Order-One Rolling Shutter Cameras

TL;DR

This work develops a rigorous algebraic framework for order-one rolling shutter cameras (RS$_1$), defining a back-projection model via a rational map $\Lambda$ and rolling-planes map $\Sigma$ to connect 3D geometry with RS imaging through a birational picture-taking map $\Phi: \mathbb{P}^3 \dashrightarrow \mathbb{P}^2$. It shows RS$_1$ cameras exist precisely when all rolling planes intersect in a line $K$ and the camera center moves along a rational curve $\mathcal{C}$, introducing three constructible RS$_1$ types (I–III) with explicit parameterizations and dimension counts. The paper then classifies 31 minimal relative pose problems for linear RS$_1$ cameras, highlighting practical problems with small degrees that enable efficient solvers, and it links RS$_1$ theory to existing models such as Straight-Cayley to demonstrate broad applicability. Collectively, the results provide a unified, algebraic foundation for RS$_1$ cameras, enabling principled design of minimal solvers and offering insights into RS models used in practice and their geometric implications.

Abstract

Rolling shutter (RS) cameras dominate consumer and smartphone markets. Several methods for computing the absolute pose of RS cameras have appeared in the last 20 years, but the relative pose problem has not been fully solved yet. We provide a unified theory for the important class of order-one rolling shutter (RS$_1$) cameras. These cameras generalize the perspective projection to RS cameras, projecting a generic space point to exactly one image point via a rational map. We introduce a new back-projection RS camera model, characterize RS$_1$ cameras, construct explicit parameterizations of such cameras, and determine the image of a space line. We classify all minimal problems for solving the relative camera pose problem with linear RS$_1$ cameras and discover new practical cases. Finally, we show how the theory can be used to explain RS models previously used for absolute pose computation.

Figures (8)

• Figure 1: (a) General rolling shutter cameras see points in space multiple times ( en.wikipedia.org/wiki/Rolling_shutter). (b) Order-one rolling shutter cameras see points in space exactly once. Their rolling planes intersect in a line. Examples include (c) perspective cameras and (d) some Straight-Cayley cameras moving on a twisted cubic.
• Figure 2: Overview of RS notation.
• Figure 3: Illustration of $\mathcal{P}_{1,3,\delta}$: E.g., the rolling plane $\Sigma(0:1)$ meets the infinity plane $H^\infty$ at a line with normal vector $B$.
• Figure 4: Illustration of all minimal problems as lines and points in $\mathbb{P}^3$. Each problem is encoded by 9 integers: The number $m$ of cameras, followed by its combinatorial signature (see \ref{['def:balanced']} ff.) with $p_\infty$ at the end. Points on dashed lines are known to be collinear in $\mathbb{P}^3$, but the image conics are not observed. Lower bounds for the degrees are shown below the sketches, see \ref{['deg-comp']}. "$\ast$": the maximum from different computational runs is shown (the actual number is close to the number shown). "$+$": interrupted runs (the actual number is higher than the number shown).
• Figure 5: (a) \ref{['ex:twistedCubicCameras1']}, (b) \ref{['ex:twistedCubicCameras2']}, (c) \ref{['ex:twistedCubicCameras3']}. The camera center ${\textcolor{cyan}{C}}$ (cyan) moves along a twisted cubic curve $\mathcal{C}$. The rolling planes $\Sigma$ (black) intersect in a line ${\textcolor{magenta}{K}}$ (magenta). See \ref{['sec:details-twistedCubicCameras']} for more details.

Theorems & Definitions (91)

• Remark 1
• Definition 2
• Example 3
• Theorem 4
• Remark 5
• Proposition 6
• Remark 7
• Proposition 8
• Proposition 9
• Proposition 10