295 © 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License Enhanced OCR Techniques for Recognizing Mathematical Expressions in Scanned Documents Yahya Baher Al-Askary 1* , Saad Al-Momen 2 1,2 Department of Mathematics, College of Sciences, University of Baghdad, Baghdad, Iraq. *Corresponding Author. Received: 2 July 2023 Accepted: 1 October 2023 Published: 20 October 2025 doi.org/10.30526/38.4.3640 Abstract When OCR systems are utilized to recognize mathematical expressions in scanned documents, they encounter numerous challenges. These challenges arise because the mathematical expressions use an extensive array of symbols, variables, numbers, and operations, each with its distinct writing style. Moreover, mathematical expressions have a hierarchical relationship and adhere to logical rules governing the grouping of their components. This paper presents a three-step approach segmentation, symbol recognition, and interpretation to overcome these issues. The segmentation process aims to identify and separate each symbol based on its spatial location. Then, various features are extracted to describe the horizontal and vertical projections of the symbol, enabling effective recognition. Two neural network-based classification methods were proposed for the recognition of the symbols. The first one achieved 96.6% recognition rates, while the second achieved 97.7%. Lastly, the paper introduces three guidelines for interpreting the mathematical meaning of the expression and accurately converting it into textual form. The study demonstrates the potential of these methods in enhancing the capabilities of OCR systems for recognizing mathematical expressions. Keywords: Classification, Mathematical Expression, Neural Network, Character Recognition, Image Processing. 1. Introduction There has been a remarkable revolution in the field of Optical Character Recognition (OCR) in recent years, leading to a complete transformation in the way scanned documents are converted into digital format (1). This advancement has made it possible to archive these documents with high accuracy and efficiency. Nevertheless, there is still a notable challenge to overcome: recognizing mathematical expressions within these documents. The complexity lies in the diverse range of symbols, variables, numbers, and mathematical operations that form these expressions. The extensive variety of components in mathematical expressions poses a significant obstacle to their recognition (2, 3). They can involve Greek symbols, subscripts, superscripts, numbers, and mathematical symbols, whereas OCR systems are primarily designed to recognize standard alphabet characters and numbers (4). As a result, there is an urgent need to enhance these systems and broaden their capabilities. https://orcid.org/0009-0002-6783-9102 mailto:yahia.baher2103m@sc.uobaghdad.edu.iq https://orcid.org/0000-0002-9715-4529 mailto:saad.m@sc.uobaghdad.edu.iq https://orcid.org/0009-0002-6783-9102 mailto:yahia.baher2103m@sc.uobaghdad.edu.iq https://orcid.org/0000-0002-9715-4529 mailto:saad.m@sc.uobaghdad.edu.iq https://orcid.org/0009-0002-6783-9102 mailto:yahia.baher2103m@sc.uobaghdad.edu.iq https://orcid.org/0000-0002-9715-4529 mailto:saad.m@sc.uobaghdad.edu.iq https://orcid.org/0009-0002-6783-9102 mailto:yahia.baher2103m@sc.uobaghdad.edu.iq https://orcid.org/0000-0002-9715-4529 mailto:saad.m@sc.uobaghdad.edu.iq https://orcid.org/0009-0002-6783-9102 mailto:yahia.baher2103m@sc.uobaghdad.edu.iq https://orcid.org/0000-0002-9715-4529 mailto:saad.m@sc.uobaghdad.edu.iq https://orcid.org/0009-0002-6783-9102 mailto:yahia.baher2103m@sc.uobaghdad.edu.iq https://orcid.org/0000-0002-9715-4529 mailto:saad.m@sc.uobaghdad.edu.iq IHJPAS. 2025,38(4) 296 Moreover, mathematical expressions are not merely an assortment of isolated symbols; they exhibit hierarchical and logical relationships governed by mathematical rules. Optical recognition systems encounter difficulty in discerning these relationships, often resulting in ambiguous or erroneous interpretations. Simultaneously, mathematical expressions are composed in various fonts and styles (5). They can be handwritten or printed in scientific and mathematical formats (6,7). Consequently, optical recognition systems must address this vast diversity and adapt their recognition mechanisms accordingly. The intricate nature of this diversity adds complexity to the recognition process. Based on the preceding facts, it becomes evident that optical recognition systems are prone to errors when it comes to recognizing mathematical expressions. Consequently, these errors can result in misinterpretations, causing a complete alteration in the meaning of a mathematical expression due to the misinterpretation of a single symbol or component within that expression. This can potentially propagate errors that originated in earlier stages, ultimately leading to critical errors in subsequent stages (8). Researchers have made notable strides in tackling the formidable challenges faced by OCR systems in recognizing mathematical expressions. Specialized algorithms and techniques have been developed, leveraging machine learning methods like deep learning (9) and neural networks (10). These advancements have significantly enhanced the accuracy and resilience of mathematical expression recognition. Various approaches have been suggested, encompassing template-based recognition, syntactic analysis, semantic analysis, and integrated text-image processing (11). 