Zenon Kulpa:
From Picture Processing to Interval Diagrams
IPPT PAN Reports (ISSN 0208-5658), 4/2003, Warsaw 2003
(ix+313 pp., 152+12 figures, 16 tables).

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Excerpts from the Preface:

The sense of sight is undoubtedly the most important sense of human beings (and many animals too) with respect to the quantity and richness of perceived information. Most of the information we receive from the world, including that communicated between human beings, is of visual nature. Hence there should be no doubt about the importance of studying this kind of information, and developing appropriate technical means - in recent times this means mostly computer hardware and software - to aid and facilitate its use.

Taking that into account, it may be surprising that despite the widespread use of visual methods in science and other areas of human activity, a serious scientific study of this representation, communication and reasoning tool has started only recently. Compare that with centuries of studying language and associated disciplines - the difference appears quite striking. Moreover, in certain areas (notably mathematics), the very use of diagrammatic representations has been (and still often is) discouraged, or even attempts were made to expel it completely. Mathematicians wrote books on geometry - undoubtedly, the most diagrammatic field of mathematics - without a single diagram in them, and were proud of that.

Fortunately, the profound importance of visual reasoning methods for the scientific work becomes recently acknowledged by philosophers of science:

Finally, the discipline of diagrammatics is born, and growing numbers of researchers and publications in this area bring hope of filling yet another gap in our knowledge about means of representing and processing complex information in humans and machines. This also means the development of new tools and methods that will boost effectiveness of knowledge and information processing, necessary to meet the challenges put forward by the emerging Information Age. This work adds another segment to that broad field of study, by discussing some basic issues of diagrammatics in a novel way, and by developing a new diagrammatic notation for interval algebra and computation.

Aims of the work. The main aim of the work is the development and presentation of the novel diagrammatic notation for interval algebra and computation developed by the author, and showing its usefulness in some areas of interval algebra. Additionally, as a background for that undertaking, a state of the art survey and partially novel systematization of basic issues of diagrammatics is attempted, including a unified framework for relating various subfields and aspects of the pictorial information handling domain.

  I.3 Discrete picture processing  14
    I.3.1 Discrete pictures  15
    I.3.2 Picture processing operations  16
    I.3.3 Number-valued pictures  18
    I.3.4 Operations on number-valued pictures  19
    I.3.5 Operations on binary pictures  21
    I.3.6 Mathematical morphology  22
    I.3.7 Sequential picture operations  25

  I.4 Discrete image analysis  29
    I.4.1 Picture segmentation  29
    I.4.2 Area measurement  32
    I.4.3 Perimeter measurement  34

  I.5 Scene interpretation and understanding  39
    I.5.1 Monocular depth perception  40
    I.5.2 "Impossible figures": errors of spatial interpretation  42
    I.5.3 Impossibility sources  46

I.6 Diagrammatics 49

Chapter II. Diagrammatics: an introduction 51

II.1 Knowledge representation 52
II.1.1 Analogical versus propositional representations 52

    II.1.2 Logical representation  56
      II.1.2.1 Reasoning with logical representation  60
      II.1.2.2 Problems with logical representation  62
      II.1.2.3 Perceptual rules  66
      II.1.2.4 Only logical framework?  67

II.1.3 Diagrammatic representation 70
II.1.4 The field of diagrammatics 73

  II.2 Visual languages  75
    II.2.1 Visual vocabulary and syntax  77
               Sample page 78  [PDF, 78 KB].
    II.2.2 Expressiveness of visual languages  80
    II.2.3 Pragmatic criteria  82

II.3 Diagrammatic representations 85

    II.3.1 Advantages of diagrammatic representations  86
      II.3.1.1 Effective visual apparatus  87
      II.3.1.2 Spatiality of diagrams  87
      II.3.1.3 Analogicity of representation  89
      II.3.1.4 Getting rid of reference labels  91
      II.3.1.5 Exploitation of symmetries  92

    II.3.2 Problems with diagrammatic representations  92
      II.3.2.1 Imprecision of diagrams  93
      II.3.2.2 Incomplete information and disjunctive knowledge  97
      II.3.2.3 Particularity  100
      II.3.2.4 Accidental alignments and general position  103
      II.3.2.5 Specificity and negation by omission  105

    II.3.3 Diagram application modes  106
      II.3.3.1 Information representation (recording)  107
      II.3.3.2 Information processing (reasoning)  107

II.4 Diagrammatic reasoning 109

    II.4.1 Quantitative and qualitative reasoning  112
      II.4.1.1 Metric reasoning  112
      II.4.1.2 Structural reasoning  115
      II.4.1.3 Discrete token counting  116

    II.4.2 Emergence  118
      II.4.2.1 False emergence  121
      II.4.2.2 Unreliable emergence  122

    II.4.3 Divergence  124
      II.4.3.1 Overlooked divergence  126
      II.4.3.2 False divergence  128

II.5 Diagrams in mathematics 132

    II.5.1 Are diagrams difficult?  133
      II.5.1.1 Individual abilities answer  134
      II.5.1.2 Skill training answer  134
      II.5.1.3 Pictorial effector answer  135

    II.5.2 Are diagrams unreliable?  135
      II.5.2.1 Are formulae reliable?  137
                    Sample pages 136-138  [PDF, 92 KB].

