In addition to determining interior orientation elements, laboratory methods of camera calibration provide an evaluation of the camera’s resolving power. As noted in Sec. 2-3, there are two common methods of specifying lens resolving power. One is a direct count of the maximum number of lines per millimeter that can be clearly reproduced by a lens; the other is the modulation transfer function (MTF) of the lens. The method of calibration employed to determine the line count consists of photographing resolution test patterns using a very high-resolution emulsion. The test patterns (an example is shown in Fig. 3-19) are comprised of numerous sets of line pairs (parallel black lines of varying thickness separated by white spaces of the same thickness). The measure of line thickness for each set is its number of line pairs per millimeter. Line thickness variations in a typical test pattern may range from 10 to 80 or more line pairs per millimeter. If the multicollimator method is used to calibrate a camera, the test patterns may be projected by the collimators simultaneously with the collimator crosses and imaged on the diagonals of the camera format. After the photograph is made, the resulting images are examined under magnification to determine the finest set of parallel lines that can be clearly resolved. The average of the four resolutions at each angular increment from the central collimator is reported in the calibration certificate. Another parameter generally reported is the area-weighted average resolution (AWAR), which is an indication of resolution over the entire format.
FIGURE 3-19 Resolution test pattern for camera calibration.
Whereas the above-described maximum-line-count method appears to be a relatively simple and effective way of quantifying resolving power, it is not without its shortcomings. In the line count procedure, with each succeedingly smaller test pattern, the sharpness of distinction between lines and spaces steadily diminishes, and the smallest pattern that can clearly be discerned becomes somewhat subjective. The preferred measure of resolution is the modulation transfer function.
A fundamental concept involved in quantifying the modulation transfer function is the notion of spatial frequency. Spatial frequency is a measure of the number of cycles of a sinusoidal wave per unit distance. An analogy can be drawn from audio signals where frequency concerns the number of sound waves per unit time. The units of audio frequency are generally specified in cycles per second, or hertz (abbreviated Hz), whereas units of spatial frequency are typically given in terms of cycles per millimeter. Spatial frequency is directly related to the count of line pairs per millimeter discussed above. A black-and-white line pair corresponds to the up-and-down pulse of a sine wave and thus can be defined as one cycle of a wave. Therefore the number of line pairs per millimeter is equivalent to cycles per millimeter, or spatial frequency. Images that contain areas of rapidly changing levels of
brightness and darkness have high spatial frequency, whereas images that contain areas of gently changing levels have low spatial frequency.
To determine modulation transfer, density scans using a photogrammetric scanner (see Sec. 4-6) are taken in a single trace across test patterns similar to those used in the line count procedure, as shown in Fig. 3-20a and c. For heavy lines with wide spacing, the actual distribution of density (brightness variations) across the object pattern would appear as the dashed lines shown in Fig. 3-20b, whereas brightness distributions measured with a densitometer across the image of this pattern would appear as the solid lines. Note that the edges of the image patterns are rounded somewhat in Fig. 3-20b, but the amplitude of brightness differences is the same as that for the original object. Thus at this spatial frequency of the pattern, modulation transfer is said to be 100 percent. Figure 3-20c shows an object pattern at a frequency four times that of the pattern shown in Fig. 3-20a. The density distributions of the object and resulting image of this higher-frequency pattern are shown in Fig. 3-20d. Note that in this figure, not only are the edges rounded, but also the amplitude of brightness differences is about one-half that of the original object. This indicates a modulation transfer of 50 percent from object to image. Actually, Fig. 3-20 is a somewhat simplified illustration of the quantification of modulation transfer. In the rigorous determination of modulation transfer, exposure values (rather than densities), which have a logarithmic relationship to density, are employed. The reader may consult references at the end of this chapter for more details on the modulation transfer function.
FIGURE 3-20 (a) Test object at low spatial frequency with density trace. (b) Density modulation of object (dashed) and image (solid). (c) Test object at high spatial frequency with density trace. (d) Density modulation of object (dashed) and image (solid). [Note that in part (b), the amplitude of the image modulation is the same as that of the object, corresponding to 100 percent modulation transfer.
