Image Divergence and Deformation from Closed Curves

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Abstract
This article describes a novel method for measuring the differ ential invariants of the image velocity field from the integral of normal image velocities around image contours. This is equiva lent to measuring the temporal changes in the area of a closed contour. This avoids having to recover a dense image velocity field and take partial derivatives. It also does not require point or line correspondences. Moreover, integration provides some immunity to image measurement noise. It is shown how an active observer making small, deliberate motions can use the estimates of the divergence and deforma tion of the image velocity field to determine the object-surface orientation and time to contact. The results of real-time ex periments are presented in which arbitrary image shapes are tracked using B-spline snakes, and the invariants are com puted efficiently as closed-form functions of the B-spline control points. This information is used to guide a robot manipula tor in obstacle collision avoidance, object manipulation, and navigation. 1. The time duration before the observer and object collide if they continue with the same relative translational motion (Hoyle 1957; Gib son 1979). 3. Koenderink (1986) defines F as a "nearness gradient," ∇(log(1/Z)). In this article F is defined as a scaled depth gradient. These two quantities differ by a sign. 4. Translational velocities appear scaled by depth, making it impossible to determine whether the effects are due to a nearby object moving slowly or a faraway object moving quickly. 5. This is somewhat related to the reliable estimation of relative depth from the relative image velocities of two nearby points—motion parallax (Longuet-Higgins and Prazdny 1980; Rieger and Lawton 1985, Cipolla and Blake 1992). Both motion parallax and the deformation of the image velocity field relate local measurements of relative image velocities to scene structure in a simple way that is uncorrupted by the rotational image velocity component. In the case of parallax, the depths are discontinuous, and differences of discrete velocities are related to the difference of in verse depths. Equation (13) on the other hand assumes a smooth and continuous surface, and derivatives of image velocities are related to derivatives of inverse depth. 6. This equation can be derived by considering the flux linking the area of the contour. This changes with time since the contour is carried by the velocity field. The flux field, g, in our example does not change with time. Similar integrals appear in fluid mechanics, e.g., the flux transport theorem (Davis and Snider 1979). 8. The decomposition is known in applied mechanics as the Cauchy— Stokes decomposition theorem (Aris 1962). 9. (cos μ, sin μ) is the eigenvector of the traceless and symmetric com ponent of the velocity gradient tensor. It corresponds to the positive eigenvalue with magnitude def v. The other eigenvector specifies the axis of contraction and is orthogonal. It corresponds to the negative eigenvalue with magnitude — def v. 10. There are three simple ways to represent surface orientation: com ponents of a unit vector, n; gradient space representation ( p q), and the spherical coordinates (σ, τ). Changing from one representation to another is trivial and is listed here for completeness:

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