An integrated grammar-based approach for mathematical expression recognition

This is the author’s version of a work that was accepted for publication in Pattern Recognition. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Pattern Recognition 51 (2016) 135–147. DOI 10.1016/j.patcog.2015.09.013.Automatic recognition of mathematical expressions is a challenging pattern recognition problem since there are many ambiguities at different levels. On the one hand, the recognition of the symbols of the mathematical expression. On the other hand, the detection of the two-dimensional structure that relates the symbols and represents the math expression. These problems are closely related since symbol recognition is influenced by the structure of the expression, while the structure strongly depends on the symbols that are recognized. For these reasons, we present an integrated approach that combines several stochastic sources of information and is able to globally determine the most likely expression. This way, symbol segmentation, symbol recognition and structural analysis are simultaneously optimized. In this paper we define the statistical framework of a model based on two-dimensional grammars and its associated parsing algorithm. Since the search space is too large, restrictions are introduced for making the search feasible. We have developed a system that implements this approach and we report results on the large public dataset of the CROHME international competition. This approach significantly outperforms other proposals and was awarded best system using only the training dataset of the competition. (C) 2015 Elsevier Ltd. All rights reserved.This work was partially supported by the Spanish MINECO under the STraDA research project (TIN2012-37475-C02-01) and the FPU Grant (AP2009-4363).Álvaro Muñoz, F.; Sánchez Peiró, JA.; Benedí Ruiz, JM. (2016). An integrated grammar-based approach for mathematical expression recognition. Pattern Recognition. 51:135-147. https://doi.org/10.1016/j.patcog.2015.09.013S1351475

Stroke order normalization for improving recognition of online handwritten mathematical expressions

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

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

Query-Driven Global Graph Attention Model for Visual Parsing: Recognizing Handwritten and Typeset Math Formulas

We present a new visual parsing method based on standard Convolutional Neural Networks (CNNs) for handwritten and typeset mathematical formulas. The Query-Driven Global Graph Attention (QD-GGA) parser employs multi-task learning, using a single feature representation for locating, classifying, and relating symbols. QD-GGA parses formulas by first constructing a Line-Of-Sight (LOS) graph over the input primitives (e.g handwritten strokes or connected components in images). Second, class distributions for LOS nodes and edges are obtained using query-specific feature filters (i.e., attention) in a single feed-forward pass. This allows end-to-end structure learning using a joint loss over primitive node and edge class distributions. Finally, a Maximum Spanning Tree (MST) is extracted from the weighted graph using Edmonds\u27 Arborescence Algorithm. The model may be run recurrently over the input graph, updating attention to focus on symbols detected in the previous iteration. QD-GGA does not require additional grammar rules and the language model is learned from the sets of symbols/relationships and the statistics over them in the training set. We benchmark our system against both handwritten and typeset state-of-the-art math recognition systems. Our preliminary results show that this is a promising new approach for visual parsing of math formulas. Using recurrent execution, symbol detection is near perfect for both handwritten and typeset formulas: we obtain a symbol f-measure of over 99.4% for both the CROHME (handwritten) and INFTYMCCDB-2 (typeset formula image) datasets. Our method is also much faster in both training and execution than state-of-the-art RNN-based formula parsers. The unlabeled structure detection of QDGGA is competitive with encoder-decoder models, but QD-GGA symbol and relationship classification is weaker. We believe this may be addressed through increased use of spatial features and global context

WordSup: Exploiting Word Annotations for Character based Text Detection

Imagery texts are usually organized as a hierarchy of several visual elements, i.e. characters, words, text lines and text blocks. Among these elements, character is the most basic one for various languages such as Western, Chinese, Japanese, mathematical expression and etc. It is natural and convenient to construct a common text detection engine based on character detectors. However, training character detectors requires a vast of location annotated characters, which are expensive to obtain. Actually, the existing real text datasets are mostly annotated in word or line level. To remedy this dilemma, we propose a weakly supervised framework that can utilize word annotations, either in tight quadrangles or the more loose bounding boxes, for character detector training. When applied in scene text detection, we are thus able to train a robust character detector by exploiting word annotations in the rich large-scale real scene text datasets, e.g. ICDAR15 and COCO-text. The character detector acts as a key role in the pipeline of our text detection engine. It achieves the state-of-the-art performance on several challenging scene text detection benchmarks. We also demonstrate the flexibility of our pipeline by various scenarios, including deformed text detection and math expression recognition.Comment: 2017 International Conference on Computer Visio

Discriminative estimation of probabilistic context-free grammars for mathematical expression recognition and retrieval

[EN] We present a discriminative learning algorithm for the probabilistic estimation of two-dimensional probabilistic context-free grammars (2D-PCFG) for mathematical expressions recognition and retrieval. This algorithm is based on a generalization of the H-criterion as the objective function and the growth transformations as the optimization method. For the development of the discriminative estimation algorithm, the N-best interpretations provided by the 2D-PCFG have been considered. Experimental results are reported on two available datasets: Im2Latex and IBEM. The first experiment compares the proposed discriminative estimation method with the classic Viterbi-based estimation method. 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