- Email: [email protected]
Spiral Light Beams and Contour Image Processing
Available online at www.sciencedirect.com
ScienceDirect Physics Procedia 86 (2017) 131 – 135
International Conference on Photonics of Nano- and Bio-Structures, PNBS-2015, 19-20 June 2015, Vladivostok, Russia and the International Conference on Photonics of Nano- and MicroStructures, PNMS-2015, 7-11 September 2015, Tomsk, Russia
Spiral light beams and contour image processing Sergey A. Kishkina, Svetlana P. Kotovaa,b* and Vladimir G. Volostnikov a,b b
a Lebedev Physical Institute, 221, Novo-Sadovaya Str., Samara, 443011, Russia Samara National Research University, 34, Moskovskoye shosse, Samara, 443086, Russia
Abstract Spiral beams of light are characterized by their ability to remain structurally unchanged at propagation. They may have the shape of any closed curve. In the present paper a new approach is proposed within the framework of the contour analysis based on a close cooperation of modern coherent optics, theory of functions and numerical methods. An algorithm for comparing contours is presented and theoretically justified, which allows convincing of whether two contours are similar or not to within the scale factor and/or rotation. The advantages and disadvantages of the proposed approach are considered; the results of numerical modeling are presented. © 2017 2016The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. Keywords: recognition of contour images; spiral light beams
1. Introduction The contours analysis is one of the key stages in the task of the images identification or recognition. The gist of it consists in the assessment of the image as a set of contours (Vizilter et. al. (2010), Furman (2003)). The paper outlines a new method of recognizing the image contours based on the use of the spiral beams (i.e. light fields retaining their structure under focusing and propagation) apparatus. The method is considered to be capable of removing a number of shortcomings of the existing methods of contour analysis. The essence of the approach proposed in the present paper is that the operations are carried out not with a planar curve, but with the spiral beam,
* Corresponding author. Tel.: +7-846-335-57-31; fax: +7-846-335-56-00. E-mail address: [email protected]
1875-3892 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. doi:10.1016/j.phpro.2017.01.034
132
Sergey A. Kishkin et al. / Physics Procedia 86 (2017) 131 – 135
determined by this curve. This is possible owing to a one-to-one correspondence between curves and beams. However, it is more reasonable to consider a spiral beam as an object, since it is more ‘rich’ from the mathematical point of view and possesses a number of useful properties. The recognition result can be based on a totality of solutions for many contours marked out on the image. This is easily achieved provided that the mechanism for the two contours comparison is available. The research described is aimed precisely at finding of the contours qualitative parameters and extracting of information on their similarity. 2. Contour description In the paper certain closed planar curves consisting of an ordered set of points are considered as the mathematical representation of the contours:
] (t )
x (t ) iy (t ), t [0, T ].
(1)
It is clear that any contour can be presented in the form of an infinite expansion in a certain complete set of orthogonal functions. The problem of a suitable basis selection is, of course, essential. The problem of expansion of the above functions is thoroughly considered in (Dedus (2004)), where the classical bases, used in image recognition, are presented. The problem is, however, two-fold. First, to provide reasonable time limits for the analysis process one has to confine himself to a finite number of basic functions. Second, the dimension of the finite set of expansion coefficients with respect to the basis is radically dependent on the reference point, where the curve originates (i.e. on defining the corresponding function within the interval [0,T] or [a, a + T]). Of course, in the context of the curve description it is of no importance, but only in the case of a complete basis set, which is hardly implemented in practice because of the limited time and computation resources. All this challenged us to seek for an alternative approach. 3. Spiral beam In the process of making the analysis of various light fields a new type of light beams referred to as spiral beams was discovered, theoretically analyzed and experimentally implemented (Abramochkin and Volostnikov (2010)). It appeared that a spiral beam represents a light field retaining its structure to within the scale factor and rotation in the course of propagation and focusing. Moreover, the structure of such light field can be rather diverse; in particular, it can have the shape of an arbitrary planar curve, including a closed one. It was found that the complex amplitude S ( z , z ) of the field of this beam is uniquely related to the corresponding curve ] (t ) and is described by the expression: S(z, z | ] (t), t [0,T]) T
° zz ½° ° ] (t)] (t) 2z] (t) °½ exp ® 2 ¾ exp ® 2 ¾u U2 U ¿° ¯° U ¿° 0 ¯°
³
(2)
° 1 t ½° uexp ® 2 ª] (W )] c(W ) ] (W )] c(W )º dW ¾ | ] c(t) | dt ¬ ¼ °¯ U 0 °¿
³
where U is the Gaussian parameter of the beam, and the the top linedenotes a complex conjugation. The example of a given curve and the corresponding spiral beam is presented in Fig. 1. Rather essential is the property of ‘quantization’ of spiral beams in the form of closed curves. If the condition Scurve
1 2 SU N , N 2
0, 1, 2 }
(3)
133
Sergey A. Kishkin et al. / Physics Procedia 86 (2017) 131 – 135
Fig. 1.(a) “Generating” curve; (b) intensity distribution and (c) phase distribution of the corresponding spiral beam.
(where S curve is the area under the curve) is met, then the complex amplitude of the beam field does not depend on the choice of the reference point at the curve (Abramochkin and Volostnikov (2004)). In other words, the spiral beam is not determined by the reference point at the contour. Therefore, any finite sum of the series S N ( z , z | ] (t ), t [a, a T ]) is also independent of the reference point to the mutual unimodular term, depending only on the parameter a. Therefore, the problem of the reference point choice in the analysis and recognition of the input contour is removed. This means that with any required precision we can put into correspondence to the spiral beam S ( z , z | ] (t ), t [ a, a T ]) a finite sum of the series S N ( z , z | ] (t ), t [a, a T ])
e
zz
U2
N
¦c z
n
(4)
n
n 0
Since under the rotation of the analyzed contour by the angle D the finite sum of the series changes as S N ( zeiD , ze iD | ] (t ), t [a, a T ])
e
zz
U2
N
¦ c e D z i n
n
n 0
n
e
zz
U2
N
¦ cc z
n
n
n 0
the problem of the contour rotation is eliminated, too, and this proves once more that the expansion coefficients can characterize the rotation angles. 4. Algorithm of contour comparison Now let us consider two contours, the input one and the reference one, stored in a database, and detect whether they correspond to each other or not. Let us construct spiral beams for both contours, keeping the necessary number of terms in the series (Volostnikov et al (2013)). On using the above scheme let us put into correspondence to the contours the two spiral beams or two sets of complex coefficients {cn(1) }nN 0 and {cn(2) }nN 0 . It is assumed that prior to the construction of beams the normalization of the area, bounded by the contours, has already been carried out. The reduction to one area allows the determination of the scale of the input object. In the case when the quantization parameter is sufficient to provide distinguishing between the two contours, the above sets of coefficients coincide to within rotation (decisively, within the framework of the fixed basis: n 1, N ,
| cn(1) | | cn(2)
|
1, Mn
(1) (2) 1 cn cn 1 ln (2) (1) , i cn cn 1
(5)
134
Sergey A. Kishkin et al. / Physics Procedia 86 (2017) 131 – 135
If Mn { const for all n, then Mn is the angle D of the relative rotation of the contours. This is easily obtained by deriving the expression for the ratio of two complex amplitudes from the representation of the spiral beams in the form of series (4) and (5). If condition (6) is not satisfied, one can conclude that the contours do not correspond to each other.
Fig. 2.General block diagram of the method.
The block diagram of the algorithm for estimation of the two contours similarity is shown in Fig.2. The recognition algorithm is characterized by the following peculiarities. The first feature is the independence of the algorithm operation on the choice of the reference point location at the contour and the scale of the contour image. The second one is that the contour object can be of an arbitrary shape; its complexity is limited by the system resolution only, not by the number of the contour sections as in other known methods. An attractive property of the proposed method is that the enumeration of possibilities typical of a generally accepted recognition method of the contour analysis based on the correlation functions, is not required. One can logically refer to the shortcomings of the offered method the exponential operation used in calculations and being rather “heavy” for computation, especially in mobile devices. Besides, currently there is no efficient technique available for estimating of the quantization parameter and the number of the residual series terms sufficient (and obligatory) to ensure the recognition results stability. 5. Accounting for noise The numerical modeling results proved a reliable performance of the proposed algorithm in comparing of the model objects. Still, the unavoidable in real systems noise and distortions are not taken into consideration. The distortions occur due to discretization errors in CCD-matrices, to malfunctions during data transmission from the image detectors to computation devices, etc. The research showed that the method is rather resistant to noise in the contour being recognized. It can be attributed, first of all, to the Gaussian exponent in the expression for complex amplitude of the spiral beam (2) that ensures a “smoothing” effect, and as the result small deformations of the contour leads to insignificant changes in the complex amplitude. It should be also noted that the integral transform proper is also a good stabilizer. In spite of the resistibility of the complex amplitude and its zeroes to contour deformations the expansion coefficients (see Vieta theorem) can vary rather notably. In this case we propose to additionally use the correlation function K ( T)
³³ S
K (T )
³³
(1)
( z , z ) S (2) ( zeiT , zeiT ) dxdy
S (1) ( z, z ) S (1) ( z, z ) dxdy
³³
S (2) ( z , z ) S (2) ( z , z ) dxdy
(6)
Sergey A. Kishkin et al. / Physics Procedia 86 (2017) 131 – 135
by the angle of mutual rotation T of the spiral beams S (1) ( z , z ) ɢ S (2) ( z , z ) . The module maximum of this function is achieved at the true angle D of mutual rotation of the contours being recognized. Fig.3 illustrates an aircraft image, a marked boundary contour for the initial object, contour for a turned out and reduced object and also images of the spiral beams obtained. It is seen that in the contour of the reduced object the sections in the tail area are lacking, while spiral beams are practically identical. The calculated correlation function has its maximum value of 0.8 for the rotation angle equal to 123 degrees. And this is the real angle of the image rotation.
