Shape-Based Features for Cat Ganglion Retinal Cells Classification

Real-Time Imaging 8, 213–226 (2002)doi:10.1006/rtim.2002.0281, available online at http://www.idealibrary.com on

Shape-Based Features for Cat GanglionRetinal Cells Classification

This article presents a quantitative and objective approach to cat ganglion cellcharacterization and classification. The combination of several biologically relevantfeatures such as diameter, eccentricity, fractal dimension, influence histogram, influence

area, convex hull area, and convex hull diameter are derived from geometrical transforms andthen processed by three different clustering methods (Ward’s hierarchical scheme, K-means andgenetic algorithm), whose results are then combined by a voting strategy. These experimentsindicate the superiority of some features and also suggest some possible biological implications.

# 2002 Published by Elsevier Science Ltd.

Regina Celia Coelho1,3, Vito Di Gesu

2, Giosue Lo Bosco

2,

Julia Sawaki Tanaka1,4

and Cesare Valenti2

1Cybernetic Vision Research Group, IFSC–University of Sao Paulo,Caixa Postal 369, Sao Carlos, SP 13560-970, Brazil E-mail: {reginac, julia}@if.sc.usp.br

2Dipartimento di Matematica ed Applicazioni, University of Palermo,Via Archirafi 34, 90123, Palermo, Italy E-mail: {digesu, lobosco, cvalenti}@math.unipa.it

3State University of Maringa, Campus Universitario, Av. Colombo, 5790 Maringa, PR 87020-900, BrazilE-mail: [email protected]

4IQ–UNESP– Sao Paulo State University, Caixa Postal 355, Araraquara, SP 14801-970, BrazilE-mail: julia@iq. unesp. br

Introduction

Although the morphological aspects of neurons andneural structures are potentially important, they havereceived relatively scant attention from neuroscientistsover the last decades. At the same time, increasingamount of works have shown that the neural shape isdirectly related to the respective function [1–6], in thesense that the behavior exhibited by neural cells is aconsequence of not only the biochemical processes inand outside the cells, but also of their respectivemorphology. While the majority of approaches toneuroscience has relied predominantly on the formeraspect (i.e. biochemical processes), the morphologyexhibited by neural cells is also essential in the sense

1077-2014/02/$35.00

that it not only constrains the biochemical processesinside the cells, but also defines the potential ofinteraction between each specific neuron and the restof the neural system. For instance, a neuron presenting acomplex dendritic tree will enhance the changes ofestablishing synaptic connections. In addition to largelydefining the number of synapses that a neuron receives,the neural shape is also important with respect to fieldinteractions between the cell and its surroundingenvironment.

Neural shape can be characterized quantitatively byobtaining morphometric measurements (e.g. diameter,spatial cover, complexity) allowing a reasonable repre-sentation of the physical constraints over the cells

r 2002 Published by Elsevier Science Ltd.

214 R. C. COELHO ET AL.

[7–12]. Thus, the determination of a meaningful set offeatures related to the neural behavior of the differentneural classes can help to segregate neurons intocoherent classes [7,10,13]. These sets of features shouldnot only be related just to the geometrical properties ofthe neural processes (i.e. dendrites and axons) of theneurons, but also to the types of fields of influencesaffecting those processes, because they are essential inorder to better understand the spatial cover exhibited bycells.

The primary cells used in many morphologicalinvestigations are retinal ganglion cells, because theirnearly planar dendritic arborization can be convenientlyreduced to two dimensions. Indeed, important evidencesof the relationship between shape and function havebeen verified for the specific case of cat retinal ganglioncells. Wassle [4] verified that their receptive fields arerelated to the convex hull area of the dendriticarborization, and related findings were verified byseveral researches [14–18], which defined a correspon-dence between the three morphological (a, b and g) andfunctional (i.e. Y, X and W) groups. Of these threefunctional groups, the W cells are the least known.

The present work considers a group of a and b catretinal ganglion cells and investigates several biologi-cally relevant morphological measures that can beparticularly helpful to characterize each of those groups.The objective is to try to find the best combinationamong the adopted features that can properly classifythese neurons inside each group with the smallestpossible mistake. The considered features are relatedto the shape and influence fields of neural cells, includingfractal dimension, symmetry, diameter, eccentricity andconvex hull. An analysis of the correlations between theconsidered features is also included.

