# Loss functions

Loss functions are generally introduced in mathematical optimization theory. The purpose is to map a certain optimization problem onto a real number. By minimizing this real number, one hopes that the obtained parameters provide a useful result for the problem. One common loss function (especially in deep learning) is simply the $L^2$ norm between measurement and prediction.

So far we provide three adapted loss functions with our package. However, it is relatively easy to incorporate custom defined loss functions or import them from packages like Flux.jl. The interface from Flux.jl is the same as for our loss functions.

## Poisson Loss

As mentioned in Noise Model, Poisson shot noise is usually the dominant source of noise. Therefore one achieves good results by choosing a loss function which considers both the difference between measurement and reconstruction but also the noise process. See [4] and [1] for more details on that. As key idea we interpret the measurement as a stochastic process. Our aim is to find a deconvolved image which describes as accurate as possible the measured image. Mathematically the probability for a certain measurement $Y$ is

$$$p(Y(r)|\mu(r)) = \prod_r \frac{\mu(r)^{Y(r)}}{\Gamma(Y(r) + 1)} \exp(- \mu(r))$$$

where $Y$ is the measurement, $\mu$ is the expected measurement (ideal measurement without noise) and $\Gamma$ is the generalized factorial function. In the deconvolution process we get $Y$ as input and want to find the ideal specimen $S$ which results in a measurement $\mu(r) = (S * \text{PSF})(r))$. Since we want to find the best reconstruction, we want to find a $\mu(r)$ so that $p(Y(r) | \mu(r))$ gets as large as possible. Because that means that we find the specimen which describes the measurement with the highest probability. Instead of maximizing $p(Y(r) | \mu(r))$ a common trick is to minimize $- \log(p(Y(r)|\mu(r)))$. Mathematically, the optimization of both functions provides same results but the latter is numerically more stable.

$$$\underset{S(r)}{\arg \min} (- \log(p(Y(r)|\mu(r)))) = \underset{S(r)}{\arg \min} \sum_r (\mu(r) + \log(\Gamma(Y(r) + 1)) - Y(r) \log(\mu(r))$$$

which is equivalent to

$$$\underset{S(r)}{\arg \min}\, L = \underset{S(r)}{\arg \min} \sum_r (\mu(r) - Y(r) \log(\mu(r))$$$

since the second term only depends on $Y(r)$ but not on $\mu(r)$. The gradient of $L$ with respect to $\mu(r)$ is simply

$$$\nabla L = 1 - \frac{Y(r)}{\mu(r)}.$$$

The function $L$ and the gradient $\nabla L$ are needed for any gradient descent optimization algorithm. The numerical evaluation of the Poisson loss can lead to issues. Since $\mu(r)=0$ can happen for a measurement with zero intensity background. However, the loss is not defined for $\mu \leq 0$. In our source code we set all intensity values below a certain threshold $\epsilon$ to $\epsilon$ itself. This prevents the evaluation of the logarithm at undefined values.

## Scaled Gaussian Loss

It is well known that the Poisson density function behaves similar as a Gaussian density function for $\mu\gg 1$. This approximation is almost for all use cases in microscopy valid since regions of interest in an image usually consists of multiple photons and not to a single measured photon. Mathematically the Poisson probability can be approximately (using Stirling's formula in the derivation) expressed as:

$$$p(Y(r)|\mu(r)) \approx \prod_r \frac{\exp \left(-\frac{(x-\mu(r) )^2}{2 \mu(r) }\right)}{\sqrt{2 \pi \mu(r) }}$$$

Applying the negative logarithm we get for the loss function:

$$$\underset{S(r)}{\arg \min}\, L = \underset{S(r)}{\arg \min} \sum_r \frac12 \log(\mu(r)) + \frac{(Y(r)-\mu(r))^2}{2 \mu(r)}$$$

$$$\nabla L = \frac{\mu(r) + \mu(r)^2 - Y(r)^2}{2 \mu^2}$$$

## Gaussian Loss

A very common loss in optimization (and Deep Learning) is a simple Gaussian loss. However, this loss is not recommended for low intensity microscopy since it doesn't consider Poisson noise. However, still combined with suitable regularizer reasonable results can be achieved. The probability is defined as

$$$p(Y(r)|\mu(r)) = \prod_r \frac1{\sqrt{2 \pi \sigma^2}} \exp\left(- \frac{(Y(r) - \mu(r))^2}{2 \sigma ^2} \right)$$$

where $\sigma$ is the standard deviation of the Gaussian.

Applying the negative logarithm we can simplify the loss to be minimized:

$$$\underset{S(r)}{\arg \min}\, L = \underset{S(r)}{\arg \min} \sum_r (Y(r) - \mu(r))^2$$$

Since we are looking for $\mu(r)$ minimizing this expression, $\sigma$ is just a constant offset being irrelevant for the solution. This expression is also called L2 loss.