Fuzzy B-Splines: A Surface Model Encapsulating Uncertainty

Graphical Models62,40–55 (2000)

doi:10.1006/gmod.1999.0512, available online at http://www.idealibrary.com on

Fuzzy B-Splines: A Surface ModelEncapsulating Uncertainty

Giovanni Gallo

Department of Mathematics, University of Catania, Viale Andrea Doria, 6-95125 Catania, Italy

Michela Spagnuolo

Institute for Applied Mathematics, CNR, Via De Marini, 6-16149 Genoa, Italy

and

Salvatore Spinello

Department of Mathematics, University of Catania, Viale Andrea Doria 6-95125 Catania, Italy

Received December 9, 1998; revised August 20, 1999; accepted October 13, 1999

In the context of surface modeling, fuzzy B-splines are proposed as an integratedapproach to uncertainty coding and data reduction. Fuzzy B-splines are suitable forrepresenting and simplifying both crisp and imprecise surface data and support inter-rogation of the model at different presumption levels. A high degree of compressioncan be achieved through a procedure that defines the most significant representa-tive among spatially clustered points. Experimental results are shown to prove theeffectiveness of the proposed approach.c© 2000 Academic Press

1. INTRODUCTION

One of the primary roles of mathematical models is to set a correspondence between real-world phenomena and formalized objects and functions describing their behavior. Generally,for a given context, one may choose among different reasonable models depending on thedata properties and model’s purpose. Mathematical models always define anidealizationof the underlying phenomenon; that is, a model only defines a formal framework useful todescribe and analyze the phenomenon. The problem becomes even more crucial in dealingwith the representation of natural objects, since the process needed to measure data obviouslyintroduces a further discrepancy between model and reality.

For these reasons, users should be aware that models and real objects may differ fordifferent kinds of uncertainty or imprecision: uncertainty deriving from the use of a specific

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FUZZY B-SPLINES 41

mathematical model of the phenomenon itself, imprecision affecting the data due to acqui-sition processes, and imprecision deriving from the embedding of the mathematical modelinto the digital environment.

As pointed out in [1], the presentation of uncertainty has been often separated from thepresentation of the model itself. Even if restricting the scope of the discussion to surfacemodeling, which is the topic of this paper, we might say that traditional geometric approachesdo not support uncertainty coding or visualization. Usually, the quality of a model may begiven in terms of numerical accuracy, such as upper bounds to error estimates, but usershave no tools for understanding or managing the uncertainty inherent in the specific model,especially when doing analysis or making decisions.

Our approach to the problem is based on the integration of classical geometric techniques(B-spline approximation) with mathematical methods for uncertainty formalization (fuzzyset theory) [2]. The resulting fuzzy B-splines (FBS) give users a framework for looking atthe surface model at different levels of reliability and tuning the model according to thedegree required by the specific application.

More precisely, the proposed modeling approach takes into account the imprecision,which might affect the surface samples, and produces a reduced data set, based on “fuzzysummaries” of the original data. The fuzzy B-spline, which approximates the reduced dataset, supports evaluation and interrogation of the model at different presumption levels, i.e.,different hypotheses of model reliability.

There exists at least one notable example in the literature that addresses the problem ofrepresenting and handling the uncertainty using a smooth surface reconstruction approach.The method is based on the ideaenvelopingor interval explicitB-splines (IBS) [3, 4] whichhas inspired the approach presented in this paper and which will be described in the nextsection. It is worth pointing out that both methods achieve a very interesting result: datareduction with guaranteed bounded approximation errors.

The remainder of the paper is organized as follows: first, the IBS method will be described;then, a brief description of the main concepts related to fuzzy numbers and arithmetic willbe given, to introduce the theoretical background of the work. In the fourth section the datareduction process will be described, while the idea of fuzzy B-splines will be defined in thefifth section. Examples and discussion will be given and, finally, some concluding remarksand future developments will be outlined.

2. THE INTERVAL-EXPLICIT B-SPLINES METHOD

Besides the loss of information due to the discretization procedure itself, there usuallyexists an imprecision associated to the samples, which can sometimes be quantified andis generally caused by the imprecision of the measurement instrument. When the data aredescribed as a set of samples on a bivariate functionz= F(x, y), uncertainty may occur inthe (x, y) as a positioning error and also in thez as a proper measurement error. In somecases the error may also depend on environmental conditions, for instance, when gatheringdata of the sea floor with echo-sounding equipment in critical areas [5]. Once the geometricmodel is constructed, the user has no tools for understanding which is the quality of themodel used and at which extent the results of spatial analysis can be considered reliable.

