## The Interpretation of Fuzziness (1996)

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Venue: | IEEE Transactions on Systems, Man, and Cybernetics |

Citations: | 26 - 13 self |

### BibTeX

@ARTICLE{Wang96theinterpretation,

author = {Charlie C. L. Wang and Terry K. K. Chang and Matthew M. F. Yuen},

title = {The Interpretation of Fuzziness},

journal = {IEEE Transactions on Systems, Man, and Cybernetics},

year = {1996},

volume = {26},

pages = {312--326}

}

### Years of Citing Articles

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### Abstract

From laser-scanned data to feature human model: a system based on

### Citations

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Fuzzy sets
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- 1965
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Citation Context ...ness and randomness can be uniformly treated. 1 Zadeh on Fuzziness Zadeh's idea of "fuzzy set" came from an observation: classes of objects in everyday thinking usually have no well-defined =-=boundary ([29]-=-). More concretely, he made the following claims: 1. For these classes, no two-valued membership function can be defined on instances, and there are always instances that standing on the boundary, suc... |

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Judgment under Uncertainty: Heuristics and Biases
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- 1974
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Citation Context ...al problem solving, multiple types of uncertainty usually co-exist and merge with each other, as shown by the mixing of fuzziness (representativeness) and randomness (probability) in human judgments (=-=[23, 20]-=-). In summary, fuzzy logic is not proposed as a pure formal system that only have some interesting mathematical properties, but as a formal model of fuzziness that happening all the time in human cogn... |

1104 |
Features of similarity
- Tversky
- 1977
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Citation Context ...ined by the degree of similarity between an instance to be judged and a prototype or a known instance, so (at least at the simplest cases) membership measurement is reduced to similarity measurement (=-=[22, 14]). Two kin-=-ds of similarities can be distinguished: those are symmetric and those are asymmetric ([22]). To avoid confusion, I'll define "inheritance relation" as an asymmetric similarity relation, and... |

959 | The concept of a linguistic variable and its application to approximate reasoning
- Zadeh
- 1975
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Citation Context ...le. But, by membership function, we can describe the compatibility between a linguistic label and a numerical value, such as to say the compatibility between the label young and the age 28 to be 0.7 (=-=[32, 34]). 3. Such a compati-=-bility have no frequency interpretation. By "The membership of John's age to `young' is 0.7", we don't mean that John's age is a random number, which takes the value "young" in 70%... |

667 |
Outline of a new approach to the analysis of complex systems and decision processes
- Zadeh
- 1973
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Citation Context ...r example, according to Zadeh, the membership function of red flower can be calculated from the membership functions of red and flower by applying the intersection operation, which is defined as min (=-=[31]-=-). Therefore, a question will be risen naturally, that is, how to determine these functions at the first place, that is, for red and flower? Zadeh suggested two ways to define a membership function ([... |

543 |
Context theory of classification learning
- Medin, Schaffer
- 1978
(Show Context)
Citation Context ...w similar it is to the prototype ([18, 19]). Another explanation, exemplar theory, assume that membership of a novel instance is evaluated by directly comparing it with given members of the category (=-=[12, 14]-=-). Generally speaking, the basic cause of the type 2 fuzziness is: the concept is not defined by sufficient/necessary conditions, but is exemplified by many objects/actions/events, which share common ... |

543 |
Family resemblances: Studies in the internal structure of categories
- Rosch, Mervis
- 1975
(Show Context)
Citation Context ... that people judge some instances to be better examples of a concept than some other instances are, and can answer category membership questions more rapidly for good examples than for poor examples (=-=[18, 19, 15]-=-). Several theories are proposed by psychologists to explain the phenomena. One explanation, prototype theory, suggests that from given members of a category, people abstract out the central tendency ... |

476 |
Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1
- Zadeh
- 1978
(Show Context)
Citation Context ...le. But, by membership function, we can describe the compatibility between a linguistic label and a numerical value, such as to say the compatibility between the label young and the age 28 to be 0.7 (=-=[32, 34]). 3. Such a compati-=-bility have no frequency interpretation. By "The membership of John's age to `young' is 0.7", we don't mean that John's age is a random number, which takes the value "young" in 70%... |

