Abstract

A novel method for evaluation of bacterial colonies number (Colony Forming Units - CFU), is described. Proposed algorithm, based on the Mellin transform, allows the CFU evaluation, invariant for the spatial orientation and scale changes. The proposed method involves image recording of bacteria grown in Petri dishes, calculation of the Fourier spectrum followed by coordinates transformation, and determination of the Mellin transform. It was proved that there is a high correlation between CFU and maxima of Mellin spectra. The method was practically implemented for evaluation of antibacterial activity of silver-based nanomaterials and the effect of an additional laser light irradiation.

© 2010 OSA

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2009 (2)

E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “System automation for a bacterial colony detection and identification instrument via forward scattering,” Meas. Sci. Technol. 20(1), 1–9 (2009).
[CrossRef]

Q. Yin, L. Shen, J. N. Kim, and Y. J. Jeong, “Scale-invariant pattern recognition using a combined Mellin radial harmonic function and the bidimensional empirical mode decomposition,” Opt. Express 17(19), 16581–16589 (2009).
[CrossRef] [PubMed]

2008 (4)

K. Wysocka, I. Buzalewicz, A. Wieliczko, K. Kowal, W. Stręk, and H. Podbielska, “Biomaterials with antibacterial activity,” Engin. Biomaterials 81–84, 117–119 (2008).

M. Venkatapathi, B. Rajwa, K. Ragheb, P. P. Banada, T. Lary, J. P. Robinson, and E. D. Hirleman, “High speed classification of individual bacterial cells using a model-based light scatter system and multivariate statistics,” Appl. Opt. 47(5), 678–686 (2008).
[CrossRef] [PubMed]

B. Rajwa, M. Venkatapathi, K. Ragheb, P. P. Banada, E. D. Hirleman, T. Lary, and J. P. Robinson, “Automated classification and recognition of bacterial particles in flow by multi-angle scatter measurement and a support-vector machine classifier,” Cytometry A 73A(4), 369–379 (2008).
[CrossRef]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time – resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13(1), 1–8 (2008).
[CrossRef]

2007 (4)

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

P. P. Banada, S. Guo, B. Bayraktar, E. W. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron. 22(8), 1664–1671 (2007).
[CrossRef]

E. Kolenović, E. Kolenović, T. Kreis, Ch. von Kopylow, and W. Jüptner, “Determination of large-scale out-of-plane displacements in digital Fourier holography,” Appl. Opt. 46(16), 3118–3125 (2007).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46(17), 3639–3648 (2007).
[CrossRef] [PubMed]

2005 (1)

M. Putman, R. Burton, and M. H. Nahm, “Simplified method to automatically count bacterial colony forming unit,” J. Immunol. Methods 302(1-2), 99–102 (2005).
[CrossRef] [PubMed]

2004 (1)

X. Liu, S. Wang, L. Sendi, and M. J. Caulfield, “High-throughput imaging of bacterial colonies grown on filter plates with application to serum bactericidal assays,” J. Immunol. Methods 292(1-2), 187–193 (2004).
[CrossRef] [PubMed]

2003 (4)

K. H. Kim, J. Yu, and M. H. Nahm, “Efficiency of a pneumococcal opsonophagocytic killing assay improved by multiplexing and by coloring colonies,” Clin. Diagn. Lab. Immunol. 10(4), 616–621 (2003).
[PubMed]

C. Y. Wu, A. R. D. Somervell, T. G. Haskell, and T. H. Barnes, “Optical Mellin transform through Haar wavelet transformation,” Opt. Commun. 227(1-3), 75–82 (2003).
[CrossRef]

M. Kawashita, S. Toda, H.-M. Kim, T. Kokubo, and N. Masuda, “Preparation of Antibacterial Silver-Doped Silica Glass Microspheres,” J. Biomed. Mater. Res. 66(2), 266–274 (2003).
[CrossRef]

L. Yaroslavsky, “Boundary effect free and adaptive discrete signal sinc-interpolation algorithms for signal and image resampling,” Appl. Opt. 42(20), 4166–4175 (2003).
[CrossRef] [PubMed]

2002 (2)

T. Irino and R. D. Patterson, “Segregating information about the size and the shape of the vocal tract using a time domain auditory model: The stabilized wavelet-Mellin transform,” Speech Commun. 36(3), 181–203 (2002).
[CrossRef]

J. Alvarez-Borrego, R. Mouriño-Pérez, G. Cristóbal, and J. Pech-Pacheco, “Invariant recognition of polychromatic image of Vibrio Cholerae O1,” Opt. Eng. 41(4), 827–833 (2002).
[CrossRef]

2001 (1)

Ch. Zoppou, “Review of urban storm water models,” Environ. Model. Softw. 16(3), 195–231 (2001).
[CrossRef]

