Abstract

A new method for deconvolution of one-dimensional and multidimensional data is suggested. The proposed algorithm is local in the sense that the deconvolved data at a given point depend only on the value of the experimental data and their derivatives at the same point. In a regularized version of the algorithm the deconvolution is constructed iteratively with the help of an approximate deconvolution operator that requires only the low-order derivatives of the data and low-order integral moments of the point-spread function. This algorithm is expected to be particularly useful in applications in which only partial knowledge of the point-spread function is available. We tested and compared the proposed method with some of the popular deconvolution algorithms using simulated data with various levels of noise.

© 2003 Optical Society of America

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References

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  1. A. N. