Glossar-English:Space and Geometry: Unterschied zwischen den Versionen
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| − | + | Sub-Item to: [[Glossar-English:Image Syntax|Image Syntax]] | |
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| − | + | Original: [[Raum und Geometrie]] | |
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| + | ==Taking space as the fundamental category of image morphology== | ||
| + | The set of the syntactic components for [[Ähnlichkeit und wahrnehmungsnahe Zeichen|perceptoid signs]] can generally be divided into two groups: On the one hand, there is an abstract relational ''basic structure''. It 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 us to perceive the underlying basic structure, which indeed is, as Kant has put it, a category.<ref>As a conceptual basis of perception (a ‘condition of possibility of experience’), categories are not perceptible in themselves.</ref> In the case of pictures, the basic structure is two-dimensional space, and the marker values are essentially [[Glossar-English:Color as an image-syntactic category|colors]] and [[Textur|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 the focus of interest as early as 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 the geometry of ancient Greece and its algebraic reformulation in the 16<sup>th</sup> century that influenced the conception of space, which also underlies pictorial syntax.<ref>That does not imply that those calculi are sufficient when dealing with semantic and pragmatic aspects: ⊳ [[Theorien des Bildraums|Theories of picture space]].</ref> | |
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| − | + | ==Geometric calculi as a formalization of space== | |
| + | Basically, geometric calculus is a vocabulary with a set of rules originally setting up the options of talking in the abstract – without reference to any concrete objects with contingent non-geometric properties – 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 into relations concerning contact and neighborhood (topological relations),<ref>Cf. also [https://en.wikipedia.org/wiki/Topological_space Wikipedia: Topological Space].</ref> relations concerning distances and extensions (metric relations),<ref>Cf. also [https://en.wikipedia.org/wiki/Metric_space Wi­ki­pe­di­a: Metric Space].</ref> and relations concerning direction and orientation (directional or projective relations).<ref>Cf. also [https://en.wikipedia.org/wiki/Projective_geometry Wikipedia: Projektive Geometry].</ref> | ||
| − | + | Having geometric concepts at one's disposal has essentially three consequences: | |
| + | * The “''ontological''” aspect of the rules appears in that they enable us to describe something as ''being spatial''.<ref>More precisely, such a description consists of utterances containing a [[Glossar-English:Proposition|proposition]] with a [[Glossar-English:Predication|predication]] using the geometric concept determined by calculus.</ref> 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).<ref>That is, ‘being geometrical’ as such does not necessarily imply ‘being spatial’.</ref> ''The world appears spatially organized if we distinguish its parts by means of geometric concepts''. | ||
| − | *'''' | + | * The “''epistemological''” aspect entails that the [[Gestalt]]s 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 ⊳ [[Gegenstand der visuellen Wahrnehmung|Objects of Spatial Perception]]): ''We see the world as geometrically 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’:<ref>In particular, the rules linking geometric concepts with each other, e.g., that »left« is the inverse of »right«, or that »in« is, under certain conditions, a transitive relation, can be used as “middle terms” in spatial syllogisms relating several [[Glossar-English:Proposition|propositions]] about geometric entities. Note, however, also the components of such deductions exceeding pure geometry as motivated in particular in <bib id='Herskovits 1986a'></bib> or <bib id='Aurnague & Vieu 1993a'></bib>, and the association of those considerations with concept-genetic arguments in <bib id='Schirra 1994a'></bib>: in particular Chap. 5.</ref> ''When discussing space rationally, 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«. However, this concept is quite abstract and rather distant to (spatial) experience. A formal approach to geometry primarily developed in the 20<sup>th</sup> century tries to do justice to cognitive principles. As a result, a family of non-standard geometric calculi has been developed. It offers interesting properties for [[Glossar-English:Image Morphology|image morphology]]: ‘mereogeometries’. |
| − | + | ===Euclidean calculus of geometry=== | |
