Skip to main content

Table 2 Optimization Model for Partitions

From: Partition-based optimization model for generative anatomy modeling language (POM-GAML)

Arg Min: \( P{\left(\sum \limits_{l=1}^N{k}_l\left|{p}_l-{p}_{Destination}\right|\right)}_t \)
Subject to:
Distij − |pi − pj| = 0 for (i,j)At (5)
\( {\mathit{\cos}}^{-1}\left(\frac{\left({p}_{io}-{p}_j\right)\bullet \left({p}_i-{p}_j\right)}{\left\Vert \left({p}_{io}-{p}_j\right)\right\Vert \times \left\Vert \left({p}_i-{p}_j\right)\right\Vert}\right)-{\theta}_{ij}<0 \) for (i,j)Bt (6)
(pio − pj)axis − (pi − pj)axis = 0 (6a)
Distij − ∆dmax − |pi − pj| < 0 for (i,j)Ct (7)
|pi − pj| − Distij − ∆dmax < 0 (8)
\( 2\operatorname{sgn}\left({p}_i{p}_j.{p}_i{v}_j\right){\times \mathit{\tan}}^{-1}\left(\frac{r}{\left\Vert {v}_j\right\Vert}\right)-{a}_{ij}<0 \) for (i,j)Dt (9)
\( \left(\frac{\tan \left({a}_{ij}\right)}{\left\Vert {v}_j\right\Vert}\right)<0 \) (10)
|pi − vj| < 0 (11a)
|pi − vi| = 0 (11b)
i, jJ and i, jM, kl > 0, A ∩ C =   ,
P(A)m ∩ P(A)n =   , vi = pi, vj = pj, vV, vA, B, C