We present a chemical extraction and search pipeline intended to support information tasks related to drug discovery. Commonly used search tools for drug discovery such as Reaxys and SciFinder do not allow users to obtain retrieval results at the passage level. To address this, we present a passage retrieval tool for chemical patents that supports queries combining text and molecule diagrams expressed in SMILES. When SMILES is provided as a part of a query, the system refines text retrieval results through matching both textual names and drawn figures based on extracted SMILES representations. Molecule matches are obtained through substructure matching and structural similarity. This functionality was motivated by a chemist's need to find synthesis pathways for specific molecules containing a substructure of interest that binds and thus inhibits specific human genes. For this demonstration, we index a collection of 131 PDF patents categorized into 12 specific genes enabling a user to search on them. There are 32,301 document pages in the collection. Our user interface can be accessed at https://unichemfinder.gccis.rit.edu/. Our source code and data is available at https://gitlab.com/dprl/unichemfinder.

We present a multimodal search tool for retrieval of chemical reactions, molecular structures, and associated text from scientific literature.Queries may combine molecular diagrams, textual descriptions, and reaction data, allowing users to connect different chemical information representations.Indexing includes chemical diagram extraction and parsing, extraction of reaction data from text in tabular form, and cross-modal linking of diagrams with their mentions in text.We describe the system's architecture and retrieval features, along with expert assessments of the system.Our demo highlights the workflow and search components.Online demo: https://www.cs.rit.edu/

Mathematical notation is a key analytical resource for science and technology. Unfortunately, current math-aware search engines require LATEX or template palettes to construct formulas, which can be challenging for non-experts. Also, their indexed collections are primarily web pages where formulas are represented explicitly in machine-readable formats (e.g., LATEX, Presentation MathML). The new MathDeck system searches PDF documents in a portion of the ACL Anthology using both formulas and text, and shows matched words and formulas along with other extracted formulas in-context. In PDF, formulas are not demarcated: a new indexing module extracts formulas using PDF vector graphics information and computer vision techniques. For non-expert users and visual editing, a central design feature of MathDeck's interface is formula 'chips' usable in formula creation, search, reuse, and annotation with titles and descriptions in cards. For experts, LATEX is supported in the text query box and the visual formula editor. MathDeck is open-source, and our demo is available online.

In math formula search, relevance is determined not only by the similarity of formulas in isolation, but also by their surrounding context. We introduce MathAMR, a new unified representation for sentences containing math. MathAMR generalizes Abstract Meaning Representation (AMR) graphs to include math formula operations and arguments. We then use Sentence-BERT to embed linearized MathAMR graphs for use in formula retrieval. In our first experiment, we compare MathAMR against raw text using the same formula representation (Operator Trees), and find that MathAMR produces more effective rankings. We then apply our MathAMR embeddings to reranking runs from the ARQMath-2 formula retrieval task, where in most cases effectiveness measures are improved. The strongest reranked run matches the best P$'[email protected] for an original run, and exceeds the original runs in nDCG$'[email protected]

In Mathematical Information Retrieval (MIR), formulae can be used in a query to match other similar formulae in documents. However, due to the structural complexity of formulae, specialized processing is needed for formula matching. Formulae may be represented by their appearance in Symbol Layout Trees (SLTs) or by their syntax in Operator Trees (OPTs). Previous approaches for formula retrieval used one or both of these representations and used unification to improve search results for inexact matches (e.g., allowing different variable names to match). On these representations, models for matching full expressions (trees), subexpressions, and paths have been used. Recently embedding models were used to represent formulae as vectors. In this paper, the effectiveness of retrieval models and formula representations are studied to identify their relative strengths and weaknesses. Then, a learning to rank model is proposed, using SVM-rank over similarity scores from different formula retrieval models as features. Experiments on the ARQMath formula retrieval task results show that the proposed learning to rank model is effective, producing new state-of-the-art results.

The recognition of mid-18th to mid-20th century piano scores presents segmentation challenges caused by touching and broken symbols produced by imprinting techniques and time degradation. We present a new notehead accidental dataset containing 2955 images from dense and damaged piano scores. We address this detection problem with very small training samples using a simple Spatial Transformer (ST)-based Convolutional Neural Network detector improved through bootstrapping and contextual information, and more powerful deep learning detectors (Faster R-CNN, R-FCN, and SSD) with transfer-learning on the COCO dataset. We trained all our detectors using 5 fold cross-validation and obtain 98.73% mean Average Precision (mAP) for an Intersection over Union (IoU) threshold of 0.75 with our best detector. Our ST-based detector obtains a slightly lower mAP of 94.81%, but runs 40 times faster, and uses 18 times less memory.

