An Image-Based Measure for Evaluation of Mathematical Expression Recognition

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-38628-2_81Mathematical expression recognition is an active research field that is related to document image analysis and typesetting. In this study, we present a novel global performance evaluation measure for mathematical expression recognition based on image matching. Using an image representation for evaluation tries to overcome the representation ambiguity as human beings do. The results of a recent competition were used to perform several experiments in order to analyze the benefits and drawbacks of this measure.This work was partially supported by the Spanish MEC under the STraDA research project (TIN2012-37475-C02-01), the MITTRAL (TIN2009-14633-C03-01) project, the FPU grant (AP2009-4363), by the Generalitat Valenciana under the grant Prometeo/2009/014, and through the EU 7th Framework Programme grant tranScriptorium (Ref: 600707)Álvaro Muñoz, F.; Sánchez Peiró, JA.; Benedí Ruiz, JM. (2013). An Image-Based Measure for Evaluation of Mathematical Expression Recognition. En Pattern Recognition and Image Analysis. Springer. 682-690. https://doi.org/10.1007/978-3-642-38628-2_81S682690Álvaro, F., Sánchez, J.A., Benedí, J.M.: Unbiased evaluation of handwritten mathematical expression recognition. In: Proceedings of ICFHR, Italy, pp. 181–186 (2012)Chan, K.F., Yeung, D.Y.: Error detection, error correction and performance evaluation in on-line mathematical expression recognition. Pattern Recognition 34(8), 1671–1684 (2001)Chou, P.A.: Recognition of equations using a two-dimensional stochastic context-free grammar. In: Pearlman, W.A. (ed.) Visual Communications and Image Processing IV. SPIE Proceedings Series, vol. 1199, pp. 852–863 (1989)Garain, U., Chaudhuri, B.B.: A corpus for OCR research on mathematical expressions. Int. Journal on Document Analysis and Recognition 7, 241–259 (2005)Keysers, D., Deselaers, T., Gollan, C., Ney, H.: Deformation models for image recognition. IEEE Trans. on Pattern Analysis and Machine Intelligence 29(8), 1422–1435 (2007)Mouchére, H., Viard-Gaudin, C., Garain, U., Kim, D.H., Kim, J.H.: ICFHR 2012 – Competition on Recognition of On-line Mathematical Expressions (CROHME 2012). In: Proceedings of ICFHR, Italy, pp. 807–812 (2012)Otsu, N.: A Threshold Selection Method from Gray-level Histograms. IEEE Transactions on Systems, Man and Cybernetics 9(1), 62–66 (1979)Sain, K., Dasgupta, A., Garain, U.: EMERS: a tree matching-based performance evaluation of mathematical expression recognition system. International Journal of Document Analysis and Recognition (2010)Toselli, A.H., Juan, A., Vidal, E.: Spontaneous Handwriting Recognition and Classification. In: Proceedings of ICPR, England, UK, pp. 433–436 (2004)Zanibbi, R., Blostein, D., Cordy, J.R.: Recognizing mathematical expressions using tree transformation. IEEE Trans. on Pattern Analysis and Machine Intelligence 24(11), 1–13 (2002)Zanibbi, R., Pillay, A., Mouchere, H., Viard-Gaudin, C., Blostein, D.: Stroke-based performance metrics for handwritten mathematical expressions. In: Proceedings of ICDAR, pp. 334–338 (2011

A Survey on Retrieval of Mathematical Knowledge

We present a short survey of the literature on indexing and retrieval of mathematical knowledge, with pointers to 72 papers and tentative taxonomies of both retrieval problems and recurring techniques.Comment: CICM 2015, 20 page

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

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

For Geometric Inference from Images, What Kind of Statistical Model Is Necessary?

In order to facilitate smooth communications with researchers in other fields including statistics, this paper investigates the meaning of "statistical methods" for geometric inference based on image feature points, We point out that statistical analysis does not make sense unless the underlying "statistical ensemble" is clearly defined. We trace back the origin of feature uncertainty to image processing operations for computer vision in general and discuss the implications of asymptotic analysis for performance evaluation in reference to "geometric fitting", "geometric model selection", the "geometric AIC", and the "geometric MDL". Referring to such statistical concepts as "nuisance parameters", the "Neyman-Scott problem", and "semiparametric models", we point out that simulation experiments for performance evaluation will lose meaning without carefully considering the assumptions involved and intended applications

Taming Gradient Variance in Federated Learning with Networked Control Variates

Federated learning, a decentralized approach to machine learning, faces significant challenges such as extensive communication overheads, slow convergence, and unstable improvements. These challenges primarily stem from the gradient variance due to heterogeneous client data distributions. To address this, we introduce a novel Networked Control Variates (FedNCV) framework for Federated Learning. We adopt the REINFORCE Leave-One-Out (RLOO) as a fundamental control variate unit in the FedNCV framework, implemented at both client and server levels. At the client level, the RLOO control variate is employed to optimize local gradient updates, mitigating the variance introduced by data samples. Once relayed to the server, the RLOO-based estimator further provides an unbiased and low-variance aggregated gradient, leading to robust global updates. This dual-side application is formalized as a linear combination of composite control variates. We provide a mathematical expression capturing this integration of double control variates within FedNCV and present three theoretical results with corresponding proofs. This unique dual structure equips FedNCV to address data heterogeneity and scalability issues, thus potentially paving the way for large-scale applications. Moreover, we tested FedNCV on six diverse datasets under a Dirichlet distribution with {\alpha} = 0.1, and benchmarked its performance against six SOTA methods, demonstrating its superiority.Comment: 14 page