# Mathematical Optimization

Deconvolution was already described as an optimization problem in the 1970s by [2], [3]. 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.