See illuminated version for the 2007 webpage of the Lab, which has lots of nice pictures - if you're interested in vision and visual science, you'll need a web link with sufficient bandwidth!
The Laboratory for Visual Mathematics investigates the connection between vision and math. This is a two-way street: visual display can explain mathematical and scientific concepts, while conversely mathematical ideas can facilitate the understanding of vision.
The disciplines involved include math, art, physics, psychology, neuroscience, computer science, sociology, law, cognitive science, medicine, ecology, and theoretical biology. It is a central tennet of this lab that mathematics provides tools to study vision and visualization which can benefit all of these disciplines.
Issues with the computer-human interface are important for air traffic controllers, image analysts, realtime network monitors, forensic scientists, tele-operators (e.g., of drones), and recently also physicians. The major portion of this user interface is mediated through vision. Currently, the human sees what the system is displaying. Soon it will be commonplace to observe the user's facial reaction to system events via cameras and also the direction of gaze - i.e., to which screen detail is the human's attention being drawn?
An example of understanding through visualization can be seen in a proof that 1 + 2 + ... + n = n(n+1)/2. Of course, the right hand side can be rewritten as (n^2)/2 + n/2, and this is realized by the figure formed by the sequence of columns of heights k as k goes from 1 to n. This is can be visualized as the sillouette of a staircase with n stairs. The figure can be decomposed into a large n x n isoceles right triangle plus a set of disjoint n area-1/2 isoceles triangles.
This is called ``proof by picture'' in the literature. For instance, one can illustrate a proof of the Pythagorean Theorem or show how the three perpendicular bisectors of the sides of a triangle are always concurrent lines - that is, all three go through a single point. If the example were, further, dynamic, then if the triangle were distorted, one would see all three of the perpendicular bisectors of the sides correspondingly varying, while their mutual intersection remains confined in a single point.
The opposite direction, a connection from mathematics to the understanding of human vision, is shown when one goes from angular coordinates for the direction of gaze of the two eyes to a mathematical model for aspects of vision which has predictive value. For instance, one might like to know whether it is better for visual attentiveness to be given a mix of short-range and long-range phenomena that should be detected or to have a focus on one or the other alone. Fitts Law would be the prototype of such theory, but much more precision is required and with respect to various measures.
Computer implementations, involving hardware and software, are required for experiments and gedanken experiments.
In addition, the field of psychophysics plays a critical role, where psychophysics is defined as the quantitative and structural study of the process by which human and other organisms respond and anticipate internally in a fashion appropriate to the externally received signal and the internally received signal of prior responses, positions, and goals.
Psychophysics includes so-called visual illusions as well as gestalt phenomena - the dalmatian in dappled shade or the cocktail-glass/bikini. User-interface expands this into the cognitive domain, so in addition to psychology, one must also consider cognitive load and affect. Using the other side of vision, that is, the ubiquitous cameras, the operator of a program can be seen (and studied!) by its designers. In addition, there is room for artists. As we all know, more visually pleasing items tend to be the ones which are selected, and art is a way to add value.
Our goal is to show the utility of math, neuroscience, art, cognitive science, computer science, and psychophysics in the study of vision - how it works and how it can contribute to understanding of the world. By studying the visual data available from webcams, the Lab4VM will try to learn how to interpret the data to determine aspects of the user's internal state.
In Spring 2012 I am giving a course in algebraic graph theory (Math 410) open to graduate students and suitable undergrads. To enable students to use very powerful program (Mathematica, version 8), I'm building a Math-User-Interface (MUI). I am also giving a module on the MUI as part of the Interdisciplinary Program in Cognitive Science (ICOS). One of the targets of Math 410 is to understand the ideas behind Saaty's Analytic Hierarchy Process (AHP) which enables more quantitatively oriented elements to be incorporated into the decision process.
In addition to the above activities, I would like to carry out methodical experiments regarding perception of topological form in quasi-periodic visual dynamic systems. Specifically, it is not currently known how paramaters determine human perception of Lissajous figure type when the parameters are only near to some rational ratio. See the pdf of my draft paper, On perception of Lissajous figures .
Mathematical questions include how to incorporate notions from machine-intelligence and mathematical learning such as graph kernels and graph-structured-data as well as how to further "enrich" the objects of study using homology theory or category theory.
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