A Tour of Mathematical Image & Vision Analysis (Miva)

Miva resides at the intersection of Mathematics, Cognitive Science, Computer Science, and Digital Technology. After nearly two decades of interdisciplinary effort by dozens of pioneering mathematicians, Miva has now become part of the contemporary landscape of mathematics. I have coined the name Miva to maximally reflect its scope and goals. In the Bible, God is often described as THE LIGHT. In physics, within a light beam flow photons or waves. THE LIGHT is THE MESSAGE, and the photons and waves also carry all sorts of information in this optical communication era. When the light comes, we see the images of the world (except the black holes). Therefore, image representation, transmission, processing, interpretation, and human perception, make up the cornerstones of Miva. Mathematicians are expected to make substantial contributions in faithful mathematical modeling, robust model analysis, and design of efficient algorithms. In this talk, first I shall convey the most general messages of Miva: its role in science and technology, its broad applications, and the role of mathematicians. Afterwards, I will act as a tour guide by singling out several beautiful scenes (such as wavelets and the PDE method) along the trip. All will be inevitably seasoned with my personal biases. Finally, I wish you will bear the following in mind when stepping into the lecture hall: while it is obvious to human eyes that a simple closed curve cuts a piece of paper into the outside and the inside, it is a highly non-trivial theorem in topology belonging to Jordan. That helps you measure the virtual distance between vision and mathematics - what do we mean by a "simple closed curve"? and the "outside" and "inside"? although we SEE them everyday...