ECEN 4616/56164/12/2013
The effect of Detector Resolution on Images
The response of an imaging system to a point source is the Point Spread Function (PSF). This is, in fact, the impulse response of system.
Considering only the (well-corrected) optical part:
- The PSF is essentially invariant over the optics’ Field of View (FOV) – this is part of the definition of “well-corrected”.
- The PSF for circularly symmetric optics (without any obscurations) is the Airy Pattern – this is the other part of “well-corrected”.
The Airy Pattern:
The radius from the center peak to the first null of the Airy Pattern is , where λ is the wavelength of the light, and the is the ratio of the focal length to the diameter of the entrance pupil.
A normal optical imaging system is linear: If one point source is input at the object plane, one PSF (Airy Pattern) is output at the image plane. If N point sources are input, N PSFs are added to the image plane. Since any possible input source can be decomposed into a (possibly infinite) number of point sources, the output image can be determined by adding up the corresponding PSFs.
What we’ve just described is the convolution operation; So the image of an optical system can be determined by the convolution of the input object distribution with the PSF:
(eq. 1)
The convolution description only is valid if the PSF is essentially the same over the image. For space-variant systems (where the PSF does vary considerably), the summation description is still valid.
The Convolution Theorem in Fourier Transforms allows us to write the (space invariant) imaging equation (eq. 1) in terms of the Fourier Transform of the image, object, and PSF:
(eq. 2)
Where are spatial frequencies in the directions and the operation between the Object and PSF transforms is an element-by-element multiplication.
Since the Optical Transfer Function (OTF) is the Fourier Transform of the psf, we have:
Then equation 2 insures that the Image is a (spatial frequency) band-limited copy of the Object, with the Fourier Transform of the psf acting as the limit. From Abbe’s criteria, we know that the maximum spatial frequency that can exist in the image of our “perfect” (i.e., no aberrations) optical system is:
(eq. 3)
And the period of that frequency is:
(eq. 4)
(which is about 80% of the diameter of the Airy Pattern to the first null ring.)
The Aerial Image:
So far, equations 3 & 4 describe the optical image – often called the “aerial image”, since it exists just above the detector.
Sampling Theorem:
The sampling theorem (which we won’t derive) demonstrates that the value of a continuous function, that has a limiting spatial frequency of , can be uniquely recovered from a set of regularly spaced samples of the function, with spacing of . This is known as the Nyqist sampling rate, and is called the Nyquist frequency for a sample spacing of Δx. (These are named for Harry Nyquist, an engineer at Bell Laboratories in the 1950’s and 1960s.)
Just because the signal can be recovered from this sampling rate, does not mean that it is easy or that the Nyquist rate should be used. The only method of recovering the frequencies near the Nyquist limit involves replacing every sampled data point with a sinc function, centered on that point and whose value is zero at every other point (we also won’t derive this):
For example, a signal, s(x), sampled at integer values of x would be reconstructed from those samples by a summation of sinc functions:
The sinc function for the sample at k=0 would be:
Note that the interpolating function is zero at every sample point except x = 0.