| + | More than 2000 years ago, the first axiomatic system of geometry was proposed by Euclid.<ref>Cf. also [https://en.wikipedia.org/wiki/Euclidean_geometry Wikipedia: Euklidean Geometry].</ref> Based on a set of basic postulates (‘axioms’), implications can be drawn in that system in order to formally (i.e., using 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 geometric objects has indeed led to several variants of such a geometric calculus forming the field of ‘synthetic geometry’.<ref>Cf. [https://en.wikipedia.org/wiki/Synthetic_geometry Wikipedia: Synthetic Geometry]. In particular, the “fifth axiom”, as it is often called, concerned with the concept of parallelity, has been relatively fertile, although the classical Non-Euclidean calculi developed in consequence are usually not relevant for picture syntax; cf. [https://en.wikipedia.org/wiki/Parallel_postulate Wikipedia: Parallel Postulate].</ref> | ||
| − | + | In the 16<sup>th</sup> century, Descartes (Essais, 1637) developed an alternative formalism based on algebraic formulae – locations are described by numbers encoding distances to a certain point of “origin” relative to coordinate axes. This ‘analytic’ geometry, which has basically led to the vector calculi used today in most technical approaches to space, has been formally proven (Hilbert) to be completely equivalent to Euclidean calculus.<ref>Cf. also [https://en.wikipedia.org/wiki/Analytic_geometry Wikipedia: Analytic Geometry].</ref> | |
| − | + | 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”. Depending on the number of independent (‘orthogonal’) coordinate axes associated with basic directions, the points can be organized in one or more dimensions. Furthermore, each axis organizes the points according to the real numbers. That implies that Euclidean points are essentially arranged in a continuum. | |
| − | + | The rules of the geometric calculus establish spatial homogeneity (invariance with respect to translations of the point of origin) and isotropy (invariance with respect to rotations of the coordinate axes), which ultimately are the basis for the [[Glossar-English:Syntactic Density|syntactic density]] of pictorial space. However, the common Euclidean formalization of geometry also leads to the “unpleasant” consequence that the most basic pixemes must be non-extended points – a concept highly abstracted from experience. Accordingly, any extended region – i.e., any pixeme – must consist of an infinite number of basic geometrical entities, quite in contrast to the [[Gestalt]]ian conceptions of (spatial) perception. | |
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| − | + | ===Mereogeometric calculi=== | |
| − | + | Some non-standard approaches to geometry offer a compelling way out of this dilemma. Mereogeometries result from a formal approach to geometry primarily developed in the 20<sup>th</sup> century and try to do justice to fundamental cognitive principles. If a point is, as Euclid thought, that “what has no parts”, then part-whole relations should be considered crucial to geometry. At least, geometric entities that ''do'' have parts may, after all, be more natural candidates for grounding pictorial space. | |
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| − | + | In contrast to Euclidean-style geometries, the family of mereogeometries is based on the concept of an extended region, which may or may not have (distinguishable) proper parts. Such regions are often called “individuals” as they form the ultimate basic elements of that calculus.<ref>Mereogeometric regions can be conceived of as undividable (‚in-dividuum‘) within the mereogeometric calculus since their properties can not be derived from a composition of more basic elements although they partake in part-whole-relations, and hence could have other regions as their parts.</ref> They do not have immediate attributes of form or position:<ref>Here, exceptions substantiate the rule: In some variants, only circular regions are considered, for instance (<bib id='Tarski 1929a'>Tars­ki 1929a</bib>).</ref> Only the relations to other individuals, which in particular could be their parts or the wholes they are a part of, determine form, extension, and (relative) location. | |
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| − | ... | + | While Euclidean geometry first introduces the continuous range of infinitely many points, some of which are then chosen to be relevant (still an infinite number in any practical relevant instance), mereogeometric descriptions start with a (usually finite) number of relevant individuals (regions).<ref>For a more detailed example of a mereogeometric calculus, see [[Exkurs:Mereogeometries|Excursus: Mereogeometries]].</ref> Such 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. Human beings do not see sets of zero-dimensional points but extended Gestalts – especially in [[Bildwahrnehmung|picture perception]]. 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 | + | Since space has been traditionally captured by point-based geometry, it must be recognized that, overall, the properties of Euclidean space (i.e., the concepts of space as described by Euclidean calculus) fit our commonsense notion of space. Thus, it should not be surprising that most mereogeometries lead to systems ‘equivalent’ to Euclidean geometry <bib id='Borgo & Masolo 2010a'></bib>. 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 to explain 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. |
