Abstract

Hereby we describe an improved aberration-free super resolution imaging system that uses binary speckle patterns (BSP) for encoding and decoding. According to the scheme, the object is multiplied by a fine random speckle pattern and subsequently the poorly resolved acquired image is multiplied by the same pattern, stored digitally. Summing many samples of similar operations restores a high quality image. In this paper we define encoding and decoding BSP that form complementary sets. Contrary to using totally random BSP, the use of complementary sets reduces the noise of the resulting image significantly. Moreover, they allow reducing the number of operations necessary to restore the image. We found out that using sparse BSP followed by a stochastic noise reduction algorithm, fitted for the decoding process, reduces the noise significantly and allows use of even a single set of complementary BSP for obtaining excellent results. Simulation results are presented both for binary images as well as for gray level objects.

©2007 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, London 7th ed., 1999).
  2. G. Martial, “Strehl ratio and aberration balancing,” J. Opt. Soc .Am. A 8,164–170 (1991).
    [Crossref]
  3. B. Tatian, “Aberration balancing in rotationally symmetric lenses,” J. Opt. Soc .Am. A 64,1083–1091 (1974).
    [Crossref]
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  6. E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “Experimental realization of an imaging system with an extended depth of field,” Appl. Opt. 44,2792–2798 (2005).
    [Crossref] [PubMed]
  7. E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “Radial mask for imaging systems that exhibit high resolution and extended depth of field,” Appl. Opt. 45,2001–2013 (2006).
    [Crossref] [PubMed]
  8. J. O. Castaneda and L. R. Berriel-Valdos, “Zone plate for arbitrary high focal depth,” Appl. Opt. 29,994–997 (1990).
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  9. J. O. Castaneda, R. Ramos, and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27,2583–2586 (1988).
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  10. E. R Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt 34,1859 (1995).
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  11. N George and W. Chi, “Computational imaging with the logarithmic asphere: theory,” J. Opt. Soc. Am. A 20,2260–2273 (2003).
    [Crossref]
  12. Z. Zalevsky and D. Mendlovic, Optical Superresolution, (Springer-Verlag Berlin, 2004).
  13. D. Mendlovic and A. W. Lohmann, “SW- adaptation and its application for super resolution- Fundamentals,” J. Opt. Soc .Am. A 14,558–562 (1997).
    [Crossref]
  14. G. Toraldo Di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59,799–804 (1969).
    [Crossref] [PubMed]
  15. M. Francon, “Amelioration de resolution d'optique,” Nuovo Cimento Suppl. 9,283–290 (1952).
  16. W. Lukosz, “Optical systems with resolving power exceeding the classical limit,” J. Opt. Soc .Am. A 56,1463–1472 (1966).
    [Crossref]
  17. A. Shemer, D. Mendlovic, Z. Zalevsky, J. Garcia, and P. Garcia Martinez, “Super resolving optical system with time multiplexing and computer decoding,” Appl. Opt. 38,7245 (1999).
    [Crossref]
  18. A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. 9,204–206 (1960).
  19. W. Gartner and A. W. Lohmann, “An experiment going beyond Abbe's limit of diffraction,” Z. Phys. 174,18 (1963).
  20. Javier García, Zeev Zalevsky, and Dror Fixler, “Synthetic aperture superresolution by speckle pattern projection,” Opt. Express 13,6073–6078 (2005).
    [Crossref] [PubMed]
  21. E. Ben-Eliezer and E. Marom, “Aberration Free Super Resolution Imaging via Binary Speckle Patterns Encoding and Processing,” J. Opt. Soc. Am. A, (to be published).

2006 (1)

2005 (2)

2003 (1)

1999 (1)

1997 (1)

D. Mendlovic and A. W. Lohmann, “SW- adaptation and its application for super resolution- Fundamentals,” J. Opt. Soc .Am. A 14,558–562 (1997).
[Crossref]

1995 (1)

E. R Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt 34,1859 (1995).
[Crossref]

1991 (1)

G. Martial, “Strehl ratio and aberration balancing,” J. Opt. Soc .Am. A 8,164–170 (1991).
[Crossref]

1990 (1)

1989 (1)

1988 (1)

1974 (1)

B. Tatian, “Aberration balancing in rotationally symmetric lenses,” J. Opt. Soc .Am. A 64,1083–1091 (1974).
[Crossref]

1971 (1)

1969 (1)

1966 (1)

W. Lukosz, “Optical systems with resolving power exceeding the classical limit,” J. Opt. Soc .Am. A 56,1463–1472 (1966).
[Crossref]

1963 (1)

W. Gartner and A. W. Lohmann, “An experiment going beyond Abbe's limit of diffraction,” Z. Phys. 174,18 (1963).

1960 (1)

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. 9,204–206 (1960).

