1998 年美国大学生数模竞赛题
Problem A:
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 postmortem 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
(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:
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.
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