sources can be used
for multi-target tracking.This method just employs 2D data; however, having
more information about the scene, the target’s appearance, and the camera
motion would be advantageous. Luo at el. [29] categorized MOT algorithms as
Online and offline tracking, based on the processing mode i.e. whether they
rely on future frames when trying to decide the object identities in a given
frame.

####  3.1.2 Online tracking:

Online tracking algorithms, also known as sequential tracking, generate
predictions about the current frame based on information from the past and the
present. This type of algorithm processes the frames in a step-wise manner. In
some applications e.g. autonomous driving and robotics navigation, this
information is prerequisite.

####  3.1.3 Batch tracking:

In order to determine the object identities in a given frame, batch tracking
(offline tracking) techniques employ information from a previous frame. They
frequently make use of global data, resulting in improved tracking quality;
but, due to computational and memory constraints, it is not always possible to
handle all of the frames at once.

##  4 Deep Learning in Multi-target Tracking

For multi-target tracking, deep learning algorithms have been shown to be
effective, but there is still potential for improvement. Deep learning is used
at various stages of the tracking process, therefore it is necessary to
analyze and summarize how deep learning functionalities are contributing
towards the performance improvement in each step. Gioele at el. [36]
identified main steps that shared by majority of algorithms are as:

###  4.1 Detection stage:

Locating targets in a sequence of frames, using bounding boxes by analyzing
input frames.

###  4.2 Motion Prediction Stage:

Analyzing detections to extract appearance,motion or interaction features.

###  4.3 Affinity Stage:

Extracted features are utilized in similarity/distance computation between
pair of detections.

###  4.4 Association Stage:

The similarity/distance measures are utilized in associating by providing the
same ID to detection that correspond to the same target.

###  4.5 Deep neural networks: Object detection Stage

Object detection algorithms that are precise and fast are crucial because they
allow computer vision applications to reach their full potential. Deep neural
networks have been contributing in overall tracking performance which also
includes specialized detection phases. Following are the examples of CNN-based
models.

Redmon at el. [45] proposed YOLO, single convolutional neural network
simultaneously predicts multiple bounding boxes and class probabilities
directly from full images in one evaluation, trained on full images and
directly optimizes detection performance, it also learns generalizable
representations of objects. However, YOLO imposes strict spatial constraints
on bounding box predictions, limiting the number of adjacent items that our
model can predict. Small items that appear in groups, such as birds, are
equally problematic for this model. Ren at el. [46] proposed faster R-CNN, a
single unified object identification network made up of fully deep CNN,
improves detection accuracy and efficiency while reducing computational
overhead. This model integrated a training method that alternates between
fine-tuning for the region proposal and fine-tuning for the region proposal,
enabling a unified, deep-learning-based object identification system to
operate at frame rates that are close to real-time, then fine-tuning for
object detection while maintaining the fixed targets. In certain surveillance
settings, the occlusion is so frequent that it is not possible to detect the
whole shape of the object like in the case of humans. To tackle this problem,
Khan et al. [47] proposed temporal consistency model that is trained to detect
only the head locations. Similarly, techniques like [48] is explored to do
tracking of only the head locations rather than the whole body shape. Bewley
at el 29 proposed framework SORT to leverage the power of CNN based detection,
in the MOT prospect and achieve best in class performance with respect to both
speed and accuracy,it focuses on frame-to-frame prediction and association. It
became capable of being rated as the best-performing by replacing the
detection obtained from Aggregated Channel Features (ACF) with the detection
computed by Faster R-CNN [46], based the architecture on Kalman filter [49]
and the Hungarian algorithm [39]. CNN’s has been used in the detection step
for purposes other than constructing bounding boxes for objects in some cases.
Min at el. [50] suggested a new strategy, for tracking multiple objects e.g.
cars, with a combination of robust detection and two classifiers,For robust
and precise identification of multiple vehicles, we propose the upgraded ViBe.
It was used by CNN to eliminate false positives when the vibe Algorithm [51]
was used to recognize the car. It efficiently restrains dynamic noise and
swiftly removes ghost shadows and object’s leftover shadows.

