3D pyramid interpolation

Linearized nonlinear problem

To estimate both missing data and the PEF, the problem becomes nonlinear. To avoid directly solving nonlinear problem, I linearize it by alternately estimate missing data and PEF, and use them to update each other. Corresponding pseudo code is as follows:

 for each iteration i{  
  estimate missing data m_i from PEF a_(i-1) using equation 5
  estimate PEF a_i from missing data m_i using equation 4
 }

I start with my guess of the PEF, $\bf {a_0}$. First, I make an operator $\bf {A_0}$ that is convolution with $\bf {a_0}$, and I use it to estimate the missing data $\bf {m_0}$. From $\bf {m_0}$, I make an operator $\bf {M_0}$ that is a convolution with $\bf {m_0}$. Then I update $\bf {a_0}$ using $\bf {M_0}$, calling the updated $\bf {a_0}$ as $\bf {a_1}$. This process makes one iteration of the linearized problem. Then I repeat this process, making $\bf {A_1}$ from $\bf {a_1}$, updating $\bf {m_0}$ to $\bf {m_1}$ using $\bf {A_1}$, making $\bf {M_1}$ from $\bf {m_1}$, updating $\bf {a_1}$ to $\bf {a_2}$ using $\bf {M_1}$, and so on... Finally the algorithm will converge to some $\bf {m}$ and $\bf {a}$; hopefully, by careful choosing of $\bf {a_0}$, I will converge to the correct $\bf {m}$ and $\bf {a}$.

3D pyramid interpolation