Reply to Pizlo, Rosenfeld, and Weiss

1. DISCUSSION It is possible that invariants are not appropriate for FCDPs. In particular, PRW do not exhibit any invariants of FCDPs The starting point for the arguments put forward by sufficient to explain shape constancy in human vision. Pizlo, Rosenfeld, and Weiss (PRW) in [3] is beguillingly The program is faced with what appears to be a contrasimple. If two different images are correctly identified as diction. Shape constancy should be explained in terms of projections of the same object in space then this recogniinvariants, but the transformations under consideration do tion should depend on properties of the image that are not appear to have the right type of invariant. The usual unchanged as the viewpoint changes. The following quesreaction of an investigator in such cases is to seek a resolution arises: In the case of human vision, can these hypothettion by constructing a more elaborate mathematical model ical invariant properties be identified with any of the classior to seek different principles on which a model can be cal projective, affine, or Euclidean invariants? PRW argue based. The reaction of PRW is uncompromising: the atthat the answer is no. Projective invariants are rejected tempt to explain shape constancy in terms of invariants because they are too coarse. The human visual system may has not succeeded, so this means science is incompatible classify two images as projections of different objects even with mathematics. This raises some interesting philosophithough the images have the same projective invariants. cal questions, but it does not bring us any nearer to an Euclidean invariants are rejected because they are too fine. understanding of shape constancy in human vision. Two images may be classified as projections of the same object even though they have different Euclidean invari2. VISUAL SPACE ants. The reasons for rejecting affine invariants are unclear from [3]. Visual space is discussed extensively in [3], but no definiIn order to save the program of explaining shape contion of the term is offered. It is an interesting exercise to stancy in terms of invariants PRW define a new set of attempt a definition which captures two aspects of vision, transformations from the object plane to the image, the namely the information available to the observer and the fixed center directional perspective transformations decisions made by the observer on the basis of that infor(FCDPs). They argue that the FCDPs correctly model the mation. This suggests that a visual space should consist of projection from space to the retina and that shape conthree items: (i) a set of objects Oi , 1 # i # n, any of which stancy must be explained in terms of invariants of the may appear in the field of view; (ii) a set of images Ii , FCDPs. 1 # i # m; and (iii) a decision algorithm A defined on The FCDPs do not fit easily into the usual hierarchy of pairs (O, I) of objects and images such that projective, affine, and Euclidean geometries. This is because the definition of an FCDP depends on the choice in A(O, I) 5 1 if I is accepted as an image of O the object plane of a coordinate system depending on the A(O, I) 5 0 if I is not accepted as an image of O. relative position of the object plane and the image plane. The requirement of a special coordinate system greatly complicates the discussion of the FCDPs. As PRW note, The decisions made by A can depend on many factors, for example, the position of I on the retina, the accuracy the inverse of an FCDP is not in general an FCDP and the composition of two FCDPs is in general not an FCDP. with which points can be located on the retina, and the prior knowledge and desires of the observer. The algorithm A need not always make a correct decision. For example, * This reply was inadvertently omitted from a previous issue. For the original dialogue, see [3]. PRW cite in [3] an example from human vision. A straight