1999 (1)

T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging 18(11), 1049–1075 (1999).
[CrossRef]

1998 (1)

A. Robinson, N. Sadr-kazemi, G. Dickason, and S. T. L. Harrison, “Morphological characterization of yeast colonies growth on solid media using image processing,” Biotechnol. Tech. 12(10), 763–767 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

D. Mukherjee, A. Pal, S. Sarma, and D. Majumder, “Bacterial colony counting using distance transform,” Int. J. Biomed. Comput. 38(2), 131–140 (1995).
[CrossRef] [PubMed]

1993 (1)

1991 (1)

M. Masuko, S. Hosoi, and T. Hayakawa, “A novel method for detection and counting of single bacteria in a wide field using an ultra-high-sensitivity TV camera without a microscope,” FEMS Microbiol. Lett. 81(3), 287–290 (1991).
[CrossRef]

1990 (1)

Y. K. Tung, “Mellin transform applied to uncertainty analysis in hydrology/ hydraulics,” J. Hydraul. Eng. 116(5), 659–674 (1990).
[CrossRef]

1976 (1)

Alvarez-Borrego, J.

J. Alvarez-Borrego, R. Mouriño-Pérez, G. Cristóbal, and J. Pech-Pacheco, “Invariant recognition of polychromatic image of Vibrio Cholerae O1,” Opt. Eng. 41(4), 827–833 (2002).
[CrossRef]

Aroonnual, A.

E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “System automation for a bacterial colony detection and identification instrument via forward scattering,” Meas. Sci. Technol. 20(1), 1–9 (2009).
[CrossRef]

Bae, E.

E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “System automation for a bacterial colony detection and identification instrument via forward scattering,” Meas. Sci. Technol. 20(1), 1–9 (2009).
[CrossRef]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time – resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13(1), 1–8 (2008).
[CrossRef]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46(17), 3639–3648 (2007).
[CrossRef] [PubMed]

Bae, E. W.

P. P. Banada, S. Guo, B. Bayraktar, E. W. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron. 22(8), 1664–1671 (2007).
[CrossRef]

Banada, P. P.

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time – resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13(1), 1–8 (2008).
[CrossRef]

B. Rajwa, M. Venkatapathi, K. Ragheb, P. P. Banada, E. D. Hirleman, T. Lary, and J. P. Robinson, “Automated classification and recognition of bacterial particles in flow by multi-angle scatter measurement and a support-vector machine classifier,” Cytometry A 73A(4), 369–379 (2008).
[CrossRef]

M. Venkatapathi, B. Rajwa, K. Ragheb, P. P. Banada, T. Lary, J. P. Robinson, and E. D. Hirleman, “High speed classification of individual bacterial cells using a model-based light scatter system and multivariate statistics,” Appl. Opt. 47(5), 678–686 (2008).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46(17), 3639–3648 (2007).
[CrossRef] [PubMed]

P. P. Banada, S. Guo, B. Bayraktar, E. W. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron. 22(8), 1664–1671 (2007).
[CrossRef]

Barnes, T. H.

C. Y. Wu, A. R. D. Somervell, T. G. Haskell, and T. H. Barnes, “Optical Mellin transform through Haar wavelet transformation,” Opt. Commun. 227(1-3), 75–82 (2003).
[CrossRef]

Bayraktar, B.

P. P. Banada, S. Guo, B. Bayraktar, E. W. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron. 22(8), 1664–1671 (2007).
[CrossRef]

Bhunia, A. K.

E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “System automation for a bacterial colony detection and identification instrument via forward scattering,” Meas. Sci. Technol. 20(1), 1–9 (2009).
[CrossRef]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time – resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13(1), 1–8 (2008).
[CrossRef]

P. P. Banada, S. Guo, B. Bayraktar, E. W. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron. 22(8), 1664–1671 (2007).
[CrossRef]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46(17), 3639–3648 (2007).
[CrossRef] [PubMed]

Burton, R.

M. Putman, R. Burton, and M. H. Nahm, “Simplified method to automatically count bacterial colony forming unit,” J. Immunol. Methods 302(1-2), 99–102 (2005).
[CrossRef] [PubMed]

Buzalewicz, I.

K. Wysocka, I. Buzalewicz, A. Wieliczko, K. Kowal, W. Stręk, and H. Podbielska, “Biomaterials with antibacterial activity,” Engin. Biomaterials 81–84, 117–119 (2008).

Casasent, D.

Caulfield, M. J.

X. Liu, S. Wang, L. Sendi, and M. J. Caulfield, “High-throughput imaging of bacterial colonies grown on filter plates with application to serum bactericidal assays,” J. Immunol. Methods 292(1-2), 187–193 (2004).
[CrossRef] [PubMed]

Cho, M. H.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Cristóbal, G.