| − | 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 | ||
| − | 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. | ||
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| − | + | ==Calculi of geometry and the morphological structures of pictures== | |
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| − | On the one hand, the search for grounding pixemes (either as primitives or | + | 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 both domains, one arrives easily at equivalent formalisms despite 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 in cognitive, evolutionary, mental, and perceptive views. The development of several geometrical systems up to mereogeometries has led to geometric approaches that, exploiting disparate primitives, naturally result in 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 to render and understand 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 (⊳ [[Bildverarbeitung, digitale|image processing, digital]]).<ref>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 <bib id='Goodman 1968a'></bib>.</ref> |
| − | as prototypical) naturally leads to a discussion that matches the debate on basic geometrical | ||
| − | entities. On the other hand, the need | ||
| − | 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, | ||
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| + | ===Pixemes as geometric entities=== | ||
| + | In the context of the morphological structures of pictures, 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 [[Glossar-English:Syntactically Incorrect Images|syntactically invalid picture]]. With a point-based calculus, pixemes are conceived of as (infinite) sets of points determined by [[Gestalt]]-organizing processes operating on the marker values (⊳ [[Glossar-English:Color as an image-syntactic category|color as a category of pictorial syntax]]). The infinitely many points contained in any pixeme are locations that are merely potentially interesting: They might become morphologically relevant when comparing the picture vehicle in question with another picture vehicle. Correspondingly, the Euclidean-type calculus of geometry has no need for a concept of resolution: Its virtual value of resolution is always infinite – a God's eye view of space. | ||
| + | With a mereogeometric calculus, pixemes are “individuals”, 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 mereogeometries deal with pure space, pictorial morphology has to take into account other elements like granularity (which may affect very basic properties such 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, which is quite obviously somewhat different from the Euclidean point, is approximately a region that has no proper parts (or rather, a region where no proper parts are considered).<ref>The point in mereogeometry is usually not identical to the minimal region but is defined as a class of all individuals (regions) of which the minimal region is a part. </ref> When the concept »point« is introduced to the calculus in that manner, there is no need in any concrete instance for using infinitely many point instances: Only the “relevant” points have to be instantiated. That also means there is always a finite resolution for describing space with such a calculus. However, N. Asher & L. Vieu [1995] have proposed a formal mechanism called “microscopization” covering a kind of zooming operation utilizing 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. |
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| − | 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 | ||
| + | ===The empty picture plane and maximal pixemes=== | ||
| + | In contrast to space according to standard geometries, pictorial space is usually externally limited: The picture plane consists of one unique maximal pixeme, which is not part of another pixeme of that picture – Saint Martin here uses the expression “basic picture plane” (<bib id='Saint-Martin 1987a'></bib>).<ref>Of course, this maximal pixeme is still always perceived as a part of its particular spatial surroundings.</ref> The distinction essential to mereogeometric approaches of the closed region – a region that includes its borders – and that maximal part of it not including the border, delivers a direct approach for dealing with frame and picture proper (⊳ [[Rahmung, Rahmen|frame]]). | ||
| − | + | [[Datei:Energetic-SaintMartin1990.jpg|thumb|Abb. 1: Rectangular maximal pixeme with “energetic phenomenon” following Saint Martin]] | |