The extraction of the visual information is a complexprocess that involves several different phases. One of themost common paradigms involves a hierarchy of imagetransformations from the low to the high levels of vision.The definition and the selection of suitable features playa fundamental role during the whole process [19,20],since this phase may affect the understanding of theimage data at higher levels of abstraction. Such featurespaces can be further transformed into new ones byapplying suitable operators. The two-dimensional (2D)Fourier transform is an example of image map from aspatial to a frequency space. The choice of the imagetransformation depends on the kind of image properties

that we want to put in evidence, depending on theproblem to be solved.

Geometrical entities (e.g. edges, comers and surfaces)and spatial relations are natural candidates to be used inshape classification problems (syntactic approach).However, both the computation of these entities andthe evaluation of their spatial relations are often toocomplex and the design of the shape classifier isconsequently affected by the correctness of syntacticrules that are not always standard or general. Therepresentation of an image through numerical featuresallows the problem to be treated from a statistical pointof view.

In the following, a quantitative approach thatrepresents neural cells in new multi-dimensional spacesis described. It must be pointed out that the newrepresentation space may introduce some drawbacksrequiring a careful analysis of the generated features.For instance, redundancy and ambiguity may emerge,which can be partially solved by a careful choice of thefeatures themselves. The classification phase will beperformed in the new feature space by considering threedifferent classifiers: Ward’s hierarchical grouping, K-means and genetic algorithm, integrated by a weightedvote technique [21].

Shape-based Features

Ganglion retinal cells can be mathematically understoodas sets of connected points in a two or three-dimensional(3D) space F, which can be approximated in a discretebinary image space. Neuron classification performeddirectly on F is a hard task that could require O(N2)comparisons, assuming that each image has N pixels.

The representation of an image can be modified byapplying suitable image transformations (IT) mappingfrom F to a new, and typically smaller, feature space F’.An IT is said to be sound if F’ is invariant fortranslation, rotation and scaling. This property reducesthe size of the search space, therefore making theclassification easier. Translation invariance is usuallysatisfied, while scale and rotation invariance are notalways fully satisfied. However, normalized features, i.e.that range in a fixed interval, are most of the time scalinginvariant. Rotational invariance needs the definition ofisotropic ITS. In the following, three image transformfamilies used in the present approach are described indetail.

SHAPE-BASEDFEATURES FORCELLSCLASSIFICATION 215

Influence area transform

Measures of the influence field of a cell, which isdirectly related to its size, complexity, and spatialcoverage, present good potential for the identificationof neural cells. A description of these features,considered in the present work, is presented in thefollowing.

Influence area. The influence area (IA) indicates the areaaround a cell, defined by the respective dendriticarborization [7,22], which is relevant to the type ofinfluence field under consideration. For instance, in casewe are interested in synaptic contacts and know thatthese occur up to a maximum distance of 1 mm aroundany point of any dendrite, the region thus defined iscalled the influence area of that cell with respect to thattype of interaction. A natural geometric way torepresent IAs is in terms of Minkowski Sausages [22].Given a specific shape corresponding to the contours ofa 2D neural cell, it is possible to obtain, by usingdilations [12,23], a series of Minkowski sausagesfor successive radius values, as illustrated in Figure 1for sausage radii equal to 3, 6, 9 and 12 for eachcell, respectively. Interesting feature vectors expressingthe influence pattern of the cell, henceforth calledinfluence area histograms, can therefore be obtained byestimating the area of each sausage for each consideredradius. Figure l(k) presents the influence area histogramfor the images. Observe that, if necessary, it is alsopossible to normalize the areas by previously mappingthe cells onto a fixed size box (or dividing by thediameter of the cell), in such a way as to obtain scaleinvariance.