J. Alvarez-Borrego, R. Mouriño-Pérez, G. Cristóbal, and J. Pech-Pacheco, “Invariant recognition of polychromatic image of Vibrio Cholerae O1,” Opt. Eng. 41(4), 827–833 (2002).
[CrossRef]

Dickason, G.

A. Robinson, N. Sadr-kazemi, G. Dickason, and S. T. L. Harrison, “Morphological characterization of yeast colonies growth on solid media using image processing,” Biotechnol. Tech. 12(10), 763–767 (1998).
[CrossRef]

Ellerbroek, B. L.

Gönner, C.

T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging 18(11), 1049–1075 (1999).
[CrossRef]

Guo, S.

P. P. Banada, S. Guo, B. Bayraktar, E. W. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron. 22(8), 1664–1671 (2007).
[CrossRef]

Harrison, S. T. L.

A. Robinson, N. Sadr-kazemi, G. Dickason, and S. T. L. Harrison, “Morphological characterization of yeast colonies growth on solid media using image processing,” Biotechnol. Tech. 12(10), 763–767 (1998).
[CrossRef]

Haskell, T. G.

C. Y. Wu, A. R. D. Somervell, T. G. Haskell, and T. H. Barnes, “Optical Mellin transform through Haar wavelet transformation,” Opt. Commun. 227(1-3), 75–82 (2003).
[CrossRef]

Hayakawa, T.

M. Masuko, S. Hosoi, and T. Hayakawa, “A novel method for detection and counting of single bacteria in a wide field using an ultra-high-sensitivity TV camera without a microscope,” FEMS Microbiol. Lett. 81(3), 287–290 (1991).
[CrossRef]

Hirleman, E. D.

E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “System automation for a bacterial colony detection and identification instrument via forward scattering,” Meas. Sci. Technol. 20(1), 1–9 (2009).
[CrossRef]

B. Rajwa, M. Venkatapathi, K. Ragheb, P. P. Banada, E. D. Hirleman, T. Lary, and J. P. Robinson, “Automated classification and recognition of bacterial particles in flow by multi-angle scatter measurement and a support-vector machine classifier,” Cytometry A 73A(4), 369–379 (2008).
[CrossRef]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time – resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13(1), 1–8 (2008).
[CrossRef]

M. Venkatapathi, B. Rajwa, K. Ragheb, P. P. Banada, T. Lary, J. P. Robinson, and E. D. Hirleman, “High speed classification of individual bacterial cells using a model-based light scatter system and multivariate statistics,” Appl. Opt. 47(5), 678–686 (2008).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46(17), 3639–3648 (2007).
[CrossRef] [PubMed]

P. P. Banada, S. Guo, B. Bayraktar, E. W. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron. 22(8), 1664–1671 (2007).
[CrossRef]

Hosoi, S.

M. Masuko, S. Hosoi, and T. Hayakawa, “A novel method for detection and counting of single bacteria in a wide field using an ultra-high-sensitivity TV camera without a microscope,” FEMS Microbiol. Lett. 81(3), 287–290 (1991).
[CrossRef]

Huff, K.

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time – resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13(1), 1–8 (2008).
[CrossRef]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46(17), 3639–3648 (2007).
[CrossRef] [PubMed]

Hwang, C. Y.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Irino, T.

T. Irino and R. D. Patterson, “Segregating information about the size and the shape of the vocal tract using a time domain auditory model: The stabilized wavelet-Mellin transform,” Speech Commun. 36(3), 181–203 (2002).
[CrossRef]

Jeong, D. H.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Jeong, Y. J.

Jüptner, W.

Kawashita, M.

M. Kawashita, S. Toda, H.-M. Kim, T. Kokubo, and N. Masuda, “Preparation of Antibacterial Silver-Doped Silica Glass Microspheres,” J. Biomed. Mater. Res. 66(2), 266–274 (2003).
[CrossRef]

Kim, H.-M.

M. Kawashita, S. Toda, H.-M. Kim, T. Kokubo, and N. Masuda, “Preparation of Antibacterial Silver-Doped Silica Glass Microspheres,” J. Biomed. Mater. Res. 66(2), 266–274 (2003).
[CrossRef]

Kim, J. H.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Kim, J. N.

Kim, J. S.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Kim, K. H.

K. H. Kim, J. Yu, and M. H. Nahm, “Efficiency of a pneumococcal opsonophagocytic killing assay improved by multiplexing and by coloring colonies,” Clin. Diagn. Lab. Immunol. 10(4), 616–621 (2003).
[PubMed]

Kim, S. H.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Kim, Y. K.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Kokubo, T.

