diff --git a/summary/main.tex b/summary/main.tex
index 718f0ba..d978f4c 100644
--- a/summary/main.tex
+++ b/summary/main.tex
@@ -1,6 +1,7 @@
 \documentclass[sigconf]{acmart}
 \usepackage{amsmath}
 \usepackage{bbm}
+\usepackage{mathtools}
 
 %%
 %% \BibTeX command to typeset BibTeX logo in the docs
@@ -69,7 +70,7 @@
 %% article.
 \begin{abstract}
   Cross-Model Pseudo-Labeling is a new Framework for generating Pseudo-labels
-  for supervised leanring tasks where only a subset of true labels is known.
+  for supervised learning tasks where only a subset of true labels is known.
   It builds upon the existing approach of FixMatch and improves it further by
   using two different sized models complementing each other.
 \end{abstract}
@@ -90,17 +91,54 @@
 
 \section{Introduction}
 For most supervised learning tasks are lots of training samples essential.
-with too less training data the model will gerneralize not well and not fit a real world task.
-Labeling datasets is in commonly seen as an expensive task and wants to be avoided as much as possible.
+With too less training data the model will gerneralize not well and not fit a real world task.
+Labeling datasets is commonly seen as an expensive task and wants to be avoided as much as possible.
 Thats why there is a machine-learning field called Semi-Supervised learning.
 The general approach is to train a model that predicts Pseudo-Labels which then can be used to train the main model.
 
 \section{Semi-Supervised learning}
-todo write stuff
+In traditional supervised learning we have a labeled dataset.
+Each datapoint is associated with a corresponding target label.
+The goal is to fit a model to predict the labels from datapoints.
+
+In traditional unsupervised learning no labels are known.
+The goal is to find patterns and structures in the data.
+
+Those two techniques combined yield semi-supervised learning.
+Some of the labels are known, but for most of the data we have only the raw datapoints.
+The basic idea is that the unlabeled data can significantly improve the model performance when used in combination with the labeled data.
 
 \section{FixMatch}\label{sec:fixmatch}
 There exists an already existing approach called FixMatch.
 This was introduced in a Google Research paper from 2020~\cite{fixmatch}.
+The key idea of FixMatch is to leverage the unlabeled data by predicting pseudo-labels out of the known labels.
+Then both, the known labels and the predicted ones are used side by side to train the model.
+The labeled samples guide the learning process and the unlabeled samples gain additional information.
+
+Not every pseudo prediction is kept to train the model further.
+A confidence threshold is defined to evaluate how `confident` the model is of its prediction.
+The prediction is dropped if the model is too less confident.
+The quantity and quality of the obtained labels is crucial and they have an significant impact on the overall accuracy.
+This means improving the pseudo-label framework as much as possible is important.
+
+\subsection{Math of FixMatch}\label{subsec:math-of-fixmatch}
+$\mathcal{L}_u$ defines the loss-function that trains the model.
+The sum over a batch size $B_u$ takes the average loss of this batch and should be straight forward.
+The input data is augmented in two different ways.
+At first there is a weak augmentation $\mathcal{T}_{\text{weak}}(\cdot)$ which only applies basic transformation such as filtering and bluring.
+Moreover, there is the strong augmentation $\mathcal{T}_{\text{strong}}(\cdot)$ which does cropouts and edge-detections.
+The interesting part is the indicator function $\mathbbm{1}(\cdot)$ which applies a principle called `confidence-based masking`.
+It retains a label only if its largest probability is above a threshold $\tau$.
+Where $p_i \coloneqq F(\mathcal{T}_{\text{weak}}(u_i))$ is a model evaluation with a weakly augmented input.
+The second part $\mathcal{H}(\cdot, \cdot)$ is a standard Cross-entropy loss function which takes two inputs.
+$\hat{y}_i$, the obtained pseudo-label and $F(\mathcal{T}_{\text{strong}}(u_i))$, a model evaluation with strong augmentation.
+The indicator function evaluates in $0$ if the pseudo prediction is not confident and the current loss evaluation will be dropped.
+Otherwise it will be kept and trains the model further.
+
+\begin{equation}
+  \label{eq:equation2}
+  \mathcal{L}_u = \frac{1}{B_u} \sum_{i=1}^{B_u} \mathbbm{1}(\max(p_i) \geq \tau) \mathcal{H}(\hat{y}_i,F(\mathcal{T}_{\text{strong}}(u_i)))
+\end{equation}
 
 \section{Cross-Model Pseudo-Labeling}
 todo write stuff \cite{Xu_2022_CVPR}
@@ -111,7 +149,7 @@ todo write stuff \cite{Xu_2022_CVPR}
   \mathcal{L}_u = \frac{1}{B_u} \sum_{i=1}^{B_u} \mathbbm{1}(\max(p_i) \geq \tau) \mathcal{H}(\hat{y}_i,F(\mathcal{T}_{\text{strong}}(u_i)))
 \end{equation}
 
-\section{Figures}
+\section{Performance}
 
 \begin{figure}[h]
   \centering