330 |
Learning structural descriptions from examples. Reprinted
- Winston
- 1975
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Citation Context ..., corresponding to the two interpretations of similarity (symmetric and asymmetric). Various measurements of similarity (and inheritance, defined as above) have been suggested for different purposes (=-=[22, 1, 14, 28, 13]). However, as long -=-as they are defined in [0,1] (with 1 for "identical", and 0 for "completely different"), and are functions of weight of evidence, these measurements share the common form w + w + +... |

290 | Interactive multi-resolution modeling on arbitrary meshes
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- 1998
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Citation Context ...od, smoothing the high frequencies, while the major shape will only be slightly degraded. (11)sThe Laplacian operators (7) and (8) can be linearly approximated at each vertex by the umbrella operator =-=[17, 18]-=-. To construct the operator, we have to choose an appropriate parameterization in the vicinity of each vertex. The umbrella-algorithm is derived by choosing a symmetric parameterization [19] ⎛ i i ⎞ π... |

278 | Piecewise smooth surface reconstruction
- Hoppe, DeRose, et al.
- 1994
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Citation Context ...t 2 Data Preparation Human Model Construction Semantic Feature Extraction There is a large quantity of work related with building the geometric model of an object from a cloud of points. Hoppe et al. =-=[4]-=- presented a general method for automatic reconstruction of accurate, concise, piecewisessmooth surface models from scattered range data. However, in their method, it is hard to preserve the topology ... |

243 | Maps: Multiresolution adaptive parameterization of surfaces
- Lee, Sweldens, et al.
- 1998
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Citation Context ... (c) original male (d) result 7.2 Parametric Design Fig. 19 Results of feature human modeling The feature lines and feature points on the human model can be applied to the parametric design algorithm =-=[23, 24]-=- to deform the human model. For example, Fig. 20a shows the original feature human model; then we make it more chubby (shown in Fig. 20b); in the following, we scale the length of its leg and neck to ... |

234 |
Decision-making in a fuzzy environment
- Bellman, Zadeh
- 1970
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Citation Context ...ship rather than the presence of random variables", and "the notion of a fuzzy set is completely nonstatistical in nature", probability theory cannot be applied here. A new theory is ne=-=eded (also see [3]-=-). Based on these intuitions, he defined the concept of fuzzy set, the relations between fuzzy sets (equal and containment), and operations on fuzzy sets (complement, union, and intersection), which b... |

188 |
On the internal structure of perceptual and semantic categories
- Rosch
- 1973
(Show Context)
Citation Context ... that people judge some instances to be better examples of a concept than some other instances are, and can answer category membership questions more rapidly for good examples than for poor examples (=-=[18, 19, 15]-=-). Several theories are proposed by psychologists to explain the phenomena. One explanation, prototype theory, suggests that from given members of a category, people abstract out the central tendency ... |

164 |
A Theory of Approximate Reasoning
- Zadeh
- 1979
(Show Context)
Citation Context ...n objective experimentation or analysis". The task of fuzzy logic is to provide rules to compute the meaning of composite terms, once the meaning of the primary terms is specified in a given cont=-=ext ([30, 35]-=-). As a result, many totally different methods are used to get membership functions when fuzzy logic is applied to practical domains ([5]), which are chosen according to the designer's preference and ... |

124 | Consistent mesh parameterizations
- Praun, Sweldens, et al.
(Show Context)
Citation Context ... (c) original male (d) result 7.2 Parametric Design Fig. 19 Results of feature human modeling The feature lines and feature points on the human model can be applied to the parametric design algorithm =-=[23, 24]-=- to deform the human model. For example, Fig. 20a shows the original feature human model; then we make it more chubby (shown in Fig. 20b); in the following, we scale the length of its leg and neck to ... |

119 |
Two Theories about Adjectives
- Kamp
- 1975
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Citation Context ...bigflea (A) = 0:9, �� flea (A) = 1, and f lea ae animal, we get �� biganimal (A) = 0:9, which is counter-intuitive ("A big flea is a big animal"). In other words, many adjectives are=-= not predicative ([9]). In AI c-=-ommunity, this problem is usually explained by saying "the membership function of `big' (as well as `young', `far', etc.) is context dependent". Of course it is, but why and how? Now let's a... |

