Smooth

In the last exercise, it is possible that multiple corners are observed, which may complicate the process of determining the chromosome's major axis. Often, these other "corners" are not of interest. They do not correspond to the ones that are farthest along chromosome's major axis. They are  important details but for now, they may be conveniently ignored in this phase of the classification. What we need is a method that removes these "extra" details, while preserving the overall shape of the chromosome.

Fourier Descriptors

From the Freeman vector code (FVC), we obtain the contour by computing for the locations of all the points along the FVC starting at seed, and then moving along the boundary, in the direction specified by the codes. Once we have the x and y-coordinates of those points, we create a Fourier descriptor for the shape by creating a complex vector from x's and y's: x + iy. Taking the discrete Fourier transform (DFT) of the complex-valued vector produces the shape's Fourier descriptor.

Fourier descriptors have properties inherited from Fourier transforms:

• Translation invariant
• Scale invariant
• Rotation and starting point

Translation invariant means that the Fourier descriptors remain the same no matter where the shape is located. Scaling the shape by a factor also scale the descriptor by the same amount. Finally, changing the origin or the orientation of the shape (through rotation) only affects the phase of the descriptors. (see Wolfram Demo site listed in the reference section)

Smooth operator

Power spectrum

In the figure above, we have an outline of a chromosome showing numerous spikes along its contour line. The power spectrum (log10 scale), show its frequency content. Notice that the power spectrum plot is mirrored and symmetric. Half of the result contain its negative frequencies content. Usually an FFTSHIFT operation is performed so that the plot is centered at f = 0. f = 0 correspond to the bias of the signal. If the contour were translated so that the origin is at (0,0) and its x and y components contains positive and negative values, we see this bias eliminated from the power spectrum.

Now, on to the business of removing the spikes. It turns out that this is straightforward process. Simply clipping, or, removing the higher frequencies by setting them to 0 and taking the inverse Fourier transform removes the spikes, i.e. making the contour smoother.

And if more frequencies are filtered out, it becomes even smoother.

Note

• The Fourier transform works for an infinite, continuous signal. For finite, sampled signals, such as those used in our computations, we use the discrete Fourier transform (DFT). All properties of the Fourier transform also applies to the DFT.

References

• Wikipedia contributors, "Discrete Fourier transform," Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Discrete_Fourier_transform


• William Sethares, "Fourier Descriptors", http://demonstrations.wolfram.com/FourierDescriptors/, Wolfram Demonstrations Project, Published: February 21, 2012


• Soriano, M (2015), Fourier Descriptors, lecture notes presented in Physics 301 - Special Topics in Experimental Physics (Advanced Signal and Image Processing) at National Institute of Physics, University of Philippines Diliman, Quezon City on 7 August 2015