J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. A 26, 1858-1867 (2009).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641-648 (2009).

[CrossRef]

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727-730 (2009).

[CrossRef]

A. Cortes, I. Velez, and J. F. Sevillano, “Radix rk FFTs: matricial representation and SDC/SDF pipeline implementation,” IEEE Trans. Signal Process. 57, 2824-2839 (2009).

[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383-2394 (2008).

[CrossRef]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “An additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599-2601 (2008).

[CrossRef]
[PubMed]

U. Gopinathan, G. Situ, T. J. Naughton, and J. T. Sheridan, “Noninterferometric phase retrieval using a fractional Fourier system,” J. Opt. Soc. Am. A 25, 108-115 (2008).

[CrossRef]

X. Liu and K.-H. Brenner, “Minimal optical decomposition of ray transfer matrices,” Appl. Opt. 47, E88-E98 (2008).

[CrossRef]
[PubMed]

J. J. Healy and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett. 33, 228-230 (2008).

[CrossRef]
[PubMed]

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825-2832 (2008).

[CrossRef]

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983-990 (2007).

[CrossRef]

T. Alieva and M. J. Bastiaans, “Properties of the linear canonical integral transformation,” J. Opt. Soc. Am. A 24, 3658-3665 (2007).

[CrossRef]

A. Nelleri, J. Joseph, and K. Singh, “Digital Fresnel field encryption for three-dimensional information security,” Opt. Eng. (Bellingham) 46, 045801 (8 pages) (2007).

[CrossRef]

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111-119 (2007).

[CrossRef]

T. J. Lundy and J. Van Buskirk, “A new matrix approach to real FFTs and convolutions of length 2k,” Computing 80, 23-45 (2007).

[CrossRef]

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).

[CrossRef]

B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).

[CrossRef]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23, 2861-2870 (2006).

[CrossRef]

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32-34 (2006).

[CrossRef]
[PubMed]

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421-1425 (2006).

[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Sci. China Ser. F, Inf. Sci. 49, 592-603 (2006).

[CrossRef]

H. M. Ozaktas, A. Koç. I. Sari, and M. Alper Kutay, “Efficient computation of quadratic-phase integrals in optics,” Appl. Opt. 31, 35-37 (2006).

[CrossRef]

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216-231 (2005).

[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22, 928-937 (2005).

[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917-927 (2005).

[CrossRef]

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. (Bellingham) 43, 2557-2563 (2004).

[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Stuttgart) 114, 251-265 (2003).

S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338-1353 (2000).

[CrossRef]

B. Barshan, M. Alper Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).

[CrossRef]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470-473 (1996).

[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).

[CrossRef]

X.-G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Trans. Signal Process. 3, 72-74 (March 1996).

[CrossRef]

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293-297 (1981).

[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).

[CrossRef]

C. M. Rader, “Discrete Fourier transforms when the number of data samples is prime,” Proc. IEEE 56, 1107-1108 (1968).

[CrossRef]

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297-301 (1965).

[CrossRef]

H. M. Ozaktas, A. Koç. I. Sari, and M. Alper Kutay, “Efficient computation of quadratic-phase integrals in optics,” Appl. Opt. 31, 35-37 (2006).

[CrossRef]

B. Barshan, M. Alper Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).

[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).

[CrossRef]

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).

[CrossRef]

B. Barshan, M. Alper Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).

[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).

[CrossRef]

T. Alieva and M. J. Bastiaans, “Properties of the linear canonical integral transformation,” J. Opt. Soc. Am. A 24, 3658-3665 (2007).

[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710-1716 (1979).

[CrossRef]

M. J. Bastiaans and K. B. Wolf, “Phase reconstruction from intensity measurements in one-parameter canonical-transform systems,” in Proceedings of Seventh International Symposium on Signal Processing and Its Applications, Vol. 1 (IEEE, 2003), pp. 589-592.

[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).

[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383-2394 (2008).

[CrossRef]

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).

[CrossRef]

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297-301 (1965).

[CrossRef]

A. Cortes, I. Velez, and J. F. Sevillano, “Radix rk FFTs: matricial representation and SDC/SDF pipeline implementation,” IEEE Trans. Signal Process. 57, 2824-2839 (2009).

[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Sci. China Ser. F, Inf. Sci. 49, 592-603 (2006).

[CrossRef]

S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338-1353 (2000).

