4) REMARKS ON CIRCULAR HORIZON ARCS
It should be plainly evident at this point that horizon arcs are not generally circular—even for a spherical celestial body. Indeed, we have already shown that spherical bodies may produce hyperbolic horizon arcs when imaged from a low altitude. Despite this, there seems to be a widespread misconception that assuming circular horizon arcs for OPNAV is a good approximation and that such an approach has the benefit of ''simplicity.'' Both claims are almost always false. The results of this tutorial clearly show there is no justification for the modern OPNAV practitioner to ever presuppose a circular horizon arc.
We begin our remarks by acknowledging that spherical bodies positioned along the camera boresight will produce a circular horizon arc. This may be intuitively deduced by taking a slice of a right circular cone with an image plane perpendicular the cone's axis of revolution (which, in the special case of a sphere, is also the direction from the camera to the body center). We may also arrive at this result algebraically by recalling that the projected conic is a circle when the coefficients of the implicit conic equation obey the constraint (A − C)2 + B2 = 0. For a spherical body, substitution from Eq. (65) shows that equation (75) This implies that XC = 0 and YC = 0, which means the vector from the camera to the celestial body center is rTC = [0, 0, ZC ] and is entirety along the camera boresight direction. Whenever the camera boresight is not pointed exactly at the body's center (for a sphere), the projected horizon arc will be something other than a circle (e.g., an ellipse).
For a spherical body having ZC > a, the conic locus of the horizon is an ellipse given by Eq. (64). Some tedious algebra will show this horizon arc to have an ellipticity (ε = ah/bh; ratio of semimajor ah to semiminor axis bh in image) of equation (76) which produces an eccentricity e of equation (77) where φ is the half-angle of the tightly bounding cone and ψ is the angle between rC and the camera boresight direction. These angle definitions are the same as illustrated in Fig. 13. The ellipticity is often more extreme for non-spherical bodies. Moreover, we find the ellipticity of the projected horizon (even if small) to be critical for navigation performance and solution stability. The navigation analyst ignores this effect at their own peril.
F. HORIZON LOCALIZATION IN A DIGITAL IMAGE
For an image of a celestial body to be useful for navigation, we must be capable of automatically extracting the pixel coordinates of points of interest. In this work, we are interested in points belonging to the body's apparent horizon. Finding these points is an image processing task.
Less than half of the apparent horizon is usable for navigation in a typical image. Except in rare special cases, images of celestial bodies will be at non-zero phase angles and the terminator (day/night line) will be visible. As a consequence, part of the horizon will be illuminated (we call this the lit limb) and part will be in shadow. Only the lit limb is useful for navigation. A review of the example images from Fig. 8 will show the shadowed limb to be nearly undetectable (when it is detectable, the signal-to-noise ratio is poor). Moreover, we generally avoid using points along the lit limb near the cusps, where the shadow of the terminator begins to encroach on the lit horizon—the result is less than 180 deg of usable horizon arc in almost all practical cases. Past studies show the best localization performance is about ±70 deg from the limb's subsolar point (for a total of 140 deg of horizon arc).
Before proceeding further, the reader is explicitly warned against using the entire lit limb (360 deg of arc) at very low phase angle, even though the celestial body's disk may qualitatively appear to be fully illuminated. In these cases, it is not uncommon for the terminator to lightly shadow (or slightly trim a few pixels off) one side of the apparent disk—even if it is indiscernible from casual inspection by the human analyst. The shadowed (or trimmed) horizon pixels will shift the apparent limb location towards the center of the body and corrupt the resulting navigation solution in a meaningful way. Outside of a simulation environment, superior performance is almost always achieved by restricting ourselves to only using horizon points belonging to the lit limb.
For an airless body (e.g., Moon, Pluto), there is a discontinuity in brightness across the lit limb. This discontinuity is spread out in a digital image due to the the optical assembly's point spread function and spatial quantization by the FPA detector. Regardless, the underlying brightness discontinuity makes limb localization for an airless body equivalent to an edge localization problem. There are a variety of widely-available edge localization algorithms with pixel-level performance (i.e., these algorithms identify edge pixels), including: Sobel, Marr-Hilderth, Canny, and many others. These pixel-level methods usually provide reasonable qualitative performance—making it possible to produce navigation solutions, compute statistics, and write papers. In practice, however, these pixel-level edge locations are inadequate for navigation. Horizon-based navigation using fits to image limb points absolutely requires subpixel horizon localization with contemporary (circa 2020) imaging sensors. Good results are achievable with the partial area effect (PAE) technique and best-in-class performance is achievable with moment-based methods. The reader is directed to the image processing literature for additional details on the edge localization problem.
Regardless of the algorithm chosen, the end result is usually a set of n pixel coordinates along the celestial body's lit limb. We can describe this set of points in homogeneous coordinates as {u ̄i}ni=1.
