Constrained tuning/Analytical solution to constrained Euclidean tunings: Difference between revisions

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Skew!
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== Otherwise normed tuning ==
== Otherwise normed tuning ==
If there is a weight, such as CTE tuning, the weight should be applied to the map and the constraint first:  
If there is a weight W and/or a skew X, such as CTWE tuning, the weight-skew should be applied to the map and the constraint first:  


<math>\displaystyle  
<math>\displaystyle  
\begin{align}
\begin{align}
V &= AW \\
V &= AWX \\
M_{\rm C} &= W^{-1}B_{\rm C}
M_{\rm C} &= (WX)^+ B_{\rm C}
\end{align}
\end{align}
</math>
</math>


Working from here, we find the weighted projection map (P<sub>C</sub>)<sub>W</sub>:  
Working from here, we find the weighted projection map (P<sub>C</sub>)<sub>WX</sub>:  


<math>\displaystyle  
<math>\displaystyle  
(P_{\rm C})_W =  
(P_{\rm C})_{WX} =  
\begin{bmatrix}  
\begin{bmatrix}  
V^+VM_{\rm C} & \begin{matrix} O \\ V_{\rm M}^+V_{\rm M} \end{matrix}
V^+VM_{\rm C} & \begin{matrix} O \\ V_{\rm M}^+V_{\rm M} \end{matrix}
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To reconstruct the original projection map, apply
To reconstruct the original projection map, apply


<math>\displaystyle P_{\rm C} = W (P_{\rm C})_W W^{-1}</math>
<math>\displaystyle P_{\rm C} = WX (P_{\rm C})_{WX} (WX)^+</math>


== Nontrivially constrained tuning ==
== Nontrivially constrained tuning ==
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<math>\displaystyle P_{\rm C} = S (P_{\rm C})_S S^{-1}</math>
<math>\displaystyle P_{\rm C} = S (P_{\rm C})_S S^{-1}</math>


Similarly, if there is a weight, it should be applied to the map and the constraint first:  
Similarly, if there is a weight and/or a skew X, it should be applied to the map and the constraint first:  


<math>\displaystyle  
<math>\displaystyle  
\begin{align}
\begin{align}
V &= AW \\
V &= AWX \\
M_{\rm C} &= W^{-1}B_{\rm C}
M_{\rm C} &= (WX)^+ B_{\rm C}
\end{align}
\end{align}
</math>
</math>


and then the basis transformation matrix should be found out in this weighted space:  
and then the basis transformation matrix should be found out in this weight-skewed space:  


<math>\displaystyle  
<math>\displaystyle  
S_W = [\begin{matrix} M_{\rm C} & M_{\rm C}^\perp \end{matrix}]
S_{WX} = [\begin{matrix} M_{\rm C} & M_{\rm C}^\perp \end{matrix}]
</math>
</math>


We should apply this S<sub>W</sub> to the weighted map and the weighted constraint to convert them into the working basis:  
We should apply this S<sub>WX</sub> to the weight-skewed map and the weight-skewed constraint to convert them into the working basis:  


<math>\displaystyle  
<math>\displaystyle  
\begin{align}
\begin{align}
V_S &= VS_W \\
V_S &= VS_{WX} \\
(M_{\rm C})_S &= S_W^{-1}M_{\rm C}
(M_{\rm C})_S &= S_{WX}^+ M_{\rm C}
\end{align}
\end{align}
</math>
</math>


Proceed as before. The projection map found this way will be weighted and in the working basis. To reconstruct the original projection map, apply
Proceed as before. The projection map found this way will be weight-skewed and in the working basis. To reconstruct the original projection map, apply


<math>\displaystyle P_{\rm C} = WS (P_{\rm C})_{WS} S^{-1} W^{-1}</math>
<math>\displaystyle P_{\rm C} = WXS (P_{\rm C})_{WXS} S^{-1} (WX)^+</math>


== Example ==
== Example ==