2. Related Work Recently there has been an increase use of Deep Neural Networks (DNNs) as feature extractors and classifiers. In 2018, Gunawan, Noor, and Kartiwi published a paper that focuses on developing an offline handwritten recognition system using DNN. To train and test the DNN, they selected two well-known English digits and letters databases: MNIST and EMNIST. These databases consist of 10 number digits and 52 English letters (capital and small). They proposed a two-stacked layer DNN system to achieve a 97.7% recognition accuracy for English digits and 88.8% for letters. The results demonstrated that the proposed system excels in accurately recognizing handwritten English digits and letters. (12). In 2019, Veres, Rishnyak, and Rishniak conducted a study to explore the feasibility of simultaneously performing structural analysis and character classification. They proposed a classification process for symbols and the development of a system based on machine learning techniques. In their study, the researchers presented a multi-classifier system that utilized five different classifiers to achieve the best possible outcome. The researchers provided evidence that this approach is highly effective in the field of structural analysis and character classification (13). Furthermore, in 2019, Nazemi, Tavakolian, Fitzpatrick, Fernando, and Suen employed deep learning techniques to recognize offline handwritten mathematical symbols. Their study focused on proposing symbol segmentation methods and achieving precise classification for a wide range of over 300 mathematical symbol classes. The paper introduces the utilization of Simple Linear Iterative Clustering for symbol segmentation and evaluates the accuracy of various classifiers for symbol recognition. The classifiers utilized in this study are kNN, salient features, Histogram of Oriented Gradient (HOG), Linear Binary Pattern (LBP), modified LeNet, and SqueezeNet (14). IHJPAS. 2025,38(4) 297 Wang and Shan in 2020 used a multiscale neural network to recognize the handwritten mathematical expressions. They highlight the challenges of recognizing handwritten mathematical expressions due to the variability in writing styles and the complexity of mathematical symbols. They proposed a model that can effectively analyze the two- dimensional structure of handwritten mathematical expressions and identify different mathematical symbols in a long expression (3). Also in 2020, Chan proposed an over-segmentation method to automatically extract strokes from textual bitmap images. The process involved breaking down the skeleton of a binarized image into junctions and segments. These segments were then merged to form strokes, and the stroke order was normalized using recursive projection and topological sort techniques. This approach demonstrated high offline accuracy when combined with conventional online recognizers. (15). In 2021, Yan, Zhang, Gao, Yuan, and Tang introduced ConvMath, a convolutional sequence modeling network, to recognize mathematical expressions. ConvMath integrates an image encoder for extracting features and a convolutional decoder for generating sequences. This study showed significantly improved efficiency compared to other methods. (16). Truong, Nguyen, Nguyen, and Nakagawa in 2021 recognized online handwritten mathematical expressions by building a symbol relation tree directly from a sequence of strokes using a bidirectional recurrent neural network (17). 3. Proposed Method Designing a system for mathematical expression recognition required careful consideration of the specific properties of these expressions. These properties arise from the extensive use of diverse symbols and their arrangement in a two-dimensional format, which are key factors that distinguish mathematical expressions from ordinary texts (18). The recognizing mathematical expressions process has three fundamental components: segmentation, symbol recognition, and mathematical expression interpretation. These three components are indispensable in developing a comprehensive and effective system for mathematical expression recognition. 