Mathematical Expression Recognition based on Probabilistic Grammars

[EN] Mathematical notation is well-known and used all over the world. Humankind has evolved from simple methods representing countings to current well-defined math notation able to account for complex problems. Furthermore, mathematical expressions constitute a universal language in scientific fields, and many information resources containing mathematics have been created during the last decades. However, in order to efficiently access all that information, scientific documents have to be digitized or produced directly in electronic formats. Although most people is able to understand and produce mathematical information, introducing math expressions into electronic devices requires learning specific notations or using editors. Automatic recognition of mathematical expressions aims at filling this gap between the knowledge of a person and the input accepted by computers. This way, printed documents containing math expressions could be automatically digitized, and handwriting could be used for direct input of math notation into electronic devices. This thesis is devoted to develop an approach for mathematical expression recognition. In this document we propose an approach for recognizing any type of mathematical expression (printed or handwritten) based on probabilistic grammars. In order to do so, we develop the formal statistical framework such that derives several probability distributions. Along the document, we deal with the definition and estimation of all these probabilistic sources of information. Finally, we define the parsing algorithm that globally computes the most probable mathematical expression for a given input according to the statistical framework. An important point in this study is to provide objective performance evaluation and report results using public data and standard metrics. We inspected the problems of automatic evaluation in this field and looked for the best solutions. We also report several experiments using public databases and we participated in several international competitions. Furthermore, we have released most of the software developed in this thesis as open source. We also explore some of the applications of mathematical expression recognition. In addition to the direct applications of transcription and digitization, we report two important proposals. First, we developed mucaptcha, a method to tell humans and computers apart by means of math handwriting input, which represents a novel application of math expression recognition. Second, we tackled the problem of layout analysis of structured documents using the statistical framework developed in this thesis, because both are two-dimensional problems that can be modeled with probabilistic grammars. The approach developed in this thesis for mathematical expression recognition has obtained good results at different levels. It has produced several scientific publications in international conferences and journals, and has been awarded in international competitions.[ES] La notación matemática es bien conocida y se utiliza en todo el mundo. La humanidad ha evolucionado desde simples métodos para representar cuentas hasta la notación formal actual capaz de modelar problemas complejos. Además, las expresiones matemáticas constituyen un idioma universal en el mundo científico, y se han creado muchos recursos que contienen matemáticas durante las últimas décadas. Sin embargo, para acceder de forma eficiente a toda esa información, los documentos científicos han de ser digitalizados o producidos directamente en formatos electrónicos. Aunque la mayoría de personas es capaz de entender y producir información matemática, introducir expresiones matemáticas en dispositivos electrónicos requiere aprender notaciones especiales o usar editores. El reconocimiento automático de expresiones matemáticas tiene como objetivo llenar ese espacio existente entre el conocimiento de una persona y la entrada que aceptan los ordenadores. De este modo, documentos impresos que contienen fórmulas podrían digitalizarse automáticamente, y la escritura se podría utilizar para introducir directamente notación matemática en dispositivos electrónicos. Esta tesis está centrada en desarrollar un método para reconocer expresiones matemáticas. En este documento proponemos un método para reconocer cualquier tipo de fórmula (impresa o manuscrita) basado en gramáticas probabilísticas. Para ello, desarrollamos el marco estadístico formal que deriva varias distribuciones de probabilidad. A lo largo del documento, abordamos la definición y estimación de todas estas fuentes de información probabilística. Finalmente, definimos el algoritmo que, dada cierta entrada, calcula globalmente la expresión matemática más probable de acuerdo al marco estadístico. Un aspecto importante de este trabajo es proporcionar una evaluación objetiva de los resultados y presentarlos usando datos públicos y medidas estándar. Por ello, estudiamos los problemas de la evaluación automática en este campo y buscamos las mejores soluciones. Asimismo, presentamos diversos experimentos usando bases de datos públicas y hemos participado en varias competiciones internacionales. Además, hemos publicado como código abierto la mayoría del software desarrollado en esta tesis. También hemos explorado algunas de las aplicaciones del reconocimiento de expresiones matemáticas. Además de las aplicaciones directas de transcripción y digitalización, presentamos dos propuestas importantes. En primer lugar, desarrollamos mucaptcha, un método para discriminar entre humanos y ordenadores mediante la escritura de expresiones matemáticas, el cual representa una novedosa aplicación del reconocimiento de fórmulas. En segundo lugar, abordamos el problema de detectar y segmentar la estructura de documentos utilizando el marco estadístico formal desarrollado en esta tesis, dado que ambos son problemas bidimensionales que pueden modelarse con gramáticas probabilísticas. El método desarrollado en esta tesis para reconocer expresiones matemáticas ha obtenido buenos resultados a diferentes niveles. Este trabajo ha producido varias publicaciones en conferencias internacionales y revistas, y ha sido premiado en competiciones internacionales.[CA] La notació matemàtica és ben coneguda i s'utilitza a tot el món. La humanitat ha evolucionat des de simples mètodes per representar comptes fins a la notació formal actual capaç de modelar problemes complexos. A més, les expressions matemàtiques constitueixen un idioma universal al món científic, i s'han creat molts recursos que contenen matemàtiques durant les últimes dècades. No obstant això, per accedir de forma eficient a tota aquesta informació, els documents científics han de ser digitalitzats o produïts directament en formats electrònics. Encara que la majoria de persones és capaç d'entendre i produir informació matemàtica, introduir expressions matemàtiques en dispositius electrònics requereix aprendre notacions especials o usar editors. El reconeixement automàtic d'expressions matemàtiques té per objectiu omplir aquest espai existent entre el coneixement d'una persona i l'entrada que accepten els ordinadors. D'aquesta manera, documents impresos que contenen fórmules podrien digitalitzar-se automàticament, i l'escriptura es podria utilitzar per introduir directament notació matemàtica en dispositius electrònics. Aquesta tesi està centrada en desenvolupar un mètode per reconèixer expressions matemàtiques. En aquest document proposem un mètode per reconèixer qualsevol tipus de fórmula (impresa o manuscrita) basat en gramàtiques probabilístiques. Amb aquesta finalitat, desenvolupem el marc estadístic formal que deriva diverses distribucions de probabilitat. Al llarg del document, abordem la definició i estimació de totes aquestes fonts d'informació probabilística. Finalment, definim l'algorisme que, donada certa entrada, calcula globalment l'expressió matemàtica més probable d'acord al marc estadístic. Un aspecte important d'aquest treball és proporcionar una avaluació objectiva dels resultats i presentar-los usant dades públiques i mesures estàndard. Per això, estudiem els problemes de l'avaluació automàtica en aquest camp i busquem les millors solucions. Així mateix, presentem diversos experiments usant bases de dades públiques i hem participat en diverses competicions internacionals. A més, hem publicat com a codi obert la majoria del software desenvolupat en aquesta tesi. També hem explorat algunes de les aplicacions del reconeixement d'expressions matemàtiques. A més de les aplicacions directes de transcripció i digitalització, presentem dues propostes importants. En primer lloc, desenvolupem mucaptcha, un mètode per discriminar entre humans i ordinadors mitjançant l'escriptura d'expressions matemàtiques, el qual representa una nova aplicació del reconeixement de fórmules. En segon lloc, abordem el problema de detectar i segmentar l'estructura de documents utilitzant el marc estadístic formal desenvolupat en aquesta tesi, donat que ambdós són problemes bidimensionals que poden modelar-se amb gramàtiques probabilístiques. El mètode desenvolupat en aquesta tesi per reconèixer expressions matemàtiques ha obtingut bons resultats a diferents nivells. Aquest treball ha produït diverses publicacions en conferències internacionals i revistes, i ha sigut premiat en competicions internacionals.Álvaro Muñoz, F. (2015). Mathematical Expression Recognition based on Probabilistic Grammars [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/51665TESI

