Wei Liu (u0614581)
weiliu@sci.utah.edu
Date: March 12, 2010
After this step we have Fourier coefficients
,
for
coordinates
, and coefficients
,
for coordinates
. DC
components
and
can also be obtained in similar way.
reconstruction: Can be done by choosing different number of Fourier harmonics components and by inverse Fourier transform in the paper.
elliptic loci: Careful observation of
and
(the
th component of reconstructed contour indicate this is a elliptical
shape. This can be shown by canceling the sine and cosine terms and get something like
![]() |
Phase shift (starting point): The change of starting point
does not change the reconstructed contour. It only change the phase of
the sine and cosine. If we use
for the forward displacement
between original starting point and new staring point, the
relationship between
, and the change of phase
of
first component is
, where
is the
period of the first harmonics.
The phase shift
can be computed by the method of the paper,
i.e. look at the derivative of the magnitude of first harmonics with
respect to
. once we get
, we can use the
relationship between original coefficients and
shifted
coefficients, and get the new coefficients
, which
is the coefficients when starting point is located at semi-major axis.
Scaling invariant: is done by dividing all coefficients by the magnitude of semi-major axis.
[vase]
![]() ![]() ![]() ![]() |
[vase]
![]() ![]() ![]() ![]() [vase] ![]() ![]() ![]() ![]() |
Changing starting point: We then test the reconstructed contour using
, which is the coefficients for the starting point at semi-major axis. The results at fig. 3. Because I did not connect the starting point and ending point, it can be clear seen that the starting points are located at the semi-major axis.
[vase]
![]() ![]() ![]() ![]() [vase] ![]() ![]() ![]() ![]() [vase] ![]() ![]() ![]() ![]() |
rotated contour: After having rotation invariant coefficients, we use these coefficients to reconstruct contours. The reconstructed ones are all in a new coordinates system such that the positive
is in the same direction with the semi-major axis of the ellipse. The results at fig. 4. Because there is ambiguity of the two possible orientation of the semi-major axis, I also plot another orientation with phase different
with fig. 4. The results at fig. 5.
[vase]
![]() ![]() ![]() ![]() [vase] ![]() ![]() ![]() ![]() [vase] ![]() ![]() ![]() ![]() |
[vase]
![]() ![]() ![]() ![]() [vase] ![]() ![]() ![]() ![]() [vase] ![]() ![]() ![]() ![]() |
Scale invariant: After dividing all the coefficients by the magnitude of semi-major axis, we get scale invariant coefficients. Fig. 6 has reconstructed contours from these rotation and scale invariant features.
[vase]
![]() ![]() ![]() ![]() [vase] ![]() ![]() ![]() ![]() [vase] ![]() ![]() ![]() ![]() |
Sum of difference: To see the difference of two contours, we can compare their Fourier coefficients. We use sum of squared difference between to set of coefficients. For the coefficients of
, I just add them together. Because for each contours, there are two possible set of Fourier coefficients, for the two possible orientations. So the difference between two contours have four possible values. The minimal value among the four is a good choice to indicate the true difference between the two contours.
Table 1 and table 2 have the comparisons. If I look for a object most similar to 'house', one component and 20 component have different results: For one component, 'duck' is most similar to 'house'. For 20 components, 'vase' is the most similar.
My conclusion is this is great tool for shape analysis, even from view of nowadays. Further, I think if we look at the image as 2-D signal, we can use many concepts and algorithms in signal processing, as DSP has long history and rich method. Fourier harmonics is only one of them.