Operating on spatial relations

examples and a formal definition are given later. As might be anticipated, the notion ofspatial relations is fundamental to the understanding ofgeometry, now regarded not as the study of 'geometrical objects' but of the relations between those objects. The point was made forcibly by the German mathematician David Hilbert, who allegedly said that the words 'point', 'line' and 'plane' could be replaced by 't~ble', 'chair' and 'beer-mug' without changing geometry In the least (Bourbaki, 1968, p 317). Although examples ofvisually plausible spatial relations may be produced easily, there are a number of problems that arise when one considers more carefully the possible kinds of spatial properties. These properties may be numerical, for example, scalars (the distance between two objects), or vectors (the distance between two objects and direction the one defines with respect to the other), or they may be predicates asserting something geometrical or topological (the closure of a curve). It would be desirable to classify the spatial relations of vision within the framework of one or more formal mathematical structures (see e.g..Bourbaki, 1968, Chapter 4), or possibly some fuzzy version of these structures (Kaufmann, 1975). But which structures are the most appropriate for vision, how they should be combined with each other, and how they should be modified to reflect the limitations of the visual apparatus are largely unknown. In addition to these foundational matters, there are u~certainties concerning the status of some commonly cited examples of spatial relations. Consider the set given by Barlow et ale (1972), here slightly modified for later reference: D. H. Foster

[1]  Paul A Kolers,et al.  Spatial and ordinal components of form perception and literacy , 1975, Cognitive Psychology.

[2]  D. W. Heeley,et al.  Meridional anisotropies of orientation discrimination for sine wave gratings , 1988, Vision Research.

[3]  F. Attneave,et al.  Discriminability of stimuli varying in physical and retinal orientation , 1967 .

[4]  J. Swets,et al.  A decision-making theory of visual detection. , 1954, Psychological review.

[5]  A. Treisman,et al.  Emergent features, attention, and object perception. , 1984, Journal of experimental psychology. Human perception and performance.

[6]  Johan Wagemans,et al.  Visual search in dot-patterns with bilateral and skewed symmetry , 1990 .

[7]  E. Essock.  The Oblique Effect of Stimulus Identification Considered with Respect to Two Classes of Oblique Effects , 1980, Perception.

[8]  Hermann von Helmholtz,et al.  Treatise on Physiological Optics , 1962 .

[9]  G. Mandler,et al.  Subitizing: an analysis of its component processes. , 1982, Journal of experimental psychology. General.

[10]  I. Rock,et al.  An experimental analysis of visual symmetry , 1963 .

[11]  R Vogels,et al.  Human orientation discrimination: changes with eccentricity in normal and amblyopic vision. , 1986, Investigative ophthalmology & visual science.

[12]  C. B. Cave,et al.  Evidence for two types of spatial representations: hemispheric specialization for categorical and coordinate relations. , 1989, Journal of experimental psychology. Human perception and performance.

[13]  M. Corballis,et al.  Winding one's ps and qs: mental rotation and mirror-image discrimination. , 1984, Journal of experimental psychology. Human perception and performance.

[14]  J VOLKMANN,et al.  The visual perception of perpendicularity. , 1958, The American journal of psychology.

[15]  F ATTNEAVE,et al.  Dimensions of similarity. , 1950, The American journal of psychology.

[16]  Marvin Minsky,et al.  Perceptrons: An Introduction to Computational Geometry , 1969 .

[17]  B Julesz,et al.  "Where" and "what" in vision. , 1985, Science.

[18]  F. Attneave.  Triangles as ambiguous figures. , 1968, The American journal of psychology.

[19]  Lotfi A. Zadeh,et al.  Fuzzy Logic and Its Application to Approximate Reasoning , 1974, IFIP Congress.

[20]  Naum Yakimoff,et al.  A concept of perceived orientation , 1979 .

[21]  R Sekuler,et al.  Perception of stimulus direction: hemispheric homology and laterality. , 1973, The American journal of psychology.

