Question: Short note on Fourier Descriptors.

Subject : Digital Image Processing

Topic: Chap 3: Image Segmentation and Representation

Difficulty: Medium

mumbai university dip(47) • 40 views
modified 29 days ago  • written 29 days ago by gravatar for tanya.tanyabarnwal tanya.tanyabarnwal0

A common method of describing the contour of an object is by using 1-Dimesnional Fourier Transform.

enter image description here

The figure shows a N-point digital boundary in the spatial Domain. Each of these edges pixels can be defined by its x and y coordinates. Starting at an arbitrary point (x0,y0),(x1,y1),(x2,y2)…….(xn-1,yn-1) points are encountered as we move in the counter clockwise direction.

Each of these points can be expressed as xr and yr for r=0,1……N-1. These coordinates values can be used to generate a complex function of the form,

f(n) = x(n) + j y(n) for n=0,1, 2,...........,N-1

Hence the x-axis is treated as real axis and y-axis as the imaginary axis. The Fourier transform of this function f(n) yields the frequency components that describe the given edge.

The discrete Fourier transform (DFT) of f(n) is

      F(u)= 1/N∑_(n=0)^(N-1)▒〖f(n)e^(-j2" " un/N) 〗

for u=0,1,2,………...,N-1

The advantage of using this equation is that it reduces the edge description problem from 2-Diemsnsional to 1-Dimension.Substituting the value of f(n) we have,

F(u)=1/N for u=0,1,2,…..,N-1

The coefficients of F(u) are called Fourier descriptors. The inverse discrete fourier transform (IDFT) of F(u) gives back f(n).


for n=0,1,2,3……N-1

However instead of using all the F(u) coefficients, we only use few of them while remaining terms are made zero,

‘ (n)=

for n=0,1,2,3……N-1

Although only M terms are used for F(u), f(n) still has 0 to N-1 values. That is the same number of points exist in the new approximated boundary but not as many terms are used in the reconstruction of each point.

written 29 days ago by gravatar for tanya.tanyabarnwal tanya.tanyabarnwal0
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