[CrossRef]

J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. dissertation (National Taiwan University, 2001).

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).

[CrossRef]

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).

[CrossRef]

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111-119 (2007).

[CrossRef]

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216-231 (2005).

[CrossRef]

W. M. Gentleman and G. Sande, “Fast Fourier transforms--for fun and profit,” in Proceedings of AFIPS Fall Joint Computer Conference, Vol. 29 (ACM, 1966), pp. 563-578.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

U. Gopinathan, G. Situ, T. J. Naughton, and J. T. Sheridan, “Noninterferometric phase retrieval using a fractional Fourier system,” J. Opt. Soc. Am. A 25, 108-115 (2008).

[CrossRef]

B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).

[CrossRef]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23, 2861-2870 (2006).

[CrossRef]

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293-297 (1981).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641-648 (2009).

[CrossRef]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “An additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599-2601 (2008).

[CrossRef]
[PubMed]

J. J. Healy and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett. 33, 228-230 (2008).

[CrossRef]
[PubMed]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “An additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599-2601 (2008).

[CrossRef]
[PubMed]

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32-34 (2006).

[CrossRef]
[PubMed]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23, 2861-2870 (2006).

[CrossRef]

B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).

[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917-927 (2005).

[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22, 928-937 (2005).

[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Stuttgart) 114, 251-265 (2003).

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).

[CrossRef]

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111-119 (2007).

[CrossRef]

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216-231 (2005).

[CrossRef]

A. Nelleri, J. Joseph, and K. Singh, “Digital Fresnel field encryption for three-dimensional information security,” Opt. Eng. (Bellingham) 46, 045801 (8 pages) (2007).

[CrossRef]

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. A 26, 1858-1867 (2009).

[CrossRef]

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32-34 (2006).

[CrossRef]
[PubMed]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23, 2861-2870 (2006).

[CrossRef]

B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).

[CrossRef]

B. Kernighan, and D. Ritchie, The C Programming Language (Prentice-Hall, 1978).

G. Kloos, Matrix Methods for Optical Layout (SPIE Press, 2007).

[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383-2394 (2008).

[CrossRef]

T. Kreis, Handbook of Holographic Interferometry, Optical and Digital Methods (Wiley-VCH, 2005).

J.-C. Kuo, C.-H. Wen, and A.-Y. Wu, “Implementation of a programmable 64~2048-point FFT/IFFT processor for OFDM-based communication systems,” in Proceedings of the 2003 International Symposium on Circuits and Systems, Vol. 2 (IEEE, 2003), pp. 121-124.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383-2394 (2008).

[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).

[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983-990 (2007).

[CrossRef]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470-473 (1996).

[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).

[CrossRef]

T. J. Lundy and J. Van Buskirk, “A new matrix approach to real FFTs and convolutions of length 2k,” Computing 80, 23-45 (2007).

[CrossRef]

A. Nelleri, J. Joseph, and K. Singh, “Digital Fresnel field encryption for three-dimensional information security,” Opt. Eng. (Bellingham) 46, 045801 (8 pages) (2007).

[CrossRef]

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727-730 (2009).

[CrossRef]

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32-34 (2006).

[CrossRef]
[PubMed]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23, 2861-2870 (2006).

[CrossRef]

B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).

[CrossRef]

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. (Bellingham) 43, 2557-2563 (2004).

[CrossRef]

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727-730 (2009).

[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383-2394 (2008).

[CrossRef]

H. M. Ozaktas, A. Koç. I. Sari, and M. Alper Kutay, “Efficient computation of quadratic-phase integrals in optics,” Appl. Opt. 31, 35-37 (2006).

[CrossRef]

B. Barshan, M. Alper Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).

[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).

[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).

[CrossRef]

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32-34 (2006).

[CrossRef]
[PubMed]

S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338-1353 (2000).

[CrossRef]

C. M. Rader, “Discrete Fourier transforms when the number of data samples is prime,” Proc. IEEE 56, 1107-1108 (1968).

[CrossRef]

W. T. Rhodes, “Light tubes, Wigner diagrams and optical signal propagation simulation,” in Optical Information Processing: A Tribute to Adolf Lohmann, H.J.Caulfield, ed. (SPIE Press, 2002) pp. 343-356 (2002).