Interval-based methods are powerful and simple approaches to the formalization of vari-ability ranges of real data. The basic idea is to represent an imprecise datum by the minimumand maximum measured values, that is, by the smallest interval enclosing the measured

42 GALLO, SPAGNUOLO, AND SPINELLO

FIG. 1. Samples are reduced to cellular data.

quantity. Interval arithmetic then offers a sound mathematical background to perform op-erations on interval data.

The interval-based approach is the core of the method defined within the general contextof underwater environment modeling at the MIT Department of Ocean Engineering [3, 4].Here, a brief description will be given of the overall modeling process, leaving details onthe mathematical aspects of this specific model to the interested reader.

The main points of the method are the reduction of the original data to a grid of intervaldata and the definition of the approximating surface as a combination of standard basisB-splines with interval coefficients (interval B-splines).

The first step of the reduction process is aimed at sorting the measured values into auniformly distributed grid. Based on this regular sorting, cellular data are derived, whichare characterized (in thez direction) by minimum and maximum measured values, asdepicted in Fig. 1.

Whenever one or more empty cells occur, then a linear interpolation is used based ona Delaunay triangulation, to derive measurements from nonempty cells. Empty cells mayexist both due to the selected cell size and due to the distribution of the data. The derivedcellular data are used to define piecewise bilinear bounding surfaces, which are used as asupport to build the final B-spline model since they make possible a discretized constraintfor the surface fitting method.

In accordance with the classical definition, an interval B-spline is defined as

FM,N(u, v) =m−1∑i=0

n−1∑j=0

Fi, j ∗ Bi,M (u) ∗ Bj,N(v),

where each of the control points,Fi, j , is represented by an interval instead of a unique realnumber.

Upper and lower bounding surfaces, in the form of interval uniform biquadratic B-splines,are then fitted by minimizing the difference between the bounding B-spline and the corre-sponding bilinear surface with the constraint that the upper (resp. lower) B-spline surfacemust lie completely above (resp. below) the upper (resp. lower) bilinear surface (see Fig. 2).

The minimization problem with linear constraints can be solved using techniques forlinear programming.

FUZZY B-SPLINES 43

FIG. 2. The IBS approach.

The resolution of the resulting surface can be controlled by the selection of the grid size:for a smaller size more resolution will be achieved at the expense of increased storage. Asthe authors suggest in [3] a more complicated question is the accuracy. Typical methodsfor surface reconstruction minimize the position error, for instance, using the least-squarestechnique. Compared with this method, the IBS reconstruction might show a higher max-imum position error. The problem in this case depends mainly on the constraint imposedto the upper enveloping IBS to be completely above the data set, and for the lower to becompletely below, since in this manner the resulting surface may suffer from the presenceof highly fluctuating data. The least-squares approximation, in contrast, allows the resultingsurface to pass on either side of the data. One solution to this effect, however, is simply tofilter data before reconstructing the surface. The IBS model appears to be extremely usefulfor applications in which knowledge of the error positioning is crucial, as in the case ofunderwater vehicle navigation.

3. THE FUZZY APPROACH

Coding imprecise values using intervals is a satisfactory choice whenever there is noadditional knowledge available on data behavior and reliability. Intervals do not give anyinformation, indeed, about the “degree of membership” in the sense that every value withinthe interval is possible. In other words, in the absence of any information, coding imprecisevalues as intervals corresponds to assuming a uniform probability distribution over theinterval, that is, a constant probability.

Fuzzy numbers constitute a very interesting alternative to intervals. Suppose indeedthat measurements are judged by an expert, whose knowledge allows us to assign a higherpresumption to certain values or a lower presumption to rare extreme values. This additionalknowledge may be perfectly fit within the concept offuzzy number, which integrates thesimple intervals with the idea of presumption level.

As stated in [6], a fuzzy number is defined by making use of the interval analysis andby relating two important concepts: confidence interval and presumption levels. As before,a confidence interval is an interval of real numbers that provides a representation for animprecise numerical value by means of its sharpest enclosing range. A presumption levelis an estimated truth value for some knowledge. Presumption levels belong to the [0, 1]interval: the maximum estimated truth is at level 1, while the minimum is at level 0. Thisleads naturally to the following [6]:

DEFINITION 1. A fuzzy numberF is an ordered set of confidence intervals, A, eachof them providing the related numerical value at a given presumption levelα ∈ [0, 1]. Theconfidence intervalsA should comply with the following relation (sometimes called the“convexity property”): ifα1<α2, thenAα1 ⊇ Aα2 (see Fig. 3).