Fig. 3.From left to right: the aircraft initial image, marked boundary contour, intensity of the emerging spiral beams.
6. Conclusion A new approach to the contour images analysis is described, based on a close cooperation of the modern coherent optics, theory of functions and numeric techniques. The algorithm for images matching and comparison is proposed and theoretically grounded. This method allows detecting whether the two contours are identical to within scale and/or rotation. The dynamics of the offered method adaptation is shown for the case when the image under examination is subject to noise and deformations. Acknowledgements The work was supported by the Ministry of Education and Science of the Russian Federation and the Educational and Research Complex of the P.N. Lebedev Physical Institute References Abramochkin, E.G., Volostnikov, V.G., 2004. Spiral Light Beams. Uspekhi Fizicheskikh Nauk, 174, 1273-1300. Abramochkin, E.G., Volostnikov, V.G., 2010. Modern Optics of Gaussian Beams. M.: Fizmatlit, Moscow, pp. 184. Dedus, F.F., 2004. Classical and Orthogonal Bases for Tasks of Signals Analytical Description and Processing. Ɇ.: MSU. Furman, Ya.A., 2003. Introduction to Contour Analysis and its Applications in Image and Signal Processing. M.: Fizmatlit. Vizilter, Yu.V. et al., 2010. Image Processing and Analysis in Tasks of Computer Vision. M.: Fizmatkniga. Volostnikov, V.G., Kishkin, S.A., Kotova, S.P., 2013. New method of contour image processing based on the formalism of spiral light beam. Quantum Electronics, 43, 646–650.
135
ScienceDirect Physics Procedia 86 (2017) 131 – 135
International Conference on Photonics of Nano- and Bio-Structures, PNBS-2015, 19-20 June 2015, Vladivostok, Russia and the International Conference on Photonics of Nano- and MicroStructures, PNMS-2015, 7-11 September 2015, Tomsk, Russia
Spiral light beams and contour image processing Sergey A. Kishkina, Svetlana P. Kotovaa,b* and Vladimir G. Volostnikov a,b b
a Lebedev Physical Institute, 221, Novo-Sadovaya Str., Samara, 443011, Russia Samara National Research University, 34, Moskovskoye shosse, Samara, 443086, Russia
Abstract Spiral beams of light are characterized by their ability to remain structurally unchanged at propagation. They may have the shape of any closed curve. In the present paper a new approach is proposed within the framework of the contour analysis based on a close cooperation of modern coherent optics, theory of functions and numerical methods. An algorithm for comparing contours is presented and theoretically justified, which allows convincing of whether two contours are similar or not to within the scale factor and/or rotation. The advantages and disadvantages of the proposed approach are considered; the results of numerical modeling are presented. © 2017 2016The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. Keywords: recognition of contour images; spiral light beams
1. Introduction The contours analysis is one of the key stages in the task of the images identification or recognition. The gist of it consists in the assessment of the image as a set of contours (Vizilter et. al. (2010), Furman (2003)). The paper outlines a new method of recognizing the image contours based on the use of the spiral beams (i.e. light fields retaining their structure under focusing and propagation) apparatus. The method is considered to be capable of removing a number of shortcomings of the existing methods of contour analysis. The essence of the approach proposed in the present paper is that the operations are carried out not with a planar curve, but with the spiral beam,
* Corresponding author. Tel.: +7-846-335-57-31; fax: +7-846-335-56-00. E-mail address: [email protected]
1875-3892 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. doi:10.1016/j.phpro.2017.01.034
132
Sergey A. Kishkin et al. / Physics Procedia 86 (2017) 131 – 135
determined by this curve. This is possible owing to a one-to-one correspondence between curves and beams. However, it is more reasonable to consider a spiral beam as an object, since it is more ‘rich’ from the mathematical point of view and possesses a number of useful properties. The recognition result can be based on a totality of solutions for many contours marked out on the image. This is easily achieved provided that the mechanism for the two contours comparison is available. The research described is aimed precisely at finding of the contours qualitative parameters and extracting of information on their similarity. 2. Contour description In the paper certain closed planar curves consisting of an ordered set of points are considered as the mathematical representation of the contours:
] (t )
x (t ) iy (t ), t [0, T ].