1952 (1)

M. Francon, “Amelioration de resolution d'optique,” Nuovo Cimento Suppl. 9,283–290 (1952).

Ben-Eliezer, E.

Berriel-Valdos, L. R.

Born, M.

M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, London 7th ed., 1999).

Castaneda, J. O.

Cathey, W. T.

E. R Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt 34,1859 (1995).
[Crossref]

Chi, W.

Diaz, A.

Dowski Jr, E. R

E. R Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt 34,1859 (1995).
[Crossref]

Fixler, Dror

Francon, M.

M. Francon, “Amelioration de resolution d'optique,” Nuovo Cimento Suppl. 9,283–290 (1952).

Garcia, J.

García, Javier

Garcia Martinez, P.

Gartner, W.

W. Gartner and A. W. Lohmann, “An experiment going beyond Abbe's limit of diffraction,” Z. Phys. 174,18 (1963).

George, N

Kartashev, A. I.

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. 9,204–206 (1960).

Konforti, N.

Lohmann, A. W.

D. Mendlovic and A. W. Lohmann, “SW- adaptation and its application for super resolution- Fundamentals,” J. Opt. Soc .Am. A 14,558–562 (1997).
[Crossref]

W. Gartner and A. W. Lohmann, “An experiment going beyond Abbe's limit of diffraction,” Z. Phys. 174,18 (1963).

Lukosz, W.

W. Lukosz, “Optical systems with resolving power exceeding the classical limit,” J. Opt. Soc .Am. A 56,1463–1472 (1966).
[Crossref]

Marom, E.

Martial, G.

G. Martial, “Strehl ratio and aberration balancing,” J. Opt. Soc .Am. A 8,164–170 (1991).
[Crossref]

Mendlovic, D.

A. Shemer, D. Mendlovic, Z. Zalevsky, J. Garcia, and P. Garcia Martinez, “Super resolving optical system with time multiplexing and computer decoding,” Appl. Opt. 38,7245 (1999).
[Crossref]

D. Mendlovic and A. W. Lohmann, “SW- adaptation and its application for super resolution- Fundamentals,” J. Opt. Soc .Am. A 14,558–562 (1997).
[Crossref]

Z. Zalevsky and D. Mendlovic, Optical Superresolution, (Springer-Verlag Berlin, 2004).

Mino, M.

Noyola-Isgleas, A.

Okano, Y.

Ramos, R.

Shemer, A.

Tatian, B.

B. Tatian, “Aberration balancing in rotationally symmetric lenses,” J. Opt. Soc .Am. A 64,1083–1091 (1974).
[Crossref]

Tepichin, E.

Toraldo Di Francia, G.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, London 7th ed., 1999).

Zalevsky, Z.

Zalevsky, Zeev

Appl. Opt (1)

E. R Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt 34,1859 (1995).
[Crossref]

Appl. Opt. (7)

J. Opt. Soc .Am. A (4)

D. Mendlovic and A. W. Lohmann, “SW- adaptation and its application for super resolution- Fundamentals,” J. Opt. Soc .Am. A 14,558–562 (1997).
[Crossref]

G. Martial, “Strehl ratio and aberration balancing,” J. Opt. Soc .Am. A 8,164–170 (1991).
[Crossref]

B. Tatian, “Aberration balancing in rotationally symmetric lenses,” J. Opt. Soc .Am. A 64,1083–1091 (1974).
[Crossref]

W. Lukosz, “Optical systems with resolving power exceeding the classical limit,” J. Opt. Soc .Am. A 56,1463–1472 (1966).
[Crossref]

J. Opt. Soc. Am. (1)

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

Nuovo Cimento Suppl. (1)

M. Francon, “Amelioration de resolution d'optique,” Nuovo Cimento Suppl. 9,283–290 (1952).

Opt. Express (1)

Opt. Spectrosc. (1)

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. 9,204–206 (1960).

Z. Phys. (1)

W. Gartner and A. W. Lohmann, “An experiment going beyond Abbe's limit of diffraction,” Z. Phys. 174,18 (1963).

Other (3)

E. Ben-Eliezer and E. Marom, “Aberration Free Super Resolution Imaging via Binary Speckle Patterns Encoding and Processing,” J. Opt. Soc. Am. A, (to be published).

