Pencil and Entropy

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O R I G I N A L

Evandro Luis Linhari RodriguesValentin Obac Roda

Methodology for classification of images based on thecharacteristics of a new interpretation of Shannons entropy

Received: 20 April 2004 / Published online: 21 January 2005 Springer-Verlag 2005

Abstract A new visual inspection method for the classification of wooden

plates used in pencil manufacturing is presented. Darker regions in woodenplates indicate the presence of growth rings which are regions of hard wood.Pencils manufactured with these plates are more difficult to sharpen and have atendency to bend and crack; therefore, these plates are classified as not beingadequate for pencil manufacturing. The proposed method is based on theextraction and analysis of the features of the wooden plates using gray levelimages. The method classifies the plates using the results obtained by anautomatic threshold determination based on Shannons entropy. The methodwas idealized aiming at low computational complexity, i.e., algorithm calcu-

lations involving only simple operations such as addition, subtraction, multi-plication and division which could be implemented in hardware using VLSItechnology or programmable logic devices. The wooden plate is mapped in anoptimal number of regions; each region is pre-classified considering some rel-evant features based on the entropy gray level distribution of the pixels.Information from all regions is combined based on heuristic decision rules,arriving in a pre-classification stage where the regions are labeled in four classes(A, B, C and X). Two decision algorithms have been investigated for the finalclassification: one based in a co-occurrence matrix considering only a uni-directional horizontal neighborhood of the regions and the second is based on a

heuristic method of information reduction considering combinations of the pre-classified regions. The final results obtained by the two algorithms were com-pared with the classification made by a human expert, demonstrating that theproposed method had very good performance.

Keywords Visual inspection Pattern classification Entropic thresholding Wooden plates

E. L. L. Rodrigues (&) V. O. RodaDepartamento de Engenharia Ele trica-EESC-USP,Escola de Engenharia de Sao Carlos, Av. Trabalhador Saocarlense 400,CEP 13566-590 Sao Carlos (SP), BrazilE-mail: [email protected] Tel.: +55-16-33739339 Fax: +55-16-33739372E-mail: [email protected] Tel.: +55-16-33739335 Fax: +55-16-33739372

Wood Sci Technol (2005) 39: 113128DOI 10.1007/s00226-004-0276-y


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0. Raw materialrenewable plantation of incense-cedar and pine trees1. Wood is cut into blocks.2. Blocks are cut into small plates.3. Plates receive a painting and wax treatment. (The plate selection procedure

used in this work is done after this stage.)4. Grooves are made for the graphite leads.5. Leads are distributed on the grooves.6. A second plate is grooved and glued to the first one as in a sandwich.7. The sandwich is machine processed to a pencil shape.8. Each pencil is cut from the sandwich and submitted to a finishing pro-

cess.9. The pencil is painted.

10. In some models, there is the placement of an eraser.

Visual inspection in wood

Many solutions have been proposed for the identification of wood defects,aiming to produce an acceptance/rejection criterion and also for automation ofwoodcutting in sawing mills (Conners and Harlow 1980; Conners et al. 1983;Wagner et al. 1991; Funt and Bryant 1987; and Koivo and Kim 1989). Theliterature of visual inspection in wood, in its great majority, presents articlesfocused on the detection of wood defects for the furniture and wood partsindustry. The solutions found in the literature try to identify, classify and selectthe most common defects such as knots (unsound, hole, arris, vight), blue stain,split, decay, warping, twist, resin pocket, hull, wane, cupping, insect hole, etc.Articles reporting investigation on wood use for pencil production and its re-sults are scarce; the results of studies and research on the subject are of extremeimportance to the pencil manufacturers and, when revealed, can help compet-itors. The level of information normally published, as presented in articles byRhemrev et al. 1993, 1991, are of minor importance to competitors.

In the pencil production cycle, illustrated in Fig. 2, the great majority of the

published works could be applied in the early stages (1 and 2), where the wood isselected/classified for the presence of common defects like knots, insect holes, etc.The methodology developed in the present work is applied in the third stage

of the pencil production cycle (Fig. 2), using the form and distribution of thegrowth rings hard wood for its classification.