| + | If the usual rectangular form of the picture plane is chosen, four regions that are points in the mereogeometric sense have to be considered in mereogeometric approaches: the four corners. The “energetic load” of the corners indicated by Fernande Saint-Martin in her description of picture morphology (cf. Fig. 1) might be related to the construction of the concept of a point in most mereogeometries: In the calculus of Tarski, for example, points are introduced as the classes of all ''concentric'' circles in the situation described (<bib id='Tarski 1929a'></bib>). Therefore, in the Tarskian description of a rectangular picture plane, corresponding circles had to be instantiated for each corner as the defining basis for the point. The parts of those circles that are inside the picture plane indeed correspond precisely to those energy lines of Saint-Martin. | ||
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| − | + | ==Further Aspects== | |
| − | + | ===Semantic aspects of geometry=== | |
| − | + | Obviously, the spatial concepts framed in geometric calculi also play a significant role in [[Glossar-English:Image Semantics|picture semantics]]: Projecting a three-dimensional scene to a two-dimensional picture plane obeys rules that are in particular (though not only) determined by the calculi of projective geometry. Those considerations are discussed in the lemma [[Perspektive und Projektion|Perspective and Projection]]. | |
| + | ===Time and Geometry=== | ||
| + | In physics, temporal extension and order are regarded as another (i.e., fourth) geometric dimension. Corresponding conceptions may be used in picture philosophy when dealing with [[film]], [[video]], [[Fernsehen|TV]], or other moving picture formats. In the strong sense of physics, the temporal “direction in space” is contrasted to the spatial “directions in space” by taking it as a (mathematically) imaginary axes of a four-dimensional complex vector space, or inversely as the only real axes complementing imaginary spatial components (‘quaternions’).<ref>Cf. [https://en.wikipedia.org/wiki/Quaternion Wikipedia: Quarternion]. </ref> In computer graphics, quaternions are often used to calculate transformations of 3D models. Rotations, in particular, can then be handled quite easily. | ||
| − | + | {{GlossarSieheE}} | |
| − | + | * [[Glossar-English:Color as an image-syntactic category|Color as an image-syntactic category]] | |
| − | + | * [[Computergraphik|Computer graphics]] | |
| − | * [[ | + | * [[Glossar-English:Context|Context]] |
| − | + | * [[Bildverarbeitung, digitale|Digital image processing]] | |
| − | * [[ | + | * [[Exkurs:Mereogeometrien|Exkursus: Mereogeometries]] |
| − | + | * [[Film]] | |
| − | * [[ | + | * [[Rahmung, Rahmen|Framing, frame]] |
| − | + | * [[Gestalt]] | |
| + | * [[Glossar-English:Image Morphology|Image morphology]] | ||
| + | * [[Bildwahrnehmung|Image perception]] | ||
| + | * [[Glossar-English:Image Semantics|Image semantics]] | ||
| + | * [[Anschauung|Intuition]] | ||
| + | * [[Gegenstand der visuellen Wahrnehmung|Objects of visual perception]] | ||
| + | * [[Perspektive und Projektion|Perspective and projection]] | ||
| + | * [[Glossar-English:Predication|Predication]] | ||
| + | * [[Glossar-English:Proposition|Proposition]] | ||
| + | * [[Ähnlichkeit und wahrnehmungsnahe Zeichen|Resemblance and “signs close to perception”]] | ||
| + | * [[Auflösung|Resolution]] | ||
| + | * [[Glossar-English:Sign, Sign Carrier, Sign System|Sign, sign carrier, sign system]] | ||
| + | * [[Glossar-English:Syntactic Density|Syntactic density]] | ||
| + | * [[Glossar-English:Syntactically Incorrect Images|Syntactically incorrect images]] | ||
| + | * [[Textur|Texture]] | ||
| + | * [[Theorien des Bildraums|Theories of image space]] | ||
| + | * [[Fernsehen|TV]] | ||
| + | * [[Video|Video]] | ||
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| − | + | ''Responsible:'' | |
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| + | * [[Benutzer:Joerg R.J. Schirra|Schirra, Jörg R.J. ]] | ||
| + | * [[Benutzer:Stefano Borgo|Borgo, Stefano]] | ||
| + | {{GlosTab4}} | ||
| + | ''Proofread:'' | ||
| + | * [[Benutzer:Tobias Schöttler|Schöttler, Tobias]] | ||
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| + | <bib id='Schirra & Borgo 2013g-a'></bib> | ||
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Aktuelle Version vom 6. Dezember 2023, 20:46 Uhr
Sub-Item to: Image Syntax
Original: Raum und Geometrie
Taking space as the fundamental category of image morphologyThe set of the syntactic components for perceptoid signs can generally be divided into two groups: On the one hand, there is an abstract relational basic structure. It 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 us to perceive the underlying basic structure, which indeed is, as Kant has put it, a category.[1] In the case of pictures, the basic structure is two-dimensional space, and 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 the focus of interest as early as 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 the geometry of ancient Greece and its algebraic reformulation in the 16th century that influenced the conception of space, which also underlies pictorial syntax.[2]
