A Basis for Estimating Digital Camera Parameters

Home , Airy disk, Angular resolution, Optical resolution

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Abstract Developing F/p 0.82 Based on the Rayleigh Criterion Matching the diffraction-limited optical resolution with the for Resolving Power of Diffraction Limited Optics appropriate detector size is a fundamental design requirement for digital imaging systems. A useful Design Function based and the Airy Disk on the Airy disk is Fp 0.82. Where is the average wave- The wave nature of light and the consequent diffraction at a length, F is the camera F-number, and p is the detector sam- circular aperture make it impossible to generate an ideal di- pling pitch (pixel size). A second metric, attributed to Schade mensionless image point. The wave energy forming a point and reported by Holst (1999), produces an often used Design image is distributed in a central disk called the Airy disk (see Figure 1). The Airy disk is defined with a radius that subtends Function: F p 1. Examples demonstrate the use of the De- sign Functions to determine basic parameters, F-number, an angle , defined by the Rayleigh criterion (Jensen, 1968, focal length, aperture diameter, and pixel size for an imaging p. 82; Kraus, 1993, p. 69; Schott, 1997, p. 326): i.e., system. Pixel size is selected from commercially available arrays and the other parameters are estimated given the re- Radius of Angular Resolution, 1.22 D (radians) (1) quired ground sampled distance (GSD) with either of the two Design Functions. One of the primary uses for the Design where is the average wavelength being utilized by the detec- Functions is to provide optical systems engineers with a sim- tor and D is the diameter of the objective lens, mirror, or ple, fast, and proven means of arriving at first-order estimates aperture. for electro-optical camera designs. It follows that estimating Multiplying both sides of Equation 1 by the camera’s focal costs for building large space camera systems should be less length (f) yields complicated. f 1.22 fD. (2) Introduction Assuming adequate signal and scene contrast, the spatial reso- Recognize that fD is the F-number and f is the linear ra- lution of an optical imaging system with digital detectors such dius of the optical resolution limit. The most popular measure as a charge coupled device (CCD) array is limited either by the of optical resolution uses Rayleigh’s criterion and defines the resolution limit of the optics or the detector sampling size. diameter of the Airy disk. Because of diffraction, what should When ground sampled distance (GSD) is used to denote the be a point source actually is a small disk of light surrounded spatial resolution required for a remote sensing system, it is by a number of light and dark rings, as shown in Figure 1. often assumed that the detector sampling is the limiting factor Approximately 84 percent of the light falls in the central disk in spatial resolution. Airborne digital camera systems such and the remainder in the outer rings (Moffitt and Mikhail, as the Kodak DCS 460 being flown by Emerge and the Contax 1980, pp. 41–42). Therefore, in the focal plane of the camera, 645 by Pictometry International specialize in spatial resolu- the Airy disk diameter (Kraus, 1993, p. 69; Schott, 1997, tions of 0.5, 1, 2 (0.15, 0.3, 0.6, or 0.9 m) or 3 ft GSD. Space p. 326; Holst, 1999) is Imaging’s Ikonos Satellite produces a 1-m GSD and Digital Globe’s latest Quickbird Satellite offers a 0.6-m GSD. Orbital dAiry 2.44 F. (3) Science plans to offer OrbView’s 3 and 4 imagery in the post- 2000 time frame with comparable resolution. These GSD’s Equation 3 expresses the diameter (d) of the Airy disk in refer to the projected pixel size on the ground and ignore any terms of and F, and d establishes the sampling pitch be- effects that the optical system may have on the spatial resolu- tween two detectors (pixels). The F-number is a measure of tion. According to Fiete (1999), even if the detector sampling the light collecting capability of the optics. For a constant (pixel size) is the limiting factor in spatial resolution, the in- focal length (f), as F increases, the aperture gets smaller. In teraction between the detector sampling (CCD) and the perfor- this derivation, d 2 is to be the detector size (pixel size) mance of the optics plays an important role in determining which the optics must resolve. This paper will use p in the the final image quality. In this paper the Design Function, equation to represent detector size to better correspond to dig- Fp 0.82, is derived using the diameter of the Airy Disk ital camera language where p stands for one pixel dimension. which is commonly used as a measure of the resolving power Matching the optical resolution (Airy disk) to the detectors is for optical elements according to the Rayleigh Criterion the fundamental essence of the design function being derived. (Schott, 1997, p. 326). The utility of the design functions for In concert with the Nyquist sampling criterion (Holst, 1998, estimating camera parameters will be explained with exam- pp. 286, 320–323), two detectors (2p) are placed within the ples. As examples, the Kodak DCS 460 Digital Camera, the area defined by the Airy disk. This means that the area de- Ikonos Sensor System, and the Quick Bird will be evaluated fined by the Airy Disk is adequately sampled (Holst, 1998, against the Design Functions Fp 0.82 and Fp 1 in order to demonstrate their usefulness for quick first-order estimates of the principal parameters for the camera. Photogrammetric Engineering & Remote Sensing Vol. 70, No. 3, March 2004, pp. 297–300. 0099-1112/04/7003–0297/$3.00/0 6 Tawney Point, Rochester, NY 14626 © 2004 American Society for Photogrammetry ([email protected]). and Remote Sensing

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size of two detectors in the focal plane. Given an adequate signal-to-noise ratio and MTF from the optics coupled with sufﬁcient scene contrast, this combination will produce quality imagery.