We present a model for recognizing typeset math formula images from connected components or symbols. In our approach, connected components are used to construct a line-of-sight (LOS) graph. The graph is used both to reduce the search space for formula structure interpretations, and to guide a classification attention model using separate channels for inputs and their local visual context. For classification, we used visual densities with Random Forests for initial development, and then converted this to a Convolutional Neural Network (CNN) with a second branch to capture context for each input image. Formula structure is extracted as a directed spanning tree from a weighted LOS graph using Edmonds' algorithm. We obtain strong results for formulas without grids or matrices in the InftyCDB-2 dataset (90.89% from components, 93.5% from symbols). Using tools from the CROHME handwritten formula recognition competitions, we were able to compile all symbol and structure recognition errors for analysis. Our data and source code are publicly available.

We summarize the tasks, protocol, and outcome for the 6th Competition on Recognition of Handwritten Mathematical Expressions (CROHME), which includes a new formula detection in document images task (+ TFD). For CROHME + TFD 2019, participants chose between two tasks for recognizing handwritten formulas from 1) online stroke data, or 2) images generated from the handwritten strokes. To compare LATEX strings and the labeled directed trees over strokes (label graphs) used in previous CROHMEs, we convert LATEX and stroke-based label graphs to label graphs defined over symbols (symbol-level label graphs, or symLG). More than thirty (33) participants registered for the competition, with nineteen (19) teams submitting results. The strongest formula recognition results were produced by the USTC-iFLYTEK research team, for both stroke-based (81%) and image-based (77%) input. For the new typeset formula detection task, the Samsung R&D Institute Ukraine (Team 2) obtained a very strong F-score (93%). System performance has improved since the last CROHME - still, the competition results suggest that recognition of handwritten formulae remains a difficult structural pattern recognition task.

Optical Music Recognition (OMR) is the challenge of understanding the content of musical scores. Accurate detection of individual music objects is a critical step in processing musical documents because a failure at this stage corrupts any further processing. So far, all proposed methods were either limited to typeset music scores or were built to detect only a subset of the available classes of music symbols. In this work, we propose an end-to-end trainable object detector for music symbols that is capable of detecting almost the full vocabulary of modern music notation in handwritten music scores. By training deep convolutional neural networks on the recently released MUSCIMA++ dataset which has symbol-level annotations, we show that a machine learning approach can be used to accurately detect music objects with a mean average precision of over 80%.

To fill a gap in online educational tools, we are working to support search in lecture videos using formulas from lecture notes and vice versa. We use an existing system to convert single-shot lecture videos to keyframe images that capture whiteboard contents along with the times they appear. We train classifiers for handwritten symbols using the CROHME dataset, and for LATEX symbols using generated images. Symbols detected in video keyframes and LATEX formula images are indexed using Line-of-Sight graphs. For search, we lookup pairs of symbols that can 'see' each other, and connected pairs are merged to identify the largest match within each indexed image. We rank matches using symbol class probabilities and angles between symbol pairs. We demonstrate how our method effectively locates formulas between typeset and handwritten images using a set of linear algebra lectures. By combining our search engine Tangent-V) with temporal keyframe metadata, we are able to navigate to where a query formula in LATEX is first handwritten in a lecture video. Our system is available as open-source. For other domains, only the OCR modules require updating.

Lecture videos are a valuable resource for students, and thanks to online sources they have become widely available. The ability to find videos based on their content could make them even more useful. Methods for automatic extraction of this content reduce the amount of manual effort required to make indexing and retrieval of such videos possible. We present a method that generates static image summaries of handwritten whiteboard content from lecture videos recorded with still cameras. We generate a spatio-temporal index for the handwritten content in the video, and we use it for temporal segmentation by detecting and removing conflicts between content regions. Resulting segments are used to produce key-frame based summaries. Our method has been tested on a video collection showing promising results for automatic lecture video summarization with good compression ratios and 96.28% recall of connected components in test videos.

