Curvature-Based Fuzzy Surface Classification - IEEE Xplore

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 4, AUGUST 2006

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Curvature-Based Fuzzy Surface Classification Soodamani Ramalingam, Zhi-Qiang Liu, and Dmitri Iourinski

Abstract—In this paper, a fuzzy surface classification paradigm, which is an extension to the conventional techniques based on the sign of the mean ( ) and Gaussian ( ) curvatures, respectively is presented. With the conventional methods, two of the major problems that limit object descriptions are: 1) Their inability to describe surfaces in a natural way, and 2) computation of curvatures being highly sensitive to noise as well as limited by resolution. Problem 1) is addressed by treating the transitional regions between distinct surface types as smoothly varying (fuzzy) surface types. Problem 2) gets partially resolved while fuzzifying the signs of the surface curvatures for surface description. The new segmentation technique is demonstrated in a model-based object recognition system and its performance is compared with a system based on conventional surface classification. Index Terms—Curvature measures, fuzzy logic, range image segmentation, three-dimensional (3-D) object recognition.

I. INTRODUCTION ECOGNITION of three-dimensional (3-D) objects involves deriving representations of objects from extracted features and matching these representations with stored models. Matching of objects is thus influenced by the specific representation strategy employed for object recognition. Most model-based recognition systems describe objects in terms of their shapes. A good shape representation scheme combines local shape primitives, which can be computed relatively easily as they are dependent on a limited number of input data. For the sake of convenient and efficient matching,each primitive should have a limited category of possible shapes. For example, surface regions might be classified as being planar, cylindrical, or hyperbolic based on surface curvatures. The range of possible shapes must be divided into such discrete categories. However, instabilities are found at the boundaries between discrete categories. With small changes in shape, a surface that is classified as planar, for example, might at other times be classified as cylindrical. One way to resolve this conflict is to allow some overlaps and redundancies among the discrete primitives. This leads to the concept of fuzzy shape primitives. For instance, applying fuzzy parameterisation of the surface curvatures allows a surface that lies near the boundary between planar and cylindrical to be represented by the two primitives at once. Thus,

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Manuscript received July 26, 2004; revised April 10, 2005 and November 17, 2005. S. Ramalingam is with the Department of Computing and Mathematics, Manchester Metropolitan University, Manchester M1 5GD, U.K. (e-mail: [email protected]). Z.-Q. Liu is with the School of Creative Media, City University of Hong Kong, Kowloon, Hong Kong, P.R. China (e-mail: [email protected]). D. Iourinski is with the Department of Computing Science, Middlesex University, London NW4 4XN, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2006.876718

fuzzy description is expected to lead to robust recognition by way of associating the extent to which a segment belongs to each category of the surface. Also, such descriptions permit recognition to fail gracefully because fuzzy shapes enable the system to describe the degree of matching. Surface curvatures of 3-D range images are a powerful tool in image description and segmentation. They are invariant under translation and rotation. Surface curvature signs are also invariant under scale change. For this reason, a segmentation procedure based on the surface curvature is opted. This forms the basis for surface classification in a 3-D object recognition system. A number of methods have been proposed in the literature for the estimation of curvatures. Refer to [1], [2] for details. Curvature representations are obtained from gradient information which tends to be noisy. Smoothing in some cases changes the geometrical profile of the surface and in other cases is prohibitively time consuming [3]. In this paper, a fuzzy paradigm of surface segmentation is proposed, in which surface curvatures are modelled by fuzzy membership functions. Fuzzy surface curvatures permit multiple descriptions, thereby modelling the uncertainty in curvature estimation, and classification of surface points are postponed until appropriate cues are obtained (see [4] and [5] for more details), however the approach offered in the referred works is quite primitive and the features extracted and are different from the ones proposed in this from paper. Typically morphological features on segmented parts as described in [6] were used for the surface classification described in [4]. In the present technique, we use fuzzy rule base to classify the surfaces, hence making it possible to treat the two approaches as different. The rest of the paper is organized as follows: In Section II, current segmentation techniques are briefly reviewed. In Section III, the use of directional derivatives in determining the surface curvature measurements is dealt with. In Section IV, the system architecture of fuzzy surface classification is dealt with in detail. In Section V, the performance of the fuzzy classification system through experimental results is discussed. In Section VI, we conclude the paper through summary for the contribution of this paper and suggested future work. II. SEGMENTATION TECHNIQUES USING CURVATURE MEASURES: A REVIEW In general, surface segmentation techniques attempt to resolve the issue of noise sensitivity of surface curvatures by either employing recursive Gaussian smoothing or performing recursive merging until consistent region properties are obtained. Other techniques avoid the computational costs of the above procedures by resigning to a restricted and broad description of the surfaces.

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TABLE I SURFACE CLASSIFICATION BASED ON

H AND K

Besl and Jain [7], [8] used an 8-sign labelling scheme for range images. At each point on the image surface the sign of or the Gaussian and mean curvatures are recorded; . Hence there are nine combinations from the two curvatures . However, one of the combinations is impossible, for and , since . Besl and Jain categorise the eight types of surfaces as Peak, Ridge, Flat, Minimal Surface, Pit, Valley, and Saddle Valley Table I). They employ a recursive merging technique for a consistent region. It is reported in [9] that this approach is sensitive to the adjustments of merging parameters. It also involves a complex control structure. Hoffman and Jain [10] proposed segmenting the images into planar, concave and convex surfaces. This reduction was considered necessary by the authors as the curvature measures are sensitive to noise. For clean image data and perfect object types, this approach can be efficient. However, if the image data contains noise, and the object’s parts’ shapes are not exactly the same as the models, errors in surface segmentation and the existing configuration of surface patches will cause incorrect part segment [11]. Similarly, Abdelmalik [12] has proposed a 3-sign curvature label scheme. A range surface point belongs to a convex, concave, or a flat patch. This depends on the sign of the mean only. Subsequent error analysis results reveal curvature that spurious segmentation occurs on the image planar and near planar regions where the normal curvatures are small. Automatic threshold levels for the calculated curvatures in terms of the corresponding upper bounds of the curvature are estimated to give slightly better segmentation results. Fan et al. [9] have proposed a surface description technique based on segmentation at the physical boundaries in terms of zero crossings and extremal values of the surface curvature measures. Such a segmentation assumes consistent region properties between the physical boundaries enforced by fitting a surface function to the patches. In their technique, noise have to be handled by multiscale tracking, a way by which different levels of signal-to-noise ratios (SNRs) are increased. Tanake and Lee [13] have proposed a shape representation scheme by defining a set of descriptors in terms of subsets of lines of curvature and their intersections. They are based on local descriptions of the Gaussian and mean curvatures of the surface. Cai has proposed a diffusion model [14] that helps maintain the position and curvature signs along the surface boundary by making the depth variation at boundaries to be consistent with that in the inner area. Trucco and Fisher [15]

adopt the Gaussian smoothing modelled with the diffusion process. Instead of imposing a fixed interaction between between surface and background during diffusion smoothing, their technique adapts to the local smoothness criteria. Pixels are then classified as belonging to the conventional surface types based on the signs of and . The technique takes care of quantization noise but not noise due to computational errors. In depth image smoothing, the reduction of depth contrast along the surface boundary leads to the surface gradually merging into the background resulting in an inconsistent depth variation. This has the adverse effect of not preserving the shape of the object. For instance, a cylindrical surface boundary can change into a flat surface by repeated Gaussian smoothing. However, techniques employing Gaussian smoothing [9], [13], [14], and [15] involve computationally expensive alternatives to avoid the side effect of drastic change in shape, while Hoffman and Jain [10] and Abdelmalik [12] derive restricted shape descriptions owing to the noise sensitivity of the surface curvatures. Wu and Levine [11] have proposed a physics-based rather than a geometrical approach to range image recognition based on simulated electrical charge distributions that is particularly useful to identify sharp changes from convexity to concavity, thereby able to identify part boundaries. The approach involves solving an integral equation rather than performing surface curvature computations, so it does not need to satisfy the smoothness criteria that are generally required for curvature computation. However, the system will not work well for part boundaries that are not sharp nor closed or dented. Given these conditions, it is doubtful at this stage how useful the system will be for general 3-D-object recognition. Hoover et al. [16] have provided an experimental comparison of range image segmentation of planar surfaces involving four different techniques from the following research groups namely, University of South Florida (USF), University of Edinburgh (UE), Washington State University (WSU), and Bern University (BN). The USF and UE are based on region segmentation by iteratively growing from seeds. The WSU is an improvement over [17] and Flynn and Jain [1]. It employs clustering for segmentation. The UE segmenter is based on the fact, that points on a scan line that belong to a planar surface form a straight 3-D line segment and all points on a 3-D segment must lie on the same planar surface. Therefore each scan line is divided into line segments on which region growing is performed. A performance evaluation is done on all of the aforementioned techniques in terms of standard metrics such as over segmentation, under segmentation, missing and noise pixels, etc. The previous work deals only with planar segmenters. It is noted that all of these techniques involve rigorous steps, especially several iterative steps of post-processing for segmentation stabilization to take place. This has led to extremely high average processing times, except for the UE algorithm. In addition, they involve quite a few parameters to be tuned during training which are not automatic and based on visual quality of segmentation. It is inferred that these segmenters are likely to miss out small regions. They also pose a problem of correct segmentation at the borders of two regions and simply treat them as

RAMALINGAM et al.: CURVATURE-BASED FUZZY SURFACE CLASSIFICATION

being “undefined.” There is no across the board winner amongst these methods. In [18], a single step process is adopted to identify all types of edges. A surface-based approach enforced by a boundary representation scheme is adopted. It consists of developing two partial derivatives in the - and -directions, which are used to determine the presence of edge points. While the technique is good for polyhedral objects, the effect of shadow smoothing is seen on curved objects with missing edges. In this section, we have carried out a survey of current segmentation techniques that form the basis of our contribution. III. CURVATURE-BASED FUZZY SURFACE CLASSIFICATION In this section, we discuss the surface classification technique based on the estimates of surface curvatures derived from the directional derivatives. The curvature of the surface at a given point varies with the direction in which it is measured. There are two principal curvatures at every point: the maximum normal curvature and the orthogonal curvature which are used to , the Gaussian calculate two main shape descriptors: , the mean Curvature. In the curvature and object recognition system proposed by Caelli [6] the surface segmentation is done by using zero crossings of the determinant of the Haussian matrix which determines convex, concave or planar regions. The surfaces are classified according to their Gaussian and mean curvatures. The ground truth for the chess database (which is later used for testing the current system) is based on [6], however the results presented there could not be directly used for fuzzy parameterizations. The work in Caelli describes morphological features that include -ary (unary, binary, etc.) feature combinations. For the purpose of surface segmentation of the chess database unary parameters were used as a rough estimate for verifying the extracted fuzzy surface curvatures [19, Ch. 6]. In this work, the technique proposed by Fan [20] is employed for computing the shape descriptors. The detailed description of the technique is given in Appendix I. Besl and Jain [7], [8] proposed a technique that uses signs of and to classify the surfaces into eight basic types, which we later refer to as it crisp surfaces. However, the approach produces spurious segmentation results while classifying pixels at the boundaries between the surface types. We address this problem by allowing imprecise definitions to the signs of and in terms of the membership in fuzzy sets. Allowing imprecision in the sign definitions results in additional classification surfaces, which are later referred as it transitional surfaces, in this framework a pixel can belong to up to four surface types with different degrees of confidence. The confidence degree in classifying a pixel is given by the fuzzy membership functions in each surface type and is derived from the membership values for the labels of the signs of and . Thus, we have two interrelated levels of uncertainty: Fuzzy and and fuzzy surface classification. labels of signs of In both cases the fuzzy membership values are within [0,1] range, where membership value 1 means total confidence in classification. Fig. 1 gives the general system architecture for the proposed surface segmentation process as well as for the traditional approach. The input to the system in both cases