FIG. 3. Fuzzy numbers and two presumption levels.

Following an alternative approach [6, 7] a fuzzy number may also be defined as a fuzzysubset of the real line:

DEFINITION 2. A fuzzy real numberF is a closed real intervalX= [a, b] together witha piecewise continuous functionm(t): X→ [0, 1], such that, for every pairα1 andα2 in[0, 1], wheneverα1<α2, thenX⊇ Aα1⊇ Aα2, with Aα j ={t ∈ X: m(t)>α j }.

Thus, fuzzy numbers have a peak or a plateau where the membership function takes value1 (elements that are completely inside the subset) and the membership function increasestoward the peak and decreases away from it. The membership function may assume differentshapes, such as triangular or trapezoidal forms or Gaussian-like curves. Triangular fuzzynumbers are a typical and simple example of fuzzy numbers and are defined by specifyingthree real values: the support interval extrema and the internal value corresponding to thepresumption levelα= 1 (see Fig. 4).

As observed in [7] the definition of a fuzzy number “follows the natural, often implicit,mechanism of human thinking in the subjective estimation of a numerical value whenreasoning in one dimension.”

A fuzzy number presents a resemblance to a probability distribution, but the two ideasare distinct: there is no condition on the integral ofm(t) and this function only describesan estimate of the presumption level attached to the points in the intervalX. The differencebetween fuzzy numbers and probability distributions is crucial in making attractive theuse of fuzzy numbers for modeling large collections of noisy data. Although the estimatesprovided are just presumption levels and not rigorous probabilities, this information is oftenperfectly adequate to most of the users’ requirements, especially when dealing with naturalsurfaces.

FIG. 4. An example of triangular fuzzy numbers.

FUZZY B-SPLINES 45

FIG. 5. An example of sum operator on triangular fuzzy numbers.

It might be desirable, of course, to have a rigorous stochastic model for these data, but thisobjective is completely unattainable in most of the cases: either the probability distributionis unknown or, even when a simple stochastic model is available, the complexity requiredto interrogate and maintain such a model is prohibitively high. The fuzzy modeling of largecollections of noisy data described in this paper, moreover, achieves a very high degree ofcompression with a relatively low computational complexity both for maintenance and forinterrogation of the model itself.

Using the extension principle [8] it is possible, in a quite standard way, to introduce aset of arithmetic operations over the set of fuzzy numbers. This principle demonstrates thatfuzzy set theory is a generalization of classical set theory and allows us to define arithmeticoperators on fuzzy numbers in terms of interval arithmetic (see Fig. 5). First of all, the usualarithmetic operators are naturally extended to real intervals as follows [9]:

DEfiNITION 3. Given two intervals of the real line [a, b] and [c, d] and a binary contin-uous arithmetic operator⊗ over pairs of real numbers, the interval [a, b]⊗ [c, d] is definedas the smallest interval that contains all the possible values computable by⊗ on itemsbelonging to [a, b] and [c, d].

It is now simple to generalize this operator to fuzzy numbers:

DEFINITION 4. Let F1={Aα}α∈[0,1] andF2={Bα}α∈[0,1] be two fuzzy numbers. Let⊗be a binary arithmetic over pairs of intervals. The fuzzy numberF1⊗ F2 is the fuzzy numberF ={Cα}α∈[0,1], where, for everyα; Cα = Aα ⊗ Bα.

Definition 4 does not lead by itself to an exact simple algorithm for computing arithmeticcombinations of fuzzy numbers. In principle, one should compute an infinite number ofintervals because the collection of intervals{Cα}α∈[0,1] defining a fuzzy number is infinite.For most practical purposes, however, the knowledge of finite (generally small) numbersof Cα is more than adequate and one can reduce the computation over fuzzy numbers to afinite set of interval operations.

Based on these concepts, two programming libraries for fuzzy computation have beendeveloped to solve several numerical problems. The first of these two libraries provides high-precision computational capabilities and demonstrates the feasibility and correctness of apractical use of fuzzy arithmetic. The second one is a powerful, friendly, and highly portabletool for supporting generic users needing to manage imprecise data [7, 10]. Moreover,to cope with increasing computational complexity when using large data sets, a parallelimplementation of several functions has been tested both on a network-based parallel-computing environment and on the CRAY-T3D [11].