Fig. 2 Stages of the pencil production process (figure extracted and reworked from http://www.pencils.com/)

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Class definitions

In a first approach, the plates can be separated into three main classes. The firstclass includes the plates which do not have hardwood (also called smooth ones);these plates are the best. The second class includes the plates obtained by aradial cut from the log; the hardwood from the growth rings are shown as darkstripes in a regular but not very dense distribution. The last class comes from atangential cut from the log; they have more hardwood than the previous cat-egories, are darker than the previous classes and have a random superficialdistribution. The mentioned classes are illustrated in Fig. 3.

Although the three classes allow differentiating the plates, the surface pat-terns of the plates are highly random and diversified. There is not a sharpborder between the classes, giving margin to ambiguous classification. In an

attempt to minimize the occurrence of ambiguous interpretations, two moreclasses were created, exactly on the border areas among the classes. This pro-cedure groups the plates whose characteristics present a transition behaviorbetween two neighboring classes, i.e., the smooth class is neighbor to the radialcut class (and vice versa), the radial cut class is neighbor to the tangential cutclass, but the smooth class is not neighbor to the tangential cut class. Therefore,the following five classes were defined:A Smooth wood plate (best class).

B Smooth and radial cut transition wood plates.

C Wood plates obtained by a radial cut.

D Radial and tangential cut transition wood plates

X Wood plates obtained by a tangential cut.

Subdivision of the pencil slats in regions (windowing)

It is impossible to classify plates using global algorithms based on thresholdbecause, although it is possible to observe grouped values by different catego-

Fig. 3 Wood cuts for pencil manufacturing. a Smooth wood plate, does not have hard wood.b Wood plate obtained by a radial cut from the log. c Wood plate obtained by a tangential cutfrom the log

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ries, the high deviation of the categories make an effective classification verydifficult without ambiguities. Therefore, it was decided to subdivide the plates ina fixed number of areas that allow an interpretation based on previousknowledge.

With each plate it is possible to make up to nine pencils, thus it was decidedto divide the plate into nine lines enabling attribution of a partial classificationof the plates. Regarding the ideal number of columns, heuristically it was foundthat the best subdivision number is the one that incurs the smallest error whenclassifying the areas. Thus, it was decided to divide the plate into four columns,totaling thirty six processing areas. The plate subdivision is shown in Fig. 4 fora randomly selected plate; this plate will be used as an example to explain themethodology.

Plate characteristics extraction

The algorithm used to extract the plate characteristics was based on an auto-matic threshold method, which uses maximization of Shannons entropy(Shannon 1948) (Eq. 1) to determine the optimal threshold value.

H Xn

i1 pi logpi 1 pi log1 pi 1

In Eq. 1, H is the entropy, n is the number of gray levels, and in this articlep is the probability of a pixel belonging to the background, while (1p) isthe probability of any existent structure not belonging to the back-ground. When all samples of the distribution have the same probability,Eq. 1 behaves as shown in Fig. 5; this is an ideal case that rarely occurs in reallife.

The automatic threshold method finds the threshold for each region of theplate, when its entropy is maximal. As presented in Fig. 5, the maximum en-tropy occurs when p=0.5 in distributions that have the same probability for allthe samples. In the present work we used the Shannons entropy equationpresented by Wong and Sahoo 1989 that deals with a posterioriprobability to abinarized image, as showed in Eq. 2.

HPt Xt

i1

Pt logPt 1 Pt log1 Pt 2

Fig. 4 Plate subdivision in36 processing areas

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In Eq. 2, Pt is the a posterioriprobability of pixels with gray level values smallerthan t when a threshold t is applied in the image (or region of image) resultingin a two-level image and is given by:

Pt Xt

i0

pi 3

where

pi fi

N04

and

N0 Xl1

i0

fi I 0; 1; . . . ; l 1 5

Here, t is the threshold value, pi is the probability of gray level ioccurrence in aregion, fi is the gray level frequency, and N0 is the number of pixels in the region.