M. Kawashita, S. Toda, H.-M. Kim, T. Kokubo, and N. Masuda, “Preparation of Antibacterial Silver-Doped Silica Glass Microspheres,” J. Biomed. Mater. Res. 66(2), 266–274 (2003).
[CrossRef]

Kolenovic, E.

Kowal, K.

K. Wysocka, I. Buzalewicz, A. Wieliczko, K. Kowal, W. Stręk, and H. Podbielska, “Biomaterials with antibacterial activity,” Engin. Biomaterials 81–84, 117–119 (2008).

Kreis, T.

Kuk, E.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Lary, T.

B. Rajwa, M. Venkatapathi, K. Ragheb, P. P. Banada, E. D. Hirleman, T. Lary, and J. P. Robinson, “Automated classification and recognition of bacterial particles in flow by multi-angle scatter measurement and a support-vector machine classifier,” Cytometry A 73A(4), 369–379 (2008).
[CrossRef]

M. Venkatapathi, B. Rajwa, K. Ragheb, P. P. Banada, T. Lary, J. P. Robinson, and E. D. Hirleman, “High speed classification of individual bacterial cells using a model-based light scatter system and multivariate statistics,” Appl. Opt. 47(5), 678–686 (2008).
[CrossRef] [PubMed]

Lee, H. J.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Lee, Y. S.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Lehmann, T. M.

T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging 18(11), 1049–1075 (1999).
[CrossRef]

Liu, X.

X. Liu, S. Wang, L. Sendi, and M. J. Caulfield, “High-throughput imaging of bacterial colonies grown on filter plates with application to serum bactericidal assays,” J. Immunol. Methods 292(1-2), 187–193 (2004).
[CrossRef] [PubMed]

Majumder, D.

D. Mukherjee, A. Pal, S. Sarma, and D. Majumder, “Bacterial colony counting using distance transform,” Int. J. Biomed. Comput. 38(2), 131–140 (1995).
[CrossRef] [PubMed]

Masuda, N.

M. Kawashita, S. Toda, H.-M. Kim, T. Kokubo, and N. Masuda, “Preparation of Antibacterial Silver-Doped Silica Glass Microspheres,” J. Biomed. Mater. Res. 66(2), 266–274 (2003).
[CrossRef]

Masuko, M.

M. Masuko, S. Hosoi, and T. Hayakawa, “A novel method for detection and counting of single bacteria in a wide field using an ultra-high-sensitivity TV camera without a microscope,” FEMS Microbiol. Lett. 81(3), 287–290 (1991).
[CrossRef]

Mouriño-Pérez, R.

J. Alvarez-Borrego, R. Mouriño-Pérez, G. Cristóbal, and J. Pech-Pacheco, “Invariant recognition of polychromatic image of Vibrio Cholerae O1,” Opt. Eng. 41(4), 827–833 (2002).
[CrossRef]

Mukherjee, D.

D. Mukherjee, A. Pal, S. Sarma, and D. Majumder, “Bacterial colony counting using distance transform,” Int. J. Biomed. Comput. 38(2), 131–140 (1995).
[CrossRef] [PubMed]

Nahm, M. H.

M. Putman, R. Burton, and M. H. Nahm, “Simplified method to automatically count bacterial colony forming unit,” J. Immunol. Methods 302(1-2), 99–102 (2005).
[CrossRef] [PubMed]

K. H. Kim, J. Yu, and M. H. Nahm, “Efficiency of a pneumococcal opsonophagocytic killing assay improved by multiplexing and by coloring colonies,” Clin. Diagn. Lab. Immunol. 10(4), 616–621 (2003).
[PubMed]

Pal, A.

D. Mukherjee, A. Pal, S. Sarma, and D. Majumder, “Bacterial colony counting using distance transform,” Int. J. Biomed. Comput. 38(2), 131–140 (1995).
[CrossRef] [PubMed]

Park, S. J.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Park, Y. H.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Park, Y. K.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Patterson, R. D.

T. Irino and R. D. Patterson, “Segregating information about the size and the shape of the vocal tract using a time domain auditory model: The stabilized wavelet-Mellin transform,” Speech Commun. 36(3), 181–203 (2002).
[CrossRef]

Pech-Pacheco, J.

J. Alvarez-Borrego, R. Mouriño-Pérez, G. Cristóbal, and J. Pech-Pacheco, “Invariant recognition of polychromatic image of Vibrio Cholerae O1,” Opt. Eng. 41(4), 827–833 (2002).
[CrossRef]

Podbielska, H.

K. Wysocka, I. Buzalewicz, A. Wieliczko, K. Kowal, W. Stręk, and H. Podbielska, “Biomaterials with antibacterial activity,” Engin. Biomaterials 81–84, 117–119 (2008).

Psaltis, D.

Putman, M.