3.1. Segmentation In our previous paper (19), the image pre-processing and image segmentation processes were extensively discussed. The pre-processing stage consisted of many steps, such as the image containing the equation being read, the conversion of the image into a binary form, noise reduction, skew correction, and the linking of broken symbols. The result of these steps is a processed image ready for the segmentation process. The segmentation stage is initiated, wherein the identification and separation of each symbol take place, relying on the horizontal and vertical distances between the components of the equation. The projections are determined by adding up the values of the binary image , alongside rows and columns . The techniques utilized essentially dissect the image into individual elements based on the spatial distribution of the pixels, as portrayed in the succeeding equations: ∑ where (1) ∑ where (2) Subsequently, the image is partitioned into multiple sub-images, were each sub-image exclusively containing one of the equations Separated component. IHJPAS. 2025,38(4) 298 3.2 Symbol Recognition The classification process typically consists of two phases: (i) The training phase where the classifier is taught to identify a specific class of reference feature vectors. Then, (ii) the testing (classification) phase, where unknown vectors are categorized based on a best match criterion. Figure 1 presents a block diagram that visualizes this process. represents the matrix resulting from the segmentation stage. This matrix contains the coordinates of the four corners of the rectangle that surrounds the symbol relatively to the original image. Figure 1. Block diagram of the proposed method. 3.2.1. Feature Extraction The feature extraction phase is key to drawing out vital information from the text image, aiding in the recognition of characters within the text. Choosing a consistent and representative set of features forms the crux of the pattern recognition system (20). To streamline the feature extraction process, the images containing symbols were standardized to a size of pixels. Subsequently, each image was partitioned into smaller squares, with dimensions of pixels per square. It is evident that the density, pixel distribution, geometric characteristics, and statistical properties vary significantly across these squares when the symbols differ. However, in the case of similar symbols, these features exhibit remarkable similarity. See Figure 2, which gives some examples Figure 2. Partitioning the binary cropped image into 100 sub-squares. Each of the squares represents a matrix of size pixels. Using Equations 1 and 2 mentioned earlier, two vectors, and , are calculated for each matrix. IHJPAS. 2025,38(4) 299 These vectors represent the horizontal and vertical projections of the respective matrix. The purpose of these two vectors is to facilitate the following calculations: The mean of the projection vector of the i th square: ∑ (3) The standard deviation of the projection vector of the i th square: √ ∑ | | (4) The variance of the projection vector of the i th square: ∑ | | (5) Kurtosis of the projection vector of the i th square: ( ) (6) This feature provides information about the peakedness or flatness of a distribution compared to a normal distribution. For Equations 3-6 (21), { } and represent the length of . Figure 3 shows the 8 features that are calculated for the vertical and horizontal projections for an arbitrary square within the image. Figure 3. An enlarged image of an arbitrary square and the eight features that have been extracted from it. In addition to the above features, the aspect ratio of the symbol has been calculated too: (7) As the different shapes of symbols in terms of height and width give this feature great importance in distinguishing between symbols. To mitigate classification errors caused by variations in fonts and formats, symbols were subjected to a thinning process to make them one pixel wide. Subsequently, the Aspect Ratio of the thinned symbol was calculated to address this issue. 3.2.2. Classification Classification serves as the critical decision-making phase in an Optical Character Recognition (OCR) system. This pivotal stage capitalizes on the features identified and isolated in the preceding step, and these are employed to discern the textual components accurately. It operates on the basis of a set of predefined rules (22). In essence, during this classification phase, the OCR system scrutinizes the extracted features in a detailed manner, identifying and segregating each piece of data. The extracted features can range from simple shapes to complex patterns. The classification process essentially IHJPAS. 2025,38(4) 300 interprets and translates these features into recognizable characters or symbols. Given its significant role, the classification phase serves as the OCR system's defining point, where it begins to demonstrate its true capability to transcribe the image-based text into a machine- readable format. By applying the predetermined rules and algorithms, the system can recognize and interpret each text segment effectively, paving the way for accurate digitization of textual information (23). Neural networks are powerful tools for the classification of features extracted from images, including those containing letters in OCR systems. A common type of neural network used in these contexts is a Convolutional Neural Network (CNN), which is designed to automatically and adaptively learn spatial hierarchies of features from the provided input images (24). Two distinct methods of neural networks are used here for the classification. 