# The use of trace transform for classification of visualized transparent polymeric foils.

1. INTRODUCTION

Properties of optical transparent polymeric foils are related to their production and therefore the evaluation of the quality of the foils is an important part of their production (Bolf et al., 1993). Optical visualization methods and especially schlieren method (Smith & Lim, 2000) allows visual quality assessment, because it permits to locate on a pre-processed picture the places with elastic strains that are marked by variations in the level brightness (Caastle et al., 1994). By the image processing it is possible to evaluate quantitative properties as well. Properties of visualized images of the foils are similar to the properties of textures. In order to recognize images of the foils methods suitable for textures recognition such as statistical, frequency methods and wavelet transform can be used. The completion of the optical visualization method with suitable characteristics calculated from the pre-processed image has found the application in the field of defectoscopy, identification and classification of polymeric foil. Numerical characteristics calculated from the image function of the region called features correspond to the object. Elements of the signature vector are formed by the aligned set of features. Position of ending point of signature vector in the n-dimensional space, where n means number of elements of signature vector, represents pattern of the object. The main problem is to create such a description of object in ideal case whose patterns belonging to the same class can be grouped into the subspace in that no pattern of object belonging to the different class is situated. Very important group of methods of region description enables to calculate characteristics that are invariant to translation, rotation and scaling of the image. Classifiers utilizing the decision rules make a decision upon the classification of the image to the corresponding class. In the simplest case it is possible to construct the decision rules analytically. In the completion case classifiers based on learning, e.g. on the artificial neural network are used (Grman, 2000). The contribution treats the principle of description of images by the trace transform method (Kadyrov & Petrou, 2001).

2. TRACE TRANSFORM

We can regard trace transform as a generalization of the principle of Radon transform. Radon transform is useful when to calculate the characteristic numbers of image especially an integral calculated along the lines crossing the image. Trace transform uses in addition to the integrals calculated along lines also other functionals calculated along lines crossing the image. It uses combination of triple different or the same functionals along the lines in the image. The advantage of transform represents the possibility to create many characteristic numbers--features, whose choice depends on the character of classification problem.

An arbitrary line crossing the image can be expressed by the function of three parameters l(r,[theta],f), t is position of point in the line, [theta] is angle between line and x axis and r represents distance of the line from the origin of coordinate system.

A functional F is real-valued function on a vector space, usually of functions. Image is described by a triple features. Procedure of the triple features construction can be described by the steps on:

* Produce the trace transform of the image by applying a Trace functional T along lines crossing the image. As a result there is the matrix with the columns containing values of functionals for the different values of line distance from the origin of coordinates. Rows contain functionals values in dependence of the rotation of lines.

* Produce the "circus function" of the image by applying a diametric functional R along the columns (along values of distances) of the trace transform,

* Produce triple feature by applying a circus functional 9 along the string (along values of rotation) of numbers produced in preceding step.

The triple features can be defined by the system of functionals

G([f]) = [PHI][R[T[[f]r, [theta], t)]]] (1)

f() is image (brightness) function.

There are some rules based on functionals with specific properties to triple features calculation used in (Turan et al., 2006). The first is the shift property. If

F[f (x + a)] = F[ [florin] (x)] [for all] a [member of] R (2)

then functional is called shift invariant functional, if

H[ [florin] (x + a)] = H[ [florin] (x)] - a [for all]a [member of] R (3)

then functional is called sensitive functional. The second is specific property to the scaling. For shift invariant functionals it is the abscissa homogeneity property

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

and ordinate homogeneity property

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

shift sensitive functionals booth condition (6) should be fulfilled

H[[f](bx)] = 1/b H[[f](x)] [conjunction] H[bf(x)] = H[[f](x)] [for all] b > 0 [[alpha].sub.H] = -1 [conjunction] [[beta].sub.H] = 0 (6)

It is possible to create triple features invariant to the translation, rotation and scaling by two combinations of transforms T, R, [PHI]:

Combination 1:

* Functional T is shift invariant (2) with abscissa homogeneity property (4)

* Functional R is shift invariant (2) with abscissa homogeneity property (4) and ordinate homogeneity property (5)

* Functional [PHI] is shift invariant (2) with abscissa homogeneity property (4) and ordinate homogeneity property (5)

Combination 2:

* Functional T has the same properties such as in case 1.

* Functional R is sensitive (3) with properties (6)

* Functional [PHI] is shift invariant (2) with ordinate homogeneity property constant (5), and is not sensitive to the first harmonic of the function f.

Triple features computed for original f and geometrically modified image fM are related by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The invariance is fulfilled for [delta] = 0 or when to put instead triple features ratio of two normalized triple features

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

3. EXPERIMENTAL RESULTS

In order to verify applicability of trace transform for classification of polymeric foils we have used six different types of foils that are in fig. 1 presented. From each type of the foil we captured 16 images of different samples. We used a digital camera with resolution 5 megapixels. Triple features were calculated from the part of images with dimensions 1400x1400 pixels.