M. Putman, R. Burton, and M. H. Nahm, “Simplified method to automatically count bacterial colony forming unit,” J. Immunol. Methods 302(1-2), 99–102 (2005).
[CrossRef] [PubMed]

Ragheb, K.

B. Rajwa, M. Venkatapathi, K. Ragheb, P. P. Banada, E. D. Hirleman, T. Lary, and J. P. Robinson, “Automated classification and recognition of bacterial particles in flow by multi-angle scatter measurement and a support-vector machine classifier,” Cytometry A 73A(4), 369–379 (2008).
[CrossRef]

M. Venkatapathi, B. Rajwa, K. Ragheb, P. P. Banada, T. Lary, J. P. Robinson, and E. D. Hirleman, “High speed classification of individual bacterial cells using a model-based light scatter system and multivariate statistics,” Appl. Opt. 47(5), 678–686 (2008).
[CrossRef] [PubMed]

Rajwa, B.

M. Venkatapathi, B. Rajwa, K. Ragheb, P. P. Banada, T. Lary, J. P. Robinson, and E. D. Hirleman, “High speed classification of individual bacterial cells using a model-based light scatter system and multivariate statistics,” Appl. Opt. 47(5), 678–686 (2008).
[CrossRef] [PubMed]

B. Rajwa, M. Venkatapathi, K. Ragheb, P. P. Banada, E. D. Hirleman, T. Lary, and J. P. Robinson, “Automated classification and recognition of bacterial particles in flow by multi-angle scatter measurement and a support-vector machine classifier,” Cytometry A 73A(4), 369–379 (2008).
[CrossRef]

P. P. Banada, S. Guo, B. Bayraktar, E. W. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron. 22(8), 1664–1671 (2007).
[CrossRef]

Robinson, A.

A. Robinson, N. Sadr-kazemi, G. Dickason, and S. T. L. Harrison, “Morphological characterization of yeast colonies growth on solid media using image processing,” Biotechnol. Tech. 12(10), 763–767 (1998).
[CrossRef]

Robinson, J. P.

E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “System automation for a bacterial colony detection and identification instrument via forward scattering,” Meas. Sci. Technol. 20(1), 1–9 (2009).
[CrossRef]

B. Rajwa, M. Venkatapathi, K. Ragheb, P. P. Banada, E. D. Hirleman, T. Lary, and J. P. Robinson, “Automated classification and recognition of bacterial particles in flow by multi-angle scatter measurement and a support-vector machine classifier,” Cytometry A 73A(4), 369–379 (2008).
[CrossRef]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time – resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13(1), 1–8 (2008).
[CrossRef]

M. Venkatapathi, B. Rajwa, K. Ragheb, P. P. Banada, T. Lary, J. P. Robinson, and E. D. Hirleman, “High speed classification of individual bacterial cells using a model-based light scatter system and multivariate statistics,” Appl. Opt. 47(5), 678–686 (2008).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46(17), 3639–3648 (2007).
[CrossRef] [PubMed]

P. P. Banada, S. Guo, B. Bayraktar, E. W. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron. 22(8), 1664–1671 (2007).
[CrossRef]

Roux, F. S.

Sadr-kazemi, N.

A. Robinson, N. Sadr-kazemi, G. Dickason, and S. T. L. Harrison, “Morphological characterization of yeast colonies growth on solid media using image processing,” Biotechnol. Tech. 12(10), 763–767 (1998).
[CrossRef]

Sarma, S.

D. Mukherjee, A. Pal, S. Sarma, and D. Majumder, “Bacterial colony counting using distance transform,” Int. J. Biomed. Comput. 38(2), 131–140 (1995).
[CrossRef] [PubMed]

Sasiela, R. J.

Sendi, L.

X. Liu, S. Wang, L. Sendi, and M. J. Caulfield, “High-throughput imaging of bacterial colonies grown on filter plates with application to serum bactericidal assays,” J. Immunol. Methods 292(1-2), 187–193 (2004).
[CrossRef] [PubMed]

Shelton, J. D.

Shen, L.

Somervell, A. R. D.

C. Y. Wu, A. R. D. Somervell, T. G. Haskell, and T. H. Barnes, “Optical Mellin transform through Haar wavelet transformation,” Opt. Commun. 227(1-3), 75–82 (2003).
[CrossRef]

Spitzer, K.

T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging 18(11), 1049–1075 (1999).
[CrossRef]

Strek, W.

K. Wysocka, I. Buzalewicz, A. Wieliczko, K. Kowal, W. Stręk, and H. Podbielska, “Biomaterials with antibacterial activity,” Engin. Biomaterials 81–84, 117–119 (2008).

Toda, S.

