01-073.qxd 2/4/04 11:33 AM Page 297 A Basis for Estimating Digital Camera Parameters Don Light Based on the Rayleigh Criterion 0.82 ؍ Abstract Developing ␭F/p 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 ␭F͞p ϭ 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 ␭ ͞ ϭ 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 ar- ␪ ϭ ␭͞ rays 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␪ ϭ 1.22 ␭f͞D. (2) Introduction Assuming adequate signal and scene contrast, the spatial reso- Recognize that f͞D 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 ϭ ␭ 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- ␭F͞p ϭ 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 ␭F͞p ϭ 0.82 and ␭F͞p ϭ 1 in order to demonstrate their usefulness for quick first-order esti- mates 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 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING March 2004 297 01-073.qxd 2/4/04 11:33 AM Page 298 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 mm 28 ؍ Length f Let average ␭ ϭ 0.6 ␮m for the visible spectrum and pixel size, p ϭ 9 ␮m. Solving Equation 7 for the F-number yields F ϭ 0.82p͞␭ (8) or F ϭ 0.82(9 ␮m)͞0.6 ␮m ϭ 12.3. Conclusion: The DCS 460 using an F-number Յ 12 will have the capability to produce good image quality assuming ade- quate lighting and proper shutter speed. Because F ϭ f͞D, then the aperture diameter, is Figure 1. Airy disk containing two pixels. D ϭ f͞F. (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 ϭ 28 mm͞12 ഡ 2.4 mm. as ͞ ␭ ϭ 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 ͞ ͞ ͞ ϭ 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 ac- commodate low light levels with appropriate shutter speed. Assuming the image motion is kept to one half pixel or less, 1.22 ␭F͞p ϭ 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. ␭F͞p ϭ 1͞1.22 ϭ 0.82. (6) Computing the GSD for the DCS 460 Camera, So, the design function is given by Equation 7: i.e., GSD ϭ (H͞f )p (10) ␭F͞p ϭ 0.82 (The Design Function) (7) where H is flying height above mean ground level; then If ␭F͞p is less than 0.82, the aperture is larger than opti- GSD ϭ (3110 ft)(0.009 mm)͞28 mm mal; if ␭F͞p is greater than 0.82, the aperture is smaller than optimal (Jones, unpublished notes, 1999). GSD ϭ 1 ft ϭ 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 ␭ ϭ 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 ␭ ϭ 0.675 ␮m, H ϭ 680 km, p ϭ 12 ␮m, and f ϭ 10 m. image motion. When ␭F͞p ϭ 0.82, the area of the Airy disk is Recall from Equation 8 that F ϭ 0.82p͞␭.