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is a range image and the output of the system is a set of decomposed images each representing pixel distributions of a surface type. Computation of the curvatures is basic to both methods, both methods require a merging process to smooth out any spurious result and reduce the effects of oversegmentation. The merging operation is normally performed based on certain smoothness criteria of the surface patches resulting in larger sized patches. It assigns to a pixel a surface type that dominates in its local neighborhood. This process is similar to convolving a Gaussian mask over the image. The main difference between the traditional and the proposed approaches lies in the way and are defined. Our classification of pixels is based and in terms of fuzzy membership on the signs of functions. Thus, fuzzy system architecture consists of two general modules namely, the fuzzy parameterisation and defuzzification. The former consists of a fuzzifier and a rule base which are based on a multiple-input–multiple-output (MIMO) mapping system, while the latter consists of fuzzy segmentation and merging based on a voting process. These modules are discussed in Sections III-A and III-B. A. The System Architecture We consider the MIMO fuzzy system shown on Fig. 1. The input to the fuzzifier module are the curvatures, and , of an input image and the output are the fuzzy representations and later fed as input to the fuzzy of the signs of inference engine. The fuzzy inference engine uses the fuzzy rule base to make the membership assignments. The general rules of classification based on the partitioning of the universe and . Finally, the defuzzifier decides on of discourse of the choice of a label by “maximum voting” from the pixels in the neighborhood of a pixel. Later, we discuss the two main modules of the system and then, in Section III-A.2, we give an overview of the defuzzification process involved in segmentation. 1) Fuzzifier: The process of fuzzification divides the input and output spaces into fuzzy regions [21], [22]. The range of input values lies within certain domain, which is divided into a number of regions, with each region identified by a label and a fuzzy membership function. The partitioning of the universe of discourse for the input spaces is done manually by extracting the mean and Gaussian curvatures of the objects in the chess database. This set of data (database of size 25) acts as training data for deriving the number of fuzzy membership functions and their shape. The resulting membership functions may be applied for segmenting objects of another database. and , there are basically Since we fuzzify the signs of three fuzzy bins for each of it (ZERO, POSITIVE, and NEGATIVE). This division can be treated as suitable for a coarse-level description of the surfaces. We need to provide a richer description, a finer division of these regions is more appropriate [23]. Each of the coarse-level divisions is further subdivided into three regions. This helps in describing, for instance, how good a surface is as a saddle ridge, based and are. It brings out the gradual on how negative change in the shape of the object as we move from one and are divided pixel to another. The input variables

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Fig. 1. Basic organization and configuration of a fuzzy system. (a) System organization: Fuzzy segmentation versus conventional segmentation. (b) Basic configuration of fuzzy system with fuzzifier and defuzzifier.

heuristically into nine fuzzy regions that are associated with linguistic labels

Very Negative Negative Negative to zero Close to zero Zero Zero to Positive Close to Positive Positive Very Positive (1) and are not distributed uniformly along The values of their ranges: Data points tend to be clustered around certain values. The regions around such values are considered to be “centers” of fuzzy bins and are assigned fuzzy membership value of 1, the regions that are hit more rarely are treated as “peripheries” of the fuzzy bins and thus have fuzzy membership . This simple analysis was performed on function values the values of the mean and Gaussian curvatures of the objects from the chess database (size 25). Fuzzy functions derived as a result of it can later be used for segmenting objects from other databases as well. The resulting fuzzy membership functions are shown in Fig. 2 along - and -axes. The values of these functions are read and . The fuzzifier can be from auxiliary axes marked seen as a coordinate system, where the labels are assigned accoordinates cording to the rules that are triggered by the

of a point. In case of only crisp surfaces the coordinate plane is divided into nine nonoverlapping regions. In case of fuzzy classification we allow for the regions to overlap. The transitional surfaces are then results of a point being mapped to a place where the regions overlap. Such assumptions leads to the following possible surface types: SaddleRidge Saddle Valley ValleySurface Pit

Minimum Surface Ridge Peak

Flat Undefined

where indicate fuzzy surfaces based on support in two surface types while is based on finding support in four surface types (support in three surface types is imposreflects sible). Quite clearly, classifying a pixel as the highest uncertainty possible in the approach, but since the overlapping parts of regions are defined according to the simple statistical analysis outlined in the previous paragraph (i.e., the overlapping parts are the parts with the lower density of data occurrences), such classification is also the least likely one. 2) Fuzzy Rule Base: To apply the fuzzy rules the inputs and must be nine-tuples with the memberships for both in each bin. It is clear from the Fig. 2 that any of such tuple has at most two nonzero entries. Such tuples are created by


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Fig. 2. Fuzzy surface classification based on H and K.

finding the support sets for each pairs of the data, e.g., represents and the point . Projecting these values to the axis of and we can see that in case of nonzero values are for CP and P and that the corresponding membership values can be read from , these values and , the auxiliary axis values for fuzzy bin memberships for are read in exactly the same fashion. Therefore, the support sets of and at are given by:

or and in vector form. A sample of fuzzy rules generated based on these support measures is illustrated in Table II. The rules are interpreted as (assuming, for simplicity, no separate label is assigned to the transition region) if

is

and

is

then Surface type is

(2)

However, considering that the fuzzy transitional regions B5 and D2 exist between the Saddle Valley and Valley Surface types, the assignment becomes more diversified. It is easy to see that if

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TABLE II FUZZY MAPPINGS FOR SURFACE CLASSIFICATION BASED ON H AND K

coordinate is CP and P (with no fixed ) all possible pixel classifications lay in any of ten regions in the right third of the plane (consider a vertical stripe across the coordinate system spanned by nonzero values of CP and P membership functions). Similarly, classifying as NZ or CZ can lead to any of ten surface types of the upper third of the plane (consider a horizontal stripe spanned across the coordinate system by all nonzero values of NZ and Z). These parts of the coordinate plane overlap in the upper right corner of the plane with six shared surface types: B2, B5, B7, D2, VS, and SV. Thus, the full fuzzy version of the rule is is and is if then Surface is either SV or VS or B2 or B5 or (3) More precisely such an assignment is a result of combining four different rules (as there are four different pairs of and ) if then if then

is CP and is NZ Surface type is B2 or D2 or SV or B5 is CP and is CZ Surface type is D2 or B5 or B7 or VS

if then if then

is P and is NZ Surface type is SV or B5 is CP and is NZ Surface type is B5 or VS

Such multiple assignment is not a sign of imprecision—it just takes into account all the possible overlaps between the crisp regions. Moreover, it is clear from the picture that some of such classifications are unlikely, say B7. This problem is resolved by assigning low confidence degree to the label as it is shown later. At this stage we do not try to assign a unique label to a pixel, we of the set of all possible labels, rather look for a subset . Any pixel classified according to the presuch that vious procedure can be affiliated with at most nine different lahave corresponding confidence degrees. bels. The entries in These confidence degrees are later used for normalization of the final label assignment and for determining the voting weights of pixels at the merging stage. B. Defuzzification for Segmentation In this section, a defuzzification technique derived from the one proposed in [4], [5] is formulated for surface classification. The newly proposed technique differs from [4],[5] in that: i)