M. Kawashita, S. Toda, H.-M. Kim, T. Kokubo, and N. Masuda, “Preparation of Antibacterial Silver-Doped Silica Glass Microspheres,” J. Biomed. Mater. Res. 66(2), 266–274 (2003).
[CrossRef]

Tung, Y. K.

Y. K. Tung, “Mellin transform applied to uncertainty analysis in hydrology/ hydraulics,” J. Hydraul. Eng. 116(5), 659–674 (1990).
[CrossRef]

Venkatapathi, M.

B. Rajwa, M. Venkatapathi, K. Ragheb, P. P. Banada, E. D. Hirleman, T. Lary, and J. P. Robinson, “Automated classification and recognition of bacterial particles in flow by multi-angle scatter measurement and a support-vector machine classifier,” Cytometry A 73A(4), 369–379 (2008).
[CrossRef]

M. Venkatapathi, B. Rajwa, K. Ragheb, P. P. Banada, T. Lary, J. P. Robinson, and E. D. Hirleman, “High speed classification of individual bacterial cells using a model-based light scatter system and multivariate statistics,” Appl. Opt. 47(5), 678–686 (2008).
[CrossRef] [PubMed]

von Kopylow, Ch.

Wang, S.

X. Liu, S. Wang, L. Sendi, and M. J. Caulfield, “High-throughput imaging of bacterial colonies grown on filter plates with application to serum bactericidal assays,” J. Immunol. Methods 292(1-2), 187–193 (2004).
[CrossRef] [PubMed]

Wieliczko, A.

K. Wysocka, I. Buzalewicz, A. Wieliczko, K. Kowal, W. Stręk, and H. Podbielska, “Biomaterials with antibacterial activity,” Engin. Biomaterials 81–84, 117–119 (2008).

Wu, C. Y.

C. Y. Wu, A. R. D. Somervell, T. G. Haskell, and T. H. Barnes, “Optical Mellin transform through Haar wavelet transformation,” Opt. Commun. 227(1-3), 75–82 (2003).
[CrossRef]

Wysocka, K.

K. Wysocka, I. Buzalewicz, A. Wieliczko, K. Kowal, W. Stręk, and H. Podbielska, “Biomaterials with antibacterial activity,” Engin. Biomaterials 81–84, 117–119 (2008).

Yaroslavsky, L.

Yin, Q.

Yu, J.

K. H. Kim, J. Yu, and M. H. Nahm, “Efficiency of a pneumococcal opsonophagocytic killing assay improved by multiplexing and by coloring colonies,” Clin. Diagn. Lab. Immunol. 10(4), 616–621 (2003).
[PubMed]

Yu, K. N.

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Zoppou, Ch.

Ch. Zoppou, “Review of urban storm water models,” Environ. Model. Softw. 16(3), 195–231 (2001).
[CrossRef]

Appl. Opt. (7)

Biosens. Bioelectron. (1)

P. P. Banada, S. Guo, B. Bayraktar, E. W. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron. 22(8), 1664–1671 (2007).
[CrossRef]

Biotechnol. Tech. (1)

A. Robinson, N. Sadr-kazemi, G. Dickason, and S. T. L. Harrison, “Morphological characterization of yeast colonies growth on solid media using image processing,” Biotechnol. Tech. 12(10), 763–767 (1998).
[CrossRef]

Clin. Diagn. Lab. Immunol. (1)

K. H. Kim, J. Yu, and M. H. Nahm, “Efficiency of a pneumococcal opsonophagocytic killing assay improved by multiplexing and by coloring colonies,” Clin. Diagn. Lab. Immunol. 10(4), 616–621 (2003).
[PubMed]

Cytometry A (1)

B. Rajwa, M. Venkatapathi, K. Ragheb, P. P. Banada, E. D. Hirleman, T. Lary, and J. P. Robinson, “Automated classification and recognition of bacterial particles in flow by multi-angle scatter measurement and a support-vector machine classifier,” Cytometry A 73A(4), 369–379 (2008).
[CrossRef]

Engin. Biomaterials (1)

K. Wysocka, I. Buzalewicz, A. Wieliczko, K. Kowal, W. Stręk, and H. Podbielska, “Biomaterials with antibacterial activity,” Engin. Biomaterials 81–84, 117–119 (2008).

