Abstract

A mathematical analysis of the frequency response of the wavefront-coding odd-symmetric quadratic phase mask is presented. An exact solution for the optical transfer function of a wavefront-coding imager using this type of mask is derived from first principles, whose result applies over all misfocus values. The misfocus-dependent spatial filtering property of this imager is described. The available spatial frequency bandwidth for a given misfocus condition is quantified. A special imaging condition that yields an increased dynamic range is identified.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. R. Dowski, Jr., and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995).
    [Crossref] [PubMed]
  2. V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F. T. Luk, ed., Proc. SPIE 5205, 348-357 (2003).
    [Crossref]
  3. S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, "Engineering the pupil phase to improve image quality," in Visual Information Processing XII, Z. Rahman, R. A. Schowengerdt, and S. E. Reichenbach, eds., Proc. SPIE 5108, 1-12 (2003).
    [Crossref]
  4. S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, "Pupil-phase optimization for extended-focus, aberration-corrected imaging systems," in Applications of Digital Image Processing XXVII, A. G. Tescher, ed., Proc. SPIE 5559, 335-345 (2004).
    [Crossref]
  5. D. Zalvidea, C. Colautti, and E. E. Sicre, "Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function," J. Opt. Soc. Am. A. 17, 867-873 (2000).
    [Crossref]
  6. D. Zalvidea and E. E. Sicre, "Phase pupil functions for focal-depth enhancement derived from a Wigner distribution function," Appl. Opt. 37, 3623-3627 (1998).
    [Crossref]
  7. W. Chi and N. George, "Electronic imaging using a logarithmic asphere," Opt. Lett. 26, 875-877 (2001).
    [Crossref]
  8. S. Mezouari and A. R. Harvey, "Phase pupil functions for reduction of defocus and spherical aberrations," Opt. Lett. 28, 771-773 (2003).
    [Crossref] [PubMed]
  9. W. Chi and N. George, "Computational imaging with the logarithmic asphere: theory," J. Opt. Soc. Am. A 20, 2260-2273 (2003).
    [Crossref]
  10. J. Ojeda-Castañeda, J. E. A. Landgrave, and H. M. Escamilla, "Annular phase-only mask for high focal depth," Opt. Lett. 30, 1647-1649 (2005).
    [Crossref] [PubMed]
  11. A. R. FitzGerrell, E. R. Dowski, Jr., and W. T. Cathey, "Defocus transfer function for circularly symmetric pupils," Appl. Opt. 36, 5796-5804 (1997).
    [Crossref] [PubMed]
  12. K. Kubala, E. Dowski, and W. T. Cathey, "Reducing complexity in computational imaging systems," Opt. Express 11, 2102-2108 (2003).
    [Crossref] [PubMed]
  13. W. T. Cathey and E. R. Dowski, "New paradigm for imaging systems," Appl. Opt. 41, 6080-6092 (2002).
    [Crossref] [PubMed]
  14. H. B. Wach, E. R. Dowski, Jr., and W. T. Cathey, "Control of chromatic focal shift through wavefront coding," Appl. Opt. 37, 5359-5367 (1998).
    [Crossref]
  15. E. R. Dowski, Jr., and G. E. Johnson, "Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems," in Current Developments in Optical Design and Optical Engineering VIII, R. E. Fischer and W. J. Smith, eds., Proc. SPIE 3779, 137-145 (1999).
    [Crossref]
  16. E. R. Dowski, Jr., R. H. Cormack, and S. D. Sarama, "Wavefront coding: jointly optimized optical and digital imaging systems," in Visual Information Processing IX, S. K. Park and Z. Rahman, eds., Proc. SPIE 4041, 114-120 (2000).
    [Crossref]
  17. R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Image Processing: Algorithms and Systems III, E. R. Dougherty, J. T. Astola, and K. O. Egiazarian, eds., Proc. SPIE 5299, 163-174 (2004).
    [Crossref]
  18. M. Somayaji and M. P. Christensen, "Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask," Appl. Opt. 45, 2911-2923 (2006).
    [Crossref] [PubMed]
  19. S. S. Sherif, W. T. Cathey, and E. R. Dowski, "Phase plate to extend the depth of field of incoherent hybrid imaging systems," Appl. Opt. 43, 2709-2721 (2004).
    [Crossref] [PubMed]
  20. A. Sauceda and J. Ojeda-Castañeda, "High focal depth with fractional-power wave fronts," Opt. Lett. 29, 560-562 (2004).
    [Crossref] [PubMed]
  21. A. Castro and J. Ojeda-Castañeda, "Asymmetric phase masks for extended depth of field," Appl. Opt. 43, 3474-3479 (2004).
    [Crossref] [PubMed]
  22. J. W. Goodman, "Frequency analysis of optical imaging systems," in Introduction to Fourier Optics, 2nd ed., L. Cox and J. M. Morriss, eds. (McGraw-Hill, 1996), pp. 146-151.