In (d) however, amplitude of the image modulation is one-half that of the object, corresponding to reduced modulation transfer.]
By measuring densities across many patterns of varying spatial frequencies, and plotting the resulting modulation transfer percentages on the ordinate versus corresponding spatial frequencies on the abscissa, a curve such as that illustrated in Fig. 3-21 is obtained. This curve is the modulation transfer function. The MTF has a number of advantages over the simple line count method. It is a very sensitive indicator of edge effects, and it also affords the capability of predicting the resolution that may be expected at any given degree of detail. Furthermore, MTF curves can be combined for different lenses, films, and film processes; thus, it is possible to estimate the combined effects of any given imaging system. For these reasons, the MTF has become the preferred method of expressing resolution.
FIGURE 3-21 Curve of modulation transfer function (MTF).
The upper limit of resolution for a digital frame camera is absolutely fixed because of sampling into discrete elements. Since a full cycle of a wave in terms of spatial frequency must consist of a dark-to-light transition (line pair), two CCD elements are the minimum number that can capture information at the highest frequency. Thus the maximum spatial frequency (best resolution) at image scale that can be detected is
(3-2)
where w = width between centers of adjacent CCD elements fmax = maximum detectable frequency
Example 3-2
A digital frame camera consists of a 5120 × 5120 array of CCD elements at a pixel size of 6 μm square. The nominal focal length of the camera is 40 mm. What is the maximum spatial frequency that can be detected (at image scale)? What is the angular field of view for this camera?
Solution By Eq. (3-2),
Format diagonal d can be calculated by
By Eq. (3-1),
References
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Anderson, J. M., and C. Lee: “Analytical In-Flight Calibration,” Photogrammetric Engineering and Remote Sensing, vol. 41, no. 11, 1975, p. 1337.
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Problems
3-1. List and briefly describe the three geometric categories of imaging devices.
3-2. List the requirements of a precision mapping camera.
3-3. What is the angular field of view of a camera having a 230-mm-square format and a 209-mm focal length?
3-4. Repeat Prob. 3-3 except that the format is 89-mm-square and the focal length is 112 mm.
3-5. For a camera having a 230-mm-square format, what range of focal lengths could it have to be classified as wide angle?
3-6. An aerial camera makes an exposure at a shutter speed of s. If aircraft speed is 390 km/h, how far does the aircraft travel during the exposure?
3-7. Repeat Prob. 3-6, except that the shutter speed is s and aircraft speed is 170 mi/h.
3-8. An aerial camera with forward-motion compensation and a 152.4-mm focal length is carried in an airplane traveling at 280 km/h. If flying height above terrain is 3200 m and if the exposure time is s, what distance (in millimeters) must the film be moved across the focal plane during exposure in order to obtain a clear image?
3-9. Repeat Prob. 3-8, except that the focal length is 305 mm and the flying height above terrain is 5500 m.
3-10. Name and briefly describe the main parts of a frame aerial mapping camera.
3-11. Discuss briefly the different types of camera shutters.
3-12. What is the purpose of the camera mount?
3-13. What is the primary benefit of gyro-stabilized camera mounts?
3-14. What is crab, and how may it be caused?
3-15. Why is camera calibration important?
3-16. What are the elements of interior orientation that can be determined in camera calibration?
3-17. List and briefly describe the various definitions of principal point.
3-18. Briefly describe the advantages of using the modulation transfer function to quantify the resolution of a lens over the simple line pairs per millimeter threshold.
3-19. Illustrate and briefly describe the concept of spatial frequency.
3-20. A digital frame camera consists of a 4096 × 4096 array of CCD elements at a pixel size of 6.5 μm square. The nominal focal length of the camera is 40 mm. What is the maximum spatial frequency that can be detected (at image scale)? What is the angular field of view for this camera?