Similarly, a posterioriprobability of all pixels with values equal or larger than

t is 1-Pt. Another consideration that must be made is that Pt log (Pt) = 0 whenPt = 0 and (1-Pt) log (1-Pt) = 0 when (1-Pt) = 0.Besides the automatic threshold, the extraction of other characteristic values,

intrinsics to the method, were used in the pre-classification algorithm. Twoimportant values are extracted from the graph, the total entropy and theasymmetry ratio. The total entropy, the integral of the entropy equation, is animportant indication of the amount of information present in the region, i.e.,the greater its value, the greater is the presence of shapes not belonging to thebackground, indicating a non uniform region. The asymmetry ratio is the ratiobetween the two areas (a and b) shown in Fig. 6 which are formed by splitting

the entropy curve for a given threshold value. As it deviates from unity, theasymmetry ratio indicates the presence of pattern modifications and alsomodifications that occupy small areas with a high contrast in the region.Asymmetry ratios smaller than unity indicate contrast regions of areas withpredominant dark areas, and asymmetry ratios greater than unity indicate re-gions with a larger number of clear areas.

The behavior of the two portions Pt log(Pt) and (1-Pt) log (1-Pt) in Eq. 2,correspond to the absolute difference between the gray levels taken from the

Fig. 5 Normalized entropyconsidering two equallyprobable possibilities p e (1-p)and the same probability forall samples of the distribution

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peaks (assigned in Figs. 7 and 8) of the two portions. High values of differenceindicate the presence of large contrasting areas in the region.

The ratio between the threshold level found in the region and the averagethreshold in all the plate is taken as a further decision element in addition to thefour characteristic values presented.

Fig. 7 Pt log(Pt) of region 9 from Fig. 4 showing the difference between the threshold found inthe region and the gray level at the maximum peak in the curve

Fig. 6 Example showing the entropy curve asymmetry of region 9 from Fig. 4 for a giventhreshold value (203)

Fig. 8 (1-Pt) log (1-Pt) of region 9 from Fig. 4 showing the difference between the threshold

found in the region and the gray level at the maximum peak in the curve

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The pencil slats classification

Classification of the plates can be divided in two stages: a pre-classification anda final classification stage. In the pre-classification stage, some values areinterpreted as a band pass decision and others as a border decision, whichmeans that under and over a fixed value they will be rejected according to thatcriteria. The limits were established after observation essays which defined fourclasses (A, B, C and X) attributed to the regions, the combination of these fourclasses would define the final class.

It must be noted that in the pre-classification of the plate areas (or processingwindows) only four classes are used while in the final part of the method there arefive classes (A, B, C, D and X, with A representing the best class and X the worst asdefined in Sect. 4). Letters (A, B, C and X) were maintained only to correspond

with the definitions presented previously, and to keep the degree of quality hier-archy in the classification of the area, because it is not possible in a single area tofind, for example, all the global characteristics that define a class X plate.

The decision algorithm presented here is based on boundary rules and acts inthe pre-classification stage, attributing a class label to each area.

The above algorithm always assigns a class to the region, not considering therejection option. An example of the result of the pre-classification, where thelabels are attributed for each area is presented in Fig. 9.

Starting from the above pre-classification, a trained human operator wouldeasily complete the classification process, attributing a global class to the plate.

The operator, in his interpretation, is capable of considering that the presenteddistribution of areas possesses information of the type: clusters position of sameclass, approximate amount of areas belonging to each class, type of presentareas, that is to say, a good notion of the behavior of the group.

It can happen that different operators, with the same degree of training, wouldoccasionally disagree about the classification and produce different final results.This is because the classification tends to be tedious and subjective, which natu-rally indicates some superposition of classes and uncertainties in its definitions.

If (total entropy) < 10.0and (difference between gray levels in the partial peaks) < 9and (region threshold ratio over the average threshold) >= 0.94

and 0.6 = 0.94and 0.7 = 0.92

and 0.65


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Certainly the final contextual understanding of the pre-classified image isrelated to the utilization of classifiers. As a solution, two methods, which

complete the classification of the plates, are proposed.

