Microworld: Metric
Mapping
Click the Hyperlink
above to visit the Microworld.
Authors:
Kate
Johnson and
J.T.
Ratnanather
The goal of this book is to explain how an algorithm for metric mapping in
1-dimension works. Only the 1-d algorithm will be explored because it is the
easiest to explain, and the 1-d approach can be applied to multiple dimensions
for 2-d or 3-d problems. A map is a function. It is used here as a transformation
function between one coordinate system and another. A 1-d map can be represented
on a set of coordinate axes for ease of viewing, as can be seen by the above
example.
The
x-axis represents the input values from one coordinate system. The y-axis
represents the output values for the target coordinate system. The same pictures
are represented, but the important thing is where they have been moved to.
The "frog"value has moved from .9 to .81, and the "car" value has moved from
.7 to .49. The pictures are only there to illustrate how the x-axis has changed.
There
are many ways to do this, but this book will focus on the concepts needed
to understand the gradient descent approach to finding the map. Furthermore,
we are not just interested in finding the one map that will translate the
template image to the target image. We are actually interested in the complete
path between the two images.
Theoretically,
there are an infinite number of paths between the template and the target,
but the path that we are interested in is the shortest path, known as the
"geodesic". The these two concepts (gradient descent and geodesic) will be
explored further in the next pages, and then an animated example will be provided.
Number
of Pages: 8
Animation: Yes
Grade Level:14-15
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| author of this website, Mathwright 2000, MindScapes, | ||
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