Note that we calculated symmetric warps
${\cal W}_{1,2p}$
$W1,2p$ and
${\cal W}_{1p,2}$
$W1p,2$, and corresponding distance matrixes
${\bm {\cal D}}_{1,2p}$
$D1,2p$ and
${\bm {\cal D}}_{1p,2}$
$D1p,2$ in the last step. However, the warps still have bias (
${\cal W}_{1,2p}$
$W1,2p$ biased toward
${\cal F}_1$
$F1$ while
${\cal W}_{1p,2}$
$W1p,2$ is biased toward
${\cal F}_2$
$F2$). To minimize the effect of biasing on alignment, we first combine
${\bm {\cal D}}_{1,2p}$
$D1,2p$ and
${\bm {\cal D}}_{1p,2}$
$D1p,2$ to make a new distance matrix
${\bm {\cal D}}_c$
$Dc$ as follows:
Display Formula8 \documentclass[12pt]{minimal}\begin{document}\begin{equation}
{\bm {\cal D}}_c = {\bm {\cal D}}_{1,2p} + {\bm {\cal D}}_{1p,2} .
\end{equation}\end{document}
$Dc=D1,2p+D1p,2.$ Once
${\bm {\cal D}}_c$
$Dc$ is obtained, global constraint based on
${\cal W}_{1,2p}$
$W1,2p$ and
${\cal W}_{1p,2}$
$W1p,2$ is added into this matrix. Denote
$W_c$
$Wc$ as the warps under the global constraint, which is restricted by the symmetric warps, as follows:
Display Formula9 \documentclass[12pt]{minimal}\begin{document}\begin{equation}
\min ({\cal W}_{1,2} ,{\cal W}_{2,1} ) \le {\cal W}_c \le \max ({\cal W}_{1,2} ,{\cal W}_{2,1} ).
\end{equation}\end{document}
$min(W1,2,W2,1)\u2264Wc\u2264max(W1,2,W2,1).$ Finally, warp
${\cal W}_c$
$Wc$, which satisfies the following equation, is chosen as the final warp.
Display Formula10 \documentclass[12pt]{minimal}\begin{document}\begin{equation}
{\cal W}_{\rm copt} = \arg _{{\cal W}_c } \min [{\rm dist}({\cal W}_c )].
\end{equation}\end{document}
$W copt =argWcmin[ dist (Wc)].$ Figure
4 shows an intuitive explanation for the proposed optimal warp calculation. The grid represents the distance matrix
${\bm {\cal D}}_c$
$Dc$. The horizontal and vertical axes represent the index of trajectories
${\cal F}_2$
$F2$ and
${\cal F}_1$
$F1$, respectively. The two bold lines represent the symmetric warps
${\cal W}_{1,2p}$
$W1,2p$ and
${\cal W}_{1p,2}$
$W1p,2$. The gray area represents the search area constrained by
${\cal W}_{1,2p}$
$W1,2p$ and
${\cal W}_{1p,2}$
$W1p,2$. The warp
${\cal W}_{\rm copt}$
$W copt $ inside the global constraint is considered as the final unbiased warp.