Finally, the reader is cautioned against making undue use of the apparent terminator location. It is true that the terminator provides a direct measurement of phase angle, sometimes making it useful as a means of determining the direction to the Sun. However, because the terminator arc occurs where the incoming sunlight is tangent to the body's ellipsoidal surface, the sensitivity of terminator location to sunlight direction is quite poor. Further, unlike the lit limb, the terminator does not correspond to a physical discontinuity that manifests as an underlying discontinuity in image intensity. Instead, the terminator arc is a locus of points on the interior of the celestial body's apparent disk. Accurate localization of the terminator in an image is difficult because the transition from sunlight to shadow is gradual and often occurs over a very large number of pixels. Moreover, real bodies have texture that further complicates the fitting of simple models to the apparent terminator intensity gradient— specifically (1) globally varying surface albedo and (2) terrain relief that casts long shadows near the terminator. Until new techniques resolve these challenges, the combination of poor sensitivity to incoming sunlight direction and poor localization in real imagery makes the terminator a poor navigation observable for most applications.
G. IMAGING CELESTIAL BODIES AT DIFFERENT WAVELENGTHS
Most space imaging systems used for navigation operate in either the visible or infrared spectrum. It is often preferable to work in the visible spectrum since these cameras have higher resolution FPAs, produce less noisy images, are cheaper, and place fewer engineering requirements on the vehicle (e.g., thermal control). Observations in the visible spectrum work exceptionally well on airless bodies where there is a crisp and predictable horizon line.
For a celestial body with an atmosphere (e.g., Earth, Venus), there is not a brightness discontinuity across the lit limb when imaged in the visible spectrum. The scattering of sunlight in the atmosphere causes a gradual change in image brightness spanning the width of the atmosphere. Thus, in the visible spectrum, we find that the atmosphere obscures the true horizon and creates an apparent horizon at an unpredictable altitude above the surface. This is a long-standing challenge for horizon-based OPNAV and the specific issues are addressed at length elsewhere. The usual solution for robotic space exploration has been to ignore bodies with atmospheres and to only use images of airless bodies for horizon-based OPNAV, which has been satisfactory for past missions. While there has been some very preliminary studies of overcoming this problem for autonomous OPNAV, this early work is incomplete and suggests no specific path forward. This is an area ripe for future research.
Past spaceflight missions have not made extensive use of infrared images for horizon-based navigation because of (1) their lack of availability and (2) poor performance stemming from their comparatively low FPA resolution. For most planetary exploration applications—especially to airless bodies— visible spectrum images are preferred. Conversely, when we must use images of Earth, sensing in the infrared becomes a more attractive option. The Earth's atmosphere is known to possess a crisp horizon in the longwave infrared (LWIR) at wavelengths of 14.0–16.3 μm. This is due to infrared radiation emitted from the atmosphere's CO2 band, which produces an apparent horizon at an altitude of about 30–50 km. For these reasons, most Earth horizon sensors used for attitude determination operate in the LWIR.
H. REMARKS ON CONIC FITTING IN AN IMAGE
Points lying on an ellipsoidal body's horizon form a conic in an image (see Sec. II-D). Thus, if we are able to find horizon points in an image (see Sec. II-F), it is sometimes useful to fit a conic to these measured horizon points. An illustrative example is shown in Fig. 14.
Horizon arcs are usually elliptical or hyperbolic (see Sec. II-E3), making circle fitting techniques (e.g., Taubin's method) inappropriate for the present application. In the case where the spacecraft is always far away from the observed celestial body, the horizon will always be an ellipse (e.g., see Fig. 13) and ellipse-specific methods may be used in these situations. Otherwise, it is better to use generic conic fitting algorithms.
There are a variety of conic and ellipse fitting algorithms that have been developed over the last few decades. While a full accounting of these various methods is beyond the scope of the present work, there are both non-iterative (e.g., [60], [62]–[64]) and iterative methods (e.g., [65]–[67]). In the case where a large number of horizon points are observed over substantial horizon arcs, we find very little practical difference between the best contemporary non-iterative methods (e.g., the ''semihyper'' least squares fit from [64] and [60]) and the latest iterative methods. The advantages of the iterative methods become more apparent when there are very few horizon points or when the arc size becomes small. These trends are supported by extensive numerical experiments throughout the remainder of this article and are consistent with the observations of [68].
Finally, note that all derivations in this tutorial work with conics in image plane coordinates instead of conics in pixel coordinates. The distinction between these two is discussed in Sec. II-D2. Since horizon localization algorithms usually provide a set of pixel coordinates {u ̄i}ni=1, there are two straight-forward ways to obtain the desired image plane conic C:
1) Transform {u ̄i}ni=1 to {x ̄i}ni=1 using Eq. (20). Apply a conic fitting algorithms to the points {x ̄ i }ni=1 and directly obtain C.
2) Apply a conic fitting algorithms to the points {u ̄i}ni=1 and directly obtain C uv . Transform C uv to C using Eq. (54).
Regardless of the method chosen, we may easily find the conic in image plane coordinates. From this point forward we assume that any ellipse fit to measured horizon points may be written as either C (conic locus) or C∗ (conic envelope).
III. OPTICAL NAVIGATION (OPNAV) WITH A WELL-KNOWN CELESTIAL BODY
The first of the three motivating scenarios from Sec. I is the case where the relative camera-to-celestial body attitude is known, but the relative position is unknown. This is exactly the situation when navigating relative to well-known celestial bodies and is the usual assumption for classical OPNAV. The objective here is to solve for the unknown location of the camera (and, hence, the location of spacecraft).
For many bodies of interest (e.g., Moon, Mars, Earth) we have excellent knowledge of both shape and orientation.