A minimal optical system consists of a lens and sensor; the eye is the basic system. A more complex system could consist of transmission, recording, analysis, and display systems. A complete analysis would include the atmosphere as part of the system. Each of these contributes to system resolution.
The Angular Resolution criteria of Lord Rayleigh are the basic measure of lens resolution. However, the quantity expressed by the Rayleigh criteria is the limiting resolution. More complete understanding of the system is expressed by the (OTF), of which the limiting resolution is but one point.
The Optical Transfer Function describes the spatial (angular) variation as a function of spatial (angular) frequency. When the image is projected onto a flat plane, such as photographic film or a solid state detector, spatial frequency is the preferred domain, but when the image is referred to the lens alone, angular frequency is preferred. OTF may be broken down into the magnitude and phase components as follows:
:
where
|
where
:
is the normalized value of the maximum (for example, the voltage or grey value of the white area)
:
is the normalized value of the minimum (for example, the voltage or grey value of the black area)
When the system can no longer resolve the bars, the black and white areas have the same value, so Contrast = 0. At very low spatial frequencies, C_max = 1 and C_min = 1 so Modulation = 1. Some modulation may be seen above the limiting resolution; these may be aliased and phase-reversed.
When using other methods, including the interferogram, sinusoid, and the edge in the ISO 12233 target, it is possible to compute the entire MTF curve. The response to the edge is similar to a
Step Response , and the Fourier Transform of the first difference of the step response yields the MTF.
An interferogram created between two coherent light sources may be used for at least two resolution-related purposes. The first is to determine the quality of a lens system (see
LUPI ), and the second is to project a pattern onto a sensor (especially photographic film) to measure resolution.
This 5 bar resolution test chart is often used for evaluation of microfilm systems and scanners. It is convenient for a 1:1 range (typically covering 1-18 cycles/mm) and is marked directly in cycles/mm. Details can be found in ISO-3334.
The resolution target consists of a pattern of 3 bar targets. Often found covering a range of 0.25 to 228 cycles/mm. Each group consists of six elements. The group is designated by a group number (-2, -1, 0, 1, 2, etc.) which is the power to which 2 should be raised to obtain the spatial frequency of the first element (e.g., group ''-2'' is 0.25 line pairs per millimeter). Each element is the 6th root of 2 smaller than the preceding element in the group (e.g. element 1 is 2^0, element 2 is 2^(-1/6), element 3 is 2(-1/3), etc.). By reading off the group and element number of the first element which cannot be resolved, the limiting resolution may be determined by inspection. The complex numbering system and use of a look-up chart can be avoided by use of a newer layout chart, which labels the groups directly in cycles/mm and is available in the links below from Applied Image.
The target is a 3 bar pattern (long bars). The spatial frequency is printed alongside each triple bar set, so the limiting resolution may be determined by inspection. This frequency is normally only as marked after the chart has been reduced in size (typically 25 times). The original application called for placing the chart at a distance 26 times the focal length of the imaging lens used. The bars above and to the left are in sequence, separated by approximately the square root of two (12, 17, 24, etc.), while the bars below and to the left have the same separation but a different starting point (14, 20, 28, etc.)
The resolution target was specifically designed to be used with television systems. The gradually expanding lines near the center are marked with periodic indications of the corresponding spatial frequency. The limiting resolution may be determined by inspection. The most important measure is the limiting horizontal resolution, since the vertical resolution is typically determined by the applicable video standard (NTSC or PAL).
The resolution target is similar to the EIA target.
The target was developed for high definition television (HDTV) applications, since HDTV spatial resolution may exceed the limitations of the older targets. It includes several knife-edge targets for the purpose of computing MTF by Fourier Transform. They are offset from the vertical by 5 degrees to avoid problems with solid state
Color Filters .
The idea is analogous to the use of a
White Noise pattern in acoustics to determine system frequency response.
The interferogram used to measure film resolution can be synthesized on personal computers and used to generate a pattern for measuring optical resolution. See especially Kodak MTF curves.
A signal is an electronic waveform used to test analog transmission, recording, and display systems. The test pattern consists of several short periods of specific frequencies. The contrast of each may be measured by inspection and recorded, giving a plot of attenuation vs. frequency. The NTSC3.58 multiburst pattern consists of 500 kHz, 1 MHz, 2 MHz, 3 MHz, and 3.58 MHz blocks. 3.58 MHz is important because it is the
Chrominance frequency for NTSC video.
It should be noted whenever using a bar target that the resulting measure is the Contrast Transfer Function (CTF) and not the MTF. The difference arises from the subharmonics of the square waves and can be easily computed.
- Gaskill, Jack D. (1978), ''Linear Systems, Fourier Transforms, and Optics'', Wiley-Interscience. ISBN 0471292885
- Goodman, Joseph W. (2004), ''Introduction to Fourier Optics (Third Edition)'', Roberts & Company Publishers. ISBN 0974707724
- Fried, David L. (1966), "Optical resolution through a randomly inhomogeneous medium for very long and very short exposures.", J. Opt. Soc. Amer. 56:1372-9
- Robin, Michael, and Poulin, Michael (2000), ''Digital Television Fundamentals (2nd edition)'', McGraw-Hill Professional. ISBN 0071355812
- Smith, Warren J. (2000), ''Modern Optical Engineering (Third Edition)'', McGraw-Hill Professional. ISBN 0071363602
- Accetta, J. S. and Shumaker, D. L. (1993), ''The Infrared and Electro-optical Systems Handbook'', SPIE/ERIM. ISBN 0819410721
- Roggeman, Michael and Welsh, Byron (1996), ''Imaging Through Turbulence'', CRC Press. ISBN 0849337879
- Tatarski, V. I. (1961), ''Wave Propagation in a Turbulent Medium'', McGraw-Hill, NY
- Applied Image produces standard and custom image test patterns. Improved layout versions of the USAF chart are available (pn T-21 (labeled in cycles/mm) and pn T-22 (labeled in cycles/mm and uses a linear pattern for easier finding and scanning)).
- Norman Koren's website includes several downloadable test patterns
- Imatest software , also developed by Norman Koren, may be used to automate testing