features extracted from, are different, and ii) the fine tuning of the fuzzy parameters has been changed. The technique is similar in principle to the one proposed by Yager [24], for labelling a pixel in the image based on the distribution of and in a local neighborhood. As a result support sets of of defuzzification the membership values for the surface types showing the level of confidence are assigned to each pixel in classifying as belonging to each of the surface types. For the sake of convenience these membership values are referred as “confidence degrees” at the segmentation stage and “belief measures” at the merging stage. The pixel level operations in a local neighborhood of size 5 5, denoted are as follows. by Fuzzy Segmentation At this step, the labels are assigned . The label assignment is based on to the pixels in and curvatures’ memberships in fuzzy bins ( , , etc.), which are aggregated to arrive at confidence degrees associated with the labels. This is achieved by determining at first. Then the fuzzy memberships voting weights within of and are updated based on the voting weights. Updated memberships are then normalized. Normalized memberships and are used to choose the dominating fuzzy bins for . Dominating fuzzy bins allow to select a unique label for the pixel, confidence degree for such label is based on the mean of max principle. The mathematical formulation for fuzzy segmentation is given in Appendix II. are now merged. The Fuzzy Merging Pixels’ labels in merging is based on confidence degrees calculated at the segmentation stage. Aggregating confidence degrees brings us the belief measure for the resulting label. At first, the voting weights are determined. Then, the importance factors of the within labels in are found. The belief measures of labels are uphaving dated according to the voting weights. A label in maximum belief measure is then chosen and assigned to the . The mathematical formulation for the fuzzy centre pixel in merging is given in Appendix III. The choice of an appropriate size of local neighborhood is an important consideration in either application of image processing [10] or information fusion. Choosing a large neighborhood size is computationally intensive while a small neighborhood is prone to noise. A 5 5 neighborhood is in general expected to give good results and hence used in this experiments. As we move from one segmented region to another, there is a gradual change in description by associating the certainty measures with the labels. Since the belief measure is derived from two previous stages of aggregation over the same local neighborhood, it should ensure a high degree of confidence for the final assignment providing an accurate and natural description. This in effect performs a dynamic label assignment to every pixel in N5. This process is not computationally intensive, as the support sets remain the same throughout the convolution process and are determined prior to segmentation and merging. The process involves only updating belief measures until final labelling is done. In this section, we have discussed through mathematical formulations the basic modules of fuzzification and defuzzification involved in the system architecture of surface classification. For clarity, a numerical example is worked out in Appendix IV.


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In the following section, we describe the experimental setup to verify these mathematical formulations. IV. RESULTS ON RANGE IMAGE SURFACE CLASSIFICATION The use of fuzzy measures for updating confidence in surface classification is validated on a set of range images that were synthetically created as well as on real range images. The results are compared with that of conventional classification scheme. The input to the system is a range image and the output is a set of images showing the distribution of classified surfaces and confidence measures associated with these surfaces projected as RGB colour components to display changes in the shape of the objects. It is noted that the outputs are like contour maps of a specific label and they do not necessarily satisfy same smoothness criteria as in other more traditional systems. In this paper we compare the performance at two different stages of the vision system. 1) Direct Measure: The results of segmentation are compared for a set of synthetic objects whose shape is very well-known: convex, concave and planar. It is established that planar objects which are very difficult to classify by traditional systems are well classified by the Fuzzy system. In addition noise was also introduced to test the system’s stability. 2) Indirect Measure: At this stage the system is tested for the object recognition. The set of modules comprising the recognition part and following segmentation is made the same for both systems. This enables us to compare the relative performances at an application level. Thus, the following tests were carried out to validate the fuzzy segmentation technique. • Controlled Environment with known shapes: Simple planar, convex and concave shapes shown in Fig. 3 have been created synthetically for evaluating the relative performance of the fuzzy technique with that of the crisp technique. The size of these three images is 256 256. The convex and concave objects are of radius 50 pixel units. The planar object has a slope 1. The percentage of pixels that have been correctly classified as planar, convex, or concave, is estimated as a relative performance measure. Fig. 4 shows the results of segmentation and merging by both the crisp and fuzzy techniques for the convex object. The same data was also produced for both planar and concave surfaces, however, the results for the convex object are displayed as the most illustrative. Missing entries at several positions mean that no pixels were classified to the corresponding type of surface. In the case of fuzzy segmentation and merging some pixels are classified into transitional classes, e.g., between SR and ridge, the case when conventional segmentation and merging techniques fail to classify them so precisely. The results may appear to contain sparsely decomposed images during surface classification. Table III sums up the results of crisp and fuzzy segmentation and merging techniques in terms of: a) the % of pixels correctly classified for known shapes, b) the deviation of pixels correctly classified (Diff) when the images are disturbed with no noise to the maximum limit of noise considered in this work. It is to be noted that the results

Fig. 3. Objects whose shapes are known a priori. (a) Convex. (b) Concave. (c) Planar.

Fig. 4. Comparative segmentation and merging results for a convex synthetic object.

of fuzzy segmentation are only approximate as they depict the overall classification of related surfaces. For instance, the results of saddle-valley, valley-surface and pit are combined to give an overall performance measure for a concave surface. This makes it possible to compare the performance with the conventional classification system. First, let us consider the noise-free situation: the conventional technique shows better results for both segmentation and merging of a convex sphere, the fuzzy technique outperforms the conventional technique on the concave sphere and the performance of both techniques on the planar surface is similar (better segmentation results for fuzzy and better merging results for the traditional one).

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Fig. 5. Instances of chess database. (a) Bishop. (b) Rook. (c) King. (d) Queen. (e) Pawn.

Fig. 6. Instances of object bishop. (a) Bishop 1. (b) Bishop 2. (c) Bishop 3. (d) Bishop 4. (e) Bishop 5.

TABLE III PERFORMANCE RESULTS FOR SIMPLE OBJECTS: CONVEX, CONCAVE, AND PLANAR

At this point, it is difficult to give preference to either of the techniques.The situation becomes clearer, when the noise is introduced into the picture: Even though the performance of both techniques does not differ much in cases of convex and concave spheres, the fuzzy technique clearly outperforms the conventional one on the planar surface (13% versus 0% during segmentation and 21% versus 0% during merging). This is dealt with in the following sections. • Controlled Environment with predetermined noise distribution: The technique is tested for its robustness to noise by introducing various levels of white Gaussian noise in the images. As it was already illustrated the technique performs better than the traditional one under extremely noisy conditions. At this point we test the robustness of the technique to noise by adding white Gaussian noise with SNRs of 0.25, 0.5, 0.75, 1.0, 1.25, 10, 50, 100, and 200 where the SNR is defined as the ratio of signal power to noise power , and the signal power is assumed to be unity:

Fig. 7. Results of segmentation and merging on the object bishop 4.

