Smooth Pycnophylactic Interpolation

Smooth Pycnophylactic Interpolation is a disaggregation where zi have minimal quadratic slopes subject to the Pycnophylactic Principle.

Mathematically: minimize f(z) := (Dxz)TDxz + (Dyz)TDyz subject to: $\forall r: \sum\limits_{i} z_i * q_i^r = Z_r$, where Dx and Dy are the linear operations that result in the partial discrete difference in the x and y directions respectively.

notes

Since

\[(\textbf{D}z) := z_{i} − z_{i−1}\]

one can derive that \(\( ((D z)^T D z) = (z^T D^T D z) \sum\limits_{i1}^n(z_i - z_{i-1})^2\sum\limits_{i1}^n(z_i^2 + z_{i-1}^2 - 2 z_i z_{i-1}) z_0^2 + \sum\limits_{i1}^n(2 z_i^2 - 2 z_i z_{i-1}) - z_n^2 \)\)

and \(\frac{\partial f(z)}{\partial z_i} = (z^T D^T D)^T_i + (D^T D z)_i = (D^T D^{TT} z)_i + (D^T D z)_i = 2(D^T D z)_i\)

this convex optimization problem can be reformulated as: \(\( \frac{\partial\[ f(z) + \sum\limits_{r} \lambda_{r} (\sum\limits_{i} z_i \*q_i^r - Z_r)}\]{\partial z_i}\)

0

\[\sum\limits_{i \in \{ x, y \} } 4 z_i - 2 z_{i-1} - 2 z_{i+1} + \sum\limits_{r} \lambda_{r} q_i^r ) \),\]

subject to \(\forall r: \sum\limits_{i} z_i \* q_i^r = Z_r\)

from which follows that \(\( z_{x,y} = {1 \over 4} (z_{x-1,y} +\) \(z_{x+1,y} + z_{x,y-1} + z_{x,y+1} ) - {1 \over 8} \sum\limits_{r}\) \(\lambda_{r} q_i^r \)\)

Links

Tobblers work, 1979