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In: Book Title Editor: Editor Name, pp. 1-4 ISBN 0000000000 c 2007 Nova Science Publishers, Inc. Chapter 1 PATTERN R ECOGNITION BY B ESSEL MASK AND O NE -D IMENSIONAL S IGNATURES Selene Solorza * Josu´ e Alvarez-Borrego ** * Facultad de Ciencias, Universidad Aut´ onoma de Baja California ** Applied Physics Division, Optics Department, CICESE Recently, invariant correlation digital systems to position, rotation, scale and illumi- nation are utilized in the pattern recognition field [1-3]. Such invariants are made of by the Fourier and Fourier-Mellin transforms in conjunction with linear or nonlinear filters (k- law). In this work a new digital system invariant to position, rotation and illumination based on Fourier transform, Bessel masks, one-dimensional signatures and linear correlations are presented. Using one-dimensional signatures instead of diffraction patterns or vectorial sig- natures of the images reduces the computational time considerably, achieving a step toward the ultimate goal, which is to develop a simple digital system that accomplishes recognition in real time at a low cost. To achieve the invariant to translation the modulus of the Fourier transform of the image is taken. And, using a Bessel binary mask of concentric rings the invariant to rotation is obtained. The discrimination between objects is done by a linear correlation of the one-dimensional signatures assigned to each image and the target, in this manner the computational cost is reduced also. The images classification range are deter- mined by the Fisher transformation statistic theory. The digital system was tested using a reference image database of 21 fossil diatoms images of gray-scale and 307 × 307 pixel. The system has a confidence level of 95.4% or greater in the classification of the 7, 560 problem images using the same illumination. Then, those problem images were altered with eight different illuminations and the system also identifies the 60, 480 images with a confidence level of 95.4% or greater. PACS 07.05.Pj. Keywords: Image processing algorithms. Key Words: Image processing, pattern recognition. AMS Subject Classification: 68U10, 94A08, 68T10. * E-mail address: [email protected]

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Page 1: P RECOGNITION BY BESSEL MASK AND -D S - CICESEjosue/pdf/chapter... · Josue Alvarez-Borrego´ Facultad de Ciencias, Universidad Autonoma de Baja California´ Applied Physics Division,

In: Book TitleEditor: Editor Name, pp. 1-4

ISBN 0000000000c© 2007 Nova Science Publishers, Inc.

Chapter 1

PATTERN RECOGNITION BY BESSEL MASK ANDONE-DIMENSIONAL SIGNATURES

Selene Solorza ∗

Josue Alvarez-Borrego∗∗

∗Facultad de Ciencias, Universidad Autonoma de Baja California∗∗Applied Physics Division, Optics Department, CICESE

Recently, invariant correlation digital systems to position, rotation, scale and illumi-nation are utilized in the pattern recognition field [1-3]. Such invariants are made of bythe Fourier and Fourier-Mellin transforms in conjunction with linear or nonlinear filters (k-law). In this work a new digital system invariant to position, rotation and illumination basedon Fourier transform, Bessel masks, one-dimensional signatures and linear correlations arepresented. Using one-dimensional signatures instead of diffraction patterns or vectorial sig-natures of the images reduces the computational time considerably, achieving a step towardthe ultimate goal, which is to develop a simple digital system that accomplishes recognitionin real time at a low cost. To achieve the invariant to translation the modulus of the Fouriertransform of the image is taken. And, using a Bessel binary mask of concentric rings theinvariant to rotation is obtained. The discrimination between objects is done by a linearcorrelation of the one-dimensional signatures assigned to each image and the target, in thismanner the computational cost is reduced also. The images classification range are deter-mined by the Fisher transformation statistic theory. The digital system was tested using areference image database of 21 fossil diatoms images of gray-scale and 307 × 307 pixel.The system has a confidence level of 95.4% or greater in the classification of the 7, 560problem images using the same illumination. Then, those problem images were alteredwith eight different illuminations and the system also identifies the 60, 480 images with aconfidence level of 95.4% or greater.

PACS 07.05.Pj. Keywords: Image processing algorithms.

Key Words: Image processing, pattern recognition.AMS Subject Classification: 68U10, 94A08, 68T10.

∗E-mail address: [email protected]

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1. The Bessel Mask

The digital system works with n×n gray-scale images. The binary mask is build using theBessel function of first kind and first order as

f(x) =J1(x− cx)

x− cx, (1)

where x = 1, . . . , n and the center pixel (cx, cx) of the image is given by

cx =

{n2 + 1, if n is even,bn2 c+ 1, if n is odd,

(2)

here bzc rounds z to the nearest integer towards −∞.The function f is symmetric in x = cx, hence

Z(x) =

{1, if f(cx) > 0,0, if f(cx) ≤ 0,

(3)

also is symmetric. Finally, the Z function is rotating 180 degrees by the symmetric axesto generate concentric cylinders centered in (cx, cx) and height one. Those cylinders aremapped to the plain to obtain the Bessel mask given in Fig. 1.

