Demodulation and phase estimation of two-dimensional patterns

DETAILED DESCRIPTION INCLUDING BEST MODE Consider a two-dimensional fringe pattern with intensity of the fringe pattern as follows f(x,y)=a(x,y)+b(x,y), cos [2π(u0{x-x0}+v0{y-y0})-χ]  (1) Position coordinates (x,y) may be continuous for an analog pattern or discrete for digital patterns.
A background level is denoted by a(x,y) while an amplitude modulation term is denoted by b(x,y), both assumed to be slowly varying functions.
Phase term χ(x0,y0) presents the local offset of the overall phase term in the square brackets [ ] of Equation (1).
Referring to FIG.
1 where a 2-dimensional fringe pattern or AM-FM pattern, located around a point (x0,y0) and formed by a number of fringes 10 is shown.
A small region of interest is defined by Ω.
Within this small region Ω, the spatial frequency components u0 and v0 are slowly varying so that they are effectively constant around the point (x0,y0), making the carrier a linear function of x and y.
A normal 12 to the local fringes 10 is at an angle β0 to the x-axis.
A noise term is omitted from Equation (1), but may be present in real fringe patterns due to the occurrence of blurring, non-linearities, quantisation errors, smudging, scratches, cuts, dust, etc.
The intensity pattern f(x,y) is (generally) a rapidly varying function of position coordinates (x,y).
The Fourier transform (FT) of the above fringe pattern region Ω is: ##EQU1## wherein ι=√{square root over (b-1)} From Equation (2) it is clear that the Fourier transform FΩ of the fringe pattern f has 3 lobe centres, the first being at the origin and another two at spectral coordinates (u0,v0) and (-u0,-v0).
The last two lobe centres are mirror images of each other with regards to the origin.
The effect of windowing the fringe pattern f over a small region Ω is to blur the otherwise sharp lobes.
The central lobe AΩ, representing the zero frequency component, can be removed, and will therefore be disregarded in the following analysis