Glossar-English:Space and Geometry
Übersetzung zu: Raum und Geometrie
Space as basic category of picture syntax/picture morphology
That any pixeme must be a geometric entity seems almost too trivial to be mentioned. That inversely any entity in flat geometry – apart from non-extended points – may also be a candidate for a pixeme is at least a good guess.
Formalization: calculi of geometry
relation with picture syntax, in particular
Geometric calculi as formalization of space
Standard approach: Euclid; (clasical Non-Euclidean usually not relevent for Picture syntax;)
Non standard: 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
Taking the common Euclidean formalization of geometry leads however to the “unpleasant” consequence that the most basic pixemes must be non-extended points – a concept highly abstracted from experience, that is.
Fortunately, some non-standard approaches to geometry offer an interesting way out. The traditional calculus of geometry develops around the fundamental concept of a zerodimensional point. In contrast, the family of mereogeometries is based on extended regions as the most elementary entities, which may or may not have (distinguishable) proper parts. The regions are often called “individuals”. Individuals do not have immediate attributes of form or position: only the relations to other individuals, in particular 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 principle 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 as the discussion on infinite resolution has shown. Therefore, the constructs of an individual calculus for the two-dimensional mereogeometry are excellent candidates for a general and exhaustive discussion of pixemes.
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, to appear]. 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.
... we can think of indviduals as being given in perception. That is, we may indeed assume that the principles governing visual perception determine the regions that are syntactically relevant, hence leading only to the essential “points” determined by the given individuals. Gestalt
Mereogeometry 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. Of course, there is much more in pictorial morphology than this as it was clear from the beginning, see for instance the seminal work of Goodman . While mereogeometry stops at the geometrical aspects of physical objects and their relationships, 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) and appearance (from which the difference between resemblance in geometry and in perception).
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  propose a formal mechanism called “microscopization” covering a kind of zooming operation by means of a modal extension to their calculus. What is a “point” on one level may be a compound of regions with several points on a microscopized level.
(Mereo-)Geometric analysis of a simple example(?)
The empty picture plain – as the simplest maximal pixeme – is particularly characterized in its most usual rectangular form by the four corner points. The “energetic field” often associated to such a maximal pixeme (cf. Fig. ) cannot easily be derived as it depends essentially on features of the perceptual mechanism not covered by the Euclidean calculus as such. Additional explanations have to be added that often employ rather mystical metaphors to physics. The mereogeometrical conception of points and limits may offer a better access to the problem of the “energetic aspects” of pixemes, and especially of the empty picture plane: As those points are only conceivable as the result of operations on extended regions, the four corner points implicitly refer to defining individuals (virtual pixemes). It is a promising hypothesis for future research to derive within the calculus of mereogeometry any “energetic effects” from those implicit pixemes.
In a minimalistic picture like Böhm's .....
Effects on other concepts
Literatur [Sammlung][Saint Martin 1990a]:
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