[22]  R. Shepard,et al.  Mental Rotation of Three-Dimensional Objects , 1971, Science.

[23]  W. R. Garner,et al.  Internal frame of reference as a determinant of the oblique effect. , 1986, Journal of experimental psychology. Human perception and performance.

[24]  William P. Banks,et al.  The asymmetry of lateral interference in visual letter identification , 1977 .

[25]  S. Appelle.  Perception and discrimination as a function of stimulus orientation: the "oblique effect" in man and animals. , 1972, Psychological bulletin.

[26]  N I Durlach,et al.  Intensity perception. I. Preliminary theory of intensity resolution. , 1969, The Journal of the Acoustical Society of America.

[27]  G. W. Strong,et al.  A solution to the tag-assignment problem for neural networks , 1989, Behavioral and Brain Sciences.

[28]  D H Foster,et al.  Visual Comparison of Random-Dot Patterns: Evidence concerning a Fixed Visual Association between Features and Feature-Relations , 1978, The Quarterly journal of experimental psychology.

[29]  G. V. Dearborn.  Recognition under objective reversal. , 1899 .

[30]  F. Campbell,et al.  The Dependence of the Visual Numerosity Limit on Orientation, Colour, and Grouping in the Stimulus , 1976, Perception.

[31]  M. Corballis,et al.  Detection of symmetry as a function of angular orientation. , 1975, Journal of experimental psychology. Human perception and performance.

[32]  Eric L. Schwartz,et al.  Computational anatomy and functional architecture of striate cortex: A spatial mapping approach to perceptual coding , 1980, Vision Research.

[33]  Alan S. Brown,et al.  Information Processing and Cognition: The Loyola Symposium , 1976 .

[34]  Irvin Rock,et al.  Orientation and form , 1974 .

[35]  D. Mitchell,et al.  Effect of orientation on the modulation sensitivity for interference fringes on the retina. , 1967, Journal of the Optical Society of America.

[36]  Z. Pylyshyn.  The role of location indexes in spatial perception: A sketch of the FINST spatial-index model , 1989, Cognition.

[37]  S. Palmer,et al.  Orientation and symmetry: effects of multiple, rotational, and near symmetries. , 1978, Journal of Experimental Psychology: Human Perception and Performance.

[38]  R. S. French.  The discrimination of dot patterns as a function of number and average separation of dots. , 1953, Journal of Experimental Psychology.

[39]  R. Shepard,et al.  Mental Images and Their Transformations , 1982 .

[40]  E. L. Kaufman,et al.  The discrimination of visual number. , 1949, The American journal of psychology.

[41]  D. W. Watkins,et al.  Grating visibility as a function of orientation and retinal eccentricity , 1975, Vision Research.

[42]  F. Attneave,et al.  What Variables Produce Similarity Grouping , 1970 .

[43]  P. Dodwell,et al.  Human Pattern and Object Perception , 1978 .

[44]  P. A. Kolers,et al.  Orientation of letters and their speed of recognition , 1969 .

[45]  H. Barlow,et al.  Visual pattern analysis in machines and animals. , 1972, Science.

[46]  B N Timney,et al.  Orientation anisotropy: incidence and magnitude in Caucasian and Chinese subjects. , 1976, Science.

[47]  J. Bradshaw,et al.  The Perception and Identification of Mirror-Reversed Patterns , 1976, The Quarterly journal of experimental psychology.

[48]  Johan Wagemans,et al.  On the possibility of correlation quadrangles as a basic primitive in perceptual grouping , 1989 .

[49]  David H. Foster.  Local and global computational factors in visual pattern recognition. , 1984 .

[50]  Fred Attneave,et al.  Cartesian organization in the immediate reproduction of spatial patterns , 1977 .

[51]  S. Ullman.  Visual routines , 1984, Cognition.

[52]  G. Orban,et al.  Human orientation discrimination tested with long stimuli , 1984, Vision Research.

