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<Paper uid="C82-1002">
  <Title>COGNITIVE MODELS FOR COMPUTER VISION</Title>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
OBJECT DESCRIPTION AND SPACE MODELLING
</SectionTitle>
    <Paragraph position="0"> The knowledge of the structure of an object is often intimately related to our capability of understanding the meaning of a spatial relat~nship; fop instance, the meaning of the sentence &amp;quot;the cat is under the car&amp;quot; is clear, even if it may depend on the state of the car, moving or parked; on the contrary, the sentence &amp;quot;the cat is under the wall&amp;quot; is not clear, unless the wall is crashed or it has a very particular shape. Every object modelling technique must deal at least with the following issues 16,7\[ : l.Objeot must be described at several levels of detail.To understand the sentence &amp;quot;put the chair near the table&amp;quot; only a rough definition of chair and table dimensions can be sufficient,while to build a model of &amp;quot;a man sitting on a chair&amp;quot; a more sophisticated knowledge abaut the structure of a chair and a man is requested.</Paragraph>
    <Paragraph position="1"> 2.The articulation of movable object parts must be properly described.</Paragraph>
    <Paragraph position="2"> The sentences&amp;quot;open the door&amp;quot; and &amp;quot;open the drawer&amp;quot; have different geometric meanings because the movements of doors and drawers usually obey different rules.</Paragraph>
    <Paragraph position="3"> 3.Characteristic features of objects must be pointed out.0ften these features are free surfaces,as the top of a table,in canonical positions. The recognition of a feature allows the generation of hypothesis about the presence of an object.</Paragraph>
    <Paragraph position="4"> 4.Typical relations between objects must be described.When we look for a pencil we do not start analyzing a wall or a window,but we look at first foP a table or some other piece of furniture in which or on which it is reasonable to find a pencil.</Paragraph>
    <Paragraph position="5"> 0ur conceptual definition language allow the definition of lines,suPface and solid objects. Solid objects are described by means of GENER-</Paragraph>
  </Section>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
ALIZED CONES 19,101,at several levels of detail.Cones can be intercon-
</SectionTitle>
    <Paragraph position="0"> nected by means of fixed or movable points,with arbitrary constraints on rotations and shifting. Specific jointing elements are defined to properly describe the surface of an articulated object;so we can correctly answer to the question: &amp;quot;is the fly on the snake?&amp;quot; indipenden~y of how the snake is actually coiled.More details can he found in Ill.</Paragraph>
    <Paragraph position="1"> From a computational point of view,the use of a system of coordinated axes represent a very natural way to describe the position of an object.</Paragraph>
    <Paragraph position="2"> If we are able to transform linguistic relations into quantitative geometrical ones,the well knows methodologies of anal~tical geometry can be used as a simple,general purpose set of inferencing rules.Hence the goal of describing objects and spatial relations by means a simple,non redundant n-tuple of coordinated axes is very appealing. Unfortunately it seems quite far from the psychology of language 121.</Paragraph>
    <Paragraph position="3"> Therefore we associate a redundant FRAME OF REFERENCE (FOR) to every object,consisting of : - an axis,Z,having direction of the &amp;quot;maOor&amp;quot; axis of the coDe.Two points aPe specified on it,Zmi n and Zmax,COrresponding to the extremities of this major axis; - a point 0,on the Z axis,which is the origin of the frame; - an axis,X,orthogonal to Z,that specifies a further privileged direction of the object;this axis is definib!e only for some objects(eg.</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
COGNITIVE MODELS FOR COMPUTER VISION
</SectionTitle>
    <Paragraph position="0"> a man) in which a front and a back can be distinguished. Objects f(r which the X axis is definable are to be called CLASS 1 objects; those for which the X. axis is not definable (eg. a pole) are to be called CLASS 2 objects; an axis,Y,orthogonal to X and Z.The Y axis is obviously not definable for class 2 objects; a radial coordinate ~ whose origin is at O; the coordinates ~ and ~ specified on the X-Y plane; a curvilinear coordinate t originating at point O.</Paragraph>
    <Paragraph position="1"> The use of cones simplifies the FOR;it allows a homogeneous representation of an object shape and of its spatial relations with the external world;it proves particularly useful in situations like &amp;quot;the ball is inside the box&amp;quot;.</Paragraph>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
SPATIAL RELATIONS BETWEEN OBJECTS
</SectionTitle>
    <Paragraph position="0"> Let's now analyze some spatial relations between objects,in order to discuss how they can be translated in terms of geometrical primitives.</Paragraph>
