John A. Christian

John A. Christian

This work introduces the problem of initial orbit determination (IOD) from only heading measurements. Such a problem occurs in practice when estimating the orbit of a spacecraft using visual odometry measurements from an optical camera. After reviewing the problem geometry, a simple solution is developed in the form of an iterative scheme on the parameters describing the orbital hodograph. Numerical results are presented for an example spacecraft in low lunar orbit. The principal intent of this brief study is to communicate the existence of a new class of IOD problem to the community and to encourage the broader study of hodographs and heading-only IOD.

Travis Driver, John A. Christian

Optical navigation is a critical component for lunar orbiter and lander missions. Image-based crater identification has emerged as a promising technology for optical navigation due to the abundance of craters on the lunar surface and the availability of extensive crater catalogs. Moreover, due to the relative morphological homogeneity among lunar craters, template matching has been identified as a promising approach for identification. In this paper, we propose EigenCrater, an automated crater template generation method based on principal component analysis of crater digital elevation maps (DEMs). We demonstrate superior detection and position estimation performance relative to hand-picked templates on simulated lunar imagery.

John A. Christian

Images of both rotating celestial bodies (e.g., asteroids) and spheroidal planets with banded atmospheres (e.g., Jupiter) can contain features that are well-modeled as a circle of latitude (CoL). The projections of these CoLs appear as ellipses in images collected by cameras or telescopes onboard exploration spacecraft. This work shows how CoL projections may be used to determine the pole orientation and covariance for a spinning asteroid. In the case of a known planet modeled as an oblate spheroid, it is shown how similar CoL projections may be used for spacecraft localization. These methods are developed using the principles of projective geometry. Numerical results are provided for simulated images of asteroid Bennu (for pole orientation) and of Jupiter (for spacecraft localization).

Sebastien Henry, John A. Christian

Images are an important source of information for spacecraft navigation and for three-dimensional reconstruction of observed space objects. Both of these applications take the form of a triangulation problem when the camera has a known attitude and the measurements extracted from the image are line of sight (LOS) directions. This work provides a comprehensive review of the history and theoretical foundations of triangulation. A variety of classical triangulation algorithms are reviewed, including a number of suboptimal linear methods (many LOS measurements) and the optimal method of Hartley and Sturm (only two LOS measurements). It is shown that the optimal many-measurement case may be solved without iteration as a linear system using the new Linear Optimal Sine Triangulation (LOST) method. Both LOST and the polynomial method of Hartley and Sturm provide the same result in the case of only two measurements. The various triangulation algorithms are assessed with a few numerical examples, including planetary terrain relative navigation, angles-only optical navigation at Uranus, 3-D reconstruction of Notre-Dame de Paris, and angles-only relative navigation.

Sébastien Henry, John A. Christian

We propose a modified normalized direct linear transform (DLT) algorithm for solving the perspective-n-point (PnP) problem with much better behavior than the conventional DLT. The modification consists of analytically weighting the different measurements in the linear system with a negligible increase in computational load. Our approach exhibits clear improvements -- in both performance and runtime -- when compared to popular methods such as EPnP, CPnP, RPnP, and OPnP. Our new non-iterative solution approaches that of the true optimal found via Gauss-Newton optimization, but at a fraction of the computational cost. Our optimal DLT (oDLT) implementation, as well as the experiments, are released in open source.

Jennifer Nolan, Travis Driver, John Christian

Image-based surface reconstruction and characterization are crucial for missions to small celestial bodies (e.g., asteroids), as it informs mission planning, navigation, and scientific analysis. Recent advances in Gaussian splatting enable high-fidelity neural scene representations but typically rely on a spherical harmonic intensity parameterization that is strictly appearance-based and does not explicitly model material properties or light-surface interactions. We introduce AstroSplat, a physics-based Gaussian splatting framework that integrates planetary reflectance models to improve the autonomous reconstruction and photometric characterization of small-body surfaces from in-situ imagery. The proposed framework is validated on real imagery taken by NASA's Dawn mission, where we demonstrate superior rendering performance and surface reconstruction accuracy compared to the typical spherical harmonic parameterization.

Sébastien Henry, John A. Christian

This work proposes a non-iterative, scalable, and statistically optimal way to triangulate called \texttt{LOSTU}. Unlike triangulation algorithms that minimize the reprojection ($L_2$) error, LOSTU will still provide the maximum likelihood estimate when there are errors in camera pose or parameters. This generic framework is used to contextualize other triangulation methods like the direct linear transform (DLT) or the midpoint. Synthetic experiments show that LOSTU can be substantially faster than using uncertainty-aware Levenberg-Marquardt (or similar) optimization schemes, while providing results of comparable precision. Finally, LOSTU is implemented in sequential reconstruction in conjunction with uncertainty-aware pose estimation, where it yields better reconstruction metrics.