Example 1: Consider the Kodak DCS 460 Digital Camera with a Frame Array (100% ﬁll factor) of 3000 by 2000 Pixels and a Lens Focal Length f 28 mm Let average 0.6 m for the visible spectrum and pixel size, p 9 m. Solving Equation 7 for the F-number yields F 0.82p (8) or F 0.82(9 m)0.6 m 12.3. Conclusion: The DCS 460 using an F-number 12 will have the capability to produce good image quality assuming adequate lighting and proper shutter speed. Because F fD, then the aperture diameter, is Figure 1. Airy disk containing two pixels.

D fF. (9)

pp. 286, 320–323). Figure 1 illustrates two pixels inside the D, the diameter of the aperture, is a major cost driver for large Airy disk. satellite cameras, but not for the DCS 460. For the DCS 460 Now Equation 3 representing Figure 1 can be re-written D 28 mm12 2.4 mm. as Obviously, a 2.4-mm aperture at F 12 seems very small. 2.44 F 2p. (4) Actually the DCS 460 with the Nikon lens has a range from Dividing both sides of Equation 4 by 2p yields F 2.8 to F 22. The F 2.8 has a D 10 mm, and this can accommodate low light levels with appropriate shutter speed. Assuming the image motion is kept to one half pixel or less, 1.22 Fp 1 (5) the Kodak DCS 460 to 760 Series cameras are providing the airborne remote sensing community with excellent (Red, Green, Blue) color imagery. The DCS 460 can take color in- and frared imagery when using a blue blocking filter. Fp 11.22 0.82. (6) Computing the GSD for the DCS 460 Camera, So, the design function is given by Equation 7: i.e., GSD (Hf )p (10) Fp 0.82 (The Design Function) (7) where H is flying height above mean ground level; then If Fp is less than 0.82, the aperture is larger than opti- GSD (3110 ft)(0.009 mm)28 mm mal; if Fp is greater than 0.82, the aperture is smaller than optimal (Jones, unpublished notes, 1999). GSD 1 ft 0.3 m. Generally speaking, for final design it may be necessary to slightly over aperture the system to provide a safety factor in order to accommodate non-optimum lighting, atmospheric Example 2: Consider an Ikonos-like Satellite Camera conditions, and lens aberrations which occur in the real (Fritz, 1996; Givens, 1998). The wavelength 0.675 m is world. chosen half way between 0.45 and 0.90 m so that Ikonos can The Design Function (Equation 7) does not take into ac- sense into the near-infrared (NIR) part of the spectrum. count such things as the signal-to-noise ratio (S/N), MTF, and Let 0.675 m, H 680 km, p 12 m, and f 10 m. image motion. When Fp 0.82, the area of the Airy disk is Recall from Equation 8 that F 0.82p. Then, for an Ikonos- approximately p2. That is, matching the area of the disk p2 like space camera, F 0.82(12 m)0.675 m 14.5. There- to the area of the two pixels 2p2 creates an optical resolution fore, D fF 0.69 m for the diameter of the camera’s primary that is 2 larger than the area of the two pixels. mirror. Finally, GSD (680,000 m10 m) 12 10 6 m 0.82 m at nadir. The GSD is 1 m at approximately 35 degrees off nadir for these parameters. Application of the Design Function The above parameters computed from the Design Func- To simplify computations, an average wavelength of (0.6 tion (Equation 7) are nearly identical to the actual Ikonos cam- 10–6 m) is used for aerial applications in the visible era parameters presently in orbit. Gerlach (1997) and Givens light spectrum. Then, if given either the pixel size p or the (1998) point out that an imaging system with Fp 0.81 is F-number for the camera, the camera designer can solve for ideal for satellite imaging. Observing the excellent Ikonos the focal length f or the aperture diameter D. The designer image quality verifies that the Design Function yields an accu- can expect this simple function, based on the Rayleigh Crite- rate estimate for the aperture size. This yields an F 14.3 for rion, to match the performance of the optics to the required the Ikonos Camera.

