Glossar-English:Space and Geometry

Translation of: Raum und Geometrie

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Space as the basic category of picture syntax/picture morphology The set of the syntactic components for perceptoid signs can generally be divided in two groups: On the one hand there is an abstract relational basic structure that usually spans several coordinated dimensions, in which the elements of the second group can be arranged. The latter are forming systems of perceptual marker values that firstly allow to perceive the underlying basic structure, which actually is, as Kant has put it, a category. In the case of pictures, the basic structure is two-dimensional space, while the marker values are essentially colors and textures that render purely spatial structures visible. Hence speaking about picture syntax implies talking about space. The rules organizing our talking about space (at least as far as it is supposed to be rationally controlled) have been in the focus of interest at least since the neolithic revolution with its introduction of architecture, farming, and the cooperative spatial manipulations involved in those activities. Today, it is essentially the calculization of geometry of ancient Greece and its algebraic reformulation in the 16th century that influences the conception of "pure space", which also underlies pictorial syntax. Geometric calculi as formalization of space Basically, a geometric calculus is a set of rules originally setting up the options of talking in the abstract about "geometric entities". Those rules form the conceptual determinations of a set of spatial concepts. The concepts for basic geometric entities are essentially determined by the relations they can enter. Those are usually divided in relations concerning contact and neighorhood (topological relations), relations concerning distances and extensions (metric relations), and relations concerning direction and orientation (directional or projective relations). Having geometric concepts has essentially three consequences: The “ontological” aspect of the rules appears in that they enable us to describe something as being spatial. Understanding something as an instance of such a concept, i.e., as a geometric entity, is to view it as a kind of purely spatial entity – apart from other characterizations that might hold of it simultaneously (e.g., being colored, being heavy). “The world appears spatially organized if we distuinguish its parts by means of geometric concepts.” The “epistemological” aspect entails that the Gestalts that appear in descriptions of visual perception can be understood as geometric (hence spatial) entities. That is, the calculus formalizes an abstract part of our understanding of visual perception (also ⊳ objects of spatial perception): “We see the world as spatially organized.” Finally the “argumentational” aspect means that the rules of the calculus describe how a given description of spatial objects can be transformed without changing the truth of the descriptions: These transformations are usually summarized as ‘Spatial reasoning’: “When we rationally discuss about space, we employ the rules of geometric concepts.” The standard approach to geometric calculi is Euclid's axiomatic system based on the concept of an unextended but uniquely located »point«: This concept is, however, quite abstract and rather distant to (spatial) experience. As a result of a formal approach to geometry that was primarily developed in the 20th century and that tries to do justices of cognitive and foundational principles, a family of non-standard geometric calculi has been developed that offers interesting properties for picture morphology: ‘mereogeometries’. Euclidean calculus of geometry More than 2000 years ago, a first axiomatic system of geometry has been proposed by Euclid. Based on a set of basic postulates (‘axioms’), implications can be drawn in that system in order to formally (i.e., by means of logical deductions) prove theorems about geometric objects and their properties or relations. The discussion of this approach and in particular of the independence of the five basic postulates about geomeric objects has indeed led to several variants of such a geometric calculus (all belonging to the field of ‘synthetic geometry’). In the 16th century, Descartes (...) developed an alternative formalism based on algebraic formulae - locations are described by numbers encoding distances to a point of "origin" relative to coordinate axes. This ‘analytic’ geometry has been formally proven completely equivalent to the Euclidean calculus (Hilbert), and it has basically lead to the vector calculi used today in most technical approaches to space (e.g., computer graphics). Essentially, all geometric entities are viewed as sets of individual locations, called ‘points’, which, as Euclid has put it, is "what does not have parts". The points can be organized in one or more dimensions – depending on the number of independent (‘orthogonal’) coordinate axes, which are associated with basic directions. Each axis furthermore organizes the points according to the real numbers. This implies that Euclidean points are essentially arranged in a continuum. The rules of the geometric calculus establish spatial homogeneity (independence from translations of the point of origin) and isotropy (independence from rotations of the coordinate axes), which are the basis for syntactic density of pictorial space. Taking the common Euclidean formalization of geometry leads however also to the “unpleasant” consequence that the most basic pixemes must be non-extended points – a concept highly abstracted from experience, that is. Any extended region – any pixeme, that is – then consists of an infinite number of basic geometrical entities, quite in contrast to the Gestaltian conceptions of (spatial) perception. Mereogeometric calculi Some non-standard approaches to geometry offer an interesting way out of this dilemma. Mereogeometries are the result of a formal approach to geometry that was primarily developed in the 20th century, and that tries to do justices of cognitive and foundational principles. If a point is, as Euclid thought, that "what does not have parts", part-whole relations can be considered as crucial to geometry, and geometric entities that do have parts may after all be more natural candidates for grounding pictorial space. In contrast to Euclidean-style geometries, the family of mereogeometries is based on extended regions as the most elementary entities, which may or may not have (distinguishable) proper parts. Such regions are often called “individuals”. Individuals do not have immediate attributes of form or position: Only the relations to other individuals, in particular to their parts, determine form and (relative) location. While Euclidean geometry first introduces the continuous range of infinitely many coordinates determining potential points some of which are then chosen to be relevant (still an infinite number in any practical relevant instance), mereogeometric calculi start with a (usually finite) number of relevant individuals (regions). An individual may quite well be thought of as a visual Gestalt – thus following the principles of perception psychology of the Gestalt school: One has to consider the perceived whole first and introduce the concepts for perceptual atoms as instruments of the explanations of the former, not the other way round. We do not see sets of zero-dimensional points but regional Gestalts. The abstract notion of a spatial entity without extension is secondarily constructed in order to explain some aspects of experienced space, but leads on the other side to severe difficulties. Since space has been traditionally captured by point-based geometry, it must be recognized that, overall, the properties of Euclidean space fit our commonsense notion of space. Thus, it should not be surprising that most mereogeometries lead to systems ‘equivalent’ to Euclidean geometry [Borgo & Masolo, 2007?]. This very fact shows how our cognitive perception of space is quite stable and precise and is not affected by the choice of geometric primitives. Indeed, the properties that commonsense space should satisfy are not an issue. The crucial point is how we cognitively attain this specific notion of space. In this perspective, the first question that mereogeometries try to answer is what primitives apply to extended objects and are expressive enough to generate the commonsense notion of space. Calculi of geometry and the morphological structures of pictures Indeed, the research of mereogeometric calculi corroborates a conclusion that has puzzled researchers in pictorial morphology: the lack of constraints on the choice of primitives. In these domains one arrives easily to equivalent formalisms starting from quite disparate assumptions. It follows that the choice of primitives cannot rely on purely formal properties, it must be supported by arguments and observations from other perspectives like those embedded into the cognitive, evolutionary, mental, and perceptive views. In geometry and mereogeometry we have observed the development of several geometrical systems which, exploiting disparate primitives, naturally lead to formal geometries of equivalent expressiveness. On the one hand, the search for grounding pixemes (either as primitives or as prototypical) naturally leads to a discussion that matches the debate on basic geometrical entities. On the other hand, the need of rendering and understanding complex images in a computational setting suggests (at least in theory) the existence of a limited number of basic pixemes that can be combined via a formal calculus of limited complexity, and thus, hopefully, being qualitative. In the context of the morphological structures of pictures, the geometric calculus describes how we talk (rationally) about the basic structure of the picture plane and the pixemes contained there. A basic structure not following the rules determined by the calculus leads to a syntactically invalid picture. With a point-based geometry, pixemes are conceived of as (infinite) sets of points, determined by Gestalt-organizing processes operating on the marker values (⊳ color as a category of pictorial syntax). The infinitely many points contained in any pixeme are locations, which are mere potentially interesting: They might become morphologically relevant when comparing the picture vehicle in question with another picture vehicle. Similarly, the Euclidean-type calculus of geometry has no need for a concept of resolution: Its virtual value of resolution is always infinit – a God's eye view on space. With a mereogeometric calculus, pixemes are indviduals, i.e., primitive entities of the calculus. Assuming that the Gestalt principles governing visual perception determine exactly those regions that are syntactically relevant, the pixemes can be quite naturally conceived of as being given in perception. There is no need to consider more points than necessary Points, resolution, zooming, and microscopization While mereogeometry deals with the pure space, pictorial morphology has to take into account other elements like granularity (which may affect very basic properties as connectedness among entities, i.e., the topology itself). In fact, the concept of a minimal region can be introduced in mereogeometry: They are usually called a “point”, but we may well use “pixel” instead. A point in this sense is a region that has no proper parts (or rather, a region where no proper parts are considered). When the concept »point« is introduced in the data structure in that manner, there is no need in any concrete instance for using infinitely many point instances: only the “relevant” points must be instantiated. This also means that there is always a finite resolution. However, N. Asher & L. Vieu [1995] propose a formal mechanism called “microscopization” covering a kind of zooming operation by means of a modal extension to their mereogeometric calculus. What is a “point” on one level may be a compound of regions with several points on a microscopized level. Figure/ground-distinction image processing, digital Basic picture plane and maximale pixeme In contrast to standard geometries, pictorial space is usually limited: the picture plane consists of one unique maximal pixeme, which is not part of another pixeme – St. Martin's basic picture plane –, and its parts (⊳ Rahmung, Rahmen) ... Semantic aspects of geometry Obviously, geometric calculi also play a major role in picture semantics: Projecting a three-dimensional scene to a two-dimensional picture plane obeies the rules determined in particular by the calculi of projective geometry (⊳ Perspective and Projection).

