Beautiful mathematical ideas abound in multimedia software! Some are not
encountered until late in undergraduate or even postgraduate study, whereas they
can be appreciated and used much earlier. This book presents a few dozen such
pearls, strung together by their ubiquity in many applications. It is based on
the course Topics in Applied Mathematics: Multimedia Algorithms, first taught at
Washington University in the Fall of 1997.
The material is divided into six roughly equal chapters, intended for a pace of
six lecture hours per chapter. Thus, the entire text may be covered in one semester,
with time left for examinations and student projects as was done at Washington
University. Alternatively, one or two of the last chapters may be omitted to fit the
material into a trimester. There is a good deal of independence among the chapters,
to permit such tailoring. Any single chapter, for example, can be a starting point
for an undergraduate thesis in applied mathematics.
Algorithms are often divided into "integer" (discrete, exact) and "real" (continuous,
approximate) types. Multimedia software is a big user of both types,
combining them as needed, so this text does not enforce any kind of segregation.
Understanding and analyzing a mix of procedures, with the assumptions, principles
and techniques commonly used in their implementations, uncovers the mathematical
gems in a more natural way than separation into "discrete mathematics" and
"applied analysis."
In some cases, the mathematical theory is the main point. Proofs crucial to
understanding are provided, at a level suited to mature undergraduates past Calculus.
An appendix on basics is available to fill gaps. Further readings in more
advanced sources are suggested at the end of each chapter. In other cases, the main
point is practical implementation. Standards, conventions and common practice are
discussed with reference to the actual standards documents. Key algorithms are
presented in pseudocode and Standard C to assist programming, experimentation,
and exercise solution. Some of the longer exercises are designed to give a taste of
applied research, though complete solutions are supplied in the answers appendix.
The shorter exercises can be easily modified for examinations.
I am grateful to my student, Dr. Wojciech Czaja, for his assistance in preparing
some of the exercise solutions and for his numerous useful comments. I am also
grateful to the National Science Foundation for the financial support, through grant
DMS-9631359, that helped me create the course and this book.