    <Paragraph position="1"> Spatial relations involving the Z axis generally use a &amp;quot;major&amp;quot; axis perpendicular to the earth surface;this is the onl~ absolute reference used in language perhaps because the concept of &amp;quot;high&amp;quot; and &amp;quot;low&amp;quot; is directly related to the line of action of the force of gravity. Therefore the sentence &amp;quot;the object A is above the obOect B&amp;quot; can be concep-</Paragraph>
    <Paragraph position="3"> Note that we can state conditions only for pairs of points whose horizontal projections are the same. In fact, even the&amp;quot;pure&amp;quot; meaning of &amp;quot;above&amp;quot; is much more constraining 14,131,this relationship is used in a number of &amp;quot;impure&amp;quot; meanings,in which we cannot say that the horizontal projection of A is included in the horizontal projection of B(Fig.</Paragraph>
    <Paragraph position="4"> la), or Z(P)~Z(0) for any pair or points P~ C0NE(A) and QeCONE(B) (Fig.lb).</Paragraph>
    <Paragraph position="5"> The preposition &amp;quot;on&amp;quot; is often synonymous of &amp;quot;above&amp;quot;,but in some cases it can mean &amp;quot;below&amp;quot;, as in &amp;quot;on the ceiling&amp;quot;, or involve horizontal relations as in &amp;quot;the lamp is on the wall&amp;quot;.Usually &amp;quot;A on B&amp;quot; requires B to support A against the action of gravity,by means of some kind of physical contact. Hence,the conceptualization of &amp;quot;a man on a chair&amp;quot; is the same as &amp;quot;a man above a cha~r~'~plus an assertion about physical contact and Supporting action :</Paragraph>
    <Paragraph position="7"> Horizontal relations ar~ much more ambigous. Sometimes FOR is expl~ity stated,as in &amp;quot;looking at the church,the post office is on your right&amp;quot;; otherwise a default assumption is to use FOR associated with the speak_ er or the listener.</Paragraph>
    <Paragraph position="8"> If we consider the sentence &amp;quot;the object A is behind the object B&amp;quot;,two interpretations are possible : a) FOR is the n-tuple associated with the object B; 10 G. ADORNI, A. BOCCALATTE and M. DI MANZO b) FOR is external to both objects A and B.</Paragraph>
    <Paragraph position="9"> Case a can be assumed only if B is a class 1 object;case b is always assumed when B is a class 2 object,but it is not usual even when B is a class 1 object. In the ease a the previous sentence is conceptualized as follows :</Paragraph>
    <Paragraph position="11"> This definition and the next one allowto handle situations asthoseshownin F~.</Paragraph>
    <Paragraph position="12"> 2a-b ; the situation of Fig.2c does not represent a proper use of &amp;quot;behind&amp;quot;;if such a preposition is used,more inferencing capabilities are needed. In the case b the previous sentence means that B is (partially) hiding A to an observer,who can be assumed,to be one of the actors in the story;hence the conceptual representation is :</Paragraph>
    <Paragraph position="14"> Let's now to consider relations as &amp;quot;on the edge of&amp;quot;,&amp;quot;on the su~faoe of;' &amp;quot;in the middle of&amp;quot; and so on. For every point P on the surface of the cone which describes the object A,its possible to find the corresponding cross-section,that is characterized by a value Z of the coordinate along the cone axis. The boundary of this section is described by a radial coordinate ~((~,~.).Therefore the sentence &amp;quot;the pen is in the middle of the table&amp;quot; can be conceptualized as follows,assuming as reference the cross section of the table cone which corresponds to the table top :</Paragraph>
    <Paragraph position="16"> max FOR c C ONE ( TABLE ) // Let's conclude looking at sentences as &amp;quot;the house is before the bridge',' &amp;quot;two miles after the lights&amp;quot; and so on. In these cases spatial relations are referred to a path,usually not straight.This type of relations can be conceptualized using a curvilinear coordinate t associated with a</Paragraph>
  </Section>
  <Section position="7" start_page="0" end_page="0" type="metho">
    <SectionTitle>
COGNITIVE MODELS FOR COMPUTER VISION II
</SectionTitle>
    <Paragraph position="0"> trajectory s originating in the center of FOR.If the analytical description of such a trajectory is unknown,the robot will be able to make inferences only about the relative positions of objects along the path;so,for instance,from the sentence &amp;quot;the house is two miles after the bridge along the ~oad t0 Florence&amp;quot; it is possible to deduce that a man wolking towards Florence will meet at first the bridge and then the house,after an evaluable time. If more informations are available (eg. the path is a road and the map of the town is known),the position relative to other FOR can be evaluated from the actual value of t,in order to infer that &amp;quot;two miles after the bridge&amp;quot; means exactly &amp;quot;on the right of the station&amp;quot;.The formal description of &amp;quot;the object A is after the object B&amp;quot;, is :</Paragraph>
    <Paragraph position="2"> Finally,we should discuss how to quantify all the inequalities which result from the previously analyzed conceptualizations. Such a quantification can be considered as a special case of spatial inference, which unfortunately we cannot introduce here because of lack of space.</Paragraph>
    <Paragraph position="3"> An attempt to classify inferences can be found in Ill.</Paragraph>
  </Section>
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