Structural Analysis of Mathematical Formulae with Verification Based on Formula Description Grammar

Abstract

In this paper, a reliable and efficient structural analysis method for mathematical formulae is proposed for practical mathematical OCR. The proposed method consists of three steps. In the first step, a fast structural analysis algorithm is performed on each mathematical formula to obtain a tree representation of the formula. This step generally provides a correct tree representation but sometimes provides an erroneous representation. Therefore, the tree representation is verified by the following two steps. In the second step, the result of the analysis step, (i.e., a tree representation) is converted into a one-dimensional representation. The third step is a verification step where the one-dimensional representation is parsed by a formula description grammar, which is a context-free grammar specialized for mathematical formulae. If the onedimensional representation is not accepted by the grammar, the result of the analysis step is detected as an erroneous result and alarmed to OCR users. This three-step organization achieves reliable and efficient structural analysis without any two-dimensional grammars.

Progressive structural analysis for dynamic recognition of on-line handwritten mathematical expressions

“…More recently, Toyota et al (Toyota et al, 2006) converted a tree representation of a mathematical structure into a one-dimensional representation and then parsed by a formula description grammar.…”

“…These include grammar-based ap-proaches (Chou, 1989;Fateman et al, 1996;Chan and Yeung, 2000a;Toyota et al, 2006), tree transformation (Zanibbi et al, 2002), Hidden Markov Models (HMMs) (Kosmala and Rigoll, 1998) and Minimum Spanning Tree Rojas, 2003, 2005). Some of the techniques such as the grammar-based approaches are slow, while others are sensitive to users' writing errors.…”

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“…Grammars may also be used as a verification step to confirm the validity of an expression recognized by some other means (e.g., [13,37]). Garain and Chaudhuri, for example, view an expression as a collection of horizontal baselines, which they call "levels".…”

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“…These include grammar-based approaches (Chou, 1989;Fateman et al, 1996;Chan and Yeung, 2000a;Toyota et al, 2006), tree transformation (Zanibbi et al, 2002), Hidden Markov Models (HMMs) (Kosmala and Rigoll, 1998) and Minimum Spanning Tree Rojas, 2003, 2005 There could also be different ways of interpreting a relationship between two specific symbols. Therefore, we need to identify the most plausible relationship between input symbols.…”