Influence histograms. While the above-presented influ-ence area histogram considers a uniform pattern ofinfluence around the cell processes, it is also possible toallow the influence to be graded, leading to the conceptof influence histograms (IH). Such graded influencesarise naturally in several relevant neural contexts, suchas the interaction of the membrane of the cell with asurrounding electric field. The IH, first proposed in [22],presents good potential for adequately characterizingthe spatial covering degree occupied by cells and hasbeen shown to be an interesting feature to classification.The first step to obtain an influence histogram is todefine the point-spread function defining the type ofinteraction under study. For instance, in case we areinterested in the electric potential around the cell, thepotential around a point charge should be used as point-spread function. Once this function is defined, it is

convolved with the neural shape in order to obtain thewhole field of influence around the cell. Now theinfluence histograms are obtained by histograming thefield intensities obtained around the cell.

In the current work, we consider Gaussian distribu-tions as point-spread functions. Let f(x, y, z) be amultivariate and symmetric Gaussian distribution,function defining the distribution of an influencefactor around cell. Considering that the shape of aneural cell is represented by binary data I(x, y, z), theinfluence histogram can be calculated by convolved f(x,y, z) with the neural cell, i.e. T(x,y,z)=f(x,y,z) n I(x,y,z),where T(x,y,z) is the resulting image. The histogram ofsuch a resulting image, which is computed only upto a maximum distance to the shape in order toallow more uniform results, has the potential to providerelevant information about the degree of spacecoverage exhibited by the cells. For instance, a morecomplex and dense cell will tend to present a histogramwith a larger average than a simple cell. The processof deriving an influence histogram is illustrated inFigure 2.

Fractal dimension. Complexity measures have been fre-quently used for shape classification, including theclassification of neural cells. Boycott and Wassle [14]were one of the first to show the differences among theganglion cells of the retina of the cat, showing that thecell complexity is an important measure to characterizethese cells. One of the most used complexity measures isthe fractal dimension (FD) [22,24–26]. In this paper, weconsidered the Box-Counting (BC) and the Minkowskisausage (Mi) [22,27] methods, but with the differencethat only the portion of the log–log curve with largestfractality in the FD estimation was considered (a furtherextension of this concept has been described in [12]).This is done in order to compensate the problem of thelimited fractality exhibited by natural objects [27]. Thecalculation of the FD by Box-Counting considered theaverage values of the numbers of boxes for inclinationsof the grid (the rotated grid in different angles and theaverage number of boxes for all inclinations wasconsidered [13]) so that calculation is more precise.The Minkowski sausages method is based on the factthat more complex shapes tend to produce Minkowskisausages with areas that increase less steeply. In otherwords, such shapes impose more constraints on thegrowing sausages. Computationally, the fractaldimension can be estimated by this method by firstobtaining the influence area histogram as described inthe on Influence area section organizing this histogram

Figure 1. Minkowsky sausages of the neuron in (a) with respect to sausage radii equal to 3 (b), 6 (c), 9 and 12 (d), and of theneuron in (b) for the same radii used in (a). Observe that the original contour can be understood as the sausage for radius 0. Theinfluence area histogram for the image (a) and (f) for radii 1–20 is presented in (k).


Figure 1. Continued


as a log–log function (i.e. logarithm of the area in termsof the logarithm of the radius), and interpolating astraight line around the region presenting the smallestderivative. The fractal dimension is taken as 2 minus theslope of this line.

Axial moments

An object is said to exhibit symmetry if the applicationof certain isometries, called symmetry operators, leavesit unchanged while parts are permuted. For instance, the

Figure 2. Influence histograms (e) obtained for the two distinct complexity neural cells (a, b). The histograms express thedistributions of the graded influence fields (assuming a Gaussian point-spread function) around these cells (c, d).



letter ‘‘A’’ remains unchanged under reflection, the letter‘‘H’’ is invariant under both reflection and half-turn, thecircle, ‘‘*’’ has an annular symmetry around its center.Symmetry operators have already been applied torepresent and describe object-parts [28] and to performimage segmentation [29].

The definition of our symmetry transform (ST)starts from the computation of a set of normalizedfirst-order axial moments (AMs) of the gray-levelintensities g around the center of gravity of each cellC [30]:

AMk ¼X

ðm;nÞ2C

jm sinjk � n cosjkjgðm; nÞ;

where k ¼ 1; . . . ;K andjk ¼ kp=K :

For the purpose of this work, we have set the number ofaxes to K=32, according to previous experimentalresults. It is easy to prove that all AMs will have thesame value in the case of an isotropic density distribu-

Figure 3. The axial moment values of an a (a) and a b (b) cell (reproduced here with permission).

tion, hence they can be used as a circular symmetrymeasure:

ST ¼ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�kAM

2k

K�

�kAMk

K

� �2s:

The ST is invariant for translation, rotation and scaling;therefore it is sound. The first property derives directlyfrom the analytical definition of the AMs. The secondproperty derives from the fact that object rotationimplies the circular permutation of the AMs. The thirdproperty is satisfied thanks to the normalization.

An example of the AM feature is shown in Figure 3.It is interesting to note that all AMs can be quicklycomputed in parallel as a sum of convolution filters.This approach can also be generalized for 3D objectrecognition [31].

Three shape parameters are derived from the AMs:their mean value (MV), their standard deviation (SD),

these neural cells figures, originally published by Saito [16], are

Table 1. Mean (m) and standard deviation (s) of MV, SD, EC

Feature mp mb sp sb

MV 0.936 0.828 0.026 0.081SD 0.007 0.017 0.003 0.011EC 0.979 0.942 0.010 0.039


and their eccentricity (EC=AMmin/AMmax), whereAMmin, and AMmax are the minimum and maximumvalues of the AMs. The MV and SD features arenormally distributed in first approximation. We as-sumed it to be true also in the case of the feature ECwhich is the ratio of two normally distributed quantities.Table 1 reports the mean (m) and standard deviation (s)parameters, while Figure 4 shows the probabilitydistribution.

Convex hull

The size of a neural cell provides an important resourcefor its identification, being often used by neuroscientistsas one of the most important features for the classifica-

Figure 4. Probability distribution for MV, SD, and EC.

tion of neurons. The size of a shape can be defined inseveral ways, including the convex hull and the shapediameter, which are addressed in the following.

Convex hull area. The convex hull (CA) is consideredas the smallest convex polygon that totally contains aspecific neuron. It has been verified experimentally [4]that the receptive field of a cell can be predicted withgood precision as being approximately 25% larger inarea than the convex hull of the dendritic arborization(Figure 5).

Diameter. The diameter (Di) is another feature used toglobally characterize the neuron size and spatialinfluence. It can be obtained from the CA of the cellas the largest distance between any of its two points, asillustrated in Figure 5.

The Classifiers

Since we are interested in investigating how themorphological cell classes are defined, an unsupervisedclassification has been adopted. Three classification

Figure 5. The convex hull and respective diameter of a neuralcell.

Figure 6. The combination tree for IA, ST and CH.

Table 2. Morphological features used to characterize cells

IT Morphologicalfeature

IA BC Box counting fractal dimensionIA Mi Minkowski sausage fractal dimensionIA IH1–11 Influence histogramsIA IA1–20 Influence areas by ratioST AM Axial momentST MV Mean valueST SD Standard deviationST EC EccentricityCH CA Convex hull areaCH Di Diameter


algorithms (Ward’s hierarchical grouping (WG), K-means clustering (KM) and genetic algorithm (GA))have been used and are briefly described in thefollowing.

This work has considered a set of 50 cat retinalganglion cells [14–16,32–36], 23 being of a type and 27 ofb type. All these cells present distance to the foveasmaller than 31 and were normalized such that they allpresent the same diameter.

The New Feature Space

Figure 6 describes the combination of the transformintroduced above as a tree, where the root is the inputfeature space F, the leaves represent the new features ofF

0

, and the internal nodes are the IT0

s.

The features considered here are listed in Table 2.They are related to measures normally used tobiologically classify these cells. The subscripts of theIH feature indicate the box number in the histograms.We have considered a histogram with 11 boxes(numbered 1–11). The subscripts of the IA featureindicate the radius of each sausage considered. We haveconsidered 20 areas of the sausage (radii varying 1–20).

Ward’s hierarchical grouping

Ward’s hierarchical grouping, which is one of the mostfrequently adopted clustering methods, relies on pro-gressively agglomerating the data in such a way as tominimize the dispersion (variance) inside each group[37,38]. It tends to produce clusters that are easily

distinguished from other clusters and which tend to betightly packed. It operates by reducing the number ofclusters one at a time starting from one cluster per objectand ending with one cluster comprising all the objects.At each cluster reduction, the method merges two ormore objects. The confusion matrices of the WGclassification are given in Table 3.