117 |
Surfaces from contours
- Meyers, Skinner, et al.
- 1992
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Citation Context .... 12 Contour line generation result of each part Surface construction from contours: In this mesh construction step, we create new polygons by sewing the neighboring copies of contour curves together =-=[14, 15]-=- (Fig. 13). After sewing the adjacent contour curves, new 3D polygonal surfaces of each part of the human model are created (Fig.14). P0 Q0 Q1 P1 Q2 P2 12 Pn-1 Qn-1 Fig. 13 Sew neighboring contours (a... |

104 |
Approximating complex surfaces by triangulation of contour lines
- Keppel
- 1975
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Citation Context .... 12 Contour line generation result of each part Surface construction from contours: In this mesh construction step, we create new polygons by sewing the neighboring copies of contour curves together =-=[14, 15]-=- (Fig. 13). After sewing the adjacent contour curves, new 3D polygonal surfaces of each part of the human model are created (Fig.14). P0 Q0 Q1 P1 Q2 P2 12 Pn-1 Qn-1 Fig. 13 Sew neighboring contours (a... |

101 | Discrete fairing
- Kobbelt
- 1997
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Citation Context ...rator [17, 18]. To construct the operator, we have to choose an appropriate parameterization in the vicinity of each vertex. The umbrella-algorithm is derived by choosing a symmetric parameterization =-=[19]-=- ⎛ i i ⎞ π (12) n n ⎠ ( , v ) = ⎜cos( 2 ), sin( 2π ) ⎟, i = 0, K, n −1 ui i with n being the valence of the center vertex p . ⎝ With the special parameterization (equation (12)), the discrete represen... |

93 |
Fuzzy Sets and Systems
- Dubois, Prade
- 1980
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Citation Context ...ing of the primary terms is specified in a given context ([30, 35]). As a result, many totally different methods are used to get membership functions when fuzzy logic is applied to practical domains (=-=[5]-=-), which are chosen according to the designer's preference and experience. 2 Why to Interpret Fuzziness Do we really need to further interpret the meaning (and origin) of membership values? Yes, we do... |

88 | Towards a unified theory of similarity and recognition
- Ashby, Perrin
- 1988
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Citation Context ...n A. This equation can be generalized to all fuzzy concepts of the type 1 by considering S(y) : C ! (\Gamma1; 1) as a measurement corresponding to the relation "R-er than", and P (x) : (\Gam=-=ma1; 1) ! [0; 1]-=- as the probability distribution of objects in C with respect to S(y). In this way, we get a function that calculates the membership of an object from a fundamental argument, as Zadeh did (see section... |

81 | The paradoxical success of fuzzy logic
- Elkan
- 1994
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Citation Context ...t successful applications of fuzzy logic happen in some control systems ([11]), rather than in natural language processing, knowledge base management, general purpose reasoning, and machine learning (=-=[6]-=-), though in the latter domains fuzziness are more notable and more closely related to the initial idea of fuzzy set. To solve the problem, we need to start by analyzing fuzziness. 3 Fuzziness from Re... |

79 |
The Logical Foundations of Statistical Inference
- Kyburg
- 1974
(Show Context)
Citation Context ...f a system (human or computer). Let's compare this issue with the case of probability. It's well known that there are various interpretations of probability, such as logical, empirical, and personal (=-=[8, 10]). Ho-=-wever, one idea is shared by the community: probability do need an interpretation, and all the operations carried on probabilities should be justified according to it. We cannot simply call it "d... |

74 |
Kinds of probability
- GOOD
- 1959
(Show Context)
Citation Context ...f a system (human or computer). Let's compare this issue with the case of probability. It's well known that there are various interpretations of probability, such as logical, empirical, and personal (=-=[8, 10]). Ho-=-wever, one idea is shared by the community: probability do need an interpretation, and all the operations carried on probabilities should be justified according to it. We cannot simply call it "d... |