Geometric calculi as a formalization of spaceBasically, geometric calculus is a vocabulary with a set of rules originally setting up the options of talking in the abstract – without reference to any concrete objects with contingent non-geometric properties – 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 into relations concerning contact and neighborhood (topological relations),[3] relations concerning distances and extensions (metric relations),[4] and relations concerning direction and orientation (directional or projective relations).[5] Having geometric concepts at one's disposal has essentially three consequences:
The standard approach to geometric calculi is Euclid's axiomatic system based on the concept of an unextended but uniquely located »point«. However, this concept is quite abstract and rather distant to (spatial) experience. A formal approach to geometry primarily developed in the 20th century tries to do justice to cognitive principles. As a result, a family of non-standard geometric calculi has been developed. It offers interesting properties for image morphology: ‘mereogeometries’. Euclidean calculus of geometryMore than 2000 years ago, the first axiomatic system of geometry was proposed by Euclid.[9] Based on a set of basic postulates (‘axioms’), implications can be drawn in that system in order to formally (i.e., using 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 geometric objects has indeed led to several variants of such a geometric calculus forming the field of ‘synthetic geometry’.[10] In the 16th century, Descartes (Essais, 1637) developed an alternative formalism based on algebraic formulae – locations are described by numbers encoding distances to a certain point of “origin” relative to coordinate axes. This ‘analytic’ geometry, which has basically led to the vector calculi used today in most technical approaches to space, has been formally proven (Hilbert) to be completely equivalent to Euclidean calculus.[11] 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”. Depending on the number of independent (‘orthogonal’) coordinate axes associated with basic directions, the points can be organized in one or more dimensions. Furthermore, each axis organizes the points according to the real numbers. That implies that Euclidean points are essentially arranged in a continuum. The rules of the geometric calculus establish spatial homogeneity (invariance with respect to translations of the point of origin) and isotropy (invariance with respect to rotations of the coordinate axes), which ultimately are the basis for the syntactic density of pictorial space. However, the common Euclidean formalization of geometry also leads to the “unpleasant” consequence that the most basic pixemes must be non-extended points – a concept highly abstracted from experience. Accordingly, any extended region – i.e., any pixeme – must consist of an infinite number of basic geometrical entities, quite in contrast to the Gestaltian conceptions of (spatial) perception. Mereogeometric calculiSome non-standard approaches to geometry offer a compelling way out of this dilemma. Mereogeometries result from a formal approach to geometry primarily developed in the 20th century and try to do justice to fundamental cognitive principles. If a point is, as Euclid thought, that “what has no parts”, then part-whole relations should be considered crucial to geometry. At least, 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 the concept of an extended region, which may or may not have (distinguishable) proper parts. Such regions are often called “individuals” as they form the ultimate basic elements of that calculus.[12] They do not have immediate attributes of form or position:[13] Only the relations to other individuals, which in particular could be their parts or the wholes they are a part of, determine form, extension, and (relative) location. While Euclidean geometry first introduces the continuous range of infinitely many points, some of which are then chosen to be relevant (still an infinite number in any practical relevant instance), mereogeometric descriptions start with a (usually finite) number of relevant individuals (regions).[14] Such 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. Human beings do not see sets of zero-dimensional points but extended Gestalts – especially in picture perception. 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 (i.e., the concepts of space as described by Euclidean calculus) fit our commonsense notion of space. Thus, it should not be surprising that most mereogeometries lead to systems ‘equivalent’ to Euclidean geometry [Borgo & Masolo 2010a]Literaturangabe fehlt.
