Sussex , Brighton BNI 9 QG , England

-This paper discusses a number of points that arise in the visual interpretation of optic flow fields. Particular attention is given to the case in which a flow field is visually ambiguous, and the paper concludes with a detailed description of the case of the moving plane. Optic flow Structure f rom motion Visual ambiguity In a "systems approach" to vision one attempts to understand how the machinery of vision enables an animal to form a useful model of its spatio-temporal environment. In this connection the phenomenon of optic flow--the motion of an optical image across the retina-reminds us that the world is not 3-dimensional but 4-dimensional, with time as the 4th dimension. In cinematography we need physical optics to help us understand how the optical image changes as the camera moves through the scene; but the theory of vision has an ulterior motive-to determine the structure of the scene from the motion of the retinal image. In this sense, vision is "inverse optics"--a thought which we owe largely to the computer vision people. In recent years, the theory of vision has benefited from an influx of ideas from computing science and artificial intelligence, the work of the late David Marr (Marr, 1982) having been particularly influential. His computational approach to vision invites us to think of the visual system as performing sophisticated computations on the optical input, in order to arrive at a representation of the visible world. According to this way of thinking, the central question about vision is: by what logical stages, exactly, does the human visual system construct a useful model of the world? In Marr's view, the progress of vision research is to be evaluated by the light that it throws on this underlying problem. In what follows I shall touch briefly on five matters relevant to the interpretation of optic flow fields, and will then summarize the main features of an important ambiguous flow field--that arising from a moving plane. The five matters are: (1) The organic implementation of motion perception. (2) The structure-from-motion problem in the presence of non-rigid deformations. (3) The merits of the Rigidity Assumption, and possible relaxations thereof. (4) The relative merits of partial versus complete, and qualitative versus quantitative, solutions of the structure-from-motion problem. (5) The description of optic flow in terms of differential invariants. My comments are as follows: (1) Possibly the algorithms used by the visual system for deriving structure from optic flow are independent of the neuronal mechanisms by which the flow field is computed in the first place. But the concept of a retinal velocity field v(x, y) is itself a hazardous abstraction from the primary input I(x,y, t), (I being the light intensity) as many authors have emphasized. The vector v may be undefined in many regions and discontinuous or even many-valued in others; the "correspondence problem" and the "aperture problem", which stand between the observation of I and the computation of the velocity v, will not go away just because one chooses to ignore them. Nevertheless it is certainly a good idea to try to clear one's mind about what the visual system might actually be computing before poking around inside it; it seems unlikely that the neuronal circuitry will somehow explain itself. (2), (3) The Rigidity Assumption is undoubtedly the most useful of all the hypotheses available to the visual system in attempting to arrive at a (3 + 1)-dimensional interpretation of the (2 + 1)-dimensional retinal image. Actually there are quite severe constraints to be satisfied by a moving image if it is to have a rigid interpretation, and most non-rigid motions are