Fourier Descriptors

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Fourier Descriptors
Historical note Much of the earliest work in FDs
was done here at OSU by Cosgriff (1960). Prof.
Hemami (1974) has also contributed to this
area. Fourier descriptors describe the shape
in terms of its spatial frequency content.
Begin by representing the boundary of the shape
as a periodic function (in one of a few
possible ways). Expand this in a Fourier
series obtain a set of coefficients that
capture shape information. When done as I will
show you... DC term --- centroid of the
region Add in the /- 1st terms --- best
fitting ellipse Add in the next pair --- best
fitting hourglass And so on. So the Fourier
descriptors offer a shape description in which
we can tune the degree of precision according
to our needs to make effective tradeoffs
among accuracy, efficiency and speed of
recognition, and the compactness of the
representation.
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We begin by imagining that the image plane is
complex.
If we make repeated traces of the contour...
For n 1, 2, 3, ... this is periodic.
Expand u(t) in a complex Fourier series
Where .... we
can pick T 2p
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The truncated series is
The period is made equal to 2p by choosing the
speed at which we circumnavigate the shape
properly. Your textbook discusses this.
The coefficients can also be normalized to
eliminate the effects of scale, translation,
and rotation -- if you wish. In our
map-reading application, we removed these effects
for spot symbol (characters, etc.)
recognition, but retained the information for
use in symbol grouping (putting letters
together to spell words -- like OHIO for
instance). We used about 40 coefficients in
all, in a decision tree that used roughly 4-6
coefficients at each node and a full check at
the leaves, with backjumping.
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Fourier Descriptors from the Crack Code
m is the number of crack code elements ø(k) is
determined from a lookup table
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Fourier Descriptors from the Chain Code
n ? 0 K is the number of chain code links T
tk is the period of the contour
The delta gamma and delta t numbers come from a
lookup table
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Fourier Descriptors from x,y Coordinates
Where...