Abstract

This Letter presents a new approach for imaging using a linear (vector) sensor. It exploits the fact that visual information within common human intelligible images may be compressed within only a partial set of radial strips of its Fourier domain. We present two imaging schemes, one coherent and the other incoherent, that capture the partial set of radial strips of the object Fourier domain. Two main advantages of the new approach are that the image is captured directly in a compressed form and that the acquisition time is shorter compared with conventional scanning imaging systems.

© 2007 Optical Society of America

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References

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  1. D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
    [CrossRef]
  2. E. J. Candes, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
    [CrossRef]
  3. E. Candes and T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
    [CrossRef]
  4. D. Takhar, J. N. Laska, M. B. Wakin, M. F. Durate, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, Proc. SPIE 6065, 606509 (2006).
    [CrossRef]
  5. A. Stern and B. Javidi, J. Display Technol. 3, 315 (2007).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  7. J. Shamir, Optical Systems and Processing (SPIE Press, 1999).
    [CrossRef]
  8. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge U. Press, 2004), Chap. 12.
  9. S. Kopeika, A System Engineering Approach to Imaging (SPIE Optical Engineering Press, 1998), pp. 169-184, 359-367.
  10. L. Levi, Applied Optics (Wiley, 1992), Vol. 1, pp. 1-430.
  11. Z.-H. Cho, J. P. Jones, and M. Singh, Foundation of Medical Imaging (Wiley, 1993), Chap. 3.

2007 (1)

2006 (4)

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

E. J. Candes, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

E. Candes and T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Durate, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

2004 (1)

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge U. Press, 2004), Chap. 12.

1999 (1)

J. Shamir, Optical Systems and Processing (SPIE Press, 1999).
[CrossRef]

1998 (1)

S. Kopeika, A System Engineering Approach to Imaging (SPIE Optical Engineering Press, 1998), pp. 169-184, 359-367.

1996 (1)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

1993 (1)

Z.-H. Cho, J. P. Jones, and M. Singh, Foundation of Medical Imaging (Wiley, 1993), Chap. 3.

1992 (1)

L. Levi, Applied Optics (Wiley, 1992), Vol. 1, pp. 1-430.

Baraniuk, R. G.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Durate, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Baron, D.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Durate, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge U. Press, 2004), Chap. 12.

Candes, E.

E. Candes and T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
[CrossRef]

Candes, E. J.

E. J. Candes, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Cho, Z.-H.

Z.-H. Cho, J. P. Jones, and M. Singh, Foundation of Medical Imaging (Wiley, 1993), Chap. 3.

Donoho, D. L.

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

Durate, M. F.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Durate, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Javidi, B.

Jones, J. P.

Z.-H. Cho, J. P. Jones, and M. Singh, Foundation of Medical Imaging (Wiley, 1993), Chap. 3.

Kelly, K. F.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Durate, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Kopeika, S.

S. Kopeika, A System Engineering Approach to Imaging (SPIE Optical Engineering Press, 1998), pp. 169-184, 359-367.

Laska, J. N.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Durate, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Levi, L.

L. Levi, Applied Optics (Wiley, 1992), Vol. 1, pp. 1-430.

Romberg, J.

E. J. Candes, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Sarvotham, S.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Durate, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Shamir, J.

J. Shamir, Optical Systems and Processing (SPIE Press, 1999).
[CrossRef]

Singh, M.

Z.-H. Cho, J. P. Jones, and M. Singh, Foundation of Medical Imaging (Wiley, 1993), Chap. 3.

Stern, A.

Takhar, D.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Durate, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Tao, T.

E. Candes and T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
[CrossRef]

E. J. Candes, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge U. Press, 2004), Chap. 12.

Wakin, M. B.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Durate, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

IEEE Trans. Inf. Theory (3)

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

E. J. Candes, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

E. Candes and T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
[CrossRef]

J. Display Technol. (1)

Proc. SPIE (1)

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Durate, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Other (6)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

J. Shamir, Optical Systems and Processing (SPIE Press, 1999).
[CrossRef]

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge U. Press, 2004), Chap. 12.

S. Kopeika, A System Engineering Approach to Imaging (SPIE Optical Engineering Press, 1998), pp. 169-184, 359-367.

L. Levi, Applied Optics (Wiley, 1992), Vol. 1, pp. 1-430.

Z.-H. Cho, J. P. Jones, and M. Singh, Foundation of Medical Imaging (Wiley, 1993), Chap. 3.

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Figures (5)

Fig. 1
Fig. 1

(a) Location of radial samples in the Fourier domain { F ( ω , θ l ) } l = 0 L 1 , L = 30 . (b) Density of Fourier-sampled points in horizontal spatial frequency direction.

Fig. 2
Fig. 2

CI system based on 4f configuration. L 1 , lens with focal length f l ; L 2 , L 3 , two perpendicular cylindrical lenses with focal lengths f l and f l 2 , respectively; S, line vector sensor.

Fig. 3
Fig. 3

(a) Original image. (b) Reconstructed image with L = 25 . (c), (d) Reconstruction form L = 25 radial Fourier slices corrupted by additive Gaussian noise, forming SNRs of 37 and 27 dB , respectively.

Fig. 4
Fig. 4

L 1 , cylindrical lens with focal length f l ; S, vector sensor located in the image plane.

Fig. 5
Fig. 5

(a) Original image. (b) Reconstructed image with L = 32 . (c) Image obtained with conventional imaging technique capturing the same amount of samples. (d) Interpolation of (c) (using spline) to have the same size as Figs. 1a, 1b.

Equations (3)

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min f ̂ n , m = 1 N 1 D f ̂ [ n , m ] subject to F ̂ ( ω , θ l ) = F ( ω , θ l ) ,
ω ω max for all { θ l } l = 0 L 1 ,
D f ̂ [ n , m ] = f ̂ [ n + 1 , m ] f ̂ [ n , m ] 2 + f ̂ [ n , m + 1 ] f ̂ [ n , m ] 2

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