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The Q of an Imaging System Stephen L. Kwiatkowski

I. Q FOR AN I DEALIZE I MAGING S YSTEM Consider the idealized imaging system shown in Figure 1 consisting of a single circular thin lens with the system stop at the lens and where the object is at infinity and the image at the focal plane is captured on a digital detector array.

percent fill factor, these rays represent the geometric region on the object that is collected and summed within a detector element causing any details within that region to be lost. The angle subtended by these rays is θp = fp .

Fig. 2. Idealized imaging system with object at infinity. The rays represent chief-rays imaged to the width of the diffraction limited impulse response. Fig. 1. Idealized imaging system with object at infinity. The rays represent chief-rays imaged onto the extreme of the detector element.

The definition of Q is: λf Q= Dp

(1)

where, p is the detector element period (equal to the detector element size for 100 percent fill-factor), f is the lens focal length, D is the diameter of the system clear aperture, and λ is the wavelength. Also shown in Figure 1 are chief-rays corresponding to the period of the detector elements. For our case of 100

The inherent diffraction of the imaging system is another cause for diminished detail in the image. Consider the impulse response, or image blur spot of the system shown in Figure 2. This blur spot is the minimum dimension in the image plane that can be created by the imaging system even for the case when the object is spatially confined point of light. This means that closely spaced features on the object will loose their individual identity because of this blurring. The region on the object associated with the diffraction blurring is found by tracing chief-rays to the edges of

the blur spot. The angular extent of the object λ blur spot is θD ∝ D . A physical meaning can be attached to Q by re-arranging the terms in Equation 1 to reveal that Q is proportional to the ratio of the diffraction angle of the system θD to the angle subtended by a detector element θp : Q =

λ D p f

Q =

θD θp

(2)

The relationship between the size of the detector element and the blur spot is illustrated in Figure 3 where the angel θp is held constant. The size of the blur spot can be adjusted by changing the size of the system clear aperture (e.g. this can be achieved by changing the aperture stop setting on a camera). We find that the image resolution for an imaging system begins to be limited by the blur spot when Q > 1 (θD > θp ), and limited by the detector element period when Q < 1 (θp > θD ). This behavior suggests that an optimum value for Q can be found by weighing system parameters against a criteria for image resolution. Another perspective of Q can be had from the spatial frequency domain of the imaging system. Using the definitions for the Nyquist 1 frequency of the sampled image as fN = 2p and the incoherent cut-off frequency for an imaging system with a circular aperture as D fc = λf , Q can be written as: Q=2

fN fc

Fig. 3. The blur spot is larger than the detector period when Q > 1 and conversely when Q < 1.

Fig. 4. The MTF for an idealized imaging system with a circular aperture.

the detector pitch, i.e. by purchasing a camera with more ’megapixels’) because the higher frequencies that could then be sampled by these more closely spaced detector elements when fN > fc never reach the detector because they are cut-off by the optical system. However, these additional detector elements could yield an indirect improvement to image quality through the de-mosaicing algorithms used by

(3)

This relationship shows that for Q > 2 the Nyquist frequency is greater than the system cut-off frequency. However, nothing is gained in image resolution when the system has been modified to reach Q > 2 (e.g. decreasing c January, 2015 Stephen L. Kwiatkowski

All Rights Reserved.

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color digital cameras (I plan to write a note on this subject later). As Q becomes smaller than 2, the detector Nyquist frequency drops below the system cut-off frequency and the frequencies between Nyquist and the cut-off are aliased. This aliasing can have a detrimental affect on perceived image quality. The cases of Q = 1 and Q = 2 are illustrated in Figure 4. II. TAKE AWAY The Q of an optical imaging system can be qualitatively characterized as; Q is the number of detector periods that fit across a blur spot. or alternately, Q is two times the ratio of the Nyquist frequency to the Cut-off frequency. End of Note.

c January, 2015 Stephen L. Kwiatkowski

All Rights Reserved.

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