Z. Zalevsky and D. Mendlovic, Optical Superresolution, (Springer-Verlag Berlin, 2004).

M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, London 7th ed., 1999).

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Figures (9)

Fig. 1.
Fig. 1. Imaging System Block diagram

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Fig. 2.
Fig. 2. One subset of five complementary BSPs with p=0.2 (N1=5). The image in the low-right corner is their summation.

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Fig. 3.
Fig. 3. The average and standard deviation of the pedestal value. Statistical variations of the theoretical vs. simulation calculation are presented. (a)- average and (b)- standard deviation of pedestal level for p=0.2 achieved with different number of elements in the BSP set, N.

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Fig. 4.
Fig. 4. Computer simulations of input object (a) and three output images (b-d). (a)- Input object. (b)- Low resolution image. (c)- Restored object using a random sequence of BSPs. (d)- Restored object using complementary BSP sequences. Note improved quality in (d) with respect to (c). In this simulation, the PSF is a circ function with diameter of 8 pixels, the BSP's had a probability p=0.25 and the number of images used was N=300.

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Fig. 5.
Fig. 5. Stochastic Pattern Noise (SPN) reduction algorithm. (a)- Low resolution image. (b)-Restored object using one complementary BSP set with application of SPN reduction algorithm for uniform intensity regions. In this simulation, the PSF was a circ function with diameter of 8 pixels, the chosen BSP's had a probability of p=1/64 with a subdivision of 8×8 pixels. Restored image (b) was achieved with N=64 only.

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Fig. 6.
Fig. 6. Mean Square Error (MSE) Vs. the number of the coding BSP patterns, N, for totally random sequence (Dash) and complementary sequence consisting of N/4 subsets (solid). PSF blurring circ function has a radius of (a) - 6 pixels; (b) - 10 pixels.

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Fig. 7.
Fig. 7. The standard deviation (STD) of the stochastic pattern noise, calculated for a uniform unity input encoded by complementary BSP sequences with p=1/64 as a function of the number L0 of complementary subsets. Each subset consists of 64 BSPs, the Gaussian PSF blur standard deviation has 3 pixels (solid), 7 pixels (solid with circles), 11 pixels (dash) and 15 pixels (dash with circles).

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Fig. 8.
Fig. 8. (a). Output image of a step function, where the decoding was done without the stochastic pattern noise (SPN) reduction method (Eq. 3). (b). Same but with the SPN reduction method (Eq. (22). For both cases p=1/64 and two complementary subsets of 64 images each were used to provide a total number of 128 BSP's.

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Fig. 9.
Fig. 9. Image restoration using Random Structured Encoding/Decoding. (a). A magnified portion of the input object “Lena”. (b). The same portion of the output image, obtained with a conventional imaging system having poor resolution, determined by a Gaussian PSF with a STD of 4 pixels as a blur kernel. (c). High resolution output image restored without noise reduction. (d). High resolution restoration with noise reduction. (e). Same as (d). but with additive Gaussian detector noise who's STD is 2% of the dynamic range. (f). Same as (e) but with two BSP sub sets. All figures obtained with a complementary BSP set, with p=1/64 and using the refinement 8×8 subdivisions. Images (c), (d) and (e) were obtained with one subset (N=64) only, while image (f) was obtained with two subsets i.e. N=128.

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Equations (34)