###  4.6 Deep neural networks: Feature Extraction and Motion Prediction

Deep models have been utilized to extract meaningful features that are later
used to associate detection outcomes, the features are learned through
classification or recognition tasks, and they are inspired by the
effectiveness of deep learning’s representational capacity. Furthermore, it
has been demonstrated e.g.Chu at el [52], Son at el [53] that when MOT
features such as temporal and spatial attention map or temporal order are
investigated, performance can be improved. Additionally, some end-to-end deep
learning based models are designed to extract the features not only for
appearance descriptors but also for motion information. Wang et al. [52]
proposed one of the earliest methods to apply DL in the MOT pipeline. This
system takes advantage of the benefits of single object trackers to tackle the
drifting difficulties caused by occlusion without compromising on the
computational power.To enhance extracted characteristics, the network used
encoders stacked in two layers, which were then used to compute affinity using
SVM. The visibility map of the target is learned and then used to infer the
spatial attention map, which is subsequently used to weight the features. In
addition, the visibility map can be used to estimate occlusion status. This is
what is known as a temporal attention process. The most commonly used CNN
based approaches can be further classified as: Classical CNNs for features
extraction and Siamese CNNs.

####  4.6.1 Classical CNN for feature extraction:

Kim et al. [54] claimed that the Multiple Hypotheses Tracking (MHT) technique
is compatible with the existing visual tracking perspective. Modern advances
in detection-based tracking and the development of efficient feature
representations for object appearance have given the MHT process new
possibilities. They improved MHT by integrating a regularized least squares
framework for online training of appearance models for each track. Wojke et
al. [55] proposed to improve SORT[56], while it achieves favorable performance
on high frame rates in terms of accuracy and precision, it yields a relatively
large number of identity shifts. Wojke et al refine it by integrating
appearance motion information, overcoming concerns by substituting the
association metric with a convolutional neural network (CNN) that has been
trained to distinguish pedestrians on a large scale person re-identification
dataset. In comparison to sort, the updated tracking system effectively
reduces the number of identity flips from 1423 to 781. This is a reduction of
about 45 percent, allowing for competitive performance while maintaining real-
time speed.

####  4.6.2 Siamese CNNS:

Siamese CNN has shown useful in MOT since the purpose of feature learning in
the tracking phase is to determine the similarity between detection and
tracks. Leal-Taixe et al. [57] implemented a two-stage strategy to match pairs
of detection methods and brings a new perspective to object association
challenges in the context of pedestrian tracking. They applied the CNN concept
to multi-person tracking in this case and proposed to learn the decision
whether two detections belong to the same trajectory,In order to avoid manual
feature design for data association.The model’s learning framework is divided
into two stages. A CNN is pre-trained in Siamese twin architecture to measure
the similarity of two equally sized image regions, and then the CNN is merged
with the collected features to produce the prediction. They address the
tracking problem well by formulating it as linear programming and combining
deep features and motion information with a gradient boosting approach.

###  4.7 Deep neural networks: Affinity computation stage:

While some implementations use deep learning models to immediately produce an
affinity score without the requirement for an explicit distance metric between
features, other approaches calculate affinity between tracklets and detections
by applying some distance measure to features obtained by a CNN. Milan et al
[58] address the difficult problem of data association and trajectory
estimation in a neural network environment. A recursive Bayesian filter,
consisting of prediction and updating from observations, can be used to
estimate the tracked object states in online MOT tasks, this model extends RNN
to model this procedure, object states, existing observations, their
corresponding matching matrix, as well as existence prospects are entered into
the network as input. The predicted states and updated results are outputted,
as well existence probabilities to decide whether the objects should be
terminated, this model obtained promising tracking results. Chen et al [59]
proposed to compute affinity between the sampled particles and the tracked
target, instead of being computed between targets and detections. Detections
that did not coincide with the tracked objects were instead used to create new
tracks and recover missing objects. Despite being an online monitoring
algorithm, it was able to get top results on MOT15 at the time of publication,
employing both public and private detections.

###  4.8 Deep neural networks: Tracking/ Association step:

Deep learning has been used in some MTT models to improve the association
step. Ma et al.[60] employed bi-directional GRU to decide where tracklets
should be terminated as they expanded the Siamese tracklet network. For each
detection, the network extracts tracklet features and sends them to a
bidirectional GRU network.The output of the bidirectional GRU network is
briefly pooled in Euclidean space to provide the overall features for
tracklets. During the tracking process, tracklets are produced and then split
into small subtracklets based on the local distance between bidirectional GRU
outputs.Finally, considering similarities between global aspects of temporal
pooling, these sub-tracklets are re-connected to long trajectories. On the
MOT16 dataset, this approach obtains results that are equivalent to the state
of the art. Ren et al. [61] propose a collaborative implementation in which
many deep RL agents were used to accomplish the association task. A prediction
network and a decision network were the two key components of the model. Using
recent tracklet trajectory, the CNN was employed as a prediction network and
was trained to anticipate target movement in new frames.

