Mathematical Contest in Modeling, 1998

The event

The Mathematical Contest in Modeling is an annual event, sponsored by the Consortium for Mathematics and Its Applications, in which students at colleges and universities all over the world are asked to develop and analyze mathematical models of open-ended, practical problems for which no direct solutions are known. Participants work in teams of three and are permitted to use libraries, computers, and other inanimate sources of knowledge and inspiration. The organizers of the contest propose three problems; over the three days of the contest, each team selects one of these problems, designs, implements, and analyzes a model, and writes a substantial report presenting its results.

The fourteenth Mathematical Contest in Modeling was held on February 6-9, 1998. Grinnell's teams this year were: Tim Andrews 1999, Shiva Kabra 2000, and Reba Schuller 1998, and Chris Graver 1998, Geoffrey Sparks 2000, and Anne Wilson 1999. The faculty sponsor was Marc Chamberland.

The problems

Here are the problems that COMAP posed this year. The team comprising Chris Graver, Geoffrey Sparks, and Anne Wilson chose to work on problem A, the team comprising Tim Andrews, Shiva Kabra, and Reba Schuller on problem B.

Problem A: MRI Scanners

Introduction

Industrial and medical diagnostic machines known as Magnetic Resonance Imagers (MRI) scan a three-dimensional object such as a brain, and deliver their results in the form of a three-dimensional array of pixels. Each pixel consists of one number indicating a color or a shade of gray that encodes a measure of water concentration in a small region of the scanned object at the location of the pixel. For instance, 0 can picture high water concentration in black (ventricles, blood vessels), 128 can picture a medium water concentration in gray (brain nuclei and gray matter), and 255 can picture a low water density in white (lipid-rich white matter consisting of myelinated axons). Such MRI scanners also include facilities to picture on a screen any horizontal or vertical slice through the three-dimensional array (slices are parallel to any of the three Cartesian coordinate axes). Algorithms for picturing slices through oblique planes, however, are proprietary. Current algorithms are limited in terms of the angles and parameter options available; are implemented only on heavily used dedicated workstations; lack input capabilities for marking points in the picture before slicing; and tend to blur and feather out sharp boundaries between the original points.

A more faithful, flexible algorithm implemented on a personal computer would be useful

for planning minimally invasive treatments,
for calibrating the MRI machines,
for investigating structures oriented obliquely in space, such as post-mortem tissue sections in animal research,
for enabling cross-sections at any angle through a brain atlas consisting of black-and-white line drawings.

Problem

Design and test an algorithm that produces sections of three-dimensional arrays by planes in any orientation in space, preserving the original gray-scale values as closely as possible.

Data Sets

The typical data set consists of a three-dimensional array A of numbers A(i, j, k) which indicates the density A(i, j, k) of the object at the location (x, y, z)_ijk. Typically, A(i, j, k) can range from 0 through 255. In most applications, the data set is quite large. Teams should design data sets to test and demonstrate their algorithms. The data sets should reflect conditions likely to be of diagnostic interest. Teams should also characterize data sets that limit the effectiveness of their algorithms.

Summary

The algorithm must produce a picture of the slice of the three-dimensional array by a plane in space. The plane can have any orientation and any location in space. (The plane can miss some or all data points.) The result of the algorithm should be a model of the density of the scanned object over the selected plane.)

Problem B: Grade Inflation

Background

Some college administrators are concerned about the grading at A Better Class (ABC) college. On average, the faculty at ABC have been giving out high grades (the average grade now given out is an A-), and it is impossible to distinguish between the good and mediocre students. The terms of a very generous scholarship only allow the top 10% of the students to be funded, so a class ranking is required.

The dean had the though of comparing each student to the other students in each class, and using this information to build up a ranking. For example, if a student obtains an A in a class in which all students obtain an A, then this student is only average in this class. On the other hand, if a student obtains the only A in a class, then that student is clearly above average. Combining information from several classes might allow students to be placed in deciles (top 10%, next 10%, etc.) across the college.

Problem

Assuming that the grades given out are (A+, A, A-, B+, ...) can the dean's idea be made to work?

Assuming that the grades given out are only (A, B, C, ...) can the dean's idea be made to work?

Can any other schemes produce a desired ranking?

A concern is that the grade in a single class could change many students' deciles. Is this possible?

Data Sets

Teams should design data sets to test and demonstrate their algorithms. Teams should characterizes data sets that limit the effectiveness of their algorithms.

Results

We received the COMAP's report of the results of the competition on May 1, 1998. The team consisting of Tim Andrews, Shiva Kabra and Reba Schuller was awarded the Meritorious rating. The team consisting of Chris Graver, Geoffrey Sparks and Anne Wilson was ranked Successful Participant.

This year, 472 teams (representing 246 academic institutions in eight countries) submitted complete entries. Seven of these entries were judged Outstanding; eighty, Meritorious. One hundred sixteen entries received Honorable Mentions in COMAP's report. The remaining 269 teams were classified as Successful Participants.