Classifier by information reduction based in combination rule

This classifier is based on the information reduction through the previousknowledge of combination of the classes. In the first stage, the concept is usedto combine four areas generating just one. A table of equivalencies was built toassociate the combinations and its resulting classes (Table 1). To reduce thenumber of combination possibilities, the permutations inside the same combi-nation were not considered. The combinations AAAB, AABA, ABAA

Table 1 Relationshipsamong combinations of fourclasses, sum of the ASCIIcodes of the combinationsand the resulting classes

Classcombinations

Sum of theASCII codes

Resultantclass

AAAA 260 AAAAB 261 AAABB 262 AAAAC 262 AAABC 263 B

BBBA 263 BAACC 264 BBBAC 264 BBBBB 264 BBBBC 265 BCCAB 265 CCCCA 266 CCCCB 267 CCCCC 268 CAAAX 283 CAABX 284 CBBAX 285 C

ABCX 286 CBBBX 286 CBBCX 287 CCCAX 287 CCCBX 288 CCCCX 289 CXX?? >=306 XXXX? >=329 XXXXX 352 X

Fig. 9 Pre-classified plate

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and BAAA, have the same resulting class A, which means that in a line anypermutation of elements do not influence the resulting classification. Also, thenumber of possible combinations in an arrangement of four positions with fourvariables results in 256 combinations, and Table 1 shows that not consideringthose permutations, there was a reduction of ten times the final number of theresulting class. To facilitate the interpretation of the classes combinations, thesum of the ASCII codes of the letters that compose the combination are usedfor making the decision. In Table 1 we can observe sequences and very definedlimits, except for a few situations which lack a complementary approach formaking the decision.

Fig. 10 Informationreduction per line

Table 2 Relationshipsamong combinations of threeclasses, sum of the ASCIIcodes of the combinationsand the resulting classes

Classcombinations Sum of theASCII codes Resultantclass

AAA 195 AAAB 196 AAAC 197 BBBA 197 BAAD 197 BABC 198 BBBB 198 BABD 199 BBBC 199 B

ACC 199 CACD 200 CBBD 200 CCCB 200 CBCD 201 CCCC 201 CADD 201 DCCD 202 CBDD 202 DCDD 203 DAAX 218 BABX 219 C

ACX 220 CBBX 220 CADX 221 DBCX 221 DBDX 222 DCCX 222 DCDX 223 DDDD 204 DXX? >= 241 X

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Combining the four labels attributed to each line, only a label per line isgenerated, culminating in a vector-column of results that follows the combi-nation of Table 1. This is illustrated in Fig. 10.

The column vector obtained transports the reduced information of the dis-tribution of the classes per line, which leaves the final extraction of the globalclass plate information.

An exact analysis of the column vector to reach the final classification impliesa decision among 262,144 possibilities of the combination of nine positions offour variables.

A second stage executes a process of reduction of the column vector gener-

ated by the previous stage, combining the labels in groups of three and gen-erating what can be seen in Table 2. Again, this procedure utilized the sum ofthe ASCII codes of the letters that compose the combination as a decisionnumber.

It is important to remember that in this classification stage, there are fivepossible classes as defined in Sect. 4, i.e., A, B, C, D and X.

The task of total reduction to reach a single label takes four combinationstages, as shown in Fig. 11, where the final classification is defined. Similar tothe previous algorithm, the considerations for the permutations inside thecombinations was adopted, but considering now only three elements.

Classifier based on co-occurrence matrix

The second proposed classifier is based on the co-occurrence matrix of a uni-directional horizontal neighborhood of the regions and uses the pre-classifica-

Fig. 11 Obtaining the final class using information reduction

Table 3 Co-occurrence percentage matrix with row and columns summation applied to theplate showed in Fig. 4

A B C X Total sum Row index

A 0.1481 0.1111 0.0741 0 0.3333 [1]B 0.1111 0.2222 0.0370 0.0741 0.4444 [2]C 0 0.1111 0 0.0370 0.1481 [3]X 0.0370 0 0 0.0370 0.0741 [4]Total sum 0.2963 0.4444 0.1111 0.1481Column index [1] [2] [3] [4]

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tion results, taking into account only the right neighbor to the region of interest.This consideration is made considering only one direction of the plates for theproduction of the pencils. This is a simple way to reduce the amount of cal-culations, since the inclusion of other directions in the calculation, as proposedby Haralick et al. 1973 and by Gotlieb and Kreyszig 1990, do not introduce any

significant improvement. The pre-classification results are scanned for eachclass and for each co-occurrence a unit is increased to the corresponding cell.Since there are four classes, the co-occurrence matrix generated has a dimensionof 44.