Michael Krause, Ava Thrasher, Priyal Soni et al.

This paper documents the results from the highly successful Lunar flashlight Optical Navigation Experiment with a Star tracker (LONEStar). Launched in December 2022, Lunar Flashlight (LF) was a NASA-funded technology demonstration mission. After a propulsion system anomaly prevented capture in lunar orbit, LF was ejected from the Earth-Moon system and into heliocentric space. NASA subsequently transferred ownership of LF to Georgia Tech to conduct an unfunded extended mission to demonstrate further advanced technology objectives, including LONEStar. From August-December 2023, the LONEStar team performed on-orbit calibration of the optical instrument and a number of different OPNAV experiments. This campaign included the processing of nearly 400 images of star fields, Earth and Moon, and four other planets (Mercury, Mars, Jupiter, and Saturn). LONEStar provided the first on-orbit demonstrations of heliocentric navigation using only optical observations of planets. Of special note is the successful in-flight demonstration of (1) instantaneous triangulation with simultaneous sightings of two planets with the LOST algorithm and (2) dynamic triangulation with sequential sightings of multiple planets.

Travis Driver, Andrew Vaughan, Yang Cheng et al.

This paper proposes the incorporation of techniques from stereophotoclinometry (SPC) into a keypoint-based structure-from-motion (SfM) system to estimate the surface normal and albedo at detected landmarks to improve autonomous surface and shape characterization of small celestial bodies from in-situ imagery. In contrast to the current state-of-the-practice method for small body shape reconstruction, i.e., SPC, which relies on human-in-the-loop verification and high-fidelity a priori information to achieve accurate results, we forego the expensive maplet estimation step and instead leverage dense keypoint measurements and correspondences from an autonomous keypoint detection and matching method based on deep learning to provide the necessary photogrammetric constraints. Moreover, we develop a factor graph-based approach allowing for simultaneous optimization of the spacecraft's pose, landmark positions, Sun-relative direction, and surface normals and albedos via fusion of Sun sensor measurements and image keypoint measurements. The proposed framework is validated on real imagery of the Cornelia crater on Asteroid 4 Vesta, along with pose estimation and mapping comparison against an SPC reconstruction, where we demonstrate precise alignment to the SPC solution without relying on any a priori camera pose and topography information or humans-in-the-loop

Travis Driver, Andrew Vaughan, Yang Cheng et al.

Image-based surface reconstruction and characterization is crucial for missions to small celestial bodies, as it informs mission planning, navigation, and scientific analysis. However, current state-of-the-practice methods, such as stereophotoclinometry (SPC), rely heavily on human-in-the-loop verification and high-fidelity a priori information. This paper proposes Photoclinometry-from-Motion (PhoMo), a novel framework that incorporates photoclinometry techniques into a keypoint-based structure-from-motion (SfM) system to estimate the surface normal and albedo at detected landmarks to improve autonomous surface and shape characterization of small celestial bodies from in-situ imagery. In contrast to SPC, we forego the expensive maplet estimation step and instead use dense keypoint measurements and correspondences from an autonomous keypoint detection and matching method based on deep learning. Moreover, we develop a factor graph-based approach allowing for simultaneous optimization of the spacecraft's pose, landmark positions, Sun-relative direction, and surface normals and albedos via fusion of Sun vector measurements and image keypoint measurements. The proposed framework is validated on real imagery taken by the Dawn mission to the asteroid 4 Vesta and the minor planet 1 Ceres and compared against an SPC reconstruction, where we demonstrate superior rendering performance compared to an SPC solution and precise alignment to a stereophotogrammetry (SPG) solution without relying on any a priori camera pose and topography information or humans-in-the-loop.

John A. Christian, Harm Derksen, Ryan Watkins

It is often necessary to identify a pattern of observed craters in a single image of the lunar surface and without any prior knowledge of the camera's location. This so-called "lost-in-space" crater identification problem is common in both crater-based terrain relative navigation (TRN) and in automatic registration of scientific imagery. Past work on crater identification has largely been based on heuristic schemes, with poor performance outside of a narrowly defined operating regime (e.g., nadir pointing images, small search areas). This work provides the first mathematically rigorous treatment of the general crater identification problem. It is shown when it is (and when it is not) possible to recognize a pattern of elliptical crater rims in an image formed by perspective projection. For the cases when it is possible to recognize a pattern, descriptors are developed using invariant theory that provably capture all of the viewpoint invariant information. These descriptors may be pre-computed for known crater patterns and placed in a searchable index for fast recognition. New techniques are also developed for computing pose from crater rim observations and for evaluating crater rim correspondences. These techniques are demonstrated on both synthetic and real images.