satisfaction. Our tests show that even a naïve implementationof this is 20%–30% better at labelling drawings ofnon-trihedral objects than previous methods.The purpose of this paper is twofold: to demonstratea deficiency in existing methods, and suggest an alternative.Sections 2 and 3 consider various approaches to theproblems of line-labelling and production of a preliminaryfrontal geometry. Sections 4 and 5 explain what K-verticesare, and examine why existing methods are inappropriatefor drawings containing K-vertices. Sections 6, 7 and 8 outlineour new approach to line-labelling and inflation, give apractical example, and give test results. Section 9 gives ourconclusions and plans for further work.2 Line Labelling ReviewWhen analysing a line drawing, we assume that the 2Dcoordinates x j , y j of each junction j are known (from theoriginal drawing—they may or may not be accurate), as arethe junction pairs joined by each line, and the loops of junctionsand lines forming each region.Line-labelling labels all lines in the drawing as convex,concave or occluding; it is a well-known, often used initialstage in the interpretation of such drawings. The standardHuffman [3] and Clowes [1] approach has the advantagesthat it requires as input no information other than the above,and that it produces as output useful information about boththe frontal geometry and the topology of the hidden part ofthe object. The non-silhouette lines of Figure 2 have beenlabelled in accordance with the Clowes-Huffman conventionas convex (+), concave (-) or occluding (arrow).Many implementations of Clowes-Huffman line labelling(e.g. [5]), use catalogue labelling. The cataloguecontains all possible junction labels, reducing line labellingto a discrete constraint satisfaction problem: alljunctions must satisfy a 1-node constraint—the label appearsin the catalogue, and all lines must satisfy a 2-nodeconstraint—the line has the same label at both ends.However, even for trihedral objects described by theClowes-Huffman catalogue, line-labelling is not truly a 1-node and 2-node constraint problem. In order to label objectscorrectly, one must also satisfy non-local (and perhapsvaguer) geometric constraints—the object must be realisable,and psychological ones—that the object must be theone the user intended. However, in the trihedral world, theadvantages of catalogue-labelling (it usually runs in O(n)time [11]), mean that occasional failures are tolerated.In the non-trihedral world, the advantages are lost butthe disadvantages remain. For example, the catalogue oftetrahedral vertices for polyhedra [15] is not sparse, so catalogue-basedlabelling is often too slow [18] to be useful.Use of the tetrahedral vertex catalogue also results inmany more possible solutions to the constraint satisfactionproblem, and choosing a good solution is difficult [18]: oftenlabellings which are valid solutions to the constraint satisfactionproblem may not be realisable geometrically.For trihedral drawings, Sugihara [14] suggested that labellingshould be used to obtain a reasonably small numberof candidate interpretations, each corresponding to a legallabelling, whose geometric realisability could then bedetermined by subsequent stages of processing. In a previouspaper [18], we explained why this idea is inappropriatefor general non-trihedral drawings. In reconsidering theidea for drawings where the only permitted non-trihedralvertices are K-vertices, we note that (i) for more than halfof the test drawings in this paper, the number of legal labellingsis no greater than five, but (ii) Figures 18 and 29have 1177 legal labellings each, Figures 15 and 19 1208each, and Figure 30 has 10398. Processing so many candidateinterpretations is impractical. The search and selectionprocesses must be aided by heuristics based on psychologicalplausibility [18].As well as geometric considerations which already existin the trihedral world (e.g. if a pocket is as deep asthe surrounding material, it should be a through hole, nota pocket), there are geometric considerations unique to K-vertices (see Section 4). For objects with K-vertices, reducingthe labelling problem to one of simple 1-node and2-node constraints is inappropriate.Other approaches exist for labelling, even for trihedralobjects (see [16] for a summary). However, they continueto treat the problem as one of local constraints, and are alsoinappropriate for labelling drawings with K-vertices.2.1 Relaxation MethodsPreviously [18], we gave a “relaxation algorithm” approachto solving the 1-node and 2-node discrete constraintsatisfaction problem by probabilistic rather than deterministicmeans. This approach is used as a benchmark againstwhich we test the new idea described in Section 6.Relaxation has two advantages over e.g. Kanatani’s algorithm[5]. Firstly, it is considerably faster when usingthe non-trihedral junction catalogue. Secondly, althoughless reliable overall for labelling than other methods, it ismore reliable for drawings with single K-vertices [16]. Thisseems to be because relaxation methods gradually accumulatedata from other parts of the drawing beyond the immediate1- and 2-node neighbourhoods, whereas heuristicbasedselection methods have no heuristics to use (one K-vertex is as good as another) and are effectively selecting atrandom.2
SumárioApresentação........................................................ 7Acesso à educação....................................... 11Salas de Recursos Multifuncionais.............................13Escola Acessível............................................................... 15Transporte Escolar Acessível – ProgramaCaminho da Escola.......................................................... 18Pronatec.............................................................................. 21Acessibilidade na Educação Superior– Incluir.................................................................................25Educação Bilíngue...........................................................27BPC na Escola...................................................................29Inclusão social.................................................37BPC Trabalho.....................................................................39Residências Inclusivas................................................... 42Centro-Dia de Referência parapessoas com deficiência.............................................. 44
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