II.5.3 Are diagrams intrinsically informal? 138
II.5.3.1 "Proofs without words" 141

    II.5.4 Visual languages of mathematics  142
      II.5.4.1 A simple style  143
      II.5.4.2 A standard textbook style  144
      II.5.4.3 A pure diagrammatic style  145
      II.5.4.4 A hybrid diagrammatic style  146
      II.5.4.5 Dynamic styles  146

II.6 Computer implementation of diagrams 151
II.6.1 Diagram input 151

    II.6.2 Internal diagram representation  152
      II.6.2.1 Diagrams on a raster  153
      II.6.2.2 Diagrams as graphs  155

II.6.3 Diagram output 156
II.6.4 Diagrammatic spreadsheet concept 157

Chapter III. Diagrammatic interval algebra 161

III.1 Interval algebra and computation 162

    III.1.1 Calculating with intervals  164
      III.1.1.1 Interval vectors and matrices  165
      III.1.1.2 Nonstandard properties of interval arithmetic 165
      III.1.1.3 Interval enclosures  166
      III.1.1.4 Overestimation  168

III.1.2 Applications of interval computation 169
III.1.3 Diagrams for interval algebra 171

  III.2 Interval space diagrams  173
    III.2.1 The E-diagram and other proposals  173
    III.2.2 The MR-diagram  174
                 Sample page 175  [PDF, 106 KB].

    III.2.3 Basic uses of the MR-diagram  176
      III.2.3.1 Interval types  177
      III.2.3.2 Extent functions  177
      III.2.3.3 Interval lattices and lozenges  180

  III.3 Interval relations  183
    III.3.1 Arrangement interval relations  184
                 Sample page 185  [PDF, 96 KB].
                 A set of LaTEX commands for relation symbols  [3 KB].
    III.3.2 The W-diagram and L-diagram  186
                 Sample page 186  [PDF, 55 KB].

    III.3.3 Convex interval relations  189
      III.3.3.1 Convexity of interval sets and relations  189
      III.3.3.2 The convex relations characterization theorem  190

    III.3.4 Pointisable interval relations  196
      III.3.4.1 Full-line relations  196
      III.3.4.2 The pointisable relations characterization theorem  196

III.3.5 Non-arrangement interval relations 201

III.4 Interval arithmetic 203

    III.4.1 Interval addition, negation and subtraction  203
      III.4.1.1 Addition of intervals  204
      III.4.1.2 Negation and subtraction of intervals  205
      III.4.1.3 The a+x = b equation  207

    III.4.2 Interval multiplication  208
      III.4.2.1 Multiplication of an interval by a number  208
      III.4.2.2 Multiplication of intervals  209
      III.4.2.3 The a*x = b equation  213

    III.4.3 Interval inverse and division  216
      III.4.3.1 Inverse of an interval  216
      III.4.3.2 Division of intervals  218

III.4.4 Kaucher arithmetic (directed intervals) 220
III.4.5 Kahan arithmetic (extervals) 221

III.5 Interval linear equations 222
III.5.1 Linear equations or relational expressions? 222

    III.5.2 The one-dimensional relational expression  223
      III.5.2.1 Solving the relation diagrammatically  224
                 Sample page 225  [PDF, 126 KB].
      III.5.2.2 Quotient sequences  226
      III.5.2.3 Basic solution types  228
      III.5.2.4 Other characterizations of solution sets  230
      III.5.2.5 The MR-diagram representation and intermediate types  233
      III.5.2.6 RR-diagrams and graphs of types  235
      III.5.2.7 Type changes from coefficient change  237

    III.5.3 The two-dimensional relational expression  239
      III.5.3.1 Boundary lines  240
      III.5.3.2 One-dimensional cuts  241
      III.5.3.3 Boundary lines selection rule  245
      III.5.3.4 Structure of solution sets  246
      III.5.3.5 Solution types in two dimensions  253
      III.5.3.6 Enumeration of two-dimensional types  253
                 Sample page 258  [PDF, 46 KB].
      III.5.3.7 Intermediate cases  259

III.5.4 Generalization to n dimensions 261

    III.5.5 Avenues for further research  264
      III.5.5.1 Systems of relations  264
      III.5.5.2 Rohn's Ayz matrices  264
      III.5.5.3 Directed (modal) intervals and generalized solution sets  264

Summary  267
Bibliography: author's publications  271
Bibliography: other publications  281
Appendix: English-Polish dictionary of basic terms  297

Sample pages:
(PDF files)

Pages 5-6 [91 KB]: Three-level conceptual model of a picture information system.
Page 78 [78 KB]: Basic graphical elements of visual languages.
Pages 136-138 [92 KB]: Unreliability of mathematical diagrams (and formulas).
Page 175 [106 KB]: The basic midpoint-radius diagram of interval space.
Page 185 [96 KB]: The table of basic interval relations with conjunction diagrams and new graphical symbols for the relations. [Plain ASCII, 3 KB] A set of LaTEX commands for producing all new symbols.
Page 186 [55 KB]: The basic W-diagram of interval arrangement relations space.
Page 225 [126 KB]: Five basic types of diagrammatic solutions to the a x = b equation.
Page 258 [46 KB]: Part of the diagrammatic vocabulary of solution types of the 2-dimensional a1 x1 + a2 x2 = b equation.

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The work can be ordered, until stock lasts, directly from the publishers (Institute of Fundamental Technological Research) by e-mailing the order (with the postal address included) to Mrs. Elzbieta Zadruzna at:
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