Math-aware search engines need to support formulae in queries. Mathematical expressions are typically represented as trees defining their operational semantics or visual layout. We propose searching both formula representations using a three-layer model. The first layer selects candidates using spectral matching over tree node pairs. The second layer aligns a query with candidates and computes similarity scores based on structural matching. In the third layer, similarity scores are combined using linear regression. The two representations are combined using retrieval in parallel indices and regression over similarity scores. For NTCIR-12 Wikipedia Formula Browsing task relevance rankings, we see each layer increasing ranking quality and improved results when combining representations as measured by Bpref and nDCG scores.

We propose a system that finds text in natural scenes using a variety of cues. Our novel data-driven method incorporates coarse-to-fine detection of character pixels using convolutional features (Text-Conv), followed by extracting connected components (CCs) from characters using edge and color features, and finally performing a graph-based segmentation of CCs into words (Word-Graph). For Text-Conv, the initial detection is based on convolutional feature maps similar to those used in Convolutional Neural Networks (CNNs), but learned using Convolutional k-means. Convolution masks defined by local and neighboring patch features are used to improve detection accuracy. The Word-Graph algorithm uses contextual information to both improve word segmentation and prune false character/word detections. Different definitions for foreground (text) regions are used to train the detection stages, some based on bounding box intersection, and others on bounding box and pixel intersection. Our system obtains pixel, character, and word detection f-measures of 93.14%, 90.26%, and 86.77% respectively for the ICDAR 2015 Robust Reading Focused Scene Text dataset, out-performing state-of-the-art systems. This approach may work for other detection targets with homogenous color in natural scenes.

This paper presents a new representation for handwritten math formulae: a Line-of-Sight (LOS) graph over handwritten strokes, computed using stroke convex hulls. Experimental results using the CROHME 2012 and 2014 datasets show that LOS graphs capture the visual structure of handwritten formulae better than commonly used graphs such as Time-series, Minimum Spanning Trees, and k-Nearest Neighbor graphs. We then introduce a shape context-based feature (Parzen window Shape Contexts (PSC)) which is combined with simple geometric features and the distance in time between strokes to obtain state-of-the-art symbol segmentation results (92.43% F-measure for CROHME 2014). This result is obtained using a simple method, without use of OCR or an expression grammar. A binary random forest classifier identifies which LOS graph edges represent stroke pairs that should be merged into symbols, with connected components over merged strokes defining symbols. Line-of-Sight graphs and Parzen Shape Contexts represent visual structure well, and might be usefully applied to other notations.

We develop a Maximum Spanning Tree (MST) based parser using Edmonds' algorithm, which extracts an MST from a directed Line-of-Sight graph in two passes. First, symbols are segmented by grouping input strokes, and then symbols and symbol pair spatial relationships are labeled. The time complexity of our MST-based parsing is lower than the time complexity of CYK parsing with 2-D Context-Free grammars. Also, our MST-based parser obtains higher formula structure and expression rates than published techniques using CYK parsing when starting from valid symbols. This parsing technique could be extended to include n-grams or other language constraints, and might be used for other notations.

This paper presents an overview of the 5th Competition on Recognition of Online Handwritten Mathematical Expressions (CROHME). As in previous years, the main task is formula recognition from handwritten strokes (Task 1). Additional tasks include classification of isolated symbols (Task 2a), classification of isolated valid and invalid symbols (Task 2b), a new task on parsing formula structure from valid handwritten symbols (Task 3), and parsing expressions with matrices (Task 4, experimental). In total, eleven (11) research labs registered for the competition, with six (6) teams submitting results. Innovations for this CROHME included providing a corpus of formulae from Wikipedia to train language models, and an online system for result submission. The highest recognition rates were obtained by MyScript corporation (Task 1. 67.65%, 2a. 92.81%, 2b. 86.77%, 3. 84.38%, and 4. 68.40%). Using only provided training data, the highest recognition rates were obtained by WIRIS corporation (Task 1. 49.61%, Task 3. 78.80%, Task 4. 56.40%), the Tokyo University of Agriculture and Technology (Task 2a. 92.28%), and RIT (Task 2b. 83.34%). The competition results suggest that recognition of handwritten formulae remains a difficult structural pattern recognition task.

When using a mathematical formula for search (query-by-expression), the suitability of retrieved formulae often depends more upon symbol identities and layout than deep mathematical semantics. Using a Symbol Layout Tree representation for formula appearance, we propose the Maximum Subtree Similarity (MSS) for ranking formulae based upon the subexpression whose symbols and layout best match a query formula. Because MSS is too expensive to apply against a complete collection, the Tangent-3 system first retrieves expressions using an inverted index over symbol pair relationships, ranking hits using the Dice coefficient; the top-k formulae are then re-ranked by MSS. Tangent-3 obtains state-of-the-art performance on the NTCIR-11 Wikipedia formula retrieval benchmark, and is efficient in terms of both space and time. Retrieval systems for other graphical forms, including chemical diagrams, flowcharts, figures, and tables, may benefit from adopting this approach.