2006 (1)

2005 (1)

2004 (5)

A. Sauceda and J. Ojeda-Castañeda, "High focal depth with fractional-power wave fronts," Opt. Lett. 29, 560-562 (2004).
[Crossref] [PubMed]

S. S. Sherif, W. T. Cathey, and E. R. Dowski, "Phase plate to extend the depth of field of incoherent hybrid imaging systems," Appl. Opt. 43, 2709-2721 (2004).
[Crossref] [PubMed]

A. Castro and J. Ojeda-Castañeda, "Asymmetric phase masks for extended depth of field," Appl. Opt. 43, 3474-3479 (2004).
[Crossref] [PubMed]

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, "Pupil-phase optimization for extended-focus, aberration-corrected imaging systems," in Applications of Digital Image Processing XXVII, A. G. Tescher, ed., Proc. SPIE 5559, 335-345 (2004).
[Crossref]

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Image Processing: Algorithms and Systems III, E. R. Dougherty, J. T. Astola, and K. O. Egiazarian, eds., Proc. SPIE 5299, 163-174 (2004).
[Crossref]

2003 (5)

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F. T. Luk, ed., Proc. SPIE 5205, 348-357 (2003).
[Crossref]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, "Engineering the pupil phase to improve image quality," in Visual Information Processing XII, Z. Rahman, R. A. Schowengerdt, and S. E. Reichenbach, eds., Proc. SPIE 5108, 1-12 (2003).
[Crossref]

S. Mezouari and A. R. Harvey, "Phase pupil functions for reduction of defocus and spherical aberrations," Opt. Lett. 28, 771-773 (2003).
[Crossref] [PubMed]

K. Kubala, E. Dowski, and W. T. Cathey, "Reducing complexity in computational imaging systems," Opt. Express 11, 2102-2108 (2003).
[Crossref] [PubMed]

W. Chi and N. George, "Computational imaging with the logarithmic asphere: theory," J. Opt. Soc. Am. A 20, 2260-2273 (2003).
[Crossref]

2002 (1)

2001 (1)

2000 (2)

E. R. Dowski, Jr., R. H. Cormack, and S. D. Sarama, "Wavefront coding: jointly optimized optical and digital imaging systems," in Visual Information Processing IX, S. K. Park and Z. Rahman, eds., Proc. SPIE 4041, 114-120 (2000).
[Crossref]

D. Zalvidea, C. Colautti, and E. E. Sicre, "Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function," J. Opt. Soc. Am. A. 17, 867-873 (2000).
[Crossref]

1999 (1)

E. R. Dowski, Jr., and G. E. Johnson, "Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems," in Current Developments in Optical Design and Optical Engineering VIII, R. E. Fischer and W. J. Smith, eds., Proc. SPIE 3779, 137-145 (1999).
[Crossref]

1998 (2)

1997 (1)

1995 (1)

Baron, A. E.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Image Processing: Algorithms and Systems III, E. R. Dougherty, J. T. Astola, and K. O. Egiazarian, eds., Proc. SPIE 5299, 163-174 (2004).
[Crossref]

Castro, A.

Cathey, W. T.

Chi, W.

Christensen, M. P.

Chumachenko, V.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Image Processing: Algorithms and Systems III, E. R. Dougherty, J. T. Astola, and K. O. Egiazarian, eds., Proc. SPIE 5299, 163-174 (2004).
[Crossref]

Colautti, C.