###  4.9 Deep neural networks: Other uses in MTT

Few of the models are briefly defined in this section, that do not fit in
those four identified steps. Jiang et al [62] used a Deep RL agent to
accomplish bounding box regression, to the employed tracking algorithms, to
increase efficiency.VGG-16 CNN was employed to to extract appearance, those
extracted features were stored to keep and use the history of last 10
movements taken by the target, and then the integrated network predicts one of
several alternative outcomes, including bounding box motion and scaling, as
well as termination action. On the MOT15 dataset, using this bounding box
regression methodology on several state-of-the-art MOT algorithms resulted in
improvements of 2 to 7 absolute MOTA points, placing it first among public
detection approaches. Xiang et al.[63] deployed MetricNet for pedestrian
tracking, combining an affinity model with trajectory estimation obtained by
Bayesian filters. The features were extracted using a VGG-16 CNN trained for
target re-identification, extracted features and performed bounding box
regression, the motion model has two parts, one takes trajectory coordinates
as input, the other combine the extracted features and a detection box that
performs the Bayesian filtering and outputs the target’s updated position On
MOT16 and MOT15, the algorithm delivered the best and second-best scores among
online approaches, respectively. Recent advances in model free single object
tracking (SOT) algorithms have significantly pushed the use of SOT in multi-
object tracking (MOT) to improve resilience and reduce dependency on external
detectors. SOT algorithms, on the other hand, are typically designed to
distinguish a target from its surroundings, and they frequently encounter
issues when a target is spatially mixed with similar artifacts, as seen in
MOT. Chu et al. [64] proposed a model to address the issue of robustness and
eliminating dependency over the external detectors. They implemented a model
using three different CNNs in their algorithm. The PafNet is integrated to
differentiate between the background and the tracked objects. The PartNet
differentiate between tracked targets.The other integrated CNN is the
convolutional layer, which determines whether or not the tracking model should
be refreshed. Using an SVM classifier and the Hungarian technique,
unassociated detections were used to recover from target occlusion. The
algorithm was tested on the MOT15 and MOT16 datasets, with the first yielding
the best overall results and the second yielding the best results among online
approaches.

##  5 MTT Evaluation

Target tracking has been an aspect of research for many years, and all of the
proposed methods are subjected to a quantitative evaluation to determine their
efficacy, which is necessary to compute the impacts of various components and
to compare them to other approaches. There are following components for MTT
evaluations.

###  5.1 Metrics:

Set of standard metrics are available to facilitate comparisons of algorithms
in experimental setup, to evaluate the performance and indicate the drawbacks.
The most relevant are Classical metrics and CLEAR MOT metrics [65][66].
Classical metrics point out faults that the algorithm can come across for-
example, Mostly Tracked (MT) trajectories,Mostly Lost (ML) trajectories, ID
switches etc. CLEAR MOT metrics are MOTA (Multiple Object Tracking Accuracy )
and MOTP (Multiple Object Tracking Precision). MOTA combines the false
positive, false negative, and mismatch rates into a single value, resulting in
a generally good overall tracking performance. Despite several flaws and
complaints, this is by far the most extensively used assessment method. MOTP
describes how precisely objects are tracked using bounding box overlap and/or
distance measurements.

Table 1: The average fragmentation score produced by an online and offline as given by Gioele at el. [36]. Mode | MOT17 | MOT16 | MOT17
---|---|---|---
Batch | 1143.8 | 1104.9 | 3188.2
Online | 1509.5 | 1820.2 | 7555.8

###  5.2 Benchmark Datasets:

Benchmark Datasets include MOTChallenge, KITTI, UA-DETRAC. The MOTChallenge
datasets are the largest and most complete datasets for pedestrian tracking
currently available, offering more data to train deep models. MOT15 [67] is
the initial MOT Challenge dataset, and it features video with a range of
attributes that the model would need to generalize better in order to achieve
good results. MOT16 [68] and MOT19 [69] are the other modified versions.

