Image Processing Fundamentals - Sampling Density for Image Processing

Sampling Density for Image Processing

Sampling aperture

To prevent the possible aliasing (overlapping) of spectral terms that is inherent in eq. two conditions must hold:

* Bandlimited A(u,v) -

* Nyquist sampling frequency -

where u_c and v_c are the cutoff frequencies in the x and y direction, respectively. Images that are acquired through lenses that are circularly-symmetric, aberration-free, and diffraction-limited will, in general, be bandlimited. The lens acts as a lowpass filter with a cutoff frequency in the frequency domain (eq. ) given by:

where NA is the numerical aperture of the lens and is the shortest wavelength of light used with the lens . If the lens does not meet one or more of these assumptions then it will still be bandlimited but at lower cutoff frequencies than those given in eq. . When working with the F-number (F) of the optics instead of the NA and in air (with index of refraction = 1.0), eq. becomes:

Sampling aperture

The aperture p(x,y) described above will have only a marginal effect on the final signal if the two conditions eqs. and are satisfied. Given, for example, the distance between samples X_o equals Y_o and a sampling aperture that is not wider than X_o, the effect on the overall spectrum--due to the A(u,v)P(u,v) behavior implied by eq.--is illustrated in Figure 16 for square and Gaussian apertures.

The spectra are evaluated along one axis of the 2D Fourier transform. The Gaussian aperture in Figure 16 has a width such that the sampling interval X_o contains +/-3 (99.7%) of the Gaussian. The rectangular apertures have a width such that one occupies 95% of the sampling interval and the other occupies 50% of the sampling interval. The 95% width translates to a fill factor of 90% and the 50% width to a fill factor of 25%. The fill factor is discussed in Section 7.5.2.

Figure 16: Aperture spectra P(u,v=0) for frequencies up to half the Nyquist frequency. For explanation of "fill" see text.