Environ. Model. Softw. (1)

Ch. Zoppou, “Review of urban storm water models,” Environ. Model. Softw. 16(3), 195–231 (2001).
[CrossRef]

FEMS Microbiol. Lett. (1)

M. Masuko, S. Hosoi, and T. Hayakawa, “A novel method for detection and counting of single bacteria in a wide field using an ultra-high-sensitivity TV camera without a microscope,” FEMS Microbiol. Lett. 81(3), 287–290 (1991).
[CrossRef]

IEEE Trans. Med. Imaging (1)

T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging 18(11), 1049–1075 (1999).
[CrossRef]

Int. J. Biomed. Comput. (1)

D. Mukherjee, A. Pal, S. Sarma, and D. Majumder, “Bacterial colony counting using distance transform,” Int. J. Biomed. Comput. 38(2), 131–140 (1995).
[CrossRef] [PubMed]

J. Biomed. Mater. Res. (1)

M. Kawashita, S. Toda, H.-M. Kim, T. Kokubo, and N. Masuda, “Preparation of Antibacterial Silver-Doped Silica Glass Microspheres,” J. Biomed. Mater. Res. 66(2), 266–274 (2003).
[CrossRef]

J. Biomed. Opt. (1)

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time – resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13(1), 1–8 (2008).
[CrossRef]

J. Hydraul. Eng. (1)

Y. K. Tung, “Mellin transform applied to uncertainty analysis in hydrology/ hydraulics,” J. Hydraul. Eng. 116(5), 659–674 (1990).
[CrossRef]

J. Immunol. Methods (2)

M. Putman, R. Burton, and M. H. Nahm, “Simplified method to automatically count bacterial colony forming unit,” J. Immunol. Methods 302(1-2), 99–102 (2005).
[CrossRef] [PubMed]

X. Liu, S. Wang, L. Sendi, and M. J. Caulfield, “High-throughput imaging of bacterial colonies grown on filter plates with application to serum bactericidal assays,” J. Immunol. Methods 292(1-2), 187–193 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “System automation for a bacterial colony detection and identification instrument via forward scattering,” Meas. Sci. Technol. 20(1), 1–9 (2009).
[CrossRef]

Nanomedicine (1)

J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S. J. Park, H. J. Lee, S. H. Kim, Y. K. Park, Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, and M. H. Cho, “Antimicrobial effects of silver nanoparticles,” Nanomedicine 3(1), 95–101 (2007).
[PubMed]

Opt. Commun. (1)

C. Y. Wu, A. R. D. Somervell, T. G. Haskell, and T. H. Barnes, “Optical Mellin transform through Haar wavelet transformation,” Opt. Commun. 227(1-3), 75–82 (2003).
[CrossRef]

Opt. Eng. (1)

J. Alvarez-Borrego, R. Mouriño-Pérez, G. Cristóbal, and J. Pech-Pacheco, “Invariant recognition of polychromatic image of Vibrio Cholerae O1,” Opt. Eng. 41(4), 827–833 (2002).
[CrossRef]

Opt. Express (1)

Speech Commun. (1)

T. Irino and R. D. Patterson, “Segregating information about the size and the shape of the vocal tract using a time domain auditory model: The stabilized wavelet-Mellin transform,” Speech Commun. 36(3), 181–203 (2002).
[CrossRef]

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A. De Sena, and D. Rocchesso, “A Fast Mellin transform with applications in DAFX,” Proceedings of 7th International Conference on Digital Audio Effects, 65–69 (2004) http://dafx04.na.infn.it/WebProc/Proc/P_065.pdf

Z. Sun, and Ch. Han, “Parameter estimation of non-Rayleigh RCS models for SAR images based on the Mellin transformation,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2009), pp. 1081–1084.

A. Derbel, F. Kalel, A. Ben Hamida, and M. Samet, “Wavelet filtering based on Mellin transform dedicated to Cochlear Prostheses”, in Proceedings of 29th Annual International Conference of the IEEE on Engineering in Medicine and Biology Society3 (IEEE, 2007), pp. 1990–1903.

Z. Tong, Y. Fusheng, and T. Qingyu, “A fast algorithm of continuous wavelet transform based on Mellin transform with biomedical application,” in Proceedings of the 20th Annual International Conference of the IEEE on Engineering in Medicine and Biology Society3 (IEEE, 1998), pp. 1142–1144.

I. Buzalewicz, K. Wysocka, and H. Podbielska, “Exploiting of optical transforms for bacteria evaluation in vitro,” Proc. SPIE 7371, 73711H–73711H–6 (2009).

J. Dongmei, and Z. Rongchun, “Speaker normalization based on the generalized time - frequency representation and Mellin transform”, in Proceedings of 5th International Conference on Signal Processing Proceedings2, (IEEE, 2000), pp. 782–785.

J. Chen, B. Xu, and T. Huang, “A novel robust feature of speech signal based on Mellin transform for speaker – independent speech recognition,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1998), pp. 629–632.

D. Casasent, and D. Psaltis, “New optical transforms for pattern recognition,” in Proceedings of the IEEE60(1), 77–84 (1977).

J. Alvarez-Borrego, R. Mouriño-Pérez, G. Cristóbal, and J. Pech-Pacheco, “Invariant optical color correlation for recognition of Vibrio cholerae O1,” in Proceedings of International IEEE Conference on Pattern Recognition, vol. 2, (IEEE, 2000), pp. 2283.