D. Zalvidea, C. Colautti, and E. E. Sicre, "Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function," J. Opt. Soc. Am. A. 17, 867-873 (2000).
[Crossref]

Cormack, R. H.

E. R. Dowski, Jr., R. H. Cormack, and S. D. Sarama, "Wavefront coding: jointly optimized optical and digital imaging systems," in Visual Information Processing IX, S. K. Park and Z. Rahman, eds., Proc. SPIE 4041, 114-120 (2000).
[Crossref]

Dowski, E.

Dowski, E. R.

Escamilla, H. M.

FitzGerrell, A. R.

George, N.

Goodman, J. W.

J. W. Goodman, "Frequency analysis of optical imaging systems," in Introduction to Fourier Optics, 2nd ed., L. Cox and J. M. Morriss, eds. (McGraw-Hill, 1996), pp. 146-151.

Greengard, A.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Image Processing: Algorithms and Systems III, E. R. Dougherty, J. T. Astola, and K. O. Egiazarian, eds., Proc. SPIE 5299, 163-174 (2004).
[Crossref]

Harvey, A. R.

Johnson, G. E.

E. R. Dowski, Jr., and G. E. Johnson, "Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems," in Current Developments in Optical Design and Optical Engineering VIII, R. E. Fischer and W. J. Smith, eds., Proc. SPIE 3779, 137-145 (1999).
[Crossref]

Kubala, K.

Landgrave, J. E. A.

Mezouari, S.

Narayanswamy, R.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Image Processing: Algorithms and Systems III, E. R. Dougherty, J. T. Astola, and K. O. Egiazarian, eds., Proc. SPIE 5299, 163-174 (2004).
[Crossref]

Ojeda-Castañeda, J.

Pauca, V.

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F. T. Luk, ed., Proc. SPIE 5205, 348-357 (2003).
[Crossref]

Pauca, V. P.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, "Pupil-phase optimization for extended-focus, aberration-corrected imaging systems," in Applications of Digital Image Processing XXVII, A. G. Tescher, ed., Proc. SPIE 5559, 335-345 (2004).
[Crossref]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, "Engineering the pupil phase to improve image quality," in Visual Information Processing XII, Z. Rahman, R. A. Schowengerdt, and S. E. Reichenbach, eds., Proc. SPIE 5108, 1-12 (2003).
[Crossref]

Plemmons, R.

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F. T. Luk, ed., Proc. SPIE 5205, 348-357 (2003).
[Crossref]

Plemmons, R. J.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, "Pupil-phase optimization for extended-focus, aberration-corrected imaging systems," in Applications of Digital Image Processing XXVII, A. G. Tescher, ed., Proc. SPIE 5559, 335-345 (2004).
[Crossref]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, "Engineering the pupil phase to improve image quality," in Visual Information Processing XII, Z. Rahman, R. A. Schowengerdt, and S. E. Reichenbach, eds., Proc. SPIE 5108, 1-12 (2003).
[Crossref]

Prasad, S.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, "Pupil-phase optimization for extended-focus, aberration-corrected imaging systems," in Applications of Digital Image Processing XXVII, A. G. Tescher, ed., Proc. SPIE 5559, 335-345 (2004).
[Crossref]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, "Engineering the pupil phase to improve image quality," in Visual Information Processing XII, Z. Rahman, R. A. Schowengerdt, and S. E. Reichenbach, eds., Proc. SPIE 5108, 1-12 (2003).
[Crossref]

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F. T. Luk, ed., Proc. SPIE 5205, 348-357 (2003).
[Crossref]

Sarama, S. D.

E. R. Dowski, Jr., R. H. Cormack, and S. D. Sarama, "Wavefront coding: jointly optimized optical and digital imaging systems," in Visual Information Processing IX, S. K. Park and Z. Rahman, eds., Proc. SPIE 4041, 114-120 (2000).
[Crossref]

Sauceda, A.

Sherif, S. S.

Sicre, E. E.

D. Zalvidea, C. Colautti, and E. E. Sicre, "Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function," J. Opt. Soc. Am. A. 17, 867-873 (2000).
[Crossref]

D. Zalvidea and E. E. Sicre, "Phase pupil functions for focal-depth enhancement derived from a Wigner distribution function," Appl. Opt. 37, 3623-3627 (1998).
[Crossref]

Somayaji, M.