###  5.3 Benchmark Results:

Gioele et al. [36] list the public results tested on MOT Challenge MOT15
datasets and MOT16 data-sets, documented from the corresponding publication,
to achieve a clear comparison of results between the approaches mentioned in
this work. The findings are classified into public and private detection-based
models since the quality of detection has an impact on performance. The
methods are divided into two categories: online and offline. The year of the
published referenced document, its operating mode, the MOTA, MOTP, IDF1,
Mostly Tracked (MT) and Mostly Lost (ML) metrics, expressed in percentages;
the absolute number of false positives (FP), false negatives (FN), ID switches
(IDS), and fragmentations (Frag); the algorithm’s speed expressed in frames
per second (Hz). For each measure, an arrow pointing up (↑) implies a higher
score, while an arrow pointing down (↓) suggests the reverse. The best
performance is emphasized among models that operate in the same mode
(batch/online), and each statistic is highlighted in bold. We only listed the
results obtained from models that were visited in this review in table. 2 and
3.

Table 2: Experimental results [36] of deep learning MOT algorithms on MOT15  | Year | Mode | MOTA $\uparrow$ | MOTP$\uparrow$ | IDF1$\uparrow$ | MT$\uparrow$ | ML$\downarrow$ | FP$\downarrow$ | FN$\downarrow$ | IDS$\downarrow$ | Frag$\downarrow$ | Hz$\uparrow$
---|---|---|---|---|---|---|---|---|---|---|---|---
[54] | 2015 | Online | 32.4 | 71.8 | 45.3 | 16.0 | 43.8 | 9064 | 32060 | 435 | 826 | 0.7
[58] | 2017 | Online | 19.0 | 71.0 | 17.1 | 5.5 | 45.6 | 11578 | 36706 | 1490 | 2081 | 165.2
[59] | 2017 | Online | 38.5 | 72.6 | 47.1 | 8.7 | 37.4 | 4005 | 33204 | 586 | 1263 | 6.7
[63] | 2019 | Online | 37.1 | 72.5 | 48.4 | 12.6 | 39.7 | 8305 | 29732 | 580 | 1193 | 1.0
[52] | 2019 | Online | 38.9 | 70.6 | 44.7 | 16.6 | 31.5 | 7321 | 29501 | 720 | 1440 | 0.3
[70] | 2019 | Batch | 28.1 | 74.3 | 38.7 |  |  | 6733 | 36952 | 477 | 790 | 16.9
Table 3: Experimental results [36] of deep learning MOT algorithms on MOT16  | Year | Mode | MOTA $\uparrow$ | MOTP$\uparrow$ | IDF1$\uparrow$ | MT$\uparrow$ | ML$\downarrow$ | FP$\downarrow$ | FN$\downarrow$ | IDS$\downarrow$ | Frag$\downarrow$ | Hz$\uparrow$
---|---|---|---|---|---|---|---|---|---|---|---|---
[15] | 2017 | Online | 46.0 | 74.9 | 50.0 | 14.6 | 43.6 | 6895 | 91117 | 473 | 1422 | 0.2
[61] | 2018 | Online | 47.3 | 74.6 |  | 17.4 | 39.9 | 6375 | 88543 |  |  |
[64] | 2019 | Online | 48.8 | 75.7 | 47.2 | 15.8 | 38.1 | 5875 | 86567 | 906 | 1116 | 1.0
[64] | 2019 | Online | 48.8 | 75.7 | 47.2 | 15.8 | 38.1 | 5875 | 86567 | 906 | 1116 | 0.1

In reality, the models that use deep learning and have online processing mode
produced the greatest results.This could, however, be the result of a stronger
emphasis on building online approaches, which is becoming more popular in the
MOT deep learning research community. The larger number of fragmentations is a
frequent problem among online approaches that is not reflected in the MOTA
score. When occlusions or detections are missing, online algorithms do not
look ahead, re-identify the missing targets, or interpolate the missing piece
of the trajectories in the video.

##  6 Conclusion:

The methods that use deep learning to tackle the MTT problem have been briefly
explored in this study. This study has discussed solutions that use deep
learning to address each of the four steps of the MTT issue, bringing the
total number of state-of-the-art MOT techniques to n. MOT algorithm
evaluations, including evaluation measures and benchmark results from
accessible data sets, were briefly discussed. Single object trackers have
recently benefited from the introduction of deep models into global graph
optimization algorithms, resulting in high-performing online trackers; batch
techniques, on the other hand, have benefited from the introduction of deep
models into global graph optimization algorithms.

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