Smooth Pycnophylactic Interpolation

Smooth Pycnophylactic Interpolation is a disaggregation where \(z_i\) have minimimal quadratic slopes subject to the Pycnophylactic Principle.

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

=notes=

Since \((Dz)_i := z_i - z_{i-1}\),

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

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

this convex optimization problem can be reformulated as: \( {{\partial [ f(z) + \sum\limits_{r} \lambda_{r} (\sum\limits_{i} z_i * q_i^r - Z_r)}] \over {\partial z_i}} =0 = \sum\limits_{i \in \{ x, y \} } 4 z_i - 2 z_{i-1} - 2 z_{i+1} + \sum\limits_{r} \lambda_{r} q_i^r ) \),

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