55 |
A fuzzy-set-theoretic interpretation of linguistic hedges
- Zadeh
- 1972
(Show Context)
Citation Context ...n objective experimentation or analysis". The task of fuzzy logic is to provide rules to compute the meaning of composite terms, once the meaning of the primary terms is specified in a given cont=-=ext ([30, 35]-=-). As a result, many totally different methods are used to get membership functions when fuzzy logic is applied to practical domains ([5]), which are chosen according to the designer's preference and ... |

43 |
Conceptual combination with prototype concepts
- Osheron, Smith
- 1988
(Show Context)
Citation Context ...eta �� high looks much more reasonable than minf�� long ; ��wide ; �� high g as big's membership function. Some psychological results are also inconsistent with the results predicted b=-=y the min rule ([16, 21]-=-). Though there are some works show that max and min can be deduced from certain axioms ([2, 7]), it is still unclear that whether human cognitions really follows these axioms or why should we follow ... |

42 |
On the analytic formalism of the theory of fuzzy sets
- Bellman, Giertz
- 1973
(Show Context)
Citation Context ...function. Some psychological results are also inconsistent with the results predicted by the min rule ([16, 21]). Though there are some works show that max and min can be deduced from certain axioms (=-=[2, 7]-=-), it is still unclear that whether human cognitions really follows these axioms or why should we follow them. ffl In his later papers ([3, 32]), Zadeh admitted that in some contexts the union/interse... |

42 |
A fuzzy-algorithmic approach to the definition of complex or imprecise concepts
- Zadeh
- 1976
(Show Context)
Citation Context ...t distinguish components of fuzzy theory, are not strongly supported by experimental evidence or theoretical consideration. They sometimes obviously lead to counter-intuitive results. For example, in =-=[33] Zadeh define big = long-=-swideshigh, but �� long \Theta ��wide \Theta �� high looks much more reasonable than minf�� long ; ��wide ; �� high g as big's membership function. Some psychological results a... |

39 | Discrete fairing and variational subdivision for freeform surface design. The Visual Computer
- Kobbelt
- 2000
(Show Context)
Citation Context ...od, smoothing the high frequencies, while the major shape will only be slightly degraded. (11)sThe Laplacian operators (7) and (8) can be linearly approximated at each vertex by the umbrella operator =-=[17, 18]-=-. To construct the operator, we have to choose an appropriate parameterization in the vicinity of each vertex. The umbrella-algorithm is derived by choosing a symmetric parameterization [19] ⎛ i i ⎞ π... |

39 |
Comparing the calibration and coherence of numerical and verbal probability judgments
- Wallsten, Budescu, et al.
- 1993
(Show Context)
Citation Context ...e "John's age is 18", since the mapping from (relative) youngfulness to (absolute) age depends on the reference class C. The concrete mapping function can be established by psychological exp=-=eriments ([17, 24]-=-). Even with these problems in mind, we can still see the possibility to extend the interpretation of fuzziness proposed in this paper to those more complicated situations, so as to provide a frequenc... |

33 | From inheritance relation to non-axiomatic logic
- Wang
- 1994
(Show Context)
Citation Context ... of positive evidence among all relevant evidence, that is, w + w . 2 Based on such an interpretation, I'm building an intelligent reasoning system, Non-Axiomatic reasoning System, or NARS for short (=-=[27, 26]-=-), where fuzziness and randomness are uniformly processed as (part of) a judgment's truth value, and referred as the frequency of the judgment. However, this doesn't mean that fuzziness and randomness... |

24 |
Fuzzy Logic
- McNeil, Freiberger
- 1993
(Show Context)
Citation Context ...n ([31]): 1. By enumeratingly assigning membership values to objects in a domain. For example, the fuzzy concept long-river can be defined in the domain fNile, Hudson, Danube, Rhine, Mississippig as (=-=[11]-=-): long-river = 1=N ile + 0:2=Hudson + 0:7=Danube + 0:4=Rhine + 0:8=Mississippi 2. By being a continuous and differentiable function of a numeral variable. For example, the membership of the fuzzy con... |