K-means clustering

K-means clustering involves an iterative scheme thatoperates over a fixed number of clusters, such that eachclass has a center which is the mean position of all thesamples in that class and each sample is in the classwhose center is closest to it [39]. The confusion matricesof the KM classification are given in Table 3.

Genetic algorithm clustering

The genetic algorithm clustering [40] is a variant ofISODATA [41]. It shows a faster convergence to thesolution by considering the clustering as a globaloptimization problem, where the optimum solutioncorresponds to the minimization of the within-clusterscatter matrix (maximization of the between-cluster

Table 3. Confusion matrices of the classifiers with respect to their best results

Feature sets K-means Ward Genetic algorithm Vote strategy

1. (AM, MV, BC, Mi, IH)19 40 27

� �22 11 26

� �22 14 23

� �22 10 27

� �

2. (AM, MV, Mi, IH)21 21 26

� �20 30 27

� �23 06 21

� �22 11 26

� �

3. (AM, BC, CA, Di)22 11 26

� �20 31 26

� �23 03 24

� �22 11 26

� �

4. (AM, SD, EC, BC, IH)22 11 26

� �20 31 26

� �22 13 24

� �22 11 26

� �

5. (SD, EC, BC, Mi, IH)22 13 24

� �22 11 26

� �20 31 26

� �22 11 26

� �

6. (AM, MV, BC, CA, Di)23 02 25

� �18 50 27

� �23 03 24

� �23 02 25

� �

7. (AM, Di)23 03 24

� �23 03 24

� �23 03 24

� �23 03 24

� �

8. (AM, MV, CA, Di)20 30 27

� �23 03 24

� �23 03 24

� �23 03 24

� �

9. (MV, SD, BC, Mi, IH)20 30 27

� �20 30 27

� �20 31 26

� �20 30 27

� �

10. (MV, SD, EC, Mi, IH)21 21 26

� �20 30 27

� �20 31 26

� �20 30 27

� �

11. (AM, MV, SD, EC, BC, Mi,

CA, Di, IH)

20 30 27

� �20 30 27

� �22 13 24

� �20 30 27

� �


scatter matrix) [42]. Initially, the sampling strategyallows us to span the solution space in a random way.The evolution is fast driven (on average by 123iterations) toward an optimal (or near-optimal) solu-tion, according to the schema theorem [43].

The feature vector of each cell and its cluster-label arecoded through a chromosome of 24 and 8 bits,respectively. This method starts by generating an initialpopulation of such chromosomes and then evolves thesebinary strings by crossover (replacing with a probabilityof 90% two of them by a pair of their offspring) and bymutation (altering with a probability of 1% some oftheir bits). The fitness function is based on the Euclideandistance computed between each chromosome and thecenter of gravity of each cluster. The evolution allows torearrange the assignment of the cells by minimizing theinternal variance of each class. The selection of the newpopulation is performed by using the ranking technique.The confusion matrices of the GA classification havebeen reported in Table 3.

The Classifier Integration

The above-described classifiers can be combined toimprove the classification accuracy. The main motiva-tion for considering combined technique is that humandecision models also use more than one evaluationparadigm, and usually a complex decision is taken bymore than one expert [21,44,45].

The integration of decisions is becoming a verypowerful approach in data analysis systems. Theapplication of a class of classifiers Cl={Cl1, Cl2, . . .Cls}on a given data set X produces a set of partitionsM={M1,M2, . . .,Ms}, whereMi is the confusion matrixof the ith classifier Cli. An indicator pi, related to theaccuracy of the method, can be assigned to eachpartition Mi, thus defining the distribution�Cl ¼ fp1;p2; . . . ;psg with 0opio=1. �Cl can beevaluated on the basis of the judgment of an expert orby means of a calibration procedure during the trainingphase; sometimes a theoretical evaluation of the

Figure 7. The parallel topology for the integration of theclassifiers.


accuracy of a classifier can be used. Intuitively, �i

weights the goodness of Cli,. Note that, without a prioriknowledge, pi=1/s.

