1998 MCM Problem A
MRI Scanners


Problem:

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 pixels.

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

            (1)  for planning minimally invasive treatments,

            (2)  for calibrating the MRI machines,

            (3)  for investigating structures oriented obliquely in space,

            such as post-mortem tissue sections in animal research, 

            (4)  for enabling cross-sections at any angle through a brain atlas

            consisting of black-and-white line drawings.

To design such an algorithm, one can access the values and locations of the pixels, but not the initial data gathered by the scanner.

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 . 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.

 

1998 MCM Problem B
Grade Inflation

 

Problem:

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 thought 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 student’s deciles.  Is this possible?

Data Sets

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