Torgersen, T.

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F. T. Luk, ed., Proc. SPIE 5205, 348-357 (2003).
[Crossref]

Torgersen, T. C.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, "Pupil-phase optimization for extended-focus, aberration-corrected imaging systems," in Applications of Digital Image Processing XXVII, A. G. Tescher, ed., Proc. SPIE 5559, 335-345 (2004).
[Crossref]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, "Engineering the pupil phase to improve image quality," in Visual Information Processing XII, Z. Rahman, R. A. Schowengerdt, and S. E. Reichenbach, eds., Proc. SPIE 5108, 1-12 (2003).
[Crossref]

van der Gracht, J.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, "Pupil-phase optimization for extended-focus, aberration-corrected imaging systems," in Applications of Digital Image Processing XXVII, A. G. Tescher, ed., Proc. SPIE 5559, 335-345 (2004).
[Crossref]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, "Engineering the pupil phase to improve image quality," in Visual Information Processing XII, Z. Rahman, R. A. Schowengerdt, and S. E. Reichenbach, eds., Proc. SPIE 5108, 1-12 (2003).
[Crossref]

Wach, H. B.

Zalvidea, D.

D. Zalvidea, C. Colautti, and E. E. Sicre, "Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function," J. Opt. Soc. Am. A. 17, 867-873 (2000).
[Crossref]

D. Zalvidea and E. E. Sicre, "Phase pupil functions for focal-depth enhancement derived from a Wigner distribution function," Appl. Opt. 37, 3623-3627 (1998).
[Crossref]

Appl. Opt. (8)

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

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

D. Zalvidea, C. Colautti, and E. E. Sicre, "Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function," J. Opt. Soc. Am. A. 17, 867-873 (2000).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Proc. SPIE (6)

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F. T. Luk, ed., Proc. SPIE 5205, 348-357 (2003).
[Crossref]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, "Engineering the pupil phase to improve image quality," in Visual Information Processing XII, Z. Rahman, R. A. Schowengerdt, and S. E. Reichenbach, eds., Proc. SPIE 5108, 1-12 (2003).
[Crossref]

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, "Pupil-phase optimization for extended-focus, aberration-corrected imaging systems," in Applications of Digital Image Processing XXVII, A. G. Tescher, ed., Proc. SPIE 5559, 335-345 (2004).
[Crossref]

E. R. Dowski, Jr., and G. E. Johnson, "Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems," in Current Developments in Optical Design and Optical Engineering VIII, R. E. Fischer and W. J. Smith, eds., Proc. SPIE 3779, 137-145 (1999).
[Crossref]

E. R. Dowski, Jr., R. H. Cormack, and S. D. Sarama, "Wavefront coding: jointly optimized optical and digital imaging systems," in Visual Information Processing IX, S. K. Park and Z. Rahman, eds., Proc. SPIE 4041, 114-120 (2000).
[Crossref]

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Image Processing: Algorithms and Systems III, E. R. Dougherty, J. T. Astola, and K. O. Egiazarian, eds., Proc. SPIE 5299, 163-174 (2004).
[Crossref]

Other (1)

J. W. Goodman, "Frequency analysis of optical imaging systems," in Introduction to Fourier Optics, 2nd ed., L. Cox and J. M. Morriss, eds. (McGraw-Hill, 1996), pp. 146-151.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Wavefront coding system incorporating a cubic phase mask. (a) A wavefront coding imager utilizes a cubic-pm element whose (b) 2D phase profile achieves an extended depth of field. (c) The corresponding OTF varies little in magnitude with misfocus. The plot in (c) shows the magnitude of the OTF for three misfocus values ( ψ = 0 , ψ = π , ψ = 5 π ) . The smooth curve in (c) represents the approximate OTF as described in Ref. [1].

Fig. 2
Fig. 2

Intensity PSF and MTF of an OSQ phase mask system. (a), (b), (c) Intensity PSFs for misfocus values of ψ = 0 , 15 π , and 30 π , respectively. (d) Normalized 1D MTF as a function of the normalized spatial frequency u. The value of α is set at 30π.