Integrated decision techniques can be described byusing graph-theoretical approaches. Graphs areweighted and directed; labeled nodes represent: theclassifiers Cl, the input data set X and the partition M.The topology of the graph is related to the decisionstrategies adopted. In particular, we adopted the paralleltopology shown in Figure 7 because our classifiers donot exchange any information during their computation.Each classifier performs the computation independently:it receives the two inputs ðX ;�ClÞ and produces theresult M.

The evolution of the integrated classifier is determinedby the following system:

M ¼ GðM1;M2; . . . ;Ms; p1; p2; . . . ; psÞ

p ¼ mðp1;p2; . . . ;psÞ

where the functions G and m are usually polynomials.For example,

M ¼P

piMiPpi

; p ¼1

s

Xpi

The parameter p is usually considered as a globalmeasure of performance. In our case, the values of pi are

computed from

Mi ¼Maa Mab

Mba Mbb

; pi ¼

ðMaaÞi þ ðMbbÞi50

since we studied 50 cells.Each entry of Mi indicates the agreement betweentwo methods. This allows to compare the decisionof the classifier Cli against a human expert: a perfectagreement holds if Mi is diagonal. Table 4 reports pifor the three classifiers we used. The weight of eachclassifier has been derived on the basis of its accuracy. Inorder to obtain the most correct grading, we applied avoting strategy among the results provided by theclassifiers. In particular, a given cell x is assigned tothe class a if

Pxipi=

Ppio0:5, to class b otherwise.

This formula is a weighted mean, where xi represents theassignment by the classifier Cli of x to a (xi=0) or tob(xi=1).

Methodology

In order to validate the potential of the adoptedmeasures for the characterization and classificationof neural cells, 50 images of cat retinal ganglioncells [14–16,33–36,39] of types a and b, all presentingdistance of the fovea smaller than 31, have beenconsidered.

Influence histograms assuming a Gaussian profilewith a standard deviation of 4, and containing 11uniformly distributed bins, were generated for eachimage. Influence areas with Dist varying from 1 to 20 arealso considered for each neuron. In order to investigatethe influence of the size of the cells, the diameter of eachcell was obtained from the respective convex hulls ofthe cells. To measure the complexity of each image, thefractal dimension was calculated by using both BoxCounting and Minkowski sausage methods. Note thatto improve the clustering results, every feature wasnormalized through a statistical transformation in sucha way as to present null average and unitary standarddeviation.

The calculation of central and axial moments hasintroduced new parameters with complementary infor-mation as compared to those obtained by other shapeindicators. Moreover, their soundness allowed us toimprove the discrimination ratio of the two types ofcells.

Table 4. Performance of the classifiers with respect to their results

Feature setsK-means(%)

Ward(%)

Geneticalgorithm (%)

Votestrategy (%)

1. (AM, MV, BC, Mi, IH) 0.92 0.96 0.90 0.982. (AM, MV, Mi, IH) 0.94 0.94 0.88 0.963. (AM, BC, CA, Di) 0.96 0.92 0.94 0.964. (AM, SD, EC, BC, IH) 0.96 0.92 0.92 0.965. (SD, EC, BC, Mi, IH) 0.92 0.96 0.92 0.966. (AM, MV, BC, CA, Di) 0.96 0.90 0.94 0.967. (AM, Di) 0.94 0.94 0.94 0.948. (AM, MV, CA, Di) 0.94 0.94 0.94 0.949. (MV, SD, BC, Mi, IH) 0.94 0.94 0.92 0.9410. (MV, SD, EC, Mi, IH) 0.94 0.94 0.92 0.9411. (AM, MV, SD, EC, BC,

Mi, CA, Di, IH)0.94 0.94 0.92 0.94


Results and Discussion

The linear interrelationship between neural features canbe analyzed by the correlation matrix shown in Table 5.This table shows only three out of the 11 bins ofinfluence histogram and five out of the 20 bins ofinfluence area. It should be recalled that value 1indicates a perfect correlation among two features,which only happens, in this case, for the same features.A negative value indicates an anti-correlation. Markedcorrelations (absolute values larger than 0.5) areconsidered strong. Classifications have been performedusing all possible combinations of 2, 3, 4, 5 and 9features, as well as by just using one and all features. Allclassification methods have been applied for suchcombinations. Although IA has presented a strongcorrelation with many other features, it turned out to bethe worst feature. Alone or combined with otherfeatures, it always tends to undermine the classification.Therefore, it is excluded from the following comments.