Multi-Scale Attention with Dense Encoder for Handwritten Mathematical Expression Recognition

Handwritten mathematical expression recognition is a challenging problem due to the complicated two-dimensional structures, ambiguous handwriting input and variant scales of handwritten math symbols. To settle this problem, we utilize the attention based encoder-decoder model that recognizes mathematical expression images from two-dimensional layouts to one-dimensional LaTeX strings. We improve the encoder by employing densely connected convolutional networks as they can strengthen feature extraction and facilitate gradient propagation especially on a small training set. We also present a novel multi-scale attention model which is employed to deal with the recognition of math symbols in different scales and save the fine-grained details that will be dropped by pooling operations. Validated on the CROHME competition task, the proposed method significantly outperforms the state-of-the-art methods with an expression recognition accuracy of 52.8% on CROHME 2014 and 50.1% on CROHME 2016, by only using the official training dataset

Recognition of on-line handwritten mathematical expressions using 2D stochastic context-free grammars and hidden Markov models

[EN] This paper describes a formal model for the recognition of on-line handwritten mathematical expressions using 2D stochastic context-free grammars and hidden Markov models. Hidden Markov models are used to recognize mathematical symbols, and a stochastic context-free grammar is used to model the relation between these symbols. This formal model makes possible to use classic algorithms for parsing and stochastic estimation. In this way, first, the model is able to capture many of variability phenomena that appear in on-line handwritten mathematical expressions during the training process. And second, the parsing process can make decisions taking into account only stochastic information, and avoiding heuristic decisions. The proposed model participated in a contest of mathematical expression recognition and it obtained the best results at different levels. 2012 Elsevier B.V. All rights reserved.Work supported by the EC (FEDER/ FSE) and the Spanish MEC/MICINN under the MIPRCV ‘‘Consolider Ingenio 2010’’ program (CSD2007-00018), the MITTRAL (TIN2009-14633-C03-01) project, the FPU Grant (AP2009-4363), and by the Generalitat Valenciana under the Grant Prometeo/2009/014.Álvaro Muñoz, F.; Sánchez Peiró, JA.; Benedí Ruiz, JM. (2014). Recognition of on-line handwritten mathematical expressions using 2D stochastic context-free grammars and hidden Markov models. Pattern Recognition Letters. 35:58-67. https://doi.org/10.1016/j.patrec.2012.09.023S58673