Fig. 3
Fig. 3

Plots of magnitude of the AF of OSQ phase mask systems. An absence of nulls inside the passband of the MTF marks the operating region of this imager. The MTFs at the boundaries of this region display a higher dynamic range than those for other misfocus values within the area where imaging is possible.

Fig. 4
Fig. 4

Effect of misfocus on the bandwidth of OSQ phase mask systems. The left column represents a system with α = 70 π and no misfocus ( ψ = 0 ) . The right column is for the same system ( α = 70 π ) , but with ψ = 30 π . (c), (d) Location where the MTF drop occurs corresponds to the zero-crossing points of b T ( u ) as seen in (a) and (b).

Fig. 5
Fig. 5

Available bandwidth of the OSQ phase mask imager and the relationship between the zero-crossing points of the function b T ( u ) and the spatial frequency bandwidth of this imager as seen from the AF magnitude plot. The expression for available spatial frequency bandwidth given by Eq. (37) is valid as long as the radial lines fall within the double-diamond pattern seen in the AF plot. The MTF at the edges of the AF magnitude plot are raised and have a normalized bandwidth of u c = 1 / 2 .

Fig. 6
Fig. 6

Imaging with an OSQ phase mask system when ψ = α . The case of ψ = α is depicted. (a) Magnitudes of the individual components of H C (u, ψ). The triangular shape of I 3 ( u , ψ ) causes an elevation in the MTF, resulting in an increased dynamic range when ψ = ± α . The horizontal dotted line indicates the stationary value of the MTF for imaging conditions of ψ < α and has a height of ( π / 8 α ) 1 / 2 . The stationary magnitude of I 2 ( u , ψ ) when ψ = α is one half the height of this dotted line. (b) MTF. (c) Corresponding radial line for ψ = α on the AF magnitude plot. The value of α was taken as 30 π .

