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Math Literate Computers Dorothea Blostein School of Computing, Queen’s University CICM 2009 M ath L iteracy : M ath L iteracy : The ability to read and write math notation.

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In people, understanding precedes literacy. Computers are fairly literate, but with shallow understanding. People learn to read before they learn to write. Computers are better at writing than reading. Math literacy relates to literacy in other diagram notations: two-dimensional, domain-specific, natural languages.

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Freedom to think with paper and pencil. Computer support for typesetting, search, automated reasoning. Goal:Smooth conversion between paper and electronic documents Four Color Theorem, Appel and Haken, 1976 Math Notation - A Tool to Support Reasoning Evolved over centuries Additional notation is invented as needed Many dialects

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Notational Conventions The mapping between information and ink. What is Math Notation, anyway? Semi-standardized. Not formally defined. Approaches to Recognizing Math Notation User Interface Issues People think about meaning, not about ink marks. Topics

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Notational conventions map between information and ink. Writing (Generation) Reading (Recognition)

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Difficult: create an aesthetically appealing diagram A solved problem Difficult. An active research area. Difficult: handle symbol recognition errors and variable layout. Writing (Generation) Reading, RecognitionReading (Recognition) Conventions geared toward generation Conventions geared toward recognition

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Many Diagrams Represent the Same Information Same use of hard conventions Varying use of Soft conventions Recognition All the diagrams lead to same information Generation One path (chosen according to user preferences) from information to diagram Hard conventions: how to encode information. Soft conventions: how to make it readable.

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Notational Conventions The mapping between information and ink. What is Math Notation, anyway? Semi-standardized. Not formally defined. Approaches to Recognizing Math Notation User Interface Issues People think about meaning, not about ink marks. Topics

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Sources of Information about Math Notation Sample Documents Math notation defined by use in society. Introspection. geared toward manual typesetting. By example. People use their judgment. Chaundy, Barrett, Batey, The Printing of Mathematics, Wick, Rules for Typesetting Mathematics, Higham, Handbook of Writing for the Math. Sciences, geared toward computational typesetting. Knuth, “Mathematical Typography,” Bulletin of the AMS, for recognizing and generating math notation. Written Descriptions Program Code Recognition Contests define datasets and evaluation metrics. Contests at ICDAR and GREC: Arc segmentation, symbol recognition, segmenting text and graphics, raster to vector conversion, signature verification, document binarization, page segmentation.

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Statistics about Math Notation: An Example Gather statistics from training data. Almost matches human performance in labeling bounding boxes. Spatial relations for pairs of bounding boxes. Top labels: most likely, based on statistics. Ambiguity due to unknown baseline [Wang&Faure, ICPR 1988]

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Notational Conventions The mapping between information and ink. What is Math Notation, anyway? Semi-standardized. Not formally defined. Approaches to Recognizing Math Notation User Interface Issues People think about meaning, not about ink marks. Topics

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Challenges in Math Recognition Symbol recognition ( C O 0 7 > S 5 / 1 l Several roles for symbols Spatial relationships Little redundancy Handwritten notation is particularly difficult Compilers easily handle math notation in programming languages. 2D math notation is harder: –Noise causes errors in segmenting and identifying symbols. –Can’t blame the user for mistakes. –Hard to capture 2D relationships effectively in a string.

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Evaluate/compare these approaches? The choice of software architecture is difficult to make and defend. Procedurally-coded math syntaxCoordinate grammar Projection profile cuttingStochastic grammars & HMMs Graph rewritingTree rewriting Math-Recognition Approaches [Survey by Blostein and Grbavec, 1997]

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Procedurally-coded math syntaxCoordinate grammar Projection profile cuttingStochastic grammars & HMMs Graph rewritingTree rewriting Math-Recognition Approaches No explicit definition of math syntax. Update code in response to recognition errors. Can get good recognition performance.

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Procedurally-coded math syntaxCoordinate grammar Projection profile cuttingStochastic grammars & HMMs Graph rewritingTree rewriting Math-Recognition Approaches Apply a rule to a set of symbols: create subsets with syntactic subgoals. A clear, well-structured representation of notational conventions. [Anderson 1969; in Fu 77 ] Attributes: xmin, ymin, xmax, ymax, xcenter m encodes meaning

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horizontal cut Procedurally-coded math syntaxCoordinate grammar Projection profile cuttingStochastic grammars & HMMs Graph rewritingTree rewriting Math-Recognition Approaches vertical cut [Okamoto and Miao, 1992] The order of cuts provides the tree-structure of the expression. A simple and efficient technique. Can be applied prior to OCR. Special handling of overlapping symbols:

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Procedurally-coded math syntaxCoordinate grammar Projection profile cuttingStochastic grammars & HMMs Graph rewritingTree rewriting Math-Recognition Approaches Hidden Markov Model [Kopec, Chou 1994] An explicit image-generation model, to drive recognition. Applied to yellow pages & music notation. 2D stochastic context-free grammar [Chou 1989] Find the most likely parse of the image, without segmentation.

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Procedurally-coded math syntaxCoordinate grammar Projection profile cuttingStochastic grammars & HMMs Graph rewritingTree rewriting Math-Recognition Approaches Rewrite rules replace one subgraph by another PROGRES language : a mix of textual and visual notation Write a graph schema to define the structure of valid graphs. The PROGRES execution environment flags violations. Build Constrain Parse [Blostein, Schürr, Software Practice and Experience, 1999]

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Math-Recognition Approaches Compiler-inspired approach, using tree rewriting [Zanibbi, Blostein, Cordy: ICPR 2002 and PAMI 2002] Separate analysis of layout, lexical, syntactic, and semantic aspects. Get partial results even if there are syntax errors. Find linear structures in the input, and create a tree from them. Operation of a compilerRecognition of math notation

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Notational Conventions The mapping between information and ink. What is Math Notation, anyway? Semi-standardized. Not formally defined. Approaches to Recognizing Math Notation User Interface Issues People think about meaning, not about ink marks. Topics

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Goal: seamless transition between - real world (stylus and paper) - electronic world Many paper documents are produced from electronic sources. Eventually include digitally-encoded contents? Methods used in digital watermarking are relevant. Electronic Paper is more advanced than Paper Electronic

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Entering math expressions How much user time? How many residual errors? How much frustration? Method 1: Use Recognition Software Scan a document image or write on a data tablet Method 2: Enter information directly Type the information (e.g. LaTeX) or use a structure-based editor User proofreads and corrects Generate math notation Recognition software Information

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User Frustration People eventually feel comfortable with irritating interfaces. The Argh is a unit of frustration. Kilarghs. Megarghs…. Arghometers need to be developed. Document recognition is frustrating because: 1.Users don’t like to correct errors made by the “stupid computer”. Better to correct errors they made themselves. 2.Users don’t like to think about the marks on the paper. They would rather think about the document contents. 3.Users don’t like unpredictable systems. Better to adapt themselves (even if inconvenient) to achieve predictability. [Talk at ICDAR 2001]

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Possible research directions Precisely define math literacy tasks. Use soft conventions in recognition. Use statistics: know about likely versus unlikely expressions. Exploit the advanced state of generation, to improve recognition. Topics:Notational ConventionsWhat is Math Notation, anyway? Math Recognition ApproachesUser Interface Issues Conclusion A group effort is required.

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