Figure 1. Bessel binary rings mask example.

2. The One-Dimensional Signature

Let I(x, y) represents the intensity of the image I in the pixel (x, y), where x, y = 1, . . . , n.The digital system works with the modulus of the Fourier transform of the image, here-after called M , because it is invariant to translation, that is, M = |F {I(x, y)} | =|F {I(x+ τ, y + χ)} |.

The Bessel mask, named B, filters the M image as

H = B ∗M, (4)

2

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where ∗ means element by element multiplication. Next, the rings in H are numbered fromthe center toward out-side to obtain the following set,

N = {ring index | ring index ∈ n} , (5)

where n = {1, 2, . . . , number of rings}.

The addition of the intensity values into each ring of H are computed to build thefunction

signature : N ×A ⊂ R,signature( ring index) =

∑H(x, y), if H(x, y) are in the ring. (6)

Because the cardinality of A always is bigger than one, the graph of the signature functionis called one-dimensional signature of the image I .

3. The pattern recognition

Let R be the set of 21 diatoms reference image (RI) shown in Fig. 2 and PI a problemimage to be classified. The PI could be translated and/or rotated in the Cartesian plane.The one-dimensional signatures of the RI and the PI are obtained as described in section 2.

Fig. 3 shows the signature of diatoms Actinocyclus ingens, Azpeitia sp, Azpeitianodulifera and Actinocyclus ellipticus, named image A, B, C and D, respectively, see Fig.2. Although the diatoms Azpeitia sp (B) and Azpeitia nodulifera (C) are so similar, theirsignatures are very different, hence we have a pattern recognition digital system.

The signatures of the PI and the k-th reference image (RIk) are compared by the linearcorrelation equation,

C(SRIk, SPI ) = F−1

{|F {SPI}| e

φPI

∣∣∣F {SRIk

}∣∣∣ e−φRIk

}, (7)

where φPI and φRIkare the phases of the Fourier transform for the signature of the PI and

the k-th RI, respectively. If the maximum value of the magnitude for the linear correlationare significant, that is similar to the autocorrelation maximum value, hence the PI containsthe RI, otherwise are different. To normalize the output results, those maximum values arescaled as

max|C(SRIk, SPI )|

(N − 1)σRIkσPI

, (8)

where N is the length of the signatures. σRIkand σPI are the standard deviations of the

k-th RI and the PI signatures, respectively.

To obtain the confidence level of the system, the 21 diatom images in Fig. 2 areused as RI, each of them were rotated degree by degree until complete the 360 degrees,hence the PI data base has 7, 560 images. The results were box plotting by the mean of

3

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4 Authors

the normalized values (eq. 8) with two standard errors (2SE) for the 360 images of eachdiatom. For example, Fig. 4 shows the box plot graph with diatom A as RI. We see in thebox plot that results are normalized and there are not overlap of the whiskers associated tothe RI, in this example diatom A, with the whiskers associated to the other diatom images.Therefore, the digital system has a confidence level of 95.4% in the diatom A identification.The same statistical analysis was done for each RI to obtained that, in general, the digitalsystem has a confidence level of 95.4%.

Once the confidence level of the system is set, to do the classification process, first ofall we have to notice that data does not has a known distribution curve. Hence, we usethe Fisher transformation to have a normal distribution. Then, the confidence intervals of a95.4% are set for identify each PI. Moreover, the PI could had different illumination of thatimages in the reference image data base and the digital system also classified it.

4. Conclusions

This work presents a low computational cost algorithm invariant to position, rotation andillumination. The digital system was tested using a reference image database of 21 fos-sil diatoms gray-scale images of 307 × 307 pixel. The system classified 7, 560 problemimages with a confidence level of 95.4% or greater. Furthermore, those problem imageswere altered with eight different illuminations background and the system also identifiesthe 60, 480 images.

AcknowledgementsThis work was partially supported by CONACyT under grant No. 102007 and 169174.

References

[1] Solorza, S. & Alvarez-Borrego, J. (2010). Digital system of invariant correlation toposition and rotation. Optics Communications, Vol. 283, No. 19, 3613-3630, ISSN0030-4018.

[2] Alvarez-Borrego, J. & Solorza, S. (2010). Comparative analysis of several digitalmethods to recognize diatoms. Hidrobiologica, Vol. 20, No. 2, 158-170, ISSN 0188-8897.

[3] Solorza, S., Alvarez-Borrego, J. and Chaparro-Magallanez, G. (2012). Pattern recog-nition of digital images by one-dimensional signatures. Chapter 13, pp. 299-316.Fourier Transform-Signal Processing. Editor Salih Mohammed Salih. Ed. Intech.

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Article Name 5

Figure 2. Reference image data base.

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6 Authors

Figure 3. Signature examples.

Figure 4. Box plot example.