Visual Interpretation of Lambertian Surface Deformation

This paper considers the interpretation (as a three-dimensional velocity field) of the changing intensity pattern induced by a smoothly deforming Lambertian surface of uniform albedo illuminated by a distant point light source. The requisite "Intensity Rate Constraint" which is derived contains no terms relating to the tangential components of surface velocity, so the determination of the velocity field is ill-posed, exhibiting a form of "Aperture Problem". A stretch-based regulariser is applied to enable estimation of the velocity field and tests with synthetic data show a requirement for high accuracy. 1 Introduction This paper considers a problem in the field of non-rigid motion interpretation. Whereas motion has often been presumed to be merely a cue as to surface structure, as perhaps implied by the phrase "structure from motion'', this work takes the stance that in a dynamic world where objects may be in relative, even non-rigid motion, the motion of a world point is itself of interest. The problem addressed here is that of the interpretation (as a three-dimensional velocity field) of the changing intensity pattern induced by a smoothly deforming Lamber-tian surface of uniform albedo illuminated by a distant point light source. Thus, the surface viewed deforming is presumed to accord with the most common assumptions in the field of shape from shading (see e.g. [Horn, 1986]), — the image intensity of a surface point is presumed to vary in a known manner with the angle between the surface normal and the light source direction. Hence, if the surface deforms such that this angle changes, the intensity of the (changing) corresponding image point will also change, and the assumption used to derive the "Mo-tion Constraint Equation", first proposed in [Horn and Schunck, 1981], and since used to underpin most work in visual motion interpretation, will be broken. However, here it is presumed that the instantaneous shape of the deforming surface region may be determined by the application of shape from shading within a bounding contour, as analysed in [Blake et a/., 1985]. Similarly the full three-dimensional velocity is presumed known at identifiable feature points on the bounding contour from combined stereo and motion. This will enable the estimation of the three-dimensional velocity along the entire bounding contour using a method such as that proposed in [Cameron-Jones, 1990], Thus, the problem remaining is that of estimating the three-dimensional velocity field over the surface region from the intensity rate, given the velocity …