We present an approach for on-line recognition of handwritten math symbols using adaptations of off-line features and synthetic data generation. We compare the performance of our approach using four different classification methods: AdaBoost. M1 with C4.5 decision trees, Random Forests and Support-Vector Machines with linear and Gaussian kernels. Despite the fact that timing information can be extracted from on-line data, our feature set is based on shape description for greater tolerance to variations of the drawing process. Our main datasets come from the Competition on Recognition of Online Handwritten Mathematical Expressions (CROHME) 2012 and 2013. Class representation bias in CROHME datasets is mitigated by generating samples for underrepresented classes using an elastic distortion model. Our results show that generation of synthetic data for underrepresented classes might lead to improvements of the average per-class accuracy. We also tested our system using the Math Brush dataset achieving a top-1 accuracy of 89.87% which is comparable with the best results of other recently published approaches on the same dataset.

We present the outcome of the latest edition of the CROHME competition, dedicated to on-line handwritten mathematical expression recognition. In addition to the standard full expression recognition task from previous competitions, CROHME 2014 features two new tasks. The first is dedicated to isolated symbol recognition including a reject option for invalid symbol hypotheses, and the second concerns recognizing expressions that contain matrices. System performance is improving relative to previous competitions. Data and evaluation tools used for the competition are publicly available.

Finding ways to help users assess relevance when they search using math expressions is critical for making Mathematical Information Retrieval (MIR) systems easier to use. We designed a study where participants completed search tasks involving mathematical expressions using two different summary styles, and measured response time and relevance assessment accuracy. The control summary style used Google's regular hit formatting where expressions are presented as text (e.g. in LaTeX), while the second summary style renders the math expressions. Participants were undergraduate and graduate students. Participants in the rendered summary style (n=19) had on average a 17.18% higher assessment accuracy than those in the non-rendered summary style (n=19), with no significant difference in response times. Participants in the rendered condition reported having fewer problems reading hits than participants in the control condition. This suggests that users will benefit from search engines that properly render math expressions in their hit summaries.

To use math expressions in search, current search engines require knowing expression names or using a structure editor or string encoding (e.g., LaTeX). For mathematical non-experts, this can lead to an "intention gap" between the query they wish to express and what the interface will allow them to express. min is a search interface that supports drawing expressions on a canvas using mouse/touch, keyboard and images. We present a user study examining whether min changes search behavior for mathematical non-experts, and to identify real-world usage scenarios for multimodal math search interfaces. Participants found query-by-expression using hand-drawn input useful, and identified scenarios in which they would like to use systems like min such as for locating, editing and sharing complex expressions (e.g., with many Greek letters), and working on complex math problems.

We consider the difficult problem of classifying spatial relationships between symbols and subexpressions in handwritten mathematical expressions. We first improve existing geometric features based on bounding boxes and center points, normalizing them using the distance between the centers of the two symbols or subexpressions in question. We then propose a novel feature set for layout classification, using polar histograms computed over points in handwritten strokes. A series of experiments are presented in which a Support Vector Machine is used with these new features to classify spatial relationships of five types in the MathBrush corpus (horizontal, superscript, subscript, below, and inside (e.g. in a square root)). The normalized geometric features provide an improvement over previously published results, while the shape-based features provide a natural representation with results comparable to those for the geometric features. Combining the features produced a very small improvement in accuracy.

This paper presents a new symbol segmentation method based on AdaBoost with confidence weighted predictions for online handwritten mathematical expressions. The handwritten mathematical expression is preprocessed and rendered to an image. Then for each stroke, we compute three kinds of shape context features (stroke pair, local neighborhood and global shape contexts) with different scales, 21 stroke pair geometric features and symbol classification scores for the current stroke and stroke pair. The stroke pair shape context features covers the current stroke and the following stroke in time series. The local neighborhood shape context features includes the current stroke and its three nearest neighbor strokes in distance while the global shape context features covers the expression. Principal component analysis (PCA) is used for dimensionality reduction. We use AdaBoost with confidence weighted predictions for classification. The method does not use any language model. To our best knowledge, there is no previous work which uses shape context features for symbol segmentation. Experiment results show the new symbol segmentation method achieves good recall and precision on the CROHME 2012 dataset.