Calculi of geometry and the morphological structures of picturesIndeed, 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 both domains, one arrives easily at equivalent formalisms despite 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 in cognitive, evolutionary, mental, and perceptive views. The development of several geometrical systems up to mereogeometries has led to geometric approaches that, exploiting disparate primitives, naturally result in 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 to render and understand 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 (⊳ image processing, digital).[15] Pixemes as geometric entitiesIn the context of the morphological structures of pictures, 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 calculus, 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 that are merely potentially interesting: They might become morphologically relevant when comparing the picture vehicle in question with another picture vehicle. Correspondingly, the Euclidean-type calculus of geometry has no need for a concept of resolution: Its virtual value of resolution is always infinite – a God's eye view of space. With a mereogeometric calculus, pixemes are “individuals”, 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 microscopizationWhile mereogeometries deal with pure space, pictorial morphology has to take into account other elements like granularity (which may affect very basic properties such 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, which is quite obviously somewhat different from the Euclidean point, is approximately a region that has no proper parts (or rather, a region where no proper parts are considered).[16] When the concept »point« is introduced to the calculus in that manner, there is no need in any concrete instance for using infinitely many point instances: Only the “relevant” points have to be instantiated. That also means there is always a finite resolution for describing space with such a calculus. However, N. Asher & L. Vieu [1995] have proposed a formal mechanism called “microscopization” covering a kind of zooming operation utilizing 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. The empty picture plane and maximal pixemesIn contrast to space according to standard geometries, pictorial space is usually externally limited: The picture plane consists of one unique maximal pixeme, which is not part of another pixeme of that picture – Saint Martin here uses the expression “basic picture plane” ([Saint-Martin 1987a]Saint-Martin, Fernande (1987).Sémiologie du langage visuel. Montréal: Presses de l’Université du Quebec, Englisch: Semiotics of Visual Language. Bloomington, Indianapolis: Indiana University Press.. Eintrag in Sammlung zeigen).[17] The distinction essential to mereogeometric approaches of the closed region – a region that includes its borders – and that maximal part of it not including the border, delivers a direct approach for dealing with frame and picture proper (⊳ frame). If the usual rectangular form of the picture plane is chosen, four regions that are points in the mereogeometric sense have to be considered in mereogeometric approaches: the four corners. The “energetic load” of the corners indicated by Fernande Saint-Martin in her description of picture morphology (cf. Fig. 1) might be related to the construction of the concept of a point in most mereogeometries: In the calculus of Tarski, for example, points are introduced as the classes of all concentric circles in the situation described ([Tarski 1929a]Literaturangabe fehlt.
Further AspectsSemantic aspects of geometryObviously, the spatial concepts framed in geometric calculi also play a significant role in picture semantics: Projecting a three-dimensional scene to a two-dimensional picture plane obeys rules that are in particular (though not only) determined by the calculi of projective geometry. Those considerations are discussed in the lemma Perspective and Projection. Time and GeometryIn physics, temporal extension and order are regarded as another (i.e., fourth) geometric dimension. Corresponding conceptions may be used in picture philosophy when dealing with film, video, TV, or other moving picture formats. In the strong sense of physics, the temporal “direction in space” is contrasted to the spatial “directions in space” by taking it as a (mathematically) imaginary axes of a four-dimensional complex vector space, or inversely as the only real axes complementing imaginary spatial components (‘quaternions’).[18] In computer graphics, quaternions are often used to calculate transformations of 3D models. Rotations, in particular, can then be handled quite easily. See also:
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Notes
[Aurnague & Vieu 1993a]:
Literaturangabe fehlt. Bitte in der Bibliographie-Sammlung einfügen als: - Buch, - Artikel in Zeitschrift, - Beitrag in Sammelband, - Sammelband, - andere Publikation, - Glossarlemma. [Borgo & Masolo 2010a]: Literaturangabe fehlt. Bitte in der Bibliographie-Sammlung einfügen als: - Buch, - Artikel in Zeitschrift, - Beitrag in Sammelband, - Sammelband, - andere Publikation, - Glossarlemma. [Goodman 1968a]: Goodman, Nelson (1968, 2. rev. Aufl. 1976). Languages of Art. Indianapolis: Hackett, dt.: Sprachen der Kunst. Suhrkamp 1998. [Herskovits 1986a]: Version 1: 2013 Responsible: Proofread: Contributions by: Joerg R.J. Schirra [30] — (Note) Quote as: [Schirra & Borgo 2013g-a]Literaturangabe fehlt. |
