PoleStack: Robust Pole Estimation of Irregular Objects from Silhouette Stacking

TL;DR

PoleStack addresses robust rotation-pole estimation for irregular space objects using silhouette stacking across hovering-camera views. It leverages reflective symmetry about the projected-pole in the silhouette stack and employs the amplitude spectrum of the 2D Fourier transform to achieve translation-invariant, noise-robust pole detection, followed by pole triangulation to recover the 3D orientation. The method demonstrates degree-level accuracy on low- to medium-resolution data under severe shadowing and registration errors, and remains effective with reduced data volume and partial longitude coverage. This approach enables early pole estimation during approach and hovering phases, with potential applicability to both natural and artificial irregular objects in proximity operations.

Abstract

We present an algorithm to estimate the rotation pole of a principal-axis rotator using silhouette images collected from multiple camera poses. First, a set of images is stacked to form a single silhouette-stack image, where the object's rotation introduces reflective symmetry about the imaged pole direction. We estimate this projected-pole direction by identifying maximum symmetry in the silhouette stack. To handle unknown center-of-mass image location, we apply the Discrete Fourier Transform to produce the silhouette-stack amplitude spectrum, achieving translation invariance and increased robustness to noise. Second, the 3D pole orientation is estimated by combining two or more projected-pole measurements collected from different camera orientations. We demonstrate degree-level pole estimation accuracy using low-resolution imagery, showing robustness to severe surface shadowing and centroid-based image-registration errors. The proposed approach could be suitable for pole estimation during both the approach phase toward a target object and while hovering.

Figures (13)

• Figure 1: Schematic of an irregular object's silhouette observed in the image plane. Key parameters used in this work are reported: perfect silhouette ($\mathcal{S}$), observed silhouette ($\mathcal{O}$), projected-pole direction $\boldsymbol{\omega}_\mathrm{proj}$, pole-projection angle ($\alpha$), image coordinate system ($u$, $v$), pole-aligned coordinate system ($u'$, $v'$). A notional image array is also reported, where the center of the $mn$-th pixel ($m$-th row, $n$-th column) has image coordinates $[u_n, v_m]^\top$. The gray area corresponds to the shadowed silhouette region.
• Figure 2: Trajectory arcs traced by a subset of surface points $\mathbf{p}\in\Omega$ rotating about the pole $\boldsymbol{\omega}$, as observed from a hovering camera at position $\mathbf{r}$ with camera-frame axes $\mathbf{i}_\mathcal{C}, \mathbf{j}_\mathcal{C}, \mathbf{k}_\mathcal{C}$. Each arc begins at a distinct camera-relative longitudinal coordinate yet spans the same total longitude range. The symmetric (blue) and asymmetric (orange, dashed) components of each trajectory arc are shown as functions of the camera-relative longitude $\phi'$.
• Figure 3: Silhouette-stack image of comet 67P-C/G observed across a full rotation ($\phi=[0,2\pi]$), with a $1^\circ$ longitude interval between consecutive images, in the spatial domain (top) and frequency domain (bottom). Note that the image lacks perfect symmetry due to using a finite number of observations.
• Figure 4: Silhouette-stack image of comet 67P-C/G observed across a quarter of rotation ($\phi=[0,\pi/2]$), with a $1^\circ$ longitude interval between consecutive images, in the spatial domain (top) and frequency domain (bottom).
• Figure 5: Centroid-aligned silhouette-stack image of comet 67P-C/G observed across a full rotation ($\phi=[0,2\pi]$), with a $1^\circ$ longitude interval between consecutive images, in the spatial domain (top) and frequency domain (bottom). Individual silhouettes are registered by aligning their brightness centroid. Despite image-registration errors originating from this procedure, a high level of symmetry is preserved in the silhouette-stack image.

Theorems & Definitions (63)

• Definition III.1
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• Lemma III.1