## The Mathematics of Mathematical Handwriting Recognition

### Stephen Watt

Faculty of Mathematics, University of Waterloo, Waterloo

Accurate computer recognition of handwritten mathematics offers to provide a natural interface for mathematical computing, document creation and collaboration. Mathematical handwriting, however, provides a number of challenges beyond what is required for the recognition of handwritten natural languages. On one hand, it is usual to use symbols from a range of different alphabets and there are many similar-looking symbols. Mathematical notation is two-dimensional and size and placement information is important. Additionally, there is no fixed vocabulary of mathematical "words" that can be used to disambiguate symbol sequences. On the other hand, there are some simplifications. For example, symbols do tend to be well-segmented. With these characteristics, new methods of character recognition are important for accurate handwritten mathematics input.

We present a geometric model that we have found useful for recognizing mathematical symbols. Characters are represented as parametric curves approximated by certain truncated orthogonal series in a coordinate or jet space. This maps symbols to a low-dimensional vector space of series coefficients in which the Euclidean distance is closely related to the variational integral between two curves. This can be used to find similar symbols very efficiently. We describe some properties of mathematical handwriting data sets when mapped into this space and compare classification methods and their confidence measures. We also show how, by choosing the functional basis appropriately, the series coefficients can be computed in real-time, as the symbol is being written and, by using integral invariant functions, orientation-independent recognition is achieved. The beauty of this theory is that a single, coherent view provides several related geometric techniques that give a high recognition rate and that do not rely on peculiarities of the symbol set.