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

η ( x , y ) = η ( x , y ) .
η ( x ) = α   sgn ( x ) x γ , 1 x 1 , α > 0 , γ 2.
sgn ( x ) = { 1 , x > 0 0 , x = 0 1 , x < 0 .
η ( x ) = { α x 2 , 1 x < 0 α x 2 , 0 x 1 .
P ( x ) = { 1 2 exp [ j ( ψ α ) x 2 ] = P ( x ) , 1 x < 0 1 2 exp [ j ( ψ + α ) x 2 ] = P + ( x ) , 0 x 1 0 , otherwise .
ψ = π L 2 4 λ ( 1 f 1 d o 1 d c ) .
H ( u , ψ ) = { u 1 u + 1 P ( x + u ) P + * ( x u ) d x , 1 u 1 2 u 1 u P ( x + u ) P * ( x u ) d x + u u P ( x + u ) P + * ( x u ) d x + u 1 + u P + ( x + u ) P + * ( x u ) d x , 1 2 u 0 u 1 u P ( x + u ) P * ( x u ) d x + u u P + ( x + u ) P * ( x u ) d x + u 1 u P + ( x + u ) P + * ( x u ) d x , 0 u 1 2 u 1 1 u P + ( x + u ) P * ( x u ) d x , 1 2 u 1 .
H ( u , ψ ) = { 1 2 u 1 u + 1 exp [ j { 4 u ψ x 2 α ( x 2 + u 2 ) } ] d x , 1 u 1 2 1 2 u 1 u exp [ j 4 u ( ψ α ) x ] d x + 1 2 u u exp [ j { 4 u ψ x 2 α ( x 2 + u 2 ) } ] d x + 1 2 u 1 + u exp [ j 4 u ( ψ + α ) x ] d x , 1 2 u 0 1 2 u 1 u exp [ j 4 u ( ψ α ) x ] d x + 1 2 u u exp [ j { 4 u ψ x + 2 α ( x 2 + u 2 ) } ] d x + 1 2 u 1 u exp [ j 4 u ( ψ + α ) x ] d x , 0 u 1 2 1 2 u 1 1 u exp [ j { 4 u ψ x + 2 α ( x 2 + u 2 ) } ] d x , 1 2 u 1 .
H ( u , ψ ) = { 1 2 u 1 u exp [ j 4 u ( ψ α ) x ] d x + 1 2 u 1 u exp [ j 4 u ( ψ + α ) x ] d x   + 1 2 u u exp [ j { 4 u ψ x + 2 α ( x 2 + u 2 ) } sgn ( u ) ] d x , 0 u 1 2 1 2 u 1 1 u exp [ j { 4 u ψ x + 2 α ( x 2 + u 2 ) } sgn ( u ) ] d x , 1 2 u 1 .
H C ( u , ψ ) = 1 2 u 1 u exp [ j 4 u ( ψ α ) x ] d x + 1 2 u 1 u exp [ j 4 u ( ψ + α ) x ] d x + 1 2 u u exp [ j { 4 u ψ x + 2 α ( x 2 + u 2 ) } × sgn ( u ) ] d x ,  0 u 1 2 ,
H T ( u , ψ ) = 1 2 u 1 1 u exp [ j { 4 u ψ x + 2 α ( x 2 + u 2 ) } × sgn ( u ) ] d x ,  1 2 u 1.
H T ( u , ψ ) = 1 2 exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sgn ( u ) ] × u 1 1 u exp [ j π 2 ( 4 α π ) ( x + ψ α u ) 2 × sgn ( u ) ] d x ,  1 2 u 1.
H T ( u , ψ ) = 1 4 ( π α ) 1 / 2 exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sgn ( u ) ] × a T ( u ) b T ( u ) exp [ j π 2 τ T 2 sgn ( u ) ] d τ T , 1 2 u 1 ,
a T ( u ) = ( 4 α π ) 1 / 2 [ u ( 1 + ψ α ) 1 ] ,
b T ( u ) = ( 4 α π ) 1 / 2 [ 1 u ( 1 ψ α ) ] .
H T ( u , ψ ) = 1 4 ( π α ) 1 / 2 exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sgn ( u ) ] × { a T ( u ) b T ( u ) cos ( π 2 τ T 2 ) d τ T + j   sgn ( u ) × a T ( u ) b T ( u ) sin ( π 2 τ T 2 ) d τ T } ,  1 2 u 1.
H T ( u , ψ ) = ( π 8 α ) 1 / 2 exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sgn ( u ) ] × 1 2 { [ C ( b T ( u ) ) C ( a T ( u ) ) ] + j   sgn ( u ) × [ S ( b T ( u ) ) S ( a T ( u ) ) ] } ,  1 2 u 1.
I 1 ( u , ψ ) = 1 2 u 1 u exp [ j 4 u ( ψ α ) x ] d x , 0 u 1 2 I 2 ( u , ψ ) = 1 2 u u exp [ j { 4 u ψ x + 2 α ( x 2 + u 2 ) } sgn ( u ) ] d x , 0 u 1 2 I 3 ( u , ψ ) = 1 2 u 1 u exp [ j 4 u ( ψ + α ) x ] d x , 0 u 1 2 .
I 1 ( u , ψ ) = 1 2 u 1 u exp [ j 4 u ( ψ α ) x ] d x = 1 2 exp [ j 4 u ( ψ α ) x ] j 4 u ( ψ α ) | u 1 u .