3.2.2.1. Neural Network Method 1 In this study, the tanh (hyperbolic tangent) activation function was employed. This function is chosen to ensure mapping any real-valued number to a range of [-1, 1]. It shares similarities with the sigmoid activation function but offers an advantage by being zero-centered. This zero-centered property aids helps prevent the network from becoming stuck during training. We used 100 neurons in hidden layers, Regularization, =30 and Maximal number of iterations is 200. L-BFGS-B (Limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm with Bound constraints) is used as a valuable tool for solving nonlinear optimization problems. It aids in refining the weights by minimizing the cost function in the backpropagation process of the neural network. L-BFGS-B is renowned for effectively handling constraints on parameters, making it particularly suitable for problems involving a large number of parameters, such as deep learning (25, 26). 3.2.2.2. Neural Network Method 2 A key part of any neural network training involves a solver, or optimization algorithm. RMSprop (Root Mean Square propagation) is one such optimization algorithm that has gained popularity for training deep neural networks. RMSprop works by keeping a moving average of the squared gradients for each weight, and then dividing the gradient by the root of this average. This method tries to resolve Adagrad's radically diminishing learning rates by using a moving average of the squared gradient. It utilizes the concept of the weighted moving average of the squared gradients to normalize the gradient itself (27, 28). Mathematically, RMSprop is expressed as follows: [ ] [ ] (8) √ [ ] (9) Where: the gradient at timestep a very small number to avoid any division by zero, usually on the order of decay rate, typically set to 0.9, learning rate. Here, represents the parameters of the model (e.g., the weights and biases in a neural network), and [ ] is the moving average of the squared gradient and Hinton suggests with a good default for as 0.001. The options used for training the model are 100 neurons in hidden layers, Max Epochs is 12, Mini Batch size is 10 and Squared Gradient Decay Factor is 0.9, the model was trained in MATLAB. IHJPAS. 2025,38(4) 301 3.3 Interpretation of Mathematical Expressions The neural network approach effectively trains the model to recognize and learn the letters and symbols present in mathematical equations. The result enables the transformation of these equations from an image-based representation into a readily understandable text format. This process facilitates interaction with the equations, supporting further computational or analytical operations. Once a letter is identified as a component of the equation, the next step is to ascertain its position within the equation whether it is inline, in superscript, or in subscript. To achieve this, three guide lines were used: one in the middle of the equation, one at the topmost part of the equation, and another at the bottom. By computing the difference in the distances between these lines and the midpoint of each character, it can establish to which line the character is closest. The closest line will determine the position of the character in the equation, as illustrated in Figures 4 and 5. Figure 4. Classification of subscripts, superscripts, and inline elements in a mathematical equation. Figure 5. Overview of the recognition process. 4. Experimental Results and Discussion The proposed system was trained using a dataset consisting of 70 distinct symbols, encompassing capital English letters (A-Z), lowercase letters (a-z), numbers (0-9), and mathematical symbols (=, +, -, *, /, >, <). Each symbol was represented by 50 distinct images varying in sizes, fonts, and styles. Approximately 25 fonts were used, including Arial, Times IHJPAS. 2025,38(4) 302 New Roman, and others. Furthermore, the system's efficacy was evaluated through the examination of 25 equations of different sizes and contents. These equations used a wide range of letters and symbols, allowing for comprehensive testing. All images of the symbols and equations used in this study are publicly available at: https://www.kaggle.com/datasets/yahiabaherjaber/print-mathematical-dataset-print-characters Table 1 presents the training data acquired through two distinct methodologies: Neural Network Method 1 and Neural Network Method 2. Table 1. Comparison of Training Data Acquired via Neural Network Method 1and Neural Network Method 2 Methods Test on Train data Train data 80% Test data 20% Repeat train/test = 5 CA F1 Prec Recall MCC Neural Network Method 1 98.8% 94.5% 94.5% 94.8% 94.8% 93% Neural Network Method 2 97.5% 89.7% 89.5% 90.2% 90.1% 87.7% The term "testing on data" refers to the practice of training the model on the entire