A better way to express the behavior of the neighborhood co-occurrence is torepresent it as a co-occurrence percentage matrix, showing the amount ofparticipation of each cell to the horizontal neighborhood of the plate, as shownin Table 3. In order to interpret this percentage information matrix simplifyingthe solution, a procedure has been adopted summing the rows and columns toobserve the behavior by the neighborhood of the classes. Two resulting vectors

were obtained, representing the total sum by row of classes and the total sum bycolumn of classes, as shown in Table 3.

From the above information an algorithm based on boundary decision levelswas used, classifying the column-vector and the row-vector. The labels havebeen obtained, using the rule shown in Table 4, which allow finding a finalclassification with five possible resulting classes, namely, A, B, C, D and X.

In the algorithm the column/row position is called element[x] and followsthe column/row indexes presented in Table 3.

Results

The proposed methods were applied to two sets of wood plates. The firstset had 104 previously classified wood plates and was used in the trainingof the pre-classification algorithm, to establish the levels of decision ofthe characteristic values. The second set, used for testing the algorithms,had 199 wood plates without previous classification. After classification of

Table 4 Combination oflabels row-column to gener-ate the resultant classes

Row/column combination Resultant class

AA AAB A

BB BAC BAD CBC CBD CCC CCD DDD DAX DBX DCX XDX X

XX X

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the plates by the algorithms, the set was also submitted to classificationby three expert technicians and the results compared as presented inTable 5.

If (element[1] of vector is the largest)If (element[1]) > 0.60and (element[2]) < 0.35and (element[3]) < 0.25and (element[4]) < 0.25then attribute label fi AIfelseIf (element[1]) > 0.35

and ((element[4]) < 0.20 or (element[2]) > 0.30)then attribute label fi BIfelse attribute label fi CIfelseIf (element[2] of vector is the largest)If (element[2]) > 0.40or (element[1]) > 0.3then attribute label fi BIfelseIf (element[3]) < 0.35then attribute label fi CIfelse

If (element[4]) > 0.25then attribute label fi DIfelse attribute label fi BIfelseIf (element[3] of vector is the largest)If (element[3]) > 0.45and ((element[4]) < 0.2 or (element[2]) > 0.3)then attribute label fi CIfelseIf (element[4]) > 0.2then attribute label fi DIfelse attribute label fi CIfelseIf (element[4]) >= 0.45and (element[3]) < 0.3then attribute label fi XIfelseIf (element[4]) >= 0.35and (element[3]) >= 0.3then attribute label fi DIfelse attribute label fi X

Table 5 Frequency of class occurrences for the pencil slats set (199 slats)

Class A Class B Class C Class D Class X Total

Inspector 1 12 33 49 49 56 199Inspector 2 16 30 52 52 49 199Inspector 3 14 37 43 54 51 199Co-occurrence 15 37 49 51 47 199Reduction 14 42 53 45 45 199Total 71 179 246 251 248 995

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Even considering the technical capacity of the inspectors discernment, thepresence of deviations in the classification accomplished by human operators isa fact that cannot be ignored, as showed by Newman and Jain 1995 and Chinand Harlow 1982. Therefore an alternative classification has been incorporated

for the plates classified by the expert technicians. In the plates where there is nota strong presence of characteristics that would determine the definitive shape ofthe class, i.e., those which would allow dubious interpretations, they are givenconsideration as the most probable classification, with also a supplementalclassification in the sense of minimizing the classification deviations that per-haps can be made by the inspectors.

To visualize the differences among the five obtained classifications (threeinspectors and two methods), the frequency of class occurrences are presentedin Table 5. In this table, Reduction refers to the information reduction algo-rithm (described in Sect. 8) and Co-occurrence refers to the co-occurrencematrix algorithm (described in Sect. 9).