23 |
Integration of fuzzy logical information
- Oden
- 1977
(Show Context)
Citation Context ...eta �� high looks much more reasonable than minf�� long ; ��wide ; �� high g as big's membership function. Some psychological results are also inconsistent with the results predicted b=-=y the min rule ([16, 21]-=-). Though there are some works show that max and min can be deduced from certain axioms ([2, 7]), it is still unclear that whether human cognitions really follows these axioms or why should we follow ... |

22 | Typicality in logically defined categories: Exemplar-similarity versus rule instantiation
- Nosofsky
- 1991
(Show Context)
Citation Context ...w similar it is to the prototype ([18, 19]). Another explanation, exemplar theory, assume that membership of a novel instance is evaluated by directly comparing it with given members of the category (=-=[12, 14]-=-). Generally speaking, the basic cause of the type 2 fuzziness is: the concept is not defined by sufficient/necessary conditions, but is exemplified by many objects/actions/events, which share common ... |

21 | Belief revision in probability theory
- Wang
- 1993
(Show Context)
Citation Context ...ation of the probability ([8, 26]). However, the same is also true for fuzziness ([26]), so randomness and fuzziness can still be similarly processed. detailed discussions on this issue, see [26] and =-=[25]-=-. 6 Discussions In this paper, a frequency interpretation of fuzziness is suggested, which has the following advantages: ffl Different types of uncertainty, such as fuzziness, randomness, ignorance, a... |

20 |
Is probability theory sufficient for dealing with uncertainty in ai? a negativ view
- Zadeh
- 1986
(Show Context)
Citation Context ...rguments are: (a) there are fuzziness in human cognition, (b) no frequency interpretation of the fuzziness has been found, and (c) some practical problems has been solved successfully by fuzzy logic (=-=[36, 11]-=-). Without a clear analysis of fuzziness in human cognition, these arguments are not enough for fuzzy logic to be accepted as a general cognitive model ([20]). By an interpretation of fuzziness, I mea... |

19 |
Fuzzy and probability uncertainty logics
- Gaines
- 1978
(Show Context)
Citation Context ...function. Some psychological results are also inconsistent with the results predicted by the min rule ([16, 21]). Though there are some works show that max and min can be deduced from certain axioms (=-=[2, 7]-=-), it is still unclear that whether human cognitions really follows these axioms or why should we follow them. ffl In his later papers ([3, 32]), Zadeh admitted that in some contexts the union/interse... |

17 |
Probabilistic versus fuzzy reasoning
- Cheeseman
- 1986
(Show Context)
Citation Context ...nd may be influenced by new knowledge. For example, "If `Mary is young' is uttered in a kindergarten or in a retirement home situation, the effect on the expected age of Mary will be very differe=-=nt" ([4]-=-). However, without a clear interpretation, there is no reasonable way to modify the memberships by new evidence, so they cannot be self-adjusted or be context-sensitive. On the other hand, it is unim... |

13 | Non-axiomatic reasoning system (version 2.2
- Wang
- 1993
(Show Context)
Citation Context ... of positive evidence among all relevant evidence, that is, w + w . 2 Based on such an interpretation, I'm building an intelligent reasoning system, Non-Axiomatic reasoning System, or NARS for short (=-=[27, 26]-=-), where fuzziness and randomness are uniformly processed as (part of) a judgment's truth value, and referred as the frequency of the judgment. However, this doesn't mean that fuzziness and randomness... |

12 |
Quantitative meanings of verbal probability expressions
- Reagan, Mosteller, et al.
- 1989
(Show Context)
Citation Context ...e "John's age is 18", since the mapping from (relative) youngfulness to (absolute) age depends on the reference class C. The concrete mapping function can be established by psychological exp=-=eriments ([17, 24]-=-). Even with these problems in mind, we can still see the possibility to extend the interpretation of fuzziness proposed in this paper to those more complicated situations, so as to provide a frequenc... |

10 |
Freehand drawing system using a fuzzy logic concept
- Chen, Xie
- 1996
(Show Context)
Citation Context ...zzy logic. It is very important to select membership functions that can reflect this intention correctly and precisely. Another criterion for selecting membership function is computational efficiency =-=[13]-=-. Thus, we choose a triangular membership function (shown in Fig. 3). The membership function considered here is the distance between two points. The function is defined as follows: ⎧ Dis tan ceAve − ... |

9 |
Binary Digital Image Processing: A Discrete Approach
- Marchand-Mailet, Sharaiha
- 1999
(Show Context)
Citation Context ...on is asked to manually delete the noisy points, his intention is to delete the points “far” from the human model. There are a lot of filtering techniques in the binary digital image-processing field =-=[12]-=-; most of them are based on the definition of neighborhood by a discrete distance function. However, when applying these techniques on the 3D point cloud, it is extremely hard to choose a distance fun... |

9 |
Verities of ignorance and the need for well-founded theories
- Smets
- 1999
(Show Context)
Citation Context ...e meaning (and origin) of membership values? Yes, we do. From the standing point of artificial intelligence and cognitive science, at least we have the following reasons to require an interpretation (=-=[5, 20]): ffl Wit-=-hout a clear interpretation, it is hard for a computer system to generate the memberships automatically or to get them either from users or from sensory device. By "hard", I mean some values... |

8 |
Analogy and Similarity in Scientific Reasoning
- Niiniluoto
- 1988
(Show Context)
Citation Context ..., corresponding to the two interpretations of similarity (symmetric and asymmetric). Various measurements of similarity (and inheritance, defined as above) have been suggested for different purposes (=-=[22, 1, 14, 28, 13]). However, as long -=-as they are defined in [0,1] (with 1 for "identical", and 0 for "completely different"), and are functions of weight of evidence, these measurements share the common form w + w + +... |

6 |
B-spline surface local updating with unorganized points. Computer-Aided Design
- Ma, He
- 1998
(Show Context)
Citation Context ...ing. Ko et al. [5] proposed a method to model a human face from a set of points. Their discussion concentrated on the re-organization of the points, facet modeling and tool path generation. Ma and He =-=[6]-=- presented an approach to shape a single B-spline surface with a cloud of points. Their approach mainly focuses on the parameterization of the unorganized points. Barhak and Fischer [7] also presented... |

6 |
A context theory of classi cation learning
- Medin, er
- 1978
(Show Context)
Citation Context ...owsimilar it is to the prototype ([18, 19]). Another explanation, exemplar theory, assume that membership of a novel instance is evaluated by directly comparing it with given members of the category (=-=[12, 14]-=-). Generally speaking, the basic cause of the type 2 fuzziness is: the concept is not de ned by su cient/necessary conditions, but is exempli ed by manyobjects/actions/events, which share common prope... |

5 |
Feature-based reserve engineering of mannequin for garment design
- Au, Yuen
- 1999
(Show Context)
Citation Context ...presented for a specific application domain; in other words, a semantic feature is an application-oriented feature defined on geometric elements. There are two approaches for building a feature model =-=[3]-=-: 1) The design by feature approach creates the feature model of an object by composing the available features in a feature library; 2) The feature recognition approach recognizes various feature from... |

5 |
Fuzziness in semantic memory, Choosing exemplars of subjective categories
- Oden
- 1977
(Show Context)
Citation Context ... that people judge some instances to be better examples of a concept than some other instances are, and can answer category membership questions more rapidly for good examples than for poor examples (=-=[18, 19, 15]-=-). Several theories are proposed by psychologists to explain the phenomena. One explanation, prototype theory, suggests that from given members of a category, people abstract out the central tendency ... |

4 |
Parameterization for reconstruction of 3D freeform objects from laser-scanned data based on a PDE method
- Barhak, Fischer
- 2002
(Show Context)
Citation Context ...ion. Ma and He [6] presented an approach to shape a single B-spline surface with a cloud of points. Their approach mainly focuses on the parameterization of the unorganized points. Barhak and Fischer =-=[7]-=- also presented a PDE based method about the parameterization for reconstruction of 3D freeform objects from laser-scanned data. Sienz et al. [8] developed a fitting technique to generate computationa... |