Table IV shows the performance of the two techniques as the percentage of possible surface types determined in these objects for different SNR.


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TABLE IV NOISE PERFORMANCE ON CONVEX OBJECT

It is observed that there exists a consistency in the fuzzy description of surfaces when the objects are disturbed by noise. In order to obtain stable descriptions of the surfaces, small changes in shape must produce small changes in description. This effect is produced by fuzzy segmentation. For instance, classification of the convex sphere by the conventional method is 81% in the case of noise-free image (upper bound), and 48% in the case of maximum noise (lower bound). The range varies linearly within these bounds. The deviation from upper to lower bounds is 33%. Merging increases the upper bound up to 92% and the lower bound to 50%. The deviation is 42%. The fuzzy technique has the same lower bound but the upper bound is lesser, 71% and deviation being only 23% during segmentation. Hence more or less, the same performance as that of the conventional system. However, the results of fuzzy merging identify only 77% and 43% of the pixels as convex. This is because a considerable percentage of pixels are distributed among the rest of the fuzzy surface types. However, the deviation is maintained low with varying noise as depicted in Table III. The %deviation during merging is 34% which is much less than that of crisp merging. This kind of classification is particularly useful in model-based machine learning where consistent properties of surfaces are the notable features of an object. This is demonstrated with the set of synthetic objects of chess pieces (Section V). In this sense, the fuzzy merging process performs better than the conventional method. In the case of the concave object, not only is the overall percentage classification by fuzzy technique better than that of the conventional method but also the deviation is maintained low. The same is observed for the planar surface. Introducing noise does not only illustrate higher robustness of the fuzzy system it also shows how the fuzzy approach overcomes some of the problems associated with the crisp one. When noise is introduced into the picture the conventional classification approach always misclassifies the planar surfaces into an undefined surface type, resulting in 0% correct classifications during both segmentation and merging stages. The fuzzy system overcomes this problem by introducing richer classification paradigm from the very beginning which results in noticeably better performance on both stages.

• Test on more complex synthetic objects: A set of chess pieces (Fig. 5) suitable for model based object recognition, that involve complex shapes is used to test the shape descriptions of the fuzzy technique with and without noise. This database is developed by the Computer Vision and Machine Intelligence Laboratory (CVMIL), The University of Melbourne, Australia. The database consists of synthetic models of chess pieces of size 256 256 which are view-dependent depth maps of 3-D objects. Typical instances of a chess object within the database appear as shown in Fig. 6. Fig. 7 illustrates the predominant types of surfaces obtained by crisp and fuzzy segmentation and merging of object Bishop-04. Just as in the case with the convex object (see Fig. 4) there are empty entries in the table. The reason for the gaps is the same as in previous case: No pixels are classified to corresponding surface types. The process produces a set of sparsely decomposed images of various crisp and transitional surfaces. The fuzzy merging process merges the transitional surfaces into crisp surfaces depending upon their credibility measures as well as by the votes of the neighbouring pixels. It is found that the results of conventional segmentation produces clear cut regions of segmentation. For the same regions, fuzzy segmentation shows regions of imprecise definitions approximated as crisp regions in conventional systems. Tests for robustness to noise were conducted on the same set of images. As before, results show that fuzzy segmentation provides better stability to noise than conventional segmentation. • Test on real range images: Tests are also carried out on a set of real range images, shown in Fig. 8, that have inherent noise in them. These images are obtained from the PRIP Laboratory, Michigan State University, East Lansing. for uniThe images were all normalized to size formity. Figs. 10–12 show a summary of the relative performance of the two techniques of surface classification in graphical form. The object database consists of 3-D polyhedral and cylindrical objects. Tests reveal that conventional merging fails to describe planar-like objects. It defines most of the pixels in these images as belonging to the Undefined class. This is due to the fact that most of the pixels lie close to the Undefined region, that is, transitional regions close to it. Such objects include Box2inch, Block1,

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Fig. 8. Instances of real range image database. (a) Hump-3. (b) Bigwye. (c) Box2Inch. (d) Block1. (e) Block2. (f) Column1. (g) Column2. (h) Grnblk. (i) Harriscup. (j) Piston. (k) Agpart2.

Fig. 9. Instances of object bigwye. (a) Bigwye1. (b) Bigwye2. (c) Bigwye3. (d) Bigwye4. (e) Bigwye5.

Harriscup, and Piston. Other surface classifications in this database have dominating types for each object which well and equally detected by both systems. Additionally, the fuzzy approach shows sparse decomposition of fuzzy surfaces as well. Since the conventional does not have the ability to decide these fuzzy regions, it tends to describe the object as a non-existing type of surface. The fuzzy merging results in a description that is consistent with the region properties of the objects. It is therefore concluded that for the sets of different objects considered in this paper, the fuzzy technique is more general than the conventional technique. The technique was also tested within a model-based system that consistently showed better results than the systems using traditional segmenting and merging approaches. For a detailed description of the experiment, see [25], [26] where a matching process (information fusion) is described. Reference [25] describes a fuzzy Hough transform, an edge segmentation technique followed by the feature extraction and object recognition. The paper presents the notion of validation, generalization and the rejection test as well as the description of the database. The same set of tests and databases were used in the present work, however the system and resulting features are different. In [26], the same database is used for testing a GA-based learning adaptor, however in both papers the features extracted are different from the ones presented previously. Thus, in this section, we have outlined all the possible tests carried out to verify the proposed fuzzy surface classification technique and compare its performance with that of conventional technique. The tests were compared at intermediate stages of segmentation and merging, as well for their stability to noise.

In the following section, we further test the techniques for their recognition performance. V. OBJECT RECOGNITION SYSTEM ARCHITECTURE In this section, we demonstrate the performance of the segmentation techniques indirectly through their object recognition performances. In conventional systems, surface based recognition techniques derive features from surface patches satisfying certain smoothness criteria. Since the set of labelled features based on fuzzy surface segmentation do not necessarily satisfy such a condition, they may be treated as surface distributions for the purpose of global feature extraction. Global feature descriptors such as moments are useful under this situation. Moment invariants [27] have been used as feature descriptors in a variety of object recognition systems [28], [29] and are frequently used for shape description as they generate values which are invariant with position, orientation, and scale changes. A fuzzy moment based recognition technique is described and tested in [30] which is employed here. Normalized feature vectors of moments are generated for each instance of an object. The technique is tested on the set of two object model bases, referred to in Section IV, namely, • Model1: A model base of five synthetic chess objects (Fig. 5) formed from five instances of each (Fig. 6). • Model2: A model base of eleven real range objects (Fig. 8) formed from five instances of each (Fig. 9). A. Performance Evaluation of Conventional Versus Fuzzy Systems Three experiments were conducted.