We report on the third international Competition on Handwritten Mathematical Expression Recognition (CROHME), in which eight teams from academia and industry took part. For the third CROHME, the training dataset was expanded to over 8000 expressions, and new tools were developed for evaluating performance at the level of strokes as well as expressions and symbols. As an informal measure of progress, the performance of the participating systems on the CROHME 2012 data set is also reported. Data and tools used for the competition will be made publicly available.

We utilize Coates' unsupervised feature learning method and AdaBoost to detect and recognize part label regions in patent drawings. Image patches are harvested from training data, and features are learned from patterns in image patches. Angle distances between samples and feature banks are computed, and used in AdaBoost classifier. We extract image patches with different sizes to counter the scale problem. An ensemble AdaBoost is used to classify pixels as text or background. Meta-Boost is introduced to improve performance. The pixel level detections are then grouped into 'Connected Components'. Several denoise methods are applied, followed by 'Tesseract' OCR. Our system achieves competitive performance without using strong prior knowledge.

This paper presents a recognition system based on Hidden Markov Model (HMM) for isolated online handwritten mathematical symbols. We design a continuous left to right HMM for each symbol class and use four online local features, including a new feature: normalized distance to stroke edge. A variant of segmental K-means is used to get initialization of the Gaussian Mixture Models' parameters which represent the observation probability distribution of the HMMs. The system obtains top-1 recognition rate of 82.9% and top-5 recognition rate of 97.8% on a dataset containing 20281 training samples and 2202 testing samples of 93 classes of symbols. For multi-stroke symbols, the top-1 recognition rate is 74.7% and the top-5 recognition rate is 95.5%. For single-stroke symbols, the top-1 recognition rate is 86.8% and the top-5 recognition rate is 98.9%. (MacLean et al., 2010) applied dynamic time warping algorithm on all the 70 classes of single-stroke symbols. Their top-1 recognition rate is 85.8%, and top-5 recognition rate is 97.0%. Our system gets top-1 recognition rate of 85.5% and top-5 recognition rate of 99.1% on the same 70 classes of single-stroke symbols.

Evaluating mathematical expression recognition involves a complex interaction of input primitives (e.g. pen/finger strokes), recognized symbols, and recognized spatial structure. Existing performance metrics simplify this problem by separating the assessment of spatial structure from the assessment of symbol segmentation and classification. These metrics do not characterize the overall accuracy of a pen-based mathematics recognition, making it difficult to compare math recognition algorithms, and preventing the use of machine learning algorithms requiring a criterion function characterizing overall system performance. To address this problem, we introduce performance metrics that bridge the gap from handwritten strokes to spatial structure. Our metrics are computed using bipartite graphs that represent classification, segmentation and spatial structure at the stroke level. Overall correctness of an expression is measured by counting the number of relabelings of nodes and edges needed to make the bipartite graph for a recognition result match the bipartite graph for ground truth. This metric may also be used with other primitive types (e.g. image pixels).

A method for locating mathematical expressions in document images without the use of optical character recognition is presented. An index of document regions is produced from recursive X-Y trees produced for each page in the corpus. Queries are provided as images of handwritten expressions, for which an X-Y tree is computed. During retrieval, the query is looked up in the document region index using features of its X-Y tree, producing a set of candidate regions. Candidate regions are ranked by the similarity of vertical pixel projections in their upper and lower halves with those of the query image, as computed using Dynamic Time Warping of the image columns. In an experiment, ten participants each wrote twenty queries from a 200-page corpus. On average, the top-10 retrieval candidates included a candidate covering 43.3% of the test query image (σ = 14.0), with the correct page being returned between 30.0% and 85.0% of the time across participants (μ = 63.2%, σ = 14.9%). When testing using the original query images, 90.0% of the queries were retrieved correctly.

We present a technique for using content-based video labeling as a CAPTCHA task. Our CAPTCHAs are generated from YouTube videos, which contain labels (tags) supplied by the person that uploaded the video. They are graded using a video's tags, as well as tags from related videos. In a user study involving 184 participants, we were able to increase the human success rate on our video CAPTCHA from roughly 70% to 90%, while keeping the success rate of a tag frequency-based attack fixed at around 13%. Through a different parameterization of the challenge generation and grading algorithms, we were able to reduce the success rate of the same attack to 2%, while still increasing the human success rate from 70% to 75%. The usability and security of our video CAPTCHA appears to be comparable to existing CAPTCHAs, and a majority of participants (60%) indicated that they found the video CAPTCHAs more enjoyable than traditional CAPTCHAs in which distorted text must be transcribed.