B. Kernighan, and D. Ritchie, The C Programming Language (Prentice-Hall, 1978).

W. M. Gentleman and G. Sande, “Fast Fourier transforms--for fun and profit,” in Proceedings of AFIPS Fall Joint Computer Conference, Vol. 29 (ACM, 1966), pp. 563-578.

H. M. Ozaktas, A. Koç. I. Sari, and M. Alper Kutay, “Efficient computation of quadratic-phase integrals in optics,” Appl. Opt. 31, 35-37 (2006).

[CrossRef]

A. Cortes, I. Velez, and J. F. Sevillano, “Radix rk FFTs: matricial representation and SDC/SDF pipeline implementation,” IEEE Trans. Signal Process. 57, 2824-2839 (2009).

[CrossRef]

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. A 26, 1858-1867 (2009).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641-648 (2009).

[CrossRef]

U. Gopinathan, G. Situ, T. J. Naughton, and J. T. Sheridan, “Noninterferometric phase retrieval using a fractional Fourier system,” J. Opt. Soc. Am. A 25, 108-115 (2008).

[CrossRef]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “An additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599-2601 (2008).

[CrossRef]
[PubMed]

J. J. Healy and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett. 33, 228-230 (2008).

[CrossRef]
[PubMed]

B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).

[CrossRef]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23, 2861-2870 (2006).

[CrossRef]

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32-34 (2006).

[CrossRef]
[PubMed]

B. M. Hennelly and J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917-927 (2005).

[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22, 928-937 (2005).

[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Stuttgart) 114, 251-265 (2003).

S. Abe and J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation an operator approach,” J. Phys. A 27, 4179-4187 (1994).

[CrossRef]

S. Abe and J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801-1803 (1994).

[CrossRef]
[PubMed]

A. Nelleri, J. Joseph, and K. Singh, “Digital Fresnel field encryption for three-dimensional information security,” Opt. Eng. (Bellingham) 46, 045801 (8 pages) (2007).

[CrossRef]

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421-1425 (2006).

[CrossRef]

A. Stern, “Why is the linear canonical transform so little known?” in Proceedings of 5th International Workshop on Information Optics, G.Cristóbal, B.Javidi, and S.Vallmitjana, eds (Springer, 2006), pp. 225-234.

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825-2832 (2008).

[CrossRef]

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983-990 (2007).

[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Sci. China Ser. F, Inf. Sci. 49, 592-603 (2006).

[CrossRef]

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297-301 (1965).

[CrossRef]

T. J. Lundy and J. Van Buskirk, “A new matrix approach to real FFTs and convolutions of length 2k,” Computing 80, 23-45 (2007).

[CrossRef]

A. Cortes, I. Velez, and J. F. Sevillano, “Radix rk FFTs: matricial representation and SDC/SDF pipeline implementation,” IEEE Trans. Signal Process. 57, 2824-2839 (2009).

[CrossRef]

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825-2832 (2008).

[CrossRef]

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983-990 (2007).

[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Sci. China Ser. F, Inf. Sci. 49, 592-603 (2006).

[CrossRef]

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. A 26, 1858-1867 (2009).

[CrossRef]

B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).

[CrossRef]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23, 2861-2870 (2006).

[CrossRef]

J.-C. Kuo, C.-H. Wen, and A.-Y. Wu, “Implementation of a programmable 64~2048-point FFT/IFFT processor for OFDM-based communication systems,” in Proceedings of the 2003 International Symposium on Circuits and Systems, Vol. 2 (IEEE, 2003), pp. 121-124.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).

M. J. Bastiaans and K. B. Wolf, “Phase reconstruction from intensity measurements in one-parameter canonical-transform systems,” in Proceedings of Seventh International Symposium on Signal Processing and Its Applications, Vol. 1 (IEEE, 2003), pp. 589-592.

[CrossRef]

S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media, 2003).

J.-C. Kuo, C.-H. Wen, and A.-Y. Wu, “Implementation of a programmable 64~2048-point FFT/IFFT processor for OFDM-based communication systems,” in Proceedings of the 2003 International Symposium on Circuits and Systems, Vol. 2 (IEEE, 2003), pp. 121-124.

X.-G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Trans. Signal Process. 3, 72-74 (March 1996).

[CrossRef]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470-473 (1996).

[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825-2832 (2008).

[CrossRef]

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).

[CrossRef]

T. J. Lundy and J. Van Buskirk, “A new matrix approach to real FFTs and convolutions of length 2k,” Computing 80, 23-45 (2007).

