Handwritten Mathematical Expression Recognition with Bidirectionally Trained Transformer

A survey on handwritten mathematical expression recognition: The rise of encoder-decoder and GNN models

Handwritten mathematical expression recognition (HMER) is a challenging task due to the complex two-dimensional structure of mathematical expressions and the similarity of handwritten texts. Most existing methods for HMER only consider single-scale features while ignoring multi-scale features that are very important to HMER. Few works have explored the fusion of multi-scale features in HMER, but exhibited an extra branch that brings more parameters and computation. In this paper, we propose an end-to-end method to integrate multi-scale features using a unified model. Specifically, we customized the Dense Atrous Spatial Pyramid Pooling (DenseASPP) to our backbone network to capture the multi-scale features of the input image meanwhile expanding the receptive fields. Moreover, we added a symbol classifier using focal loss to better discriminate and recognize similar symbols, to further improve the performance of HMER. Experiments on the Competition on Recognition of Online Handwritten Mathematical Expressions (CROHME) 2014, 2016 and 2019 shows that the proposed method achieves superior performance to most state-of-the-art methods, demonstrating the effectiveness of the proposed method.

Handwritten mathematical expression recognition (HMER) poses a formidable challenge due to the intricate two-dimensional structures and diverse handwriting styles. This paper introduces a novel approach to improve HMER accuracy by employing an integrated, high-capacity architecture that combines Transformer and Convolutional Neural Network (CNN) models, along with a denoising diffusion probabilistic model (DDPM)-based data augmentation technique. We explore three combination strategies for an attention-based encoder-decoder (AED) HMER model: 1) The “Tandem” strategy, which harnesses CNN features within a Transformer encoder to capture global interdependencies; 2) The “Parallel” strategy, which integrates Transformer encoder outputs with CNN outputs to achieve comprehensive feature fusion; 3) The “Mixing” strategy, which introduces multi-head self-attention (MHSA) at the final stage of the CNN. We evaluate these methods using the CROHME benchmark dataset and conduct a detailed comparative analysis. All three approaches significantly enhance model performance. Notably, the “Tandem” approach achieves expression recognition rates (ExpRate) of 54.85% and 58.56% on the CROHME 2016 and 2019 test sets, respectively, while the “Parallel” method attains 55.63% and 57.39% on the same test sets. Furthermore, we introduce an innovative data augmentation approach that utilizes DDPM to generate synthetic training samples. The DDPM, conditioned on LaTeX-rendered images, bridges the gap between printed and handwritten expressions, enabling the creation of realistic, stylistically diverse handwriting samples. This augmentation boosts the ExpRates of all strategies on both CROHME 2016 and 2019 test sets, yielding improvements of 1.6-4.6% relative to the unaugmented dataset.

This paper presents the competition on Offline Recognition and Spotting of Handwritten Mathematical Expressions (OffRaSHME) held at the 17th International Conference on Frontiers in Handwriting Recognition (ICFHR 2020). Handwritten Mathematical Expression Recognition (HMER) has wide potential applications and is capturing increasing attention in recent years. Previous HMER competitions mainly focused on online datasets or offline datasets that are converted from online data. In this competition, we have collected a dataset of offline handwritten mathematical expressions by scanning papers that contain expressions. Moreover, we labeled the offline dataset at symbol level, i.e., the bounding boxes of each symbol are also provided, to facilitate the research of HMER. At last, 19,749 offline handwritten mathematical expressions are collected for training, and 2,000 ones are provided for evaluating the participating systems. In the competition, 7 teams submitted 8 systems for the task of offline HMER, among which 5 systems only use the provided datasets without any extra data while 3 systems use extra data. The winner team achieved a recognition accuracy of 79.85% (without extra data) and 81.85% (with extra data) on the offline formula recognition task.