4. DATA REDUCTION BASED ON FUZZY NUMBERS

Geophysical data are spatially varying, require large amounts of computer memory forstorage, and quite often are obtained from sensors (e.g., sonar) which introduce uncertainties.

Data reduction aims at the recognition of an appropriate subset of measured data, whichmaintains the same informational content as the original set. The simplification of the dataputs into evidence the relevant information contained in the original set and, at the sametime, discards the irrelevant details. In this phase, moreover, the unavoidable approximationerrors should be kept below a prefixed level. Thus, the purpose of the data reduction processis twofold: on one hand it provides a reduction of the model complexity, and on the otherhand it supports model construction based on knowledge of the object augmented withrespect to the raw original data.

Our approach to data simplification follows the procedural scheme used for the IBSconstruction; that is, data are first reduced to a set of regularly spaced “cellular” data, butcellular data are defined as fuzzy numbers.

First of all, each measured data set is coded as a triangular fuzzy number whose support(α= 0) is defined by the minimum and maximum measured values, plus a “privileged”central value which corresponds to the most reliable value for that measure; that is, itcorresponds to theα= 1 presumption level. This choice has been suggested by the firstapplication context of the proposed method, which deals with bathymetric measures of thesea floor. However, this choice is not restrictive and crisp measures, if any, can be easilycoded in the same manner.

The initial fuzzy data now have to be sorted into a regular grid of cells, whose dimensiondetermines the compression rate and resulting resolution of the model. After the fuzzy dataare sorted, a suitable fuzzy representative is chosen for all the data falling within eachcell. Keeping in mind the intuitive notion of fuzzy number, we can naturally define thefuzzy representative for the presumption levelα= 0 as the minimum interval enclosingall theα= 0 levels of the fuzzy data. This choice corresponds to using the minimum ofadditional information available, and in this sense it corresponds exactly to the IBS reductionstep.

While the choice of the representative’s value for theα= 0 presumption level is quitenatural, the definition of the membership function for the other levels is more critical. Thisdefinition, indeed, is the core of our approach and is the most substantial difference fromthe IBS method.

The simplest approach which can be considered is to define the fuzzy representative asa triangular number whose support is the range of the data in the cell and whose vertex isthe average value of theα= 1 fuzzy data (see Fig. 6b).

Another approach, which has been considered, is the construction of a prototypical ele-ment for each cell minimizing a suitable objective function derived from Yager’s mountainfunction [12]. If the details are left to the cited paper, which fully describes the topic, themembership function defining the representative will have the following intuitive meaning:narrow (more reliable) data intervals will influence the membership more than wider (lessreliable) data intervals. This choice yields membership functions with polygonal shapes asin Fig. 6c.

The choice of the most appropriate membership function, however, is a general problemthat is strongly related to the theoretical foundation of fuzzy clustering [12–14]. Fuzzy lit-erature reports many techniques for obtaining such functions. Methods used to designa membership function range from subjective evaluations to elicitation procedures, or

FUZZY B-SPLINES 47

FIG. 6. Different choices for the membership function.

statistical-like approaches as well as physically based measurements. In our context thetwo simple approaches used may be considered to be statistical type. Even so, elicitationprocesses, which take into account the experts’ knowledge, are probably the best ways toexploit the descriptive power of fuzzy numbers. In this case, indeed, a knowledge-baseddesign can be used to define a membership function in an ad hoc form specifically intendedfor a given context. Another important point regards the location of the fuzzy representativewithin each cell. While this choice is unimportant for the construction of the prototypeitself, it turns out to be crucial for the construction of the approximating surface. For thisreason the solution adopted is to locate the fuzzy representative in the center of mass of thedata within the cell.

5. FUZZY B-SPLINES

The fuzzy B-splines may now be naturally introduced extending the traditional definitionof B-splines (see [15, 16]) as follows:

DEFINITION 4. A fuzzy B-splineF(t) relative to the crisp knot sequence (t0, t1, . . . , tn),m= k+ 2(h− 1) is a functionF from the real line to the set of real fuzzy numbers

F(t) =k+h−1∑

i=0

Fi Bi,h(t),

where theFi , the control coefficients, are fuzzy numbers and theBi,h(t) are the crisp B-splinebasis functions of orderh.

First of all, we may observe that Definition 4 is consistent with the definition of “fuzzynumber.” In other words, for everyt, F(t) is a genuine fuzzy number; i.e., it verifies theconvexity property: the presumption levelF(t)α contains the presumption levelF(t)β if


FIG. 7. An example of FBS: Several levels are depicted using different and corresponding grey levels.