The features were grouped into the three majorfeatures groups: (1) symmetry features (AM, MV, SD,and EC); (2) complexity features (BC and Mi); and (3)size features (CA, Di, and IH), that considered the sizeof the neuron, Except for group (Di, IH), that presentedsatisfactory results (error�10% by all classificationmethods), any combination using only features of thesame groups led to a poor result. This is a consequenceof the fact that both features are related to the cell size(mainly Di), one of the features used by neuroscientist toclassify neural cells. The group (AM, Di) showed asbeing suitable for the classification (presented errorequal to 6%). This was the only group with two or threefeatures that presented such a small error. Another

interesting result was obtained by using all the features,except IA. The error presented was 6%.

Most of the best classifications were obtained usinggroupings of 4 or 5 features together. Good results wereobtained when we took at least two symmetry features,at least one complexity feature and IH. The IHcombined with others also showed to be a good feature.

Most of the worst results were obtained for groups of1, 2 or 3 features. In such cases, the groups containedonly symmetry features, or symmetries and complexityor still only a symmetry feature. The error in these caseswas larger than 20%.

The number of cells assigned to each class withrespect to the best classification results is shown in Table3. In all these cases, there was at least 1 wronglyclassified cell.

It should also be observed that the relatively highmisclassification rates obtained are very likely explainedby the fact that the class assignment used as acomparison standard was produced subjectively bydiverse authors, using different criteria.

Conclusions

A quantitative approach to neural cell classification,concentrating on cat ganglion cells, has been reportedconsidering several geometrical features, and theircombinations have been investigated with respect tothree clustering algorithms. The obtained results in-dicated the superiority of some specific features, as wellas a few incompatibilities with the original classifications

Table

5.Correlationmatrixamongthefeaturesconsidered

N=50(Casewisedeletionofmissingdata)

AM

MV

SD

EC

BC

Mi

CA

Di

IH1

IH6

IH11

IA1

IA5

IA10

IA15

IA20

AM

1.0000

MV

�0.8133

1.0000

SD

0.5913

�0.7140

1.0000

EC

�0.6594

0.7897

�0.9905

1.0000

BC

0.7347

�0.8536

0.6901

�0.73071.0000

Mi

�0.1694

�0.0046

�0.1239

0.13940.15181.0000

CA

�4585

0.6310

�0.5170

0.5316

�0.5875

�0.00101.0000

Di

�0.5882

0.7752

�0.6265

0.6518

�0.76290.0082

0.93511.0000

IH1

�4200

0.5893

�0.4766

0.4902

�0.51390.0394

0.9508

0.90071.0000

IH6

�0.1605

0.1389

�0.2214

0.2326

�0.19560.39510.06250.1640

�0.06181.0000

IH11

�0.4269

0.5638

�0.4335

0.4579

�0.59570.0181

0.6450

0.7384

0.57720.13541.0000

IA1

�0.5006

0.6772

�0.5663

0.5872

�0.60200.1701

0.9377

0.9560

0.91150.2414

0.68171.0000

IA5

�0.4842

0.6657

�0.5487

0.5674

�0.59070.1124

0.9632

0.9587

0.95870.1266

0.6632

0.98981.0000

IA10

�0.4753

0.6569

�0.5376

0.5550

�0.59680.0603

0.9747

0.9593

0.97390.0847

0.6626

0.9759

0.99571.0000

IA15

�0.4770

0.6584

�0.5370

0.5543

�0.60920.0314

0.9777

0.9619

0.97360.0774

0.6647

0.9692

0.9912

0.99891.0000

IA20

�0.4810

0.6626

�0.5397

0.5570

�0.62000.0160

0.9783

0.9647

0.97130.0771

0.6672

0.9661

0.98840.9975

0.99971.0000


made by human operators. The axial moment anddiameter, in particular, resulted to be particularlyeffective. Such results indicate that the propertiesquantified by these geometrical features likely havespecial relevance to the behavior of the respective classesof cells. Moreover, the features and the methodologiesintroduced here are general and can be easily extendedto solve different kinds of object recognition problems.

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Documents

Shape-Based Features for Cat Ganglion Retinal Cells Classification