R. N. Bracewell, The Fourier Transform and Its Applications, Third edition, (McGraw-Hill, 2000).

H. Stark, ed., Applications of Optical Fourier Transforms, (Academic Press, 1982).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, Second edition, (World Scientific, 2006).

J. W. Goodman, Introduction to Fourier Optics, Third edition, (Robert & Company Publishers, 2005).

N. Götz, S. Drüe, and G. Hartmann, “Invariant object recognition with discriminant features based on local fast-Fourier Mellin transform, “in Proceedings of 15th International Conference of Pattern Recognition1, (IEEE, 2000), pp. 948-951.

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Figures (10)

Fig. 1
Fig. 1

Conception of normalized Fourier spectrum Soutput along f x (N is a modulation background correlated with n, m(fx) is a modulation factor in a form of cosines sum associated with the spatial object configuration).

Fig. 2
Fig. 2

The function Soutput(fx, fy) along fx for n = 5 square apertures.

Fig. 3
Fig. 3

Dependence between the value N of the modulation background and the objects number n.

Fig. 4
Fig. 4

Schematic representation of the Mellin transform algorithm.

Fig. 5
Fig. 5

Log-polar transformation of the input Fourier spectrum of rectangular aperture: (a) aperture with the size 8x8, (b) aperture with the size 16x16. The logarithmic component was sampled in range from ln ρ min to ln ρ max by 256 points.

Fig. 6
Fig. 6

1D normalized Mellin spectrum for three square apertures.

Fig. 7
Fig. 7

Simulation results: (a) maximal value of the Mellin spectrum versus objects number, (b) exemplary Mellin spectrum of 120 objects.

Fig. 8
Fig. 8

Exemplary images: a) control sample - colonies of Escherichia coli on MacConkey agar, b) colonies treated by antibacterial agent.

Fig. 9
Fig. 9

Log-polar transform of the initial Fourier spectrum: (a) control sample, (b) bacteria treated by colloidal solution of Ag–doped silica nanoparticles (0.25:1).

Fig. 10
Fig. 10

Comparison of maximum values of 2D Mellin spectra and number of manually counted bacteria colonies (dots – samples treated with antibacterial agents and irradiated starting from the highest concentration of antibacterial agent, triangles – samples treated with antibacterial agents, non-irradiated, square – control sample).

Tables (1)

Tables Icon

Table 1 Comparison of colonies number. CFUE - evaluated, basing on Eq. (15), CFUMC counted manually.

Equations (17)

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U n ( x , y ) = + + Γ ( ξ , η ) t 0 ( x ξ , y η ) d ξ d η ,
Γ ( ξ , η ) = i = 1 n δ ( ξ x i , η y i ) .
{ U n ( x , y ) } = { t 0 ( x , y ) } { Γ ( x , y ) } ,
F Γ ( f x , f y ) = { Γ ( x , y ) } = i = 1 n exp { 2 π i ( x i f x + y i f y ) } .
F n ( f x , f y ) = F 0 ( f x , f y ) i = 1 n exp { 2 π i ( x i f x + y i f y ) } .
S ( f x , f y ) = | F n ( f x , f y ) | 2 = F n ( f x , f y ) F n ( f x , f y ) = | F 0 ( f x , f y ) | 2 [ n + m ( f x , f y ) ] .
m ( f x , f y ) = i = 1 n 1 [ j = i n 1 2 cos { 2 π [ ( x j + 1 x i ) f x + ( y j + 1 y i ) f y ] } ] .
S o u t p u t ( f x , f y ) = [ N + m ( f x , f y ) ] ,
N 1 2 [ S o u t p u t M A X ( f x , f y ) + S o u t p u t M I N ( f x , f y ) ] .
N = n e r r o r ( n ) .
M ( s ) = 0 g ( ξ ) ξ s 1 d ξ ,
{ g ( m r ) } = 1 m F g ( ln [ ρ m ] ) = 1 m F g ( ln ρ ln m ) .
M ( ω ρ , θ ) = + F ( e ρ ˜ , θ ) exp ( i ρ ˜ ω ρ ) d ρ ˜ ,
M ( ω ρ , θ ) = + S o u t p u t ( e ρ ˜ , θ ) exp { i ( ρ ˜ ω ρ ) } d ρ ˜
= + N exp { i ( ρ ˜ ω ρ ) d ρ ˜ + + m ( e ρ ˜ , θ ) exp { i ( ρ ˜ ω ρ ) } d ρ ˜
= N δ ( 0 , 0 ) + + m ( e ρ ˜ , θ ) exp { i ( ρ ˜ ω ρ ) } d ρ ˜ .
a 2 ( M M S ) 2 + a 1 ( M M S ) + a 0

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