α <β. This is guaranteed by the convex hull property of the B-spline together with thepositiveness of the B-spline basis functions: the “strip” obtained choosing a presumptionlevelβ <α is granted to contain the “strip” obtained forα.

In Fig. 7, an example of differentα-levels of a fuzzy B-spline in the two-dimensionalcase is depicted. The generalization to higher dimensions is completely straightforward.

Given a finite set of fuzzy dataS, representing the original set reduced to a regular grid offuzzy numbers, finding an approximating fuzzy B-spline means determining a set of fuzzycontrol points that optimize the approximation process with respect to some constraints.

Since computation on fuzzy numbers are usually performed considering a finite set ofα-levels, the optimization process nicely reduces to an interval arithmetic problem, havingfixed for the FBS approximation, an arbitrary finite number of presumption levelsα1,

α2, . . . , αp. As in the IBS approach, the derived linear optimization problem can hence besolved by means of standard linear programming tools. There are two strategies to this aim,which are briefly described below.

A first simple way of computing a finite number of presumption levels of the fuzzyB-spline is to start from the interval spline of minimal area/volume that approximatesthe intervals relative to the highest required presumption. Successively the interval splinesrelative to lower values are computed providing that:

• they include the data at the prescribed presumption level;• they contain the interval splines previously computed;• they have minimal area/volume.

This approach reduces the computation of an approximating fuzzy B-spline to a sequenceof linear optimization problems. The form of these problems is completely similar to thelinear optimization problem analyzed in [3]. Alternatively, one could set up only one largelinear programming problem requiring, simultaneously for all the computed splines, that:

• they include the data at the prescribed presumption level;

FUZZY B-SPLINES 49

FIG. 8. The original data set for Mount Etna.

• they respect all the inclusion relations among the presumption levels of all the controlpoints in order to guarantee the convexity property of the fuzzy numbers;• they have minimal area/volume.

The first approach is less expensive in terms of computer resources, and for this reason wehave chosen this strategy in order to be able to process larger data sets. This choice, moreover,guarantees a time computation complexity that is linear in the number of presumption levels.If the number of data points to approximate isN and the required number of presumptionlevels is p, then the first strategy requires the solution of a sequence ofp optimizationproblems, each involving 2N linear parameters with 4N linear constraints and one objectivefunction. On the other hand, the second approach requires the solution of an optimizationproblem relative top objective functions simultaneously and with 2N linear parametersand 2N linear constraints.

6. RESULTS AND DISCUSSION

In this section results are given of the FBS construction over two geographical data sets.The first example refers to data collected on Mount Etna, near Catania in Italy, kindly

provided by the CNR. Data have been digitized from a contour map at scale 1:50,000. The


FIG. 9. The FBS constructed using a large cell size.

original data set contained 72891 points, i.e., triplets of coordinates, covering an area ofapproximately 6000× 5400 m (see Fig. 8).

This data set has been reduced using two different resolutions, given by the size of thegrid’s cell. The number of cells in the grid, moreover, can be considered to be the numberof points to be coded to store the resulting FBS model. In other words, the number of cellsis related to the obtained compression rate.

In the first example, a very high reduction was made, using a grid of 11× 10 cells, whoseapproximate size was 545× 540 meters. On the average, the number of data contained in agrid cell is 662.

The resulting FBS is shown in Fig. 9: in Figs. 9a and 9b the layers corresponding totheα-levels 0, 0.5, and 1 are depicted by different points of view, using colors to highlighteachα-level; theα-level 1 is shown in Fig. 9c. In this case, the data reduction is probablytoo strong and the reconstructed surface reproduces the original surface’s morphology at ageneralization level which is not acceptable.

For the second test a cell size of 171× 174 meters was used for the same data set, whichmeans a grid of 35× 31 cells. The resulting FBS is depicted in Fig. 10, with the samestructuring as in Fig. 9. It can be seen that the surface morphology starts to be delineated ata good level of generalization.

The second data set refers to an area in the neighborhood of Genoa, called Val Vobbia,and contains 7788 points digitized from a contour map (see Fig. 11). The FBS resulting

FUZZY B-SPLINES 51

FIG. 10. The FBS constructed using a smaller cell size.

from the application of two different grids are shown in Fig. 12: in Fig. 12a the grid is madeof 15× 9 cells, while in Fig. 12b it is of 42× 25 cells. Here, the compression rate is quitebalanced with respect to the simplification of the surface morphology.