Considering the classification made by the methods and by the inspectors, itcan be noticed that when there is a disagreement of class attribution betweenthe five samples, it occurs mainly in the neighboring classes, i.e., it never hap-pens to have a very different classification. For example, a sample attributingclass A to a plate and another attributing class D to the same plate, or a sampleattributing class B to a plate and another sample attributing class X to the sameplate, never it will occur

It is possible to say that, even considering the alternative classifications made

by the inspectors, there are not significant differences between the plates clas-sified by the five classes (A, B, C, D and X) for the five samples of classification.To determine the significance of the differences in the classification samples of

the inspectors, the co-occurrence and reduction algorithms were analyzed asfive independent samples. Then, the number of plates that each sample deter-mines to be in each of the five classes were analyzed through the use of the v2

(chi-square) non-parametric statistical method, as presented in Siegel 1977 andBeiguelman 1996 following Eq. 6, admitting as null hypothesis (H0) that thesamples do not differ significantly from each other.

The application of the v2 method to the contingency (Table 6) must follow

Eq. 6 with a degree of freedom expressed by (k)1)(r)1), where k and r representthe number of columns and rows of the table, respectively.

Table 6 Observed (o) and expected (e) frequencies of the classes

Class A Class B Class C Class D Class X Total rows

o e o e o e o e o e

Inspector 1 12 14.2 33 35.8 49 49.2 49 50.2 56 49.6 199Inspector 2 16 14.2 30 35.8 52 49.2 52 50.2 49 49.6 199Inspector 3 14 14.2 37 35.8 43 49.2 54 50.2 51 49.6 199Co-occurrence 15 14.2 37 35.8 49 49.2 51 50.2 47 49.6 199Reduction 14 14.2 42 35.8 53 49.2 45 50.2 45 49.6 199Total columns 71 71.0 179 179.0 246 246.0 251 251.0 248 248.0 995

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v2 Xr

i1

Xk

j1

oij eij2

eij6

where oij is the number ofobserved cases classified in row iof column j; eij is thenumber of expected cases under H0 in row i of column j; and

Pri1

Pkj1

indicates the summation over all cells.Applying the method to the data in Table 6 a value of v2= 6.5361 is ob-

tained. The number of freedom degrees calculated by (k)1)(r)1), where k=5and r=5, is 16, and the level of significance considered is = 0.05. In this way weobtain through the chi-square table, a critical value for the chi-square ofvc

2=26.296.Considering that the obtained value v2= 6.5361 is smaller than vc

2=26.296,the null hypothesis that the samples dont differ from each other with rela-tionship to the proportions observed for each class can be accepted. That isequivalent to saying that the five samples of classifications could be gathered tocompose only one.

Consulting the table of v2, the above statement can be summarized by thefollowing expression:v(16)

2= 6.5361; 0.98 < p < 0.99, where p is the probability of acceptance of9899% of the classifications similar to the ones in Table 6. It can be consideredthat the classifications do not differ significantly.

Conclusions

Despite the limitations of the images used, e.g., being in gray level, acquiredwith a poor resolution video commercial system, or based only in the extractedcharacteristics of the method of entropic threshold, the visual inspection systemproposed in this work achieved very good results.

The criteria used in the combination of the characteristic values in the pre-

classification, as well as their own values, proved to be enough to consistentlyattribute classes to the areas of the plates, translating with plenty of fidelity thevisual appearance of the surfaces.

Results of the application of the two methods in the final classification,comparing them through the non-parametric statistical test with classificationsaccomplished by specialist technicians, showed that even with the small clas-sification disagreements observed in Table 5, classification of the methods canbe accepted within a relatively safe success margin.

It has been demonstrated in the work that the classification is based oncharacteristics that present highly random behavior (presence and disposition

of the growth rings), which is different from what normally appears in relatedliterature.

Finally, the relevance of this work should be taken note of because themethodology proposed would efficiently aid the pencil manufacturing industry.It has been proven by the comparative results presented that the proposedmethods operate in a similar way to the human inspectors but with unques-tionable advantages such as repeatability, product standardization and estab-lishment of more solid approaches for quality control.

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