We present a technique based on stroke order normalization for improving recognition of online handwritten mathematical expressions (ME). The stroke order dependent system has less time complexity than the stroke order free system, but it must incorporate special grammar rules to cope with stroke order variations. The stroke order normalization technique solves this problem and also the problem of unexpected stroke order variations without increasing the time complexity of ME recognition. In order to normalize stroke order, the X–Y cut method is modified since its original form causes problems when structural components in ME overlap. First, vertically ordered strokes are located by detecting vertical symbols and their upper/lower components, which are treated as MEs and reordered recursively. Second, unordered strokes on the left side of the vertical symbols are reordered as horizontally ordered strokes. Third, the remaining strokes are reordered recursively. The horizontally ordered strokes are reordered from left to right, and the vertically ordered strokes are reordered from top to bottom. Finally, the proposed stroke order normalization is combined with the stroke order dependent ME recognition system. The evaluations on the CROHME 2014 database show that the ME recognition system incorporating the stroke order normalization outperforms all other systems that use only CROHME 2014 for training while the processing time is kept low.

We present a system for recognizing online mathematical expressions (ME). Symbol recognition is based on a template elastic matching distance between pen direction features. The structural analysis of the ME is based on extracting the baseline of the ME and then classifying symbols into levels above and below the baseline. The symbols are then sequentially analyzed using six spatial relations and a respective 2d structure is processed to give the resulting MathML representation of the ME. The system was evaluated on the Competition on Recognition of Online Handwritten Mathematical Expressions (CROHME) 2011 datasets and demonstrates promising results.

Numerical expression evaluation from sentences using Natural Language processing is a well-known problem domain. Evaluation of noise-free mathematical expressions has been done previously to assess the ability of the recurrent neural network to solve numerical expressions. However, the evaluation of numerical expressions when present in noisy format, i.e., operator and operands are represented “in-word” and mixed numerical format, are not well nourished till now. This paper presents a comparative study to analyze the ability of sequence-to-sequence neural network based Encoder-Decoder models to calculate the numerical output of the noisy expressions. We have treated the problem as a sequence-to-sequence based classification problem instead of a regression problem as considered earlier, to deal with the noisy data. Performance of three different Encoder-Decoder models namely Unidirectional, Bidirectional, and Attention based Encoder-Decoder LSTM models are used to evaluate the noisy expressions. It has been experimentally observed that Bidirectional and Attention based Encoder-Decoder LSTM models perform better than Unidirectional EncoderDecoder model and the sequence-to-sequence model.

We summarize the tasks, protocol, and outcome for the 6th Competition on Recognition of Handwritten Mathematical Expressions (CROHME), which includes a new formula detection in document images task (+ TFD). For CROHME + TFD 2019, participants chose between two tasks for recognizing handwritten formulas from 1) online stroke data, or 2) images generated from the handwritten strokes. To compare LATEX strings and the labeled directed trees over strokes (label graphs) used in previous CROHMEs, we convert LATEX and stroke-based label graphs to label graphs defined over symbols (symbol-level label graphs, or symLG). More than thirty (33) participants registered for the competition, with nineteen (19) teams submitting results. The strongest formula recognition results were produced by the USTC-iFLYTEK research team, for both stroke-based (81%) and image-based (77%) input. For the new typeset formula detection task, the Samsung R&D Institute Ukraine (Team 2) obtained a very strong F-score (93%). System performance has improved since the last CROHME - still, the competition results suggest that recognition of handwritten formulae remains a difficult structural pattern recognition task.

Recognition of spatial relations between pairs of subexpressions is a key problem of recognition of handwritten mathematical expressions. Most methods for spatial relation classification are based on handcrafted rules and geometric indices extracted from the subexpression bounding boxes. In this work, we propose new spatial relation features that combine subexpression bounding box and intra-subexpression information, along with prior knowledge about the general position and size of symbols. Instead of handcrafting features, we train artificial neural networks to learn the useful features from two kinds of histograms. The first type captures the relative positions and sizes of the subexpression bounding boxes. The second captures the relative positions and shape of a pair of symbols, called dominant symbols, extracted from the main baselines of the evaluated subexpressions. We evaluate and compare our features with two state-of-the-art features on a benchmark dataset. Experimental results show that our features obtain better accuracy than these two features.