[CrossRef]

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727-730 (2009).

[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383-2394 (2008).

[CrossRef]

S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338-1353 (2000).

[CrossRef]

X.-G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Trans. Signal Process. 3, 72-74 (March 1996).

[CrossRef]

A. Cortes, I. Velez, and J. F. Sevillano, “Radix rk FFTs: matricial representation and SDC/SDF pipeline implementation,” IEEE Trans. Signal Process. 57, 2824-2839 (2009).

[CrossRef]

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111-119 (2007).

[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).

[CrossRef]

B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).

[CrossRef]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23, 2861-2870 (2006).

[CrossRef]

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. A 26, 1858-1867 (2009).

[CrossRef]

U. Gopinathan, G. Situ, T. J. Naughton, and J. T. Sheridan, “Noninterferometric phase retrieval using a fractional Fourier system,” J. Opt. Soc. Am. A 25, 108-115 (2008).

[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22, 928-937 (2005).

[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917-927 (2005).

[CrossRef]

T. Alieva and M. J. Bastiaans, “Properties of the linear canonical integral transformation,” J. Opt. Soc. Am. A 24, 3658-3665 (2007).

[CrossRef]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470-473 (1996).

[CrossRef]

S. Abe and J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation an operator approach,” J. Phys. A 27, 4179-4187 (1994).

[CrossRef]

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297-301 (1965).

[CrossRef]

B. Barshan, M. Alper Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).

[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).

[CrossRef]

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293-297 (1981).

[CrossRef]

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. (Bellingham) 43, 2557-2563 (2004).

[CrossRef]

A. Nelleri, J. Joseph, and K. Singh, “Digital Fresnel field encryption for three-dimensional information security,” Opt. Eng. (Bellingham) 46, 045801 (8 pages) (2007).

[CrossRef]

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32-34 (2006).

[CrossRef]
[PubMed]

S. Abe and J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801-1803 (1994).

[CrossRef]
[PubMed]

J. J. Healy and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett. 33, 228-230 (2008).

[CrossRef]
[PubMed]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “An additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599-2601 (2008).

[CrossRef]
[PubMed]

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Stuttgart) 114, 251-265 (2003).

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216-231 (2005).

[CrossRef]

C. M. Rader, “Discrete Fourier transforms when the number of data samples is prime,” Proc. IEEE 56, 1107-1108 (1968).

[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Sci. China Ser. F, Inf. Sci. 49, 592-603 (2006).

[CrossRef]

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983-990 (2007).

[CrossRef]

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825-2832 (2008).

[CrossRef]

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421-1425 (2006).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641-648 (2009).

[CrossRef]

A. Stern, “Why is the linear canonical transform so little known?” in Proceedings of 5th International Workshop on Information Optics, G.Cristóbal, B.Javidi, and S.Vallmitjana, eds (Springer, 2006), pp. 225-234.

G. Kloos, Matrix Methods for Optical Layout (SPIE Press, 2007).

[CrossRef]

J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. dissertation (National Taiwan University, 2001).

M. J. Bastiaans and K. B. Wolf, “Phase reconstruction from intensity measurements in one-parameter canonical-transform systems,” in Proceedings of Seventh International Symposium on Signal Processing and Its Applications, Vol. 1 (IEEE, 2003), pp. 589-592.

[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

T. Kreis, Handbook of Holographic Interferometry, Optical and Digital Methods (Wiley-VCH, 2005).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).

B. Kernighan, and D. Ritchie, The C Programming Language (Prentice-Hall, 1978).

MATLAB, Users Guide (The MathWorks, Inc., 1998).

S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media, 2003).

W. T. Rhodes, “Light tubes, Wigner diagrams and optical signal propagation simulation,” in Optical Information Processing: A Tribute to Adolf Lohmann, H.J.Caulfield, ed. (SPIE Press, 2002) pp. 343-356 (2002).

J.-C. Kuo, C.-H. Wen, and A.-Y. Wu, “Implementation of a programmable 64~2048-point FFT/IFFT processor for OFDM-based communication systems,” in Proceedings of the 2003 International Symposium on Circuits and Systems, Vol. 2 (IEEE, 2003), pp. 121-124.

W. M. Gentleman and G. Sande, “Fast Fourier transforms--for fun and profit,” in Proceedings of AFIPS Fall Joint Computer Conference, Vol. 29 (ACM, 1966), pp. 563-578.