Fig. 10. Performance results on real range images.

1) Evaluating recognition rate with seen instances as test objects. These instances were originally used for aggregating the model base. All instance are tested for recognition, i.e., 25 instances against five models for case Model1 and against 11 models in case of Model2. 2) Evaluating recognition rate with unseen instances. This reflects on the generalisation capability of the system. Given a model, the system must be tested for its capacity to recognise intermediate views of an object not seen before. To achieve this, the original memory aggregates were constructed with a leave one out strategy and the left out object is used as the test object. Thus only four instances of an object is considered for each model aggregation. The size of test objects are 25 and 55 for Model1 and Model2, respectively. 3) Evaluating rejection rate. The system should be able to effectively reject test objects that do not belong to any of its model objects. Since we have two databases, we used each database against the other for this rejection test. Complete model bases are used. Test object size are 55 and 25 for Model1 and Model2, respectively. Table V gives the results of the above experiments. The following inferences are made.

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Fig. 11. Performance results on real range images.

• Classification of synthetic objects: For seen objects, conventional system has a performance measure of only 72% as against 100% in the case of the fuzzy system. The overall performance of fuzzy system of unseen instances is better than the conventional system. Recognition using fuzzy surface descriptions is much higher (over 80%) than that with crisp surface descriptions (less than 70%). Both systems reject alien objects with an equally high percentage. This may be attributed to the concept of the fuzzy conceptual graphs for memory aggregation. • Classification of real range objects: The fuzzy system has a steady recognition rate of 90% while conventional system has fluctuating results in the range 70%–80%. • Fuzzy systems show consistently high recognition rates for both models. Conventional systems show better recognition performance in case of real range objects than the synthetic noise-less objects. It is be noted that this is in contradiction to one’s expectation that the system performs well for the model base for which the fuzzy parametrization was originally designed for. It is inferred that the fuzzy conceptual graphs (FCGs) have good generalization and discriminating power leading to high recognition rates [30]. • The rejection rate is equally high by both systems for both models. In general, it is easy for object recognition systems

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TABLE V PERFORMANCE EVALUATION OF TWO MODEL BASES

Fig. 12. Performance results of real range images.

to perform well in terms of recognition but not necessarily in terms of rejecting unknown objects. This is an important credit to this system-once again due to the effectiveness of the FCGs. In this section, the performances of the proposed and conventional systems have been compared with two complex image databases. The results are promising in applying the proposed technique to larger databases.

and demonstrate its usefulness for a model-based object recognition system. It is novel in that it uses fuzzified parameters of the curvature measures, thereby giving the flexibility of representing a pixel by more than one type of surface and associating a credibility measure with a surface type. A decision on the exact type of surface that the pixel belongs to is made at the time of merging the pixels in a neighbourhood. During merging, the credibility measures in the neighbourhood of a pixel are appropriately aggregated by standard fuzzy techniques, thereby deciding a label and associating a degree of belief with it. This degree of belief that the pixel belongs to a specific surface type allows a natural description to the objects in an image, especially at the transitional regions between different surface types. The advantage of such a description is that impreciseness of the surface features is well modelled reducing discrepancies in surface descriptions. It stabilises the surface description even under varying noise levels. This is because of the fact that spurious segmentation at the transitional regions are represented by separate fuzzy surface types making the description more robust to noise. An efficient technique for model-based object recognition is employed using fuzzy paradigms which is used as a basis for comparing the performance of the proposed technique with conventional technique of surface classification. The system is demonstrated to be particularly useful for a model based object recognition system, which requires the discriminating power to identify notable features that are common amongst various instances of an image and those that make them different. This feature is made possible by means of the proposed fuzzy conceptual graphs which is well suited for inductive learning and recognition by a simple fuzzy projection technique. Because of the fact that the surface descriptors and the fuzzy feature vectors from these surfaces are designed heuristically, there is no automatic adaptability for modelling a new set of objects, and the performance results may turn out to be unpredictable. In order to overcome this drawback, we are currently designing a strategy that learns and adapts the membership functions from feedback from the recognition stage, in terms of a cost function. Further, the choice of the number of fuzzy bins and their shape are adapted through an entropy measure [31] to select the most informative fuzzy attribute. The details of this technique are a subject of another paper.

VI. CONCLUSION AND FUTURE WORK

APPENDIX I CURVATURE ESTIMATION USING DIRECTIONAL DERIVATIVES

In this paper, we present a technique of fuzzy measure aggregation for surface segmentation based on curvature properties

This appendix details the curvature estimation process using directional derivatives used in Section II.


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TABLE VI DIRECTIONAL MASKS FOR COMPUTING DIFFERENCES. (A) 0 , (B) 45 , (C) 90 , (D) 135

The input image is first smoothed by averaging pixel values . in a local neighborhood, the size of which is chosen to be The technique estimates the principal curvatures of the surface numerically from an ensemble of directional derivatives. Typically, the first-order finite differences in four directions: 0 , 45 , 90 , and 135 are computed by convolving the smoothed image with directional masks of size as shown in Table VI. The second-order difference in each direction is computed by convolving the mask and the first-order difference at that direction. The output of these masks is normalized by the sum of the absolute values of weights in the masks. From these derivatives, in direction is computed by the surface curvature

class. Corresponding fuzzy memberships need to be aggregated to the centre pixel . For appropriately to assign a label this purpose, histograms of the fuzzy memberships are determined that associate a real-valued frequency count for each of the fuzzy regions of and weighted over the sum of supports . These weighted membership values of all fuzzy regions in indicate the likelihood of the occurrence of and in various bins and they bias the associated fuzzy memberships (7) and (8) (7) (8) where

(4) and are the first- and second-order derivatives in where the direction , and and denote the first and second-order derivatives in the horizontal and vertical directions, respectively. and denote the directional derivaGiven that tives, and are then computed according to

(9) (10) The weighted membership values are then normalized according to:

(11) (12) and support sets are ready to be mapped into Now pixel’s support set

APPENDIX II FUZZY SEGMENTATION This appendix gives the mathematical formulation for the fuzzy segmentation technique discussed in Section II. the central pixel of under consideration, every Let peripheral pixel of should be assigned a label from the (see Fig. 2 for the surface types used). set The labels are assigned based on a pixel’s memberships in fuzzy and , there are nine possible fuzzy bins for both bins for curvatures (see (1). The fuzzy bin membership information is given in form of support sets defined by: (5) (6) Thus, every pixel in is assigned a pair of support sets . Since both and sets may have more than one nonzero entry more than one fuzzy rule could be applied resulting in nonzero memberships in more than one

(13) where the assignments are made based on fuzzy rules . Note that described in Section III-A.2 applied to the fuzzy rules generated are a many to one mapping in the sense that for two different input pairs the fuzzy rule maps to the same output (surface classification as shown in Table II), but with . Since varying degrees of belief reflected in values of both and are equally important in generating the rule and any combination of and results in a label assignment and as the we take the arithmetic mean of degree of belief for the assignment . For example, the rule in (2) results in assigning , this is an intuitively reasonable assignment—since in both cases the membership degrees are just 0.2 the resulting assignment should not be too far from these values either. Similarly, the rule from (4) results in

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generating four different confidence degrees (for each possible combination of fuzzy bins of and ). After the fuzzy rules are applied we have created the pixel’s support set that has several nonzero entries describing the confidence in assigning different labels. At the end of segmentation . we, however, need a Unique label assigned to each pixel in Such a label is chosen by simply taking the element with the maximum membership from the support set, or in case of several labels with the same memberships, by taking the label that is the intersection of all the label assignments

set). We instead look at the possibility measures induced by assigning weights to each source. Grabisch et al. [32] show that and are particular cases of Sugeno integrals weighted with respect to the possibility measure induced by pixels’ voting weights. Thus assigning weights to each pixel and then taking the weighted maximum of their opinions will solve the problem of information fusion at the stage of labelling the centre pixel as

(16)

(14) where

APPENDIX III FUZZY MERGING

(17)

This appendix gives the mathematical formulation for the fuzzy merging technique in Section III. At this stage, we have a set of labels and associated supports for each pixel in as given by (13). Information about peripheral pixels is used . Since every to label the central pixel with coordinates peripheral pixel may be assigned any label and the set of possible labels is the same for both peripheral and central pixels, the problem of labelling the central pixel becomes the one of information fusion and may be solved by using multi-classifier approach (first proposed by Tahani–Keller [32]). The term “information fusion” here is restricted to fusion at decision level where we need to make a decision about the central pixel’s membership in a certain class. The fusion performed is of consensus type: each pixel (expert) already has an opinion on the matter (the result of the previous step) and now these opinions must be merged in order to arrive at the decision on the higher level. The contributions of individual pixels are not equivalent and therefore some voting weights for each opinion should be taken into consideration. All of the above can be done by applying an appropriate aggregation operator. Dubois and Prade give the formal description of the requirements that an aggregation operator has to meet and explain how to construct such operator [32]. corIn our setup, we have a set of labels responding to pixels and associated with degrees of confidence . We use Sugeno integral for aggregating individual opinions (repre) in order to arrive to global decision for sented by (Tahani–Keller [32]). We thus need to evaluate

(18)

(15) is a Sugeno integral w.r.t. fuzzy measure on the set where of sources. However, taking arbitrary fuzzy measure on the set of sources leads to combinatorial explosion in terms of complexity (assuming that each sample is an -dimensional vector, we need coefficients to describe a fuzzy measure on the

The operations defined by (7)–(18) are iteratively performed over the entire image producing segmented and merged regions in the image.

APPENDIX IV NUMERICAL EXAMPLE This section illustrates the mathematical formulations of fuzzy segmentation and merging in Appendices II and III with window. a numerical example by considering an Let us assume that the central pixel of the neighborhood has , then the top left corner in any of the correcoordinates sponding tables refers to the pixel with coordinates and the lower right on to the pixel with coordinates etc. At first, we look at the support sets of the curvatures and , which describe parameters’ memberships in nine different fuzzy bins. To save space in the tables we use bin numbers rather than the names. The correspondence between numbers and names is given in Table XI. Initially, we have the data about the support sets of and for every pixel in the neighborhood. The entries in and Tables VII and VIII show which entries of the sets respectively are non-zero. For example, the first entry in the first row Table VII is (7,6) means that the only non-zero are the entries for the sixth and seventh entries in curvature bins for curvatures, i.e., that for pixel is either “P” or “CP”, the corresponding entry in Table VIII, means that the pixel’s curvature is classified only as being in the second bin, i.e., “NZ.” Tables IX and X give the corresponding fuzzy membership curvature is “P” with membervalues, so pixel’s and that it is “CP” with membership ship , similarly, the support set of has only . one nonzero entry At this point, the weighted fuzzy support measures are to be calculated according to (7), the weight biases for each of the fuzzy bins are given in Table XI, the resulting updated fuzzy


TABLE VII SUPPORT SET BINS OF

S~

S~

TABLE VIII SUPPORT SET BINS OF

TABLE IX SUPPORT SET MEASURES FOR

TABLE X SUPPORT SET MEASURES FOR

IN

N5

IN

N5

H IN N 5

K IN N 5

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TABLE XII UPDATED SUPPORT MEASURES FOR

H

TABLE XIII UPDATED SUPPORT MEASURES FOR

K

TABLE XIV NORMALISED SUPPORT MEASURES FOR

H

TABLE XV NORMALISED SUPPORT MEASURES FOR

K

TABLE XI EVIDENCE WEIGHTS OF SUPPORT SETS TABLE XVI LABEL ASSIGNMENTS. (A) DISTRIBUTION OF PIXEL LABELS IN N5

memberships are given in Table XII and Table XIII. After the weighting is done the fuzzy membership values are normalized [see (12)], the normalization results are given in Table XIV and Table XV. The data is now ready for filling in pixels’ support . As it was pointed out in the corresponding section, the sets support set of a pixel has more than one nonzero entry reflecting all the possible label assignments resulting from applying fuzzy rule base to different fuzzy bin memberships of and . To see the mechanics of constructing a pixel’s support set lets consider . The entries corresponding to this pixel are the pixel given in bold in all tables. of the pixel is classified to the bins Curvature 6(CP) and 7(P) (see Table XI), with corresponding values is classified to the bins 2(NZ) and 1(N) given in Table IX, with corresponding values in Table X. The results of weighting can be seen in Table XII and Table XIII. Normalized values are in Table XIV and Table XV. For the label assignments we use

the latter: and , similarly for and . Input to the fuzzy rules for labelling is a pair of bin laand one for , as both curvature have bels—one for two bins assigned there are four possible pairs: (6,2), (6,1), (7,2), (7,1). These four pairs trigger the multiple assignment rules listed befpre Table II. Note that the last assignment is a crisp assignment as both bins are the ones used in conventional segmentation methods. It is also straightforward to see that different pairs result in assigning the same label . since there may be only one entry for every label in Such multiple assignments are aggregated by means of a simple max operator. If more than one pair of values assigns the same label the maximal one is recorded. Thus (6,2) ; (6,1) gives gives