From images it is seen, that different foils have different characteristic features that are well observed, images of different foils can be distinguished and on the contrary images of different samples of the same foil show some similarity. Features show some regularity of stochastic character.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

In fig. 2 is the tendency of clustering of signature vectors corresponding to the same types of foils and the tendency of clustering of signature vectors corresponding to the different types of foils presented. Functionals F and S were used.

The signature vector consists of triple features [PHI][]=F, [R.sub.1]=F, [T.sub.1]=F and [[PHI].sub.2]=F, [R.sub.2]=F, [T.sub.2]=S

F = median{ [florin] (x)} (9)

S = [integral][absolute value of f(x )] dx (10)

4. CONCLUSIONS

Experimental results present suitability of the trace transform for calculation of features of images of visualized polymeric foils that are necessary for their recognition and for the detection of different foils damages. In general it is important that features would be invariant to the rotation and scaling. In case of visualized images of transparent polymeric foils the requirement of invariance to the scaling is not necessary to be fulfilled in regard to the technique of their images scanning. On contrary this method is not sensitive to the rotation of foil sample that can be the most advantage.

ACKNOWLEDGMENT

We gratefully acknowledge that the results presented here were solved as partial goals of the research task VEGA No 1/3099/06.

5. REFERENCES

Bolf, J.; Bajcsy, J. & Bolf, P. (1993). Methods of the Properties Testing of Optically Transparent Materials. Journal of Electrical Engineering, Vol. 44, No. 8, pp. 244-247

Castle, D. A.; Gibbins, B. & Hamer, P. S. (1994). Physical Methods for Examining and Comparing Transparent Plastic Bags and Cling Films, Journal of Forensic Science Society, Vol. 34, pp. 61-68

Grman, J. (2000). Neural Network Application in the Defectoscopy. Proceedings of XVI. IMEKO World Congress IMEKO 2000, pp. 205-210, Vienna, 25.-28.9.2000.

Kadyrov, A. & Petrou, M. (2001). The Trace transform and its applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23, No. 8, pp. 811-828.

Smith, A.J. & Lim, T.T. (2000). Flow visualization, Imperial College Press, London

Turan, J.; et al. (2006). Trace Transform and KLT Based Invariant Features and Image Recognition System. Acta Electrotechnica et Informatica, Vol. 6, No. 3, pp. 1- 11

Properties of optical transparent polymeric foils are related to their production and therefore the evaluation of the quality of the foils is an important part of their production (Bolf et al., 1993). Optical visualization methods and especially schlieren method (Smith & Lim, 2000) allows visual quality assessment, because it permits to locate on a pre-processed picture the places with elastic strains that are marked by variations in the level brightness (Caastle et al., 1994). By the image processing it is possible to evaluate quantitative properties as well. Properties of visualized images of the foils are similar to the properties of textures. In order to recognize images of the foils methods suitable for textures recognition such as statistical, frequency methods and wavelet transform can be used. The completion of the optical visualization method with suitable characteristics calculated from the pre-processed image has found the application in the field of defectoscopy, identification and classification of polymeric foil. Numerical characteristics calculated from the image function of the region called features correspond to the object. Elements of the signature vector are formed by the aligned set of features. Position of ending point of signature vector in the n-dimensional space, where n means number of elements of signature vector, represents pattern of the object. The main problem is to create such a description of object in ideal case whose patterns belonging to the same class can be grouped into the subspace in that no pattern of object belonging to the different class is situated. Very important group of methods of region description enables to calculate characteristics that are invariant to translation, rotation and scaling of the image. Classifiers utilizing the decision rules make a decision upon the classification of the image to the corresponding class. In the simplest case it is possible to construct the decision rules analytically. In the completion case classifiers based on learning, e.g. on the artificial neural network are used (Grman, 2000). The contribution treats the principle of description of images by the trace transform method (Kadyrov & Petrou, 2001).

2. TRACE TRANSFORM

We can regard trace transform as a generalization of the principle of Radon transform. Radon transform is useful when to calculate the characteristic numbers of image especially an integral calculated along the lines crossing the image. Trace transform uses in addition to the integrals calculated along lines also other functionals calculated along lines crossing the image. It uses combination of triple different or the same functionals along the lines in the image. The advantage of transform represents the possibility to create many characteristic numbers--features, whose choice depends on the character of classification problem.

An arbitrary line crossing the image can be expressed by the function of three parameters l(r,[theta],f), t is position of point in the line, [theta] is angle between line and x axis and r represents distance of the line from the origin of coordinate system.

A functional F is real-valued function on a vector space, usually of functions. Image is described by a triple features. Procedure of the triple features construction can be described by the steps on:

* Produce the trace transform of the image by applying a Trace functional T along lines crossing the image. As a result there is the matrix with the columns containing values of functionals for the different values of line distance from the origin of coordinates. Rows contain functionals values in dependence of the rotation of lines.