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TABLE 1. PIXEL SIZE, F-NUMBER, AND APERTURE DIAMETER FOR FP TABLE 3. COMPARING COMPUTED DESIGN WITH THE PUBLISHED ((AVG) 0.675 m) QUICK BIRD I PARAMETERS

Fp Pixel Size p F-Number Primary Aperture D Pixel Size p Computed (m) (m) Fp (m) F-Number F Aperture D (m)

0.82 12 14.5 0.69 0.82 12 14.57 0.60 0.82 9 10.9 0.92 1.0 12 17.7 0.50 0.82 8 9.7 1.03 Quick Bird 12 14.63 0.60 as above

With Fp 0.81, the Ikonos Camera is very slightly over apertured, and this can compensate for non-ideal imaging con- Example 3: Consider the EarthWatch Quick Bird I Camera ditions. Table 1 tabulates some other feasible combinations Now that two Design Functions have been derived, it seems that can be derived from Equation 7, the Design Function. instructive to look at the published characteristics (Fritz, The parameters in Table 1 are very realistic for an initial 1999) of the Earth Watch (Digital Globe) Quick Bird I Pan cut at satellite camera aperture sizing. Notice in Table 1, as Camera even though it failed to achieve orbit and, therefore, pixel size decreases, the aperture D must get larger and the no imagery is available to observe. Parameters are computed larger aperture would generally increase the cost. using both Design Functions and compared with the published parameters planned for the cameras. Assume the following parameters: 0.675 m (Average A Second Design Function of 0.45 to 0.90 m), p 12 m, H 600 km, f 8.7805 m A second and slightly different metric for system performance focal length, F 14.63 (F-number), and GSD 1 m at 30 de- is attributed to Schade and Sendall and reported by Holst grees off nadir. (1999). Their modified metric for diameter is Inserting the Quick Bird values from Table 3 into Equa- R 1.845 F (11) tion 7 indicates that Quick Bird’s Design Function was proba- optics bly equivalent to Fp (0.675 m)(14.63)12 m 0.82. The conclusion is that the Quick Bird I design met the Airy where Roptics is the diameter of the disk defined by Equa- tion 11. Note that this approach (Equation 11) provides a disk criterion Fp 0.82; therefore, assuming appropriate value that is smaller than the Rayleigh Criterions Airy disk MTF, S/N, etc., the lost Quick Bird I should have provided excellent image quality. Because Quick Bird I was lost shortly given by Equation 3. If Roptics is sampled by two pixels (2p), the disk will be appropriately sampled as shown in Figure 1. after launch, real imagery proof of the Design Function will Now, Equation 11 can be re-written as have to wait for another opportunity. 1.845 F 2p (12) Satellite Cameras for a GSD 0.5 m Dividing both sides of Equation 12 by 2p yields 1.845F In December of 2000, the Department of Commerce granted li- 2p 1 and Fp 10.9225 1.08. censes to both Space Imaging Inc. (SII), and EarthWatch (Digi- So the Design Function based on Schades slightly smaller tal Globe) for commercial remote sensors that can achieve a disk can be stated as GSD equal to 0.5 m. It may be reasonable to assume that at nadir this GSD may approach 0.41 m and go to 0.5 m at 30 de- Fp 1(Design Function No 2) (13) grees off nadir. It is instructive to use the Design Functions, Equation 7 and Equation 10, for GSD to estimate a set of cam- In practice, Fp 1 is used by some optical systems en- era parameters that could be a good first cut (ROM) for the gineers for quick rough order-of-magnitude (ROM) estimates of camera’s characteristics. The four variables that the designer the F-number and the size of the primary mirror (aperture). It has to work with are f, D, H, and p. Table 4, using the equa- is interesting to note, however, that both the Ikonos and the tions below, shows some reasonable combinations. Quick Bird that did not make it into orbit both conformed to the Airy Disk criterion in Equation 7. Equation (7), Fp 0.82 Table 2 shows the same Ikonos parameters (p, F, D) as Equation (10), GSD (Hf)p 0.41 m shown in Table 1 except that they are based on F p 1, the Equation (9), D fF No 2 design function, not Fp 0.82. Notice again that 0.675 m is chosen half way be- Assume 0.675 m for the space system. tween 0.45 and 0.9 m. This bandwidth is common among satellite cameras in order to encompass the NIR response at a Numerous other combinations are feasible considering CCD panchromatic sensor. This is somewhat analogous to ex- different orbital heights, swath widths, and focal lengths. tended red film. Notice in Table 2 that the estimate for the pri- Bates (2001) wrote that EarthWatch would launch their Quick mary aperture computes to be smaller because the Fp 1 Bird into an approximate 450-km orbit to achieve a 0.61-m disk is resolving a smaller circle. Of course, building smaller GSD. Table 4 shows that the Option B1 should provide good GSD apertures and telescopes for the same GSD can be a cost saver image quality ( of 0.61 m) when orbited at approximately and, therefore, a real cost driver for large satellite cameras 450 km. costing several million dollars. Practical Use for the Design Function TABLE 2. PIXEL SIZE, F-NUMBER, AND APERTURE DIAMETER ( 0.675 m) Camera designers are well aware that there are a number of quantifiable factors that affect the usefulness and inter- Fp Pixel Size p F-Number Primary Aperture D pretability of the image. These include image sharpness, (m) (m) radiometric fidelity, dynamic range of the imagery, and precise geo-location. When the total optical transfer function 11217.7 0.56 is considered, the geometric projection is enlarged primarily 1913.3 0.75 1811.8 0.85 by the point-spread function. The key contributors to the transfer function are , F, and p.Itistrue that most image

