Persistent topological structures in noisy images

The proposed research is on the interface between topology and algorithms.The methods are topological coming from the recently developed theory of persistent topology.The key idea of persistent topology is identifying features of geometric objects that remain stable for a wide range of varying parameters. The research objectives are applied and involve designing novel learning algorithms to recognise topological graphs in noisy images.A natural application is identifying a text-based advanced CAPTCHA (Completely Automated Public Turing test to tell Computers and Humans Apart) widely used on the web to prevent automatic posting to blogs.A desired learning algorithm will be parallel like the human brain analysing local persistent properties of pixels in a noisy image.Hence it is the next step towards reverse-engineering the human brain.Other potential applications include recognising handwritten mail addresses and designing computer glasses reading signs of directions for blind people.

We plan to complete at least three research papers. Their extended versions including more explanations, examples and computer experiments will be posted in the arXiv and on the university web page. Additionally, we will implement the final learning algorithm as a Java applet and make it publicly available for testing. The input can be any noisy image as a jpg file, e.g. a shop sign or a geographical map or a sign showing directions. The output is a text produced by the algorithm. We will seek to present our results at reputable international conferences such as COLT (computational learning theory), ALT (algorithmic learning theory), ICML (machine learning) and at various university seminars. Topology is a rather abstract branch of pure mathematics studying very general geometric objects and their properties. Successful applications of topology to real-life problems such as learning algorithms, classification of patterns, computer vision will certainly stimulate more inter-disciplinary collaboration. Many pure mathematicians feel that their research is far away from commercial developments, probably because there are few examples of the opposite. Most practical algorithms including the neural network program beating human champions in backgammon are based on very elementary facts, e.g. any continuous function can be approximated by a linear combination of sufficiently many step functions. So mathematicians will be delighted in sharing their really deep knowledge with developers of learning algorithms. The proposed parallelised learning algorithm based on the prior knowledge about topological graphs is a step towards reverse-engineering the human brain. Our brains have very slow neural connections (about 200 cps, computations per second) in comparison with the latest supercomputer Cray XT5 Jaguar (1759 teraflops cps). However, the human brain is massively parallel and can identify a familiar face in about 150 milliseconds. Recognising handwritten addresses by conventional kernel methods is widely used in the American mail system. We could reduce costs of the British Royal Mail implementing more advanced algorithms. Other applications are automatic converting papers with handwritten data into electronic files and designing computer glasses that can read signs of directions for blind people.