We present a document understanding system in which the arrangement of lines of text and block separators within a document are modeled by stochastic context free grammars. A grammar corresponds to a document genre; our system may be adapted to a new genre simply by replacing the input grammar. The system incorporates an optical character recognition system that outputs characters, their positions and font sizes. These features are combined to form a document representation of lines of text and separators. Lines of text are labeled as tokens using regular expression matching. The maximum likelihood parse of this stream of tokens and separators yields a functional labeling of the document lines. We describe business card and business letter applications.

Document recognition involves many kinds of hypotheses: segmentation hypotheses, classification hypotheses, spatial relationship hypotheses, and so on. Many recognition strategies generate valid hypotheses, which are eventually rejected, but current evaluation methods consider only accepted hypotheses. As a result, we have no way to measure errors associated with rejecting valid hypotheses. We propose describing hypothesis generation in more detail, by collecting the complete set of generated hypotheses and computing the recall and precision of this set: we call these the 'historical recall' and 'historical precision.' Using table cell detection examples, we demonstrate how historical recall and precision along with the complete set of generated hypotheses assist in the evaluation, debugging, and design of recognition strategies.

The table recognition literature contains many strategies specified informally as a sequence of operations, obscuring both models of table structure and the effects of individual decisions. Decision making is more transparent in formal model based approaches (e.g. grammar based) but these approaches are less flexible than informal ones. We propose an intermediate level of formalization, defining strategies as a sequence of basic graph transformations that correspond to recognition operations (e.g. classification, segmentation). Transformations are parameterized by logical types and decision functions, which together define structure models and executable strategies for interpreting input graphs. We provide an overview of our first attempt at this intermediate level of formalization, the Recognition Strategy Language (RSL). As a proof of concept, we reimplement two informally specified table recognition strategies from the literature in RSL. The RSL implementations capture descriptions of the formerly implicit table structure models, and automatically capture all decision making.

Handwritten mathematical notation contains ambiguities of various kinds. Here we focus on ambiguity in spatial relationships; in particular, we use fuzzy logic to treat ambiguity in subscript-or-inline and inline-or-superscript spatial relationships. We extend an existing system for recognizing handwritten mathematical notation, adding the capability of producing a ranked list of interpretations rather than a single top-choice interpretation. Fuzzy membership values are assigned to each spatial relationship; a given pair of symbols can have non-zero membership in fuzzy sets subscript and inline, or in fuzzy sets inline and superscript. These fuzzy membership values are combined to produce an overall confidence value for the entire interpretation. We have modified the user interface of our system so that a user can quickly view and select from the ranked interpretations when the highest confidence interpretation is incorrect.

We describe a robust and efficient system for recognizing typeset and handwritten mathematical notation. From a list of symbols with bounding boxes the system analyzes an expression in three successive passes. The Layout Pass constructs a Baseline Structure Tree (BST) describing the two-dimensional arrangement of input symbols. Reading order and operator dominance are used to allow efficient recognition of symbol layout even when symbols deviate greatly from their ideal positions. Next, the Lexical Pass produces a Lexed BST from the initial BST by grouping tokens comprised of multiple input symbols; these include decimal numbers, function names, and symbols comprised of nonoverlapping primitives such as "=". The Lexical Pass also labels vertical structures such as fractions and accents. The Lexed BST is translated into L/sup A/T/sub E/X. Additional processing, necessary for producing output for symbolic algebra systems, is carried out in the Expression Analysis Pass. The Lexed BST is translated into an Operator Tree, which describes the order and scope of operations in the input expression. The tree manipulations used in each pass are represented compactly using tree transformations. The compiler-like architecture of the system allows robust handling of unexpected input, increases the scalability of the system, and provides the groundwork for handling dialects of mathematical notation.

The structure of mathematics notation is particularly difficult to recognize in handwritten notation because irregular symbol placements are common. We present an efficient and robust method of parsing handwritten and typeset mathematics notation without backtracking. The system is designed to be easily adaptable to various dialects of mathematics notation. The following strategies are used: (1) separate the analysis of layout, syntax, and semantics, (2) recursively apply search functions and image partitioning to recognize dominant and nested baselines, and (3) use tree transformations to express computations in a compact, efficiently executable form.