* Produce the "circus function" of the image by applying a diametric functional R along the columns (along values of distances) of the trace transform,

* Produce triple feature by applying a circus functional 9 along the string (along values of rotation) of numbers produced in preceding step.

The triple features can be defined by the system of functionals

G([f]) = [PHI][R[T[[f]r, [theta], t)]]] (1)

f() is image (brightness) function.

There are some rules based on functionals with specific properties to triple features calculation used in (Turan et al., 2006). The first is the shift property. If

F[f (x + a)] = F[ [florin] (x)] [for all] a [member of] R (2)

then functional is called shift invariant functional, if

H[ [florin] (x + a)] = H[ [florin] (x)] - a [for all]a [member of] R (3)

then functional is called sensitive functional. The second is specific property to the scaling. For shift invariant functionals it is the abscissa homogeneity property

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

and ordinate homogeneity property

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

shift sensitive functionals booth condition (6) should be fulfilled

H[[f](bx)] = 1/b H[[f](x)] [conjunction] H[bf(x)] = H[[f](x)] [for all] b > 0 [[alpha].sub.H] = -1 [conjunction] [[beta].sub.H] = 0 (6)

It is possible to create triple features invariant to the translation, rotation and scaling by two combinations of transforms T, R, [PHI]:

Combination 1:

* Functional T is shift invariant (2) with abscissa homogeneity property (4)

* Functional R is shift invariant (2) with abscissa homogeneity property (4) and ordinate homogeneity property (5)

* Functional [PHI] is shift invariant (2) with abscissa homogeneity property (4) and ordinate homogeneity property (5)

Combination 2:

* Functional T has the same properties such as in case 1.

* Functional R is sensitive (3) with properties (6)

* Functional [PHI] is shift invariant (2) with ordinate homogeneity property constant (5), and is not sensitive to the first harmonic of the function f.

Triple features computed for original f and geometrically modified image fM are related by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The invariance is fulfilled for [delta] = 0 or when to put instead triple features ratio of two normalized triple features

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

3. EXPERIMENTAL RESULTS

In order to verify applicability of trace transform for classification of polymeric foils we have used six different types of foils that are in fig. 1 presented. From each type of the foil we captured 16 images of different samples. We used a digital camera with resolution 5 megapixels. Triple features were calculated from the part of images with dimensions 1400x1400 pixels.

From images it is seen, that different foils have different characteristic features that are well observed, images of different foils can be distinguished and on the contrary images of different samples of the same foil show some similarity. Features show some regularity of stochastic character.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

In fig. 2 is the tendency of clustering of signature vectors corresponding to the same types of foils and the tendency of clustering of signature vectors corresponding to the different types of foils presented. Functionals F and S were used.

The signature vector consists of triple features [PHI][]=F, [R.sub.1]=F, [T.sub.1]=F and [[PHI].sub.2]=F, [R.sub.2]=F, [T.sub.2]=S

F = median{ [florin] (x)} (9)

S = [integral][absolute value of f(x )] dx (10)

4. CONCLUSIONS

Experimental results present suitability of the trace transform for calculation of features of images of visualized polymeric foils that are necessary for their recognition and for the detection of different foils damages. In general it is important that features would be invariant to the rotation and scaling. In case of visualized images of transparent polymeric foils the requirement of invariance to the scaling is not necessary to be fulfilled in regard to the technique of their images scanning. On contrary this method is not sensitive to the rotation of foil sample that can be the most advantage.

ACKNOWLEDGMENT

We gratefully acknowledge that the results presented here were solved as partial goals of the research task VEGA No 1/3099/06.

5. REFERENCES

Bolf, J.; Bajcsy, J. & Bolf, P. (1993). Methods of the Properties Testing of Optically Transparent Materials. Journal of Electrical Engineering, Vol. 44, No. 8, pp. 244-247

Castle, D. A.; Gibbins, B. & Hamer, P. S. (1994). Physical Methods for Examining and Comparing Transparent Plastic Bags and Cling Films, Journal of Forensic Science Society, Vol. 34, pp. 61-68

Grman, J. (2000). Neural Network Application in the Defectoscopy. Proceedings of XVI. IMEKO World Congress IMEKO 2000, pp. 205-210, Vienna, 25.-28.9.2000.

Kadyrov, A. & Petrou, M. (2001). The Trace transform and its applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23, No. 8, pp. 811-828.

Smith, A.J. & Lim, T.T. (2000). Flow visualization, Imperial College Press, London

Turan, J.; et al. (2006). Trace Transform and KLT Based Invariant Features and Image Recognition System. Acta Electrotechnica et Informatica, Vol. 6, No. 3, pp. 1- 11

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Author: | Syrova, Livia; Ravas, Rudolf |
---|---|

Publication: | Annals of DAAAM & Proceedings |

Article Type: | Report |

Date: | Jan 1, 2008 |

Words: | 1417 |

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