dataset and then selecting a subset of it for testing purposes, effectively evaluating the model on the same data it was trained on. Conversely, the "80% training data and 20% test data" approach involves splitting the total dataset into two parts: 80% is used for training the model, and the remaining 20% is reserved for testing its performance. This train/test procedure is repeated five times to enhance the robustness of the results. Classification accuracy (CA) refers to the percentage of instances that have been classified correctly by the model, while it is calculated using Equation 10: (10) Recall, also known as sensitivity, represents the proportion of all positive cases in the dataset that were correctly identified as such, and it is determined based on Equation 11: (11) Precision is the proportion of true positives among cases classified as positive, and it is determined based on Equation 12: (12) F1 is a harmonic mean between precision and recall, and it is determined based on Equation 13: (13) The Matthews correlation coefficient (MCC), which considers both true and false positives and negatives, is typically viewed as a balanced metric. This measure remains effective even when the sizes of the classes under consideration are significantly different, and it is determined based on Equation 14: √ (14) In Equations 10-14, True Positive (TP) and True Negative (TN) are correct classifications. False Positive (FP) happens when a sample that is truly negative is predicted as positive. Also, False Negative (FN) happens when a positive sample is predicted as negative (29,30). Table 2 presents the recognition rates for each mathematical character in a dataset that consists of 25 simple printed mathematical equations. Moreover, Table 3 illustrates an assortment of strategies and methodologies employed for the recognition of mathematical symbols and expressions. It details the recognition rates, training samples, and testing samples as reported by various researchers. https://www.kaggle.com/datasets/yahiabaherjaber/print-mathematical-dataset-print-characters IHJPAS. 2025,38(4) 303 Table 2. Recognition Rate of Dataset. Mathematical Expression Name Total no. of Characters Recognized Characters Recognition Rate METHOD1 METHOD2 METHOD1 METHOD2 EQ1 6 6 6 100% 100% EQ2 6 6 6 100% 100% EQ3 13 13 13 100% 100% EQ4 6 6 6 100% 100% EQ5 11 11 11 100% 100% EQ6 4 4 4 100% 100% EQ7 7 7 7 100% 100% EQ8 7 7 7 100% 100% EQ9 5 4 4 80% 80% EQ10 5 5 5 100% 100% EQ11 8 8 8 100% 100% EQ12 4 3 3 75% 75% EQ13 7 6 7 85% 100% EQ14 9 9 9 100% 100% EQ15 9 9 9 100% 100% EQ16 6 4 4 66% 66% EQ17 8 8 8 100% 100% EQ18 8 8 8 100% 100% EQ19 8 8 8 100% 100% EQ20 10 10 10 100% 100% EQ21 10 9 10 90% 100% EQ22 6 6 6 100% 100% EQ23 6 6 6 100% 100% EQ24 6 6 6 100% 100% EQ25 5 5 5 100% 100% 180 174 176 96.6% 97.7% Table 3. Comparative study for various classification methods. Methods Feature Extraction Classifier Recognition Rate Details Proposed Method 1 See section 3.2.1. Neural Network 96.6% activation function= tanh solver = L-BFGS-B Proposed Method 2 See section 3.2.1. Neural Network 97.7% Solver= RMSprop (31) Zoning KNN 72.54% For single character Zoning SVM 73.02% Polynomial Kernel Transition KNN 86.57% (20) ZD ,Distance Profile Projection Histograms, BDD SVM, KNN 95.01% 190 samples (32) projection profile histogram SVM 97.58% 124 no. of mathematical symbols (33) ___ CNN 83% ReLU as the activation function Neural Network 91% neural net uses both softmax and sigmoid as the activation functions According to the above results, it is clear that Method 1 exceeds Method 2 in terms of recognition rates and evaluation metrics. Since it achieved an overall recognition rate of 96.6% compared to 97.7% achieved by Method 2. Both methods have a high level of competitiveness compared with similar studies. IHJPAS. 2025,38(4) 304 5. Conclusion The paper introduces a three-step approach aimed at enhancing the recognition of mathematical expressions in scanned documents using OCR systems. The approach involves extracting various features from the vertical and horizontal projection of the segmented symbols. For symbol recognition, the paper suggested two neural network-based classification methods. The first method employs the tanh activation function with the L- BFGS-B optimization method to achieve recognition rates of 96.6%. While the second method utilizes the RMSprop optimization algorithm to achieve recognition rates of 97.7%. Additionally, the paper presents three guidelines for accurately interpreting the mathematical meaning of the expression and converting it into textual form. The experimental results of the paper demonstrate improvements in recognition rates and the accurate interpretation of mathematical expressions for their conversion from image format to textual format. Acknowledgment None. Conflict of Interest The authors declare that they have no conflicts of interest. Funding No funding. References 1. Memon J, Sami M, Khan RA. 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