Specular flow is the motion field induced on the image plane by the movement of points reflected by a curved, mirror-like surface. This flow provides information about surface shape, and when the camera and surface move as a fixed pair, shape can be recovered by solving linear differential equations along integral curves of flow. Previous analysis has shown that two distinct motions (i.e., two flow fields) are generally sufficient to guarantee a unique solution without externally-provided initial conditions. In this work, we show that we can often succeed with only one flow. The key idea is to exploit the fact that smooth surfaces induce integrability constraints on the surface normal field. We show that this induces a new differential equation that facilitates the propagation of shape information between integral curves of flow, and that combining this equation with known methods often permits the recovery of unique shape from a single specular flow given only a single seed point.

An image of a specular (mirror-like) object is nothing but a distorted reflection of its environment. When the environment is unknown, reconstructing shape from such an image can be very difficult. This reconstruction task can be made tractable when, instead of a single image, one observes relative motion between the specular object and its environment, and therefore, a motion field-or specular flow-in the image plane. In this paper, we study the shape from specular flow problem and show that observable specular flow is directly related to surface shape through a nonlinear partial differential equation. This equation has the key property of depending only on the relative motion of the environment while being independent of its content. We take first steps toward understanding and exploiting this PDE, and we examine its qualitative properties in relation to shape geometry. We analyze several cases in which the surface shape can be recovered in closed form, and we show that, under certain conditions, specular shape can be reconstructed when both the relative motion and the content of the environment are unknown. We discuss numerical issues related to the proposed reconstruction algorithms, and we validate our findings using both real and synthetic data.

When a curved mirror-like surface moves relative to its environment, it induces a motion field-or specular flow- on the image plane that observes it. This specular flow is related to the mirror's shape through a non-linear partial differential equation, and there is interest in understanding when and how this equation can be solved for surface shape. Existing analyses of this `shape from specular flow equation' have focused on closed-form solutions, and while they have yielded insight, their critical reliance on externally-provided initial conditions and/or specific motions makes them difficult to apply in practice. This paper resolves these issues. We show that a suitable reparameterization leads to a linear formulation of the shape from specular flow equation. This formulation radically simplifies the reconstruction process and allows, for example, both motion and shape to be recovered from as few as two specular flows even when no externally-provided initial conditions are available. Our analysis moves us closer to a practical method for recovering shape from specular flow that operates under arbitrary, unknown motions in unknown illumination environments and does not require additional shape information from other sources.

The inference of specular (mirror-like) shape is a particularly difficult problem because an image of a specular object is nothing but a distortion of the surrounding environment. Consequently, when the environment is unknown, such an image would seem to convey little information about the shape itself. It has recently been suggested (Adato et al., ICCV 2007) that observations of relative motion between a specular object and its environment can dramatically simplify the inference problem and allow one to recover shape without explicit knowledge of the environment content. However, this approach requires solving a non-linear PDE (the dasiashape from specular flow equationpsila) and analytic solutions are only known to exist for very constrained motions. In this paper, we consider the recovery of shape from specular flow under general motions. We show that while the dasiashape from specular flowpsila PDE for a single motion is non-linear, we can combine observations of multiple specular flows from distinct relative motions to yield a linear set of equations. We derive necessary conditions for this procedure, discuss several numerical issues with their solution, and validate our results quantitatively using image data.

The image of a curved, specular (mirror-like) surface is a distorted reflection of the environment. The goal of our work is to develop a framework for recovering general shape from such distortions when the environment is neither calibrated nor known. To achieve this goal we consider far-field illumination, where the object-environment distance is relatively large, and we examine the dense specular flow that is induced on the image plane through relative object-environment motion. We show that under these very practical conditions the observed specular flow can be related to surface shape through a pair of coupled nonlinear partial differential equations. Importantly, this relationship depends only on the environment's relative motion and not its content. We examine the qualitative properties of these equations, present analytic methods for recovery of the shape in several special cases, and empirically validate our results using captured data. We also discuss the relevance to both computer vision and human perception.