# Cascade Network with Deformable Composite Backbone for Formula Detection in Scanned Document Images
## 1 Introduction  
Information extraction from document images is a primary need in various domains such as banking, archiving, or academia and industry in general. Research in document analysis has been trying to develop precise information extraction systems for several years [1, 2, 3, 4]. Although state-of-the-art optical character recognition (OCR) systems [5, 6] recognize regular text with high accuracy, they are vulnerable to recognize information from page objects (tables, figures, mathematical formulas) in document images [7, 8]. Figure 1 illustrates the problem in which an open-source OCR, Tesseract [4] (we use the LSTM-based version 4.1.1 available at [[https://github.com/tesseract-ocr/tesseract](https://github.com/tesseract-ocr/tesseract)]([https://github.com/tesseract-ocr/tesseract](https://github.com/tesseract-ocr/tesseract)) accessed on 5 July 2021), is applied to extract the content from a document image. Besides recognizing the textual content, the OCR fails to extract the information from mathematical formulas. This shows that formula detection is a crucial preliminary step for information extraction in such document images. Mathematical formulas are an integral part of documents because they allow us to represent complex information concisely by exploiting mathematics capabilities. Formulas present in the documents are categorized into isolated formulas (mentioned in a separate line) and embedded formulas (inline mathematical symbols). Figure 2 exhibits the problem of detecting isolated and embedded formulas in document images. We state and prove next the new work decomposition laws. Theorem 5.1 (Wock decomposition laws). : _Under any dynamic scheduling policy, and for any desired \(S\subseteq M\) of job classes, (a)_  
\[\sum_{i\in S}\
u_{i}^{\mathrm{P}}(s)+\frac{1}{1-\rho^{\mathrm{t}}(S)}\sum_{i\in S }\sum_{i\in S}\lambda_{i}\
u_{i}^{\mathrm{P}}(s_{i}^{\mathrm{P}})+\frac{1-\rho }{1-\rho^{\mathrm{t}}(S)}\sum_{i\in S}\
u_{i}^{\mathrm{P}}.\]  
_(b) Identity (30) can be reformulated as_  
\[ \begin{split}& E[\
u^{\mathrm{T}}]-f(S)-\sum_{i\in S}\rho(B_{i}-r)- \frac{\
u^{\mathrm{t}}(S)}{1-\rho^{\mathrm{t}}(S)}\sum_{i\in S}\
u_{i},\\ &\quad\quad\quad+\frac{1}{1-\rho^{\mathrm{t}}(S)}\sum_{i\in S} \left(\lambda_{i}\
u_{i}^{\mathrm{T}}-\rho\right)E^{\mathrm{t}}(\
u^{\mathrm{T }})+\frac{1-\rho(S)}{1-\rho^{\mathrm{t}}(S)}E[\
u^{\mathrm{T}}]B^{\mathrm{t}}= 0].\end{split}  \tag{31}\]  
Proof. : (a) In what follows we see the following notation: if \(S,T\subseteq S,z=(z_{i})_{i,i}\), then \(\lambda=(z_{i})_{i,i}\) is an \(\times\) matrix, we shall write  
\[z_{i}=(z_{i})_{i,i}\quad\text{and}\quad\Lambda_{i\
eq i}=(z_{i})_{i,i}\]  
Let \(\
u\) denote the \(s\)-vector  
\[\
u=\left( \begin{smallmatrix}\
u_{i}^{\mathrm{T}}\\ \theta\end{smallmatrix} \right).\]  
The task of detecting both isolated and embedded formulas in document images is a difficult problem because of the underlying low inter-class and high intra-class variance [10]. The hurdles involved in detecting isolated and embedded formulas are exhibited in Figure 2. The isolated formulas present in a document image can easily be misclassified with other page objects due to low inter-class variance with tables, algorithms, and figures. The embedded formulas contain mathematical functions \((log,exp,tan)\), operators \((\times,+,\sigma,\%)\), and variables \((i,j,k)\). generic datasets. Later, statistical learning, mainly machine learning-based methods, advanced the performance of formula identification systems [14; 15; 16]. The recent success of deep learning-based methods on computer vision within the last decade also had an impact on the task of formula detection in scanned document images. Several deep learning-based formula detection approaches [17; 18; 19; 20] have been presented in the past two years. They are mainly equipped with object detection algorithms such as Faster R-CNN [21], YOLO [22], SSD [23], and FPNs [24]. In recent work, [PERSON] et al.