Theory of Fourier Transform Technique
The intensity distribution of an interferogram or fringe patterns can be expressed as
I(x,y)=Im(x,y)+Ia(x,y)cos[f(x,y)] (1)
where I is the intensity distribution of the interferogram, Im is the mean intensity, Ia is the intensity modulation amplitude, f is the angular phase information of the interferogram, and (x,y) represents all the points in the x-y plane of the object and the interferogram. The intensity function can also be expressed as
I(x,y)=Ia(x,y)+c(x,y)+c*(x,y) (2)
with
c(x,y)=(1/2)Im(x,y)exp[i f(x,y)] (3)
where * denotes complex conjugate.
After a two-dimensional discrete Fourier transform (DFT), the spatial frequency domain representation of the pattern becomes
F(z, h)=A(z, h)+C(z, h)+C*(z, h) (4)
where A(z, h) is the transform of Im(x, y), and C(z, h) and C*(z, h) are the positive and negative frequency spectra of the modulated carrier fringes. z and h are the spatial frequencies that represent intensity changes with respect to spatial distances. If the image size is 2n, the fast Fourier transform (FFT) can be used, which is much faster than the ordinary Fourier transform.
At the frequency domain, if C(z, h) can be isolated from A(z, h) and C*(z, h) in equation (4), then an inverse Fourier transform can be performed for C(z, h). Finally, c(x, y) can be obtained at the spatial domain and the phase information f can be calculated from
f=arctan[Im(c(x,y))/Re(c(x,y))] (5)
where Im[ ], Re[ ] represent the imaginary and real part of c(x, y), respectively. The phase f obtained from the above equation ranges from -p to +p, and it does not reflect the fringe order. Phase unwrapping is required to make the phase represent the fractional fringe order and it will be discussed later in this chapter.
In order to be able to isolate C(z, h) from A(z, h) and C*(z, h) at the frequency domain, the intensity function should have continuous and monotonically changing derivatives across the field. Unfortunately, this condition is often violated for the fringe pattern representing engineering deformations.
The problems in Fourier transform
The Fourier transform may be not siutable to measure the complex objects accurately and on real-time. There might be some other ways to solve it.
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What a detailed description? I like it very much. Try it but just after my mcitp preparation. I hope it will be a good combination.
I like the idea of this site.
I like the idea of this site. Would hope for more references. For this one, i can provide the reference, i think:
Takeda, M, Ina, H, Kobayashi, S., (1984) Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry
now to my question: You use 2D-FFT in the code provided. Since i'm not used to image-FFTs, it would be nice, if you could provide a simple tutorial as to which corners, i should be moving in the FFT to filter out the low spatial frequencies.
Also, if anybody is interested, i could provide the matlab-code for the 1D-FFT-profilometry method including an example.