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TABLE 4. PARAMETERS FOR CAMERAS TO YIELD GSDs IN THE APPROVED 0.5 m CLASS

Satellite Options Fp Pixel Size (m) F# f (m) H (m) Aperture (m) GSD (m)

A1 0.82 12 14.57 17.56 600,000 1.20 0.41 to 0.50 A2 0.82 9 10.93 13.17 600,000 1.20 0.41 to 0.50 A3 0.82 8 9.72 13.27 680,000 1.37 0.41 to 0.50 A4 0.82 8 9.72 8.94 458,000 0.92 0.41 to 0.50 B1 0.82 12 14.57 8.85 450,000 0.61 0.61 to 0.74 B2 0.82 9 10.93 10.93 740,810 1.00 0.61 to 0.74 B3 0.82 8 9.72 8.75 448,440 0.90 0.41 to 0.50

interpreters and users do not design the optics and detectors Acknowledgment for their cameras. Interpretation is a skill of its own, and the The author gratefully acknowledges the many useful discus- ordinary user is concerned more with image quality and sions regarding the rigor and usefulness of the design function good positional accuracy of the image. The real benefit of with Mr. Peter Jones of Eastman Kodak Company. these simple expressions comes to the optical systems engineers who estimate first-order parameters and then the References cost to build these large cameras. The Design Functions are based on physics and optical theory and are simple and, Bates, J., 2001. EarthWatch alters Quick Bird 2 plans, Space News therefore, easy to use. They provide quick, yet sufficiently (26 March):1. accurate, parameter estimates for the principal and costly Fiete, R.D., 1999. Image quality and FN p for remote sensing, SPIE elements in electro-optical camera systems. These parameters Optical Engineering, 38(7):1229–1240. are the optics for the camera’s telescope, its focal length f, Fritz, L., 1999. The era of commercial Earth observation satellites, F-number, and the pixel size p. Once the parameters are Photogrammetric Engineering & Remote Sensing, 62(1):39–45. determined using the chosen Design Function, the builder Gerlach, F.W., 1997. Satellite Reconnaissance—Utility of One Meter must also consider the physical size of the camera, and GSD Commercial Satellite Systems, Space Imaging, Inc., the available space in the spacecraft. The Design Functions’ Thornton, Colorado, 18 p. simplicity make them most useful for preliminary Givens, F.L., 1998. The technology and business of high resolution time and cost estimates. imaging, Launchspace (December):41–45 and 52. Holst, G.C., 1998. CCD Arrays, Cameras, and Displays, SPIE, Bellingham, Washington, 378 p. Summary ______, 1999. Image quality: Does your detector match your optics? Although the two Design Functions are based on the well- Photonics, 33(1):144–146. known Rayleigh Criterion, the Airy disk, and Schades modi- Jensen, N., 1968. Optical and Photographic Reconnaissance Systems, fied resolution metrics, they are best used as a “rule of John Wiley and Sons, Inc, New York, N.Y., 211 p. thumb” for a first-order cut at estimating camera parameters. Kraus, Karl, 1993. Photogrammetry, Fundamentals and Standard Processes, Volume 1, Ferd. Dummlers Verlag, Bonn, Germany, Simulation of the total problem considering MTF and S/N is 397 p. necessary to refine the final imaging system parameters. On the other hand, it has been shown that Fp 0.82 matches Moffitt, F., and E. Mikhail, 1980. Photogrammetry, Third Edition, Harper & Row, New York, N.Y., 648 p. proven working systems. In summary, the design function is particularly useful for estimating the size of the camera’s Schott, John R., 1997. Remote Sensing: The Image Chain Approach, Oxford University Press, New York, N.Y., 394 p. optical/detector parameters such as the diameter of the primary mirror. Proper use of the functions can be a (Received 25 June 2001; accepted 25 January 2002; revised 10 March valuable assist to designers. 2003)

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