I 1 ( u , ψ ) = 1 2 j 1 4 u ( ψ α ) { exp [ j 4 u ( ψ α ) ( τ C 1 1 / 2 ) ] exp [ j 4 u ( ψ α ) ( τ C 1 1 / 2 ) ] } ,
I 1 ( u , ψ ) = exp [ j 2 u ( ψ α ) ] 4 u ( ψ α ) × { exp [ j 4 u ( ψ α ) τ C 1 ] exp [ j 4 u ( ψ α ) τ C 1 ] 2 j } .
I 1 ( u , ψ ) = exp [ j 2 u ( ψ α ) ] { sin [ 4 u ( ψ α ) τ C 1 ] 4 u ( ψ α ) } .
I 1 ( u , ψ ) = ( 1 2 u ) sinc [ 4 u π ( ψ α ) ( 1 2 u ) ] × exp [ j 2 u ( ψ α ) ] .
I 2 ( u , ψ ) = 1 2 exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sgn ( u ) ] × u u exp [ j π 2 ( 4 α π ) ( x + ψ α u ) 2 sgn ( u ) ] d x .
I 2 ( u , ψ ) = 1 4 ( π α ) 1 / 2 exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sgn ( u ) ] × a C ( u ) b C ( u ) exp [ j π 2 τ C 2 sgn ( u ) ] d τ C ,
a C ( u ) = ( 4 α π ) 1 / 2 u ( ψ α 1 ) ,
b C ( u ) = ( 4 α π ) 1 / 2 u ( ψ α + 1 ) .
I 2 ( u , ψ ) = 1 4 ( π α ) 1 / 2 exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sgn ( u ) ] × { a C ( u ) b C ( u ) cos ( π 2 τ C 2 ) d τ C + j   sgn ( u ) × a C ( u ) b C ( u ) sin ( π 2 τ C 2 ) d τ C } .
I 2 ( u , ψ ) = ( π 8 α ) 1 / 2 exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sgn ( u ) ] × 1 2 { [ C ( b C ( u ) ) C ( a C ( u ) ) ] + j   sgn ( u ) × [ S ( b C ( u ) ) S ( a C ( u ) ) ] } .
I 3 ( u , ψ ) = 1 2 | u | 1 | u | exp [ j 4 u ( ψ + α ) x ] d x = 1 2 exp [ j 4 u ( ψ + α ) x ] j 4 u ( ψ + α ) | | u | 1 | u | .
I 3 ( u , ψ ) = 1 2 j 1 4 u ( ψ + α ) { exp [ j 4 u ( ψ + α ) ( τ C 3 + 1 / 2 ) ] exp [ j 4 u ( ψ + α ) ( τ C 3 + 1 / 2 ) ] } ,
I 3 ( u , ψ ) = exp [ j 2 u ( ψ + α ) ] 4 u ( ψ + α ) × { exp [ j 4 u ( ψ + α ) τ C 3 ] exp [ j 4 u ( ψ + α ) τ C 3 ] 2 j } .
I 3 ( u , ψ ) = exp [ j 2 u ( ψ + α ) ] { sin [ 4 u ( ψ + α ) τ C 3 ] 4 u ( ψ + α ) } .
I 3 ( u , ψ ) = ( 1 2 u ) sinc [ 4 u π ( ψ + α ) ( 1 2 u ) ] × exp [ j 2 u ( ψ + α ) ] .
H C ( u , ψ ) = ( 1 2 u ) sinc [ 4 u π ( ψ α ) ( 1 2 u ) ] × exp [ j 2 u ( ψ α ) ] + ( π 8 α ) 1 / 2 × exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sgn ( u ) ] × 1 2 { [ C ( b C ( u ) ) C ( a C ( u ) ) ] + j   sgn ( u ) × [ S ( b C ( u ) ) S ( a C ( u ) ) ] } + ( 1 2 u ) × sinc [ 4 u π ( ψ + α ) ( 1 2 u ) ] × exp [ j 2 u ( ψ + α ) ] ,  0 u 1 2 .
H C ( 0 , ψ ) = 1.
H ( u , ψ ) = { ( 1 2 u ) sinc [ 4 u π ( ψ α ) ( 1 2 u ) ] exp [ j 2 u ( ψ α ) ] + ( π 8 α ) 1 / 2 exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sgn ( u ) ]   × 1 2 { [ C ( b C ( u ) ) C ( a C ( u ) ) ] + j   sgn ( u ) [ S ( b C ( u ) ) S ( a C ( u ) ) ] }   + ( 1 2 u ) sinc [ 4 u π ( ψ + α ) ( 1 2 u ) ] exp [ j 2 u ( ψ + α ) ] , 0 u 1 2 ( π 8 α ) 1 / 2 exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sgn ( u ) ]   × 1 2 { [ C ( b T ( u ) ) C ( a T ( u ) ) ] + j   sgn ( u ) [ S ( b T ( u ) ) S ( a T ( u ) ) ] } , 1 2 u 1 0 , otherwise ,
a C ( ± 1 2 ) = a T ( ± 1 2 ) = ( 4 α π ) 1 / 2 ( ψ 2 α 1 2 ) ,
b C ( ± 1 2 ) = b T ( ± 1 2 ) = ( 4 α π ) 1 / 2 ( ψ 2 α + 1 2 ) .
u c = α α + ψ ,  ψ α ,  α > 0 .

Metrics