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I en ( x ; n ) = I obj ( x ) S n ( x )
I F ( x ; n ) = κ ∫ I en ( x ´ M ; n ) ∣ h ( x − x ´ ) ∣ 2 dx ´ + ξ n ( x )
I out ( x ) = 1 N ∑ n = 1 N I F ( x ; n ) S n ( x M )
I out ( x ) = κ ∫ I obj ( x ´ M ) ∣ h ( x − x ´ ) ∣ 2 [ 1 N ∑ n = 1 N S n ( x ´ M ) S n ( x M ) ] dx ´ + r ( x )
r ( x ) ≡ 1 N ∑ n = 1 N ξ n ( x ) S n ( x M )
lim N → ∞ = ( 1 N ∑ n = 1 N S n ( x ´ M ) S n ( x M ) ) = Γ ( x − x ´ ) = p 2 + p ( 1 − p ) Λ ( x − x ´ B )
I blur ( x ) = lim N → ∞ 1 N ∑ n = 1 N I F ( x , n ) = pκ ∫ I obj ( x ´ M ) ∣ h ( x − x ´ ) ∣ 2 dx ´
I hr ; ∞ ( x ) = I out ( x ) − p I blur ( x ) = ( 1 − p ) pκ ∫ I obj ( x ´ M ) ∣ h ( x − x ´ ) ∣ 2 Λ ( x − x ´ B ) dx ´ + r ( x )
〈 r 2 ( x ) 〉 = 1 N 2 ∑ n ∑ n ´ 〈 S n ( x M ) S n ´ ( x M ) 〉 〈 ξ n ( x ) ξ n ´ ( x ) 〉 =
= 1 N 2 ∑ n = n ´ 〈 S n ( x M ) S n ´ ( x M ) 〉 〈 ξ n ( x ) ξ n ´ ( x ) 〉 ≤ 1 N 2 ∑ n 〈 ξ n 2 ( x ) 〉 = σ 2 N
I hr ; ∞ ( x ) = I out ( x ) − p I blur ( x ) = ( 1 − p ) pκ ∫ I obj ( x ´ M ) ∣ h ( x − x ´ ) ∣ 2 Λ ( x − x ´ B ) dx ´
S N ( x ) = 1 N ∑ n = 1 N S n ( x ) ; n ∈ { 1 … N }
p k = N k p k ( 1 − p ) N − k ; where N k ≡ N ! ( N − k ) ! k ! and k ∈ { 0 ‥ N }
〈 S N ( x ) 〉 = ∑ k = 0 N A k p k = p
〈 S N 2 ( x ) 〉 = ∑ k = 0 N A k 2 p k = p 2 N − 1 N + p N
V N = 〈 S N 2 ( x ) 〉 − 〈 S N ( x ) 〉 2 = p − p 2 N = p ( 1 − p ) N
σ N = V N = p ( 1 − p ) N
Γ ( x ; x ´ ) = 1 N ∑ n = 1 N S n ( x ´ M ) S n ( x M ) = Δ ( x , x ´ ) ∣ x − x ´ ∣ > B + Δ ∣ x − x ´ ∣ ≤ B Λ ( x − x ´ B )
Δ ( x , x ´ ) ∣ x − x ´ ∣ ≤ B ≡ 1 N ∑ n = 1 N S n ( x M ) S n ( x M ) ,
Δ ( x , x ´ ) ∣ x − x ´ ∣ > B ≡ 1 N ∑ n = 1 N S n ( x ´ M ) S n ( x M ) ,
〈 Δ ( x , x ´ ) ∣ x − x ´ ∣ > B 〉 = p 2
σ ( x ≠ x ´ ) N = p 2 ( 1 − p 2 ) N
S N ( x ) = 1 L 0 N 1 ∑ n = 1 L 0 ∑ k = 1 N 1 S n ; k ( x )
∑ k = 1 N 1 S n ; k ( x ) = 1 ( x ) ≡ 1 everywhere .
S N ( x ) = 1 L 0 N 1 ∑ n = 1 L 0 1 ( x ) = 1 N 1 = p
I blur ( x ) = 1 N ∑ n = 1 N I F ( x , n ) = pκ ∫ I obj ( x ´ M ) ∣ h ( x − x ´ ) ∣ 2 dx ´
Γ ( x ; x ´ ) = Δ ( x , x ´ ) ∣ x − x ´ ∣ > B + p Λ ( x − x ´ B )
I hr ( x ) = κ ∫ I obj ( x ´ M ) ∣ h ( x − x ´ ) ∣ 2 [ ( 1 − p ) p Λ ( x − x ´ B ) + spn ( x , x ´ ) ∣ x − x ´ ∣ > B ] dx ´
I hr ( x ) = I hr ; ∞ ( x ) + spn ( x )
I hr ; ∞ ( x ) ≡ κ ( 1 − p ) p ∫ I obj ( x ´ M ) ∣ h ( x − x ´ ) ∣ 2 Λ ( x − x ´ B ) dx ´
spn ( x ) ≡ κ ∫ I obj ( x ´ M ) spn ( x , x ´ ) ∣ x − x ´ ∣ > B ∣ h ( x − x ´ ) ∣ 2 dx ´
S F ( x , n ) ≡ κ ∫ S n ( x ´ M t ) ∣ h ( x − x ´ ) ∣ 2 dx ´
S ´ F ( x , n ) = { S n ( x M ) S F ( x , n ) if S n ( x M ) ≠ 0 0 otherwise
I out ( x ) = 1 N ∑ n = 1 N I F ( x ; n ) S ´ F ( x , n )

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