(1)
It is clear that any contour can be presented in the form of an infinite expansion in a certain complete set of orthogonal functions. The problem of a suitable basis selection is, of course, essential. The problem of expansion of the above functions is thoroughly considered in (Dedus (2004)), where the classical bases, used in image recognition, are presented. The problem is, however, two-fold. First, to provide reasonable time limits for the analysis process one has to confine himself to a finite number of basic functions. Second, the dimension of the finite set of expansion coefficients with respect to the basis is radically dependent on the reference point, where the curve originates (i.e. on defining the corresponding function within the interval [0,T] or [a, a + T]). Of course, in the context of the curve description it is of no importance, but only in the case of a complete basis set, which is hardly implemented in practice because of the limited time and computation resources. All this challenged us to seek for an alternative approach. 3. Spiral beam In the process of making the analysis of various light fields a new type of light beams referred to as spiral beams was discovered, theoretically analyzed and experimentally implemented (Abramochkin and Volostnikov (2010)). It appeared that a spiral beam represents a light field retaining its structure to within the scale factor and rotation in the course of propagation and focusing. Moreover, the structure of such light field can be rather diverse; in particular, it can have the shape of an arbitrary planar curve, including a closed one. It was found that the complex amplitude S ( z , z ) of the field of this beam is uniquely related to the corresponding curve ] (t ) and is described by the expression: S(z, z | ] (t), t [0,T]) T
° zz ½° ° ] (t)] (t) 2z] (t) °½ exp ® 2 ¾ exp ® 2 ¾u U2 U ¿° ¯° U ¿° 0 ¯°
³
(2)
° 1 t ½° uexp ® 2 ª] (W )] c(W ) ] (W )] c(W )º dW ¾ | ] c(t) | dt ¬ ¼ °¯ U 0 °¿
³
where U is the Gaussian parameter of the beam, and the the top linedenotes a complex conjugation. The example of a given curve and the corresponding spiral beam is presented in Fig. 1. Rather essential is the property of ‘quantization’ of spiral beams in the form of closed curves. If the condition Scurve
1 2 SU N , N 2
0, 1, 2 }
(3)
133
Sergey A. Kishkin et al. / Physics Procedia 86 (2017) 131 – 135
Fig. 1.(a) “Generating” curve; (b) intensity distribution and (c) phase distribution of the corresponding spiral beam.
(where S curve is the area under the curve) is met, then the complex amplitude of the beam field does not depend on the choice of the reference point at the curve (Abramochkin and Volostnikov (2004)). In other words, the spiral beam is not determined by the reference point at the contour. Therefore, any finite sum of the series S N ( z , z | ] (t ), t [a, a T ]) is also independent of the reference point to the mutual unimodular term, depending only on the parameter a. Therefore, the problem of the reference point choice in the analysis and recognition of the input contour is removed. This means that with any required precision we can put into correspondence to the spiral beam S ( z , z | ] (t ), t [ a, a T ]) a finite sum of the series S N ( z , z | ] (t ), t [a, a T ])
e
zz
U2
N
¦c z
n
(4)
n
n 0
Since under the rotation of the analyzed contour by the angle D the finite sum of the series changes as S N ( zeiD , ze iD | ] (t ), t [a, a T ])
e
zz
U2
N
¦ c e D z i n
n
n 0
n
e
zz
U2
N
¦ cc z
n
n
n 0
the problem of the contour rotation is eliminated, too, and this proves once more that the expansion coefficients can characterize the rotation angles. 4. Algorithm of contour comparison Now let us consider two contours, the input one and the reference one, stored in a database, and detect whether they correspond to each other or not. Let us construct spiral beams for both contours, keeping the necessary number of terms in the series (Volostnikov et al (2013)). On using the above scheme let us put into correspondence to the contours the two spiral beams or two sets of complex coefficients {cn(1) }nN 0 and {cn(2) }nN 0 . It is assumed that prior to the construction of beams the normalization of the area, bounded by the contours, has already been carried out. The reduction to one area allows the determination of the scale of the input object. In the case when the quantization parameter is sufficient to provide distinguishing between the two contours, the above sets of coefficients coincide to within rotation (decisively, within the framework of the fixed basis: n 1, N ,
| cn(1) | | cn(2)
|
1, Mn
(1) (2) 1 cn cn 1 ln (2) (1) , i cn cn 1
(5)
134
Sergey A. Kishkin et al. / Physics Procedia 86 (2017) 131 – 135
If Mn { const for all n, then Mn is the angle D of the relative rotation of the contours. This is easily obtained by deriving the expression for the ratio of two complex amplitudes from the representation of the spiral beams in the form of series (4) and (5). If condition (6) is not satisfied, one can conclude that the contours do not correspond to each other.