Augmented incremental recognition of online handwritten mathematical expressions

This paper presents an augmented incremental recognition method for online handwritten mathematical expressions (MEs). If an ME is recognized after all strokes are written (batch recognition), the waiting time increases significantly when the ME becomes longer. On the other hand, the pure incremental recognition method recognizes an ME whenever a new single stroke is input. It shortens the waiting time but degrades the recognition rate due to the limited context. Thus, we propose an augmented incremental recognition method that not only maintains the advantage of the two methods but also reduces their weaknesses. The proposed method has two main features: one is to process the latest stroke, and the other is to find the erroneous segmentations and recognitions in the recent strokes and correct them. In the first process, the segmentation and the recognition by Cocke-Younger-Kasami (CYK) algorithm are only executed for the latest stroke. In the second process, all the previous segmentations are updated if they are significantly changed after the latest stroke is input, and then, all the symbols related to the updated segmentations are updated with their recognition scores. These changes are reflected in the CYK table. In addition, the waiting time is further reduced by employing multi-thread processes. Experiments on our dataset and the CROHME datasets show the effectiveness of this augmented incremental recognition method, which not only maintains recognition rate even compared with the batch recognition method but also reduces the waiting time to a very small level

Features and Algorithms for Visual Parsing of Handwritten Mathematical Expressions

Math expressions are an essential part of scientific documents. Handwritten math expressions recognition can benefit human-computer interaction especially in the education domain and is a critical part of document recognition and analysis. Parsing the spatial arrangement of symbols is an essential part of math expression recognition. A variety of parsing techniques have been developed during the past three decades, and fall into two groups. The first group is graph-based parsing. It selects a path or sub-graph which obeys some rule to form a possible interpretation for the given expression. The second group is grammar driven parsing. Grammars and related parameters are defined manually for different tasks. The time complexity of these two groups parsing is high, and they often impose some strict constraints to reduce the computation. The aim of this thesis is working towards building a straightforward and effective parser with as few constraints as possible. First, we propose using a line of sight graph for representing the layout of strokes and symbols in math expressions. It achieves higher F-score than other graph representations and reduces search space for parsing. Second, we modify the shape context feature with Parzen window density estimation. This feature set works well for symbol segmentation, symbol classification and symbol layout analysis. We get a higher symbol segmentation F-score than other systems on CROHME 2014 dataset. Finally, we develop a Maximum Spanning Tree (MST) based parser using Edmonds\u27 algorithm, which extracts an MST from the directed line of sight graph in two passes: first symbols are segmented, and then symbols and spatial relationship are labeled. The time complexity of our MST-based parsing is lower than the time complexity of CYK parsing with context-free grammars. Also, our MST-based parsing obtains higher structure rate and expression rate than CYK parsing when symbol segmentation is accurate. Correct structure means we get the structure of the symbol layout tree correct, even though the label of the edge in the symbol layout tree might be wrong. The performance of our math expression recognition system with MST-based parsing is competitive on CROHME 2012 and 2014 datasets. For future work, how to incorporate symbol classifier result and correct segmentation error in MST-based parsing needs more research

Intelligent Combination of Structural Analysis Algorithms: Application to Mathematical Expression Recognition

Structural analysis is an important step in many document based recognition problem. Structural analysis is performed to associate elements in a document and assign meaning to their association. Handwritten mathematical expression recognition is one such problem which has been studied and researched for long. Many techniques have been researched to build a system that produce high performance mathematical expression recognition. We have presented a novel method to combine multiple structural recognition algorithms in which the combined result shows better performance than each individual recognition algorithms. In our experiment we have applied our method to combine multiple mathematical expression recognition parsers called DRACULAE. We have used Graph Transformation Network (GTN) which is a network of function based systems in which each system takes graphs as input, apply function and produces a graph as output. GTN is used to combine multiple DRACULAE parsers and its parameter are tuned using gradient based learning. It has been shown that such a combination method can be used to accentuate the strength of individual algorithms in combination to produce better combination result which higher recognition performance. In our experiment we were able to obtain a highest recognition rate of 74% as compared to best recognition result of 70% from individual DRACULAE parsers. Our experiment also resulted into a maximum of 20% reduction of parent recognition errors and maximum 37% reduction in relation recognition errors between symbols in expressions