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TABLE XVII FUZZY SUPPORTS MEASURES FOR LABELS. (A) AGGREGATED FUZZY SUPPORT 5 FOR LABELS IN

N

every pixel in the neighborhood under consideration serves as a central pixel in some other 5 5 neighborhood and thus receives a label assigned based on the vote of surrounding pixels. This final merging results are given in Table XIX. REFERENCES

TABLE XVIII MAPPING OF NUMBERS AND LABELS

TABLE XIX MERGING RESULTS. (A) SUPPORT SET OF

gives

; (7,2) gives . These result in

P

. (B) MERGING RESULTS

; (7,1)

. Therefore, all the label assignments have the membership value of in the support set of 0.96. A unique label to the pixel should be assigned at the end of the segmentation process. To choose such a label, we take the intersection of all the label assignments, SV in the case above. It should be clear from Fig. 2 that such a unique label exists for any multiple label assignment (it corresponds to the overlap of all the regions bounded by different segments on and axes), so we omit the rigorous proof of its existence and uniqueness (done through a simple check of a finite number of possibilities for multiple assignments). Table XVII(a) shows the setup of N5 at the beginning of the merging process. Even though there are several different surface types represented within the neighborhood, the data in Table XVIII shows that most of the pixels belong to neighboring surface types and transitional regions between them. Intuitively, it is reasonable to expect that at the merging stage the most of N5 will be classified into one surface type, moreover surface type 2 (Saddle Valley) is a strong candidate as it is the label occurring most often. The merging is done by assigning the label to the central pixel of the neighborhood and is performed iteratively over all the is regions of the image. The label assignment to the pixel done according to (16) and (18). The resulting support set of is given in Table XIX and the final label assignment to the central pixel of the neighborhood is 2 or “Saddle Valley.” As the same operation is performed over different N5 windows

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[24] R. Y. Yager, “Element selection from a fuzzy subset using the fuzzy integral,” IEEE Trans. Syst., Man, Cybern., vol. 23, no. 2, pp. 467–477, Mar./Apr. 1993. [25] R. Soodamani and Z. Q. Liu, “A fuzzy Hough transform approach to shape description,” Int. J. Image Graph., vol. 2, no. 4, pp. 603–616, 2002. [26] R. Soodamani and Z. Q. Liu, “Ga-based learning for a model based object recognition system,” Int. J. Approx. Reason., vol. 23, no. 2, pp. 95–109, 2000. [27] A. K. Jain, Fundamentals of Digital Processing. Englewood Cliffs, NJ: Prentice-Hall, 1989. [28] M. J. Magee, Y. F. Wang, and J. K. Aggarwal, “Matching three-dimensional objects using silhouettes,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, no. 4, pp. 513–518, Jul. 1984. [29] A. P. Reeves and R. W. Taylor, “Identification of three-dimensional objects using range information,” IEEE Trans. Pattern Anal. Machine Intell., vol. 11, no. 4, pp. 403–410, Apr. 1989. [30] R. Soodamani and Z. Q. Liu, “Object recognition by fuzzy modelling and matching,” in Fuzzy Systems Proc., IEEE World Congr. Computational Intelligence 1998 IEEE Int. Conf. Fuzzy Systems, 1998, vol. 1, pp. 165–170. [31] I. Bratko, PROLOG Programming for Artificial Intelligence, 2nd ed. Reading, MA: Addison-Wesley, 1990. [32] E. Walker, M. Grabisch, and H. Nguyen, Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference. Dordrecht, The Nethederlands: Kluwer, 1995.

Soodamani Ramalingam received the B.Engg. degree in electyronics and communication and the M.Engg. degree in computer science, both from PSG College of Technology, Coimbatore, India, in 1983 and 1985, respectively. She received the Ph.D. degree from the University of Melbourne, Melbourne, Australia, in 1997. She was working as a Lecturer in Computer Science and Engineering at Bharathiar University, Coimbatore, India, during 1989–2002, as a Research Fellow at Nanyang Technological University, Singapore, and as Research Scientist, Associate at the Laboratories for Information

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Technology in Singapore during 2002–2002. She also served as a Senior Lecturer in the School of Computing Science at Middlesex University, London, U.K., during 2002–2005. She is currently a Senior Lecturer in the Department of Computing and Mathematics, Manchester Metropolitan University, Manchester, U.K. Her research interests include computer vision, biometrics, fuzzy systems, soft computing, computer supported collaborative work (CSCW), stereo-based face recognition, multimodal systems for smart spaces.

Zhi-Qiang Liu (S’82–M’86–SM’91) received the M.A.Sc. degree in aerospace engineering from the Institute for Aerospace Studies, The University of Toronto, Toronto, ON, Canada, and the Ph.D. degree in electrical engineering from The University of Alberta, Alberta, Canada. He is currently a Professor with the School of Creative Media, City University of Hong Kong. He has taught computer architecture, computer networks, artificial intelligence, programming languages, machine learning, pattern recognition, computer graphics, and art and technology. His interests are neural-fuzzy systems, machine learning, media computing, human-media systems, computer vision, and computer networks.

Dmitri Iourinski received the Engineer’s Diploma in design and technology of electronic computing tools from Moscow State Technical University, Moscow, Russia, in 1995, the M.S. degree in mathematics from the University of Texas at El Paso (UTEP), in 1999. Since 2004, he has been working toward the Ph.D. degree in computing science at Middlesex University, London, U.K. He was a Mathematics Lecturer at UTEP during 2000–2003. His research interests include fuzzy logic, algebraic combinatorics, multicriteria decision making, nonprobabilistic approaches to describing and interpreting uncertain information.