optimal after epipolar

since $$\begin{gathered} ||n||\cdot ||t||-||n\cdot t|| \\ =\Sigma n_x^2\Sigma t_x^2-\Sigma (n_x{t}_x{)}^2 \\ =(n_y^2+n_z^2)t_x^2+(n_x^2+n_z^2)t_y^2+(n_x^2+n_y^2)t_z^2 \end{gathered}$$

so ∣ ∣ n ∣ ∣ ⋅ ∣ ∣ X − t ∣ ∣ − ∣ ∣ n ⋅ ( X − t ) ∣ ∣ = ( n y 2 + n z 2 ) ( x − t x ) 2 + ( n x 2 + n z 2 ) ( y − t y ) 2 + ( n x 2 + n y 2 ) ( z − t z ) 2 = [ x   y   z   1 ] [ n y 2 + n z 2 0 0 − ( n y 2 + n z 2 ) t x 0 n x 2 + n z 2 0 − ( n x 2 + n z 2 ) t y 0 0 n x 2 + n y 2 − ( n x 2 + n y 2 ) t z − ( n y 2 + n z 2 ) t x − ( n x 2 + n z 2 ) t y − ( n x 2 + n y 2 ) t z Σ x ( n y 2 + n z 2 ) t x 2 ] [ x y z 1 ] ||n||\cdot||X-t||-||n\cdot (X-t)|| \\=(n_y^2+n_z^2)(x-t_x)^2+(n_x^2+n_z^2)(y-t_y)^2+(n_x^2+n_y^2)(z-t_z)^2 \\=\begin{bmatrix}x\ y\ z\ 1\end{bmatrix}\begin{bmatrix}n_y^2+n_z^2 & 0 & 0 & -(n_y^2+n_z^2)t_x \\0&n_x^2+n_z^2&0&-(n_x^2+n_z^2)t_y\\0&0&n_x^2+n_y^2&-(n_x^2+n_y^2)t_z\\ -(n_y^2+n_z^2)t_x &-(n_x^2+n_z^2)t_y&-(n_x^2+n_y^2)t_z&\Sigma_x(n_y^2+n_z^2)t_x^2\end{bmatrix}\begin{bmatrix}x\\y\\z\\1\end{bmatrix} ∣∣n∣∣⋅∣∣X−t∣∣−∣∣n⋅(X−t)∣∣=(ny2​+nz2​)(x−tx​)2+(nx2​+nz2​)(y−ty​)2+(nx2​+ny2​)(z−tz​)2=[x y z 1​] ​ny2​+nz2​00−(ny2​+nz2​)tx​​0nx2​+nz2​0−(nx2​+nz2​)ty​​00nx2​+ny2​−(nx2​+ny2​)tz​​−(ny2​+nz2​)tx​−(nx2​+nz2​)ty​−(nx2​+ny2​)tz​Σx​(ny2​+nz2​)tx2​​ ​ ​xyz1​ ​

then
Σ i = 0 3 d i 2 = Σ i ∣ ∣ n i ∣ ∣ ⋅ ∣ ∣ X − t i ∣ ∣ − ∣ ∣ n ⋅ ( X − t i ) ∣ ∣ = [ x   y   z   1 ] [ Σ i ( n y i ) 2 + ( n z i ) 2 0 0 − Σ i ( ( n y i ) 2 + ( n z i ) 2 ) t x i 0 Σ i ( n x i ) 2 + ( n z i ) 2 0 − Σ i ( ( n x i ) 2 + ( n z i ) 2 ) t y i 0 0 Σ i ( n x i ) 2 + ( n y i ) 2 − Σ i ( ( n x i ) 2 + ( n y i ) 2 ) t z i − Σ i ( ( n y i ) 2 + ( n z i ) 2 ) t x i − Σ i ( ( n x i ) 2 + ( n z i ) 2 ) t y i − Σ i ( ( n x i ) 2 + ( n y i ) 2 ) t z i Σ i Σ x ( ( n y i ) 2 + ( n z i ) 2 ) t x 2 ] [ x y z 1 ] \Sigma_{i=0}^{3} d_i^2 \\=\Sigma_i||n_i||\cdot||X-t_i||-||n\cdot (X-t_i)|| \\=\begin{bmatrix}x\ y\ z\ 1\end{bmatrix}\begin{bmatrix}\Sigma_i(n_y^i)^2+(n_z^i)^2 & 0 & 0 & -\Sigma_i((n_y^i)^2+(n_z^i)^2)t_x^i \\0&\Sigma_i(n_x^i)^2+(n_z^i)^2&0&-\Sigma_i((n_x^i)^2+(n_z^i)^2)t_y^i \\0&0&\Sigma_i(n_x^i)^2+(n_y^i)^2&-\Sigma_i((n_x^i)^2+(n_y^i)^2)t_z^i \\ -\Sigma_i((n_y^i)^2+(n_z^i)^2)t_x^i &-\Sigma_i((n_x^i)^2+(n_z^i)^2)t_y^i &-\Sigma_i((n_x^i)^2+(n_y^i)^2)t_z^i &\Sigma_i\Sigma_x((n_y^i)^2+(n_z^i)^2)t_x^2\end{bmatrix}\begin{bmatrix}x\\y\\z\\1\end{bmatrix} Σi=03​di2​=Σi​∣∣ni​∣∣⋅∣∣X−ti​∣∣−∣∣n⋅(X−ti​)∣∣=[x y z 1​] ​Σi​(nyi​)2+(nzi​)200−Σi​((nyi​)2+(nzi​)2)txi​​0Σi​(nxi​)2+(nzi​)20−Σi​((nxi​)2+(nzi​)2)tyi​​00Σi​(nxi​)2+(nyi​)2−Σi​((nxi​)2+(nyi​)2)tzi​​−Σi​((nyi​)2+(nzi​)2)txi​−Σi​((nxi​)2+(nzi​)2)tyi​−Σi​((nxi​)2+(nyi​)2)tzi​Σi​Σx​((nyi​)2+(nzi​)2)tx2​​ ​ ​xyz1​ ​