Notes Literature [Aurnague & Vieu 1993a]Aurnague, Michel & Vieu, Laure (1993): A Three-Level Approach to the Semantics of Space. In: Zelinsky-Wibbelt, C. (Hg.), The Semantics of Prepositions – From Mental Processing to Natural Language Processing. Mouton de Gruyter, Berlin.[Borgo & Masolo 2010a]Borgo, Stefano & Masolo, Claudio (2010): Full Mereogeometries. The Review of Symvbolic Logic, 3, 4.[Goodman 1968a]Goodman, Nelson (1968): Languages of Art. Hackett, Indianapolis.[Herskovits 1986a]Herskovits, Annette (1986): Language and Cognition – An Interdisciplinary Study of the Prepositions in English. Cambridge Univ. Press, Cambridge.[Saint-Martin 1987a]Saint-Martin, Fernande (1987): Sémiologie du langage visuel. Presses de l’Université du Quebec, Montréal.[Schirra 1994a]Schirra, Jörg R.J. (1994): Bildbeschreibung als Verbindung von visuellem und sprachlichem Raum. DISKI, St. Augustin.[Tarski 1929a]Tarski, Alfred (1929): Les fondement de la géometrie des corps. Société polonaise de mathématique, Cracovie. Responsible: Borgo, Stefano Schirra, Jörg R.J. Contributions by: Joerg R.J. Schirra und Marcel Lemmes (remark)