Handwritten mathematical expression recognition (HMER) is an essential task in the OCR community, which consists of two sub-tasks, i.e., symbol recognition and structure parsing. Modern literature treats HMER as a LaTeX sequence predicting problem that simultaneously recognizes symbols and parses the structures of MEs. Although deep learning-based HMER methods have been achieving promising results on public benchmarks, it is admitted that the misclassification error between visually similar symbols still prevents these approaches from more generalized scenes. In this paper, we try to solve this issue from three aspects. 1) We enhanced the feature extraction progress by introducing path signature features, which incorporates local writing details and global spatial information. 2) We developed a language model that uses contextual information to correct the symbols misclassified by vision-only-based recognition models. 3) We solved the misalignment problem in existing ensemble method by designing a dynamic time warping (DTW) based algorithm. By combining the above improvements, our method achieved state-of-the-art results on three CROHME benchmarks, outperforming previous methods by a large margin.

Identifying strong handwritten characters is a difficult task in the field of medical field and it is tedious process on decoding handwritten medicines. Recognition of handwritten mathematical expressions is a complex issue. The distribution and classification of specific characters makes the task more difficult. In our project, handwritten numbers and symbols are read and further addition, subtraction and multiplication operations are performed. The project includes a study about model deployment using convoluted neural networks and flasks. We use the CNN to classify specific characters. Tracking of character string operations are used to solve equations. The maximum accuracy of the proposed model is 99.12% recall is 95%, sensitivity is 89% and specificity is 68%. Effectiveness of our proposed system is helpful for students who want to get handwritten answers. The equation can be extended to more complex equations and more user data can be trained to improve correction and accuracy.

We present a temporal classification constraint as an auxiliary learning method for improving the recognition of Handwritten Mathematical Expression (HME). Connectionist temporal classification (CTC) is used to learn the temporal alignment of the input feature sequence and corresponding symbol label sequence. The CTC alignment is trained with the encoder-decoder alignment through a combination of CTC loss and encoder-decoder loss to improve the feature learning of the encoder in the encoder-decoder model. We show the effectiveness of the approach in improving symbol classification and expression recognition on the CROHME datasets.

Mathematical expression recognition is still a very challenging task for the research community mainly because of the two-dimensional (2d) structure of mathematical expressions (MEs). In this paper, we present a novel approach for the structural analysis between two on-line handwritten mathematical symbols of a ME, based on spatial features of the symbols. We introduce six features to represent the spatial affinity of the symbols and compare two multi-class classification methods that employ support vector machines (SVMs): one based on the “one-against-one” technique and one based on the “one-against-all”, in identifying the relation between a pair of symbols (i.e. subscript, numerator, etc). A dataset containing 1906 spatial relations derived from the Competition on Recognition of Online Handwritten Mathematical Expressions (CROHME) 2012 training dataset is constructed to evaluate the classifiers and compare them with the rule-based classifier of the ILSP-1 system participated in the contest. The experimental results give an overall mean error rate of 2.61% for the “one-against-one” SVM approach, 6.57% for the “one-against-all” SVM technique and 12.31% error rate for the ILSP-1 classifier.

Convolutional Neural Networks have achieved great performance in computer vision and pattern recognition applications. In this work, we propose the use of Convolutional Neural Networks in order to improve the handwritten mathematical symbol classification rate. We trained a Convolutional Neural Networks to classifying mathematical symbols into a proper class using Competition on Recognition of Online Handwritten Mathematical Expressions (CROHME) 2014 dataset. The results show that Convolutional Neural Networks outperforms the previous works in terms of accuracy.

Collecting handwritten mathematical expressions (HMEs) generally requires a lot of time and effort for data preparation, data collection, annotation and so on. In this paper, we present a method for generating realistic HMEs in a wide variety of structures and styles from a LaTeX sequence or a MathML script using online isolated symbol patterns. Our method firstly positions all symbols in a symbol relation tree constructed from the input LaTeX or MathML script. Then, it places normalized online symbol patterns into the corresponding locations. A questionnaire-based experiment shows that the synthetic patterns are as clear and natural as the real patterns. Therefore, we can use the generated synthetic HME patterns for research on HME recognition and clustering.