Mathematical Optimization

Deconvolution was already described as an optimization problem in the 1970s by L.~B. Lucy (1974), William Hadley Richardson (1972). Since then, many variants and different kinds of deconvolution algorithms were presented, but mainly based on the concept of Lucy-Richardson. We try to formulate convolution as an inverse physical problem and solve it using a convex optimization loss function so that we can use fast optimizers to find the optimum. The variables we want to optimize for, are the pixels of the reconstruction $S(r)$. Therefore our reconstruction problem consists of several thousands to billion variables. Mathematically the optimization can be written as:

\[\underset{S(r)}{\arg \min}\, L(\text{Fwd}(S(r))) + \text{Reg}(S(r))\]

where $\text{Fwd}$ represents the forward model (in our case convolution of $S(r)$ with the $\text{PSF}$), $S(r)$ is ideal reconstruction, $L$ the loss function and $\text{Reg}$ is a regularizer. The regularizer puts in some prior information about the structure of the object. See the following sections for more details about each part.

Map Functions

In some cases we want to restrict the optimizer to solutions with $S(r) \geq 0$. Usually one uses boxed optimizer or penalties to prevent negativity. However, in some cases, a $S(r) < 0$ can lead to issues during the optimization process. For that purpose we can introduce a mapping function. Instead of optimizing for $S(r)$ we can optimize for some $\hat S(r)$ where $M$ is the mapping function connection

\[S(r)= M(\hat S(r)).\]

A simple mapping function leading to $S(r) \geq 0$ is

\[M(\hat S(r)) = \hat S(r)^2\]

The optimization problem is then given by

\[\underset{\hat S(r)}{\arg \min}\, L(\text{Fwd}(M(\hat S(r)))) + \text{Reg}(M(\hat S(r)))\]

After the optimization we need to apply $M$ on $\hat S$ to get the reconstructed sample

\[S(r) = M(\hat S(r))\]

One could also choose different functions $M$ to obtain reconstruction in certain intensity intervals.