Fig. 2.General block diagram of the method.
The block diagram of the algorithm for estimation of the two contours similarity is shown in Fig.2. The recognition algorithm is characterized by the following peculiarities. The first feature is the independence of the algorithm operation on the choice of the reference point location at the contour and the scale of the contour image. The second one is that the contour object can be of an arbitrary shape; its complexity is limited by the system resolution only, not by the number of the contour sections as in other known methods. An attractive property of the proposed method is that the enumeration of possibilities typical of a generally accepted recognition method of the contour analysis based on the correlation functions, is not required. One can logically refer to the shortcomings of the offered method the exponential operation used in calculations and being rather “heavy” for computation, especially in mobile devices. Besides, currently there is no efficient technique available for estimating of the quantization parameter and the number of the residual series terms sufficient (and obligatory) to ensure the recognition results stability. 5. Accounting for noise The numerical modeling results proved a reliable performance of the proposed algorithm in comparing of the model objects. Still, the unavoidable in real systems noise and distortions are not taken into consideration. The distortions occur due to discretization errors in CCD-matrices, to malfunctions during data transmission from the image detectors to computation devices, etc. The research showed that the method is rather resistant to noise in the contour being recognized. It can be attributed, first of all, to the Gaussian exponent in the expression for complex amplitude of the spiral beam (2) that ensures a “smoothing” effect, and as the result small deformations of the contour leads to insignificant changes in the complex amplitude. It should be also noted that the integral transform proper is also a good stabilizer. In spite of the resistibility of the complex amplitude and its zeroes to contour deformations the expansion coefficients (see Vieta theorem) can vary rather notably. In this case we propose to additionally use the correlation function K ( T)
³³ S
K (T )
³³
(1)
( z , z ) S (2) ( zeiT , zeiT ) dxdy
S (1) ( z, z ) S (1) ( z, z ) dxdy
³³
S (2) ( z , z ) S (2) ( z , z ) dxdy
(6)
Sergey A. Kishkin et al. / Physics Procedia 86 (2017) 131 – 135
by the angle of mutual rotation T of the spiral beams S (1) ( z , z ) ɢ S (2) ( z , z ) . The module maximum of this function is achieved at the true angle D of mutual rotation of the contours being recognized. Fig.3 illustrates an aircraft image, a marked boundary contour for the initial object, contour for a turned out and reduced object and also images of the spiral beams obtained. It is seen that in the contour of the reduced object the sections in the tail area are lacking, while spiral beams are practically identical. The calculated correlation function has its maximum value of 0.8 for the rotation angle equal to 123 degrees. And this is the real angle of the image rotation.
Fig. 3.From left to right: the aircraft initial image, marked boundary contour, intensity of the emerging spiral beams.
6. Conclusion A new approach to the contour images analysis is described, based on a close cooperation of the modern coherent optics, theory of functions and numeric techniques. The algorithm for images matching and comparison is proposed and theoretically grounded. This method allows detecting whether the two contours are identical to within scale and/or rotation. The dynamics of the offered method adaptation is shown for the case when the image under examination is subject to noise and deformations. Acknowledgements The work was supported by the Ministry of Education and Science of the Russian Federation and the Educational and Research Complex of the P.N. Lebedev Physical Institute References Abramochkin, E.G., Volostnikov, V.G., 2004. Spiral Light Beams. Uspekhi Fizicheskikh Nauk, 174, 1273-1300. Abramochkin, E.G., Volostnikov, V.G., 2010. Modern Optics of Gaussian Beams. M.: Fizmatlit, Moscow, pp. 184. Dedus, F.F., 2004. Classical and Orthogonal Bases for Tasks of Signals Analytical Description and Processing. Ɇ.: MSU. Furman, Ya.A., 2003. Introduction to Contour Analysis and its Applications in Image and Signal Processing. M.: Fizmatlit. Vizilter, Yu.V. et al., 2010. Image Processing and Analysis in Tasks of Computer Vision. M.: Fizmatkniga. Volostnikov, V.G., Kishkin, S.A., Kotova, S.P., 2013. New method of contour image processing based on the formalism of spiral light beam. Quantum Electronics, 43, 646–650.
135