[53]  Irwin Pollack,et al.  Detection of changes in spatial position: III. Dot number or dot density? , 1972 .

[54]  Masaharu Mizumoto,et al.  Some Properties of Fuzzy Sets of Type 2 , 1976, Inf. Control..

[55]  David H. Foster,et al.  Visual Comparison of Rotated and Reflected Random-Dot Patterns as a Function of Their Positional Symmetry and Separation in the Field* , 1981 .

[56]  D. Foster,et al.  Horizontal-vertical structure in the visual comparison of rigidly transformed patterns. , 1986, Journal of experimental psychology. Human perception and performance.

[57]  Michael C. Corballis,et al.  On the perception of symmetrical and repeated patterns , 1974 .

[58]  Judy F. Rosenblith,et al.  Discrimination of direction of line and the effect of stimulus alignment , 1964 .

[59]  H. Barlow,et al.  The versatility and absolute efficiency of detecting mirror symmetry in random dot displays , 1979, Vision Research.

[60]  O. Reiser,et al.  Principles Of Gestalt Psychology , 1936 .

[61]  A M ROCHLIN,et al.  The effect of tilt on the visual perception of parallelness. , 1955, The American journal of psychology.

[62]  W F Bischof,et al.  Selective internal operations in the recognition of locally and globally point-inverted patterns. , 1985, Spatial vision.

[63]  F. Campbell,et al.  The effect of orientation on the visual resolution of gratings , 1966, The Journal of physiology.

[64]  J. R. Pomerantz,et al.  Emergent features, attention, and perceptual glue in visual form perception. , 1989, Journal of experimental psychology. Human perception and performance.

[65]  W. Chase,et al.  Visual information processing. , 1974 .

[66]  Jeffrey A. Sloan,et al.  Spatial frequency analysis of the visual environment: Anisotropy and the carpentered environment hypothesis , 1978, Vision Research.

[67]  B. Frost,et al.  Human Visual Ecology and Orientation Anisotropies in Acuity , 1973, Science.

[68]  F. Attneave,et al.  Perception of place in a circular field. , 1955, The American journal of psychology.

[69]  J. B.,et al.  The Power of Numerical Discrimination , 1871, Nature.

[70]  Z W Pylyshyn,et al.  Tracking multiple independent targets: evidence for a parallel tracking mechanism. , 1988, Spatial vision.

[71]  H Egeth,et al.  Subitizing: Direct apprehension or serial processing? , 1988, Perception & psychophysics.

[72]  Anne Treisman,et al.  Emergent features, attention, and object perception. , 1984 .

[73]  P. Jolicoeur.  The time to name disoriented natural objects , 1985, Memory & cognition.

[74]  J. Beck.  Similarity grouping and peripheral discriminability under uncertainty. , 1972, The American journal of psychology.

[75]  W. R. Garner,et al.  Effect of line orientation on various information-processing tasks. , 1983, Journal of experimental psychology. Human perception and performance.

[76]  G. Chastain,et al.  Identification asymmetry of parafoveal stimulus pairs , 1979 .

[77]  H. Barlow.  Vision: A computational investigation into the human representation and processing of visual information: David Marr. San Francisco: W. H. Freeman, 1982. pp. xvi + 397 , 1983 .

[78]  Vicky G Bruce,et al.  Violations of Symmetry and Repetition in Visual Patterns , 1975 .

[79]  M. D. Arnoult.  Shape discrimination as a function of angular orientation of the stimuli. , 1954, Journal of experimental psychology.

[80]  Stephen E. Palmer,et al.  Orientation and symmetry: effects of multiple, rotational, and near symmetries. , 1978, Journal of experimental psychology. Human perception and performance.

[81]  T Radil,et al.  Influence of local orientation cues on estimating the tilt of two-dimensional patterns. , 1988, Spatial vision.

[82]  R. Shepard,et al.  CHRONOMETRIC STUDIES OF THE ROTATION OF MENTAL IMAGES , 1973 .