As is clearly shown by the examples, the choice of cell size strongly influences theresolution of the model. Large cells have a high smoothing effect and the resulting FBS isable to reproduce only the global morphological details (global features). As the cell sizedecreases, further details are reproduced in the FBS. In this sense, the data reduction processtrivially behaves as a filtering process, where the cell size determines the resolution of theresult. Moreover, the use of B-spline surfaces by itself has a smoothing effect, as pointedout in [18]. It is also well known that any smoothing technique induces a multiresolutionanalysis [19, 20]. The analysis induced by the FBS cannot be related, as one of the refereespointed out to the authors, to any linear basis, like the wavelets, because it grows out of anonlinear optimization procedure.


FIG. 11. The contour lines defining the second data set.

The core of the FBS approach, however, is not comparable in a straightforward mannerto multiresolution methods. The FBS layers at the minimum presumption level have thesame content, in terms of frequency, as the maximum presumption level (see the examplesin Figs. 9, 10, and 12). In other words, the grid having been fixed, the level of detail coded inthe surface model is the same. What is interesting here is that users might choose the FBS’slayer pair according to the presumption level which is judged appropriate to the application:For example, if we have to plan the navigation of an underwater autonomous vehicle nearthe sea floor, we might prefer to consider the route on theα= 0 FBS layer, which ensuresa safe distance from the real sea floor.

FUZZY B-SPLINES 53

FIG. 12. The FBS model, atα= 1, shown using two different grid sizes.

Another interesting issue concerns the interrogation of the FBS model with typicalqueries, such as “given a crisp levelk, where does the FBS surface take valuek?” Theanswer to such a query will produce, in general, not a simple contour but a strip of pointswithin which the relation is satisfied (see Fig. 13). For eachα-level, indeed, the correspond-ing upper and lower FBS’s layers have to be matched against thek-value. The change ofα-levels from 1 to 0 will cause not only the enlarging of the solution strips, but also theappearance of new solution areas. This effect can be clearly seen in Fig. 13: in Fig. 13a,the FBS, is interrogated at presumption levelα= 0.5, while in Fig. 13b the same surface isinterrogated atα= 0.

Such a typical query is translated into a system of nonlinear equations and for this reasoninsights into the investigated phenomena can be obtained only by solving such systemsefficiently. A parallel implementation of the adopted method, which is based on a Monte


FIG. 13. Results of the FBS interrogation at differentα-levels.

Carlo approach, has been proved to be an efficient and simple solution to the inherentcomputational problems [21].

7. CONCLUSIONS AND FUTURE DEVELOPMENTS

In this paper, a new approach to modeling surface data has been presented, which takesinto account the uncertainty which might be associated to the original data set. The resultingmodel has very good properties: first of all, uncertainty is explicitly embedded in the geo-metric model, which can be interrogated at different presumption levels. This means thatusers can visualize and reason about the reconstructed model at different levels and use thedifferent layers of the model according to the accuracy required by the specific analysis. Thesecond point regards the very high rate of data reduction that the method supports whichkeeps information on the distribution of data points and values in local neighborhoods.

While the FBS seems to provide a good balance between data reduction and resolu-tion level, one problem concerns the visualization of the different presumption levels. Thevisualization of a fuzzy B-spline is quite simple in the one-dimensional data: “Bands”of different colors that the user can select from a suitable palette represent the differentpresumption levels as in Fig. 8. When the data set to explore is three-dimensional, visual-ization is not a simple task. The technique which can be used is to draw “cuts” on a set ofsuperimposed shells, each representing a different presumption level. This representation is

FUZZY B-SPLINES 55

very intuitive, but it quickly leads to confused images when more than a few presumptionlevels are represented.

Future developments will mainly concern this point and the improvement of the graphicalinteraction with the model, as well as the investigation of simple porting of the FBS methodto a distributed environment of computation.

ACKNOWLEDGMENTS

This work has been developed within the Coordinate CNR Project on “Modeling and Analysis of Digital TerrainModels using Fuzzy Arithmetic Algorithms.” Thanks are given to all the members of the groups working withinthe projects, especially to Marcello Anile and Bianca Falcidieno. Thanks are also given to Gruppo Nazionale diVulcanologia in Pisa for the Mount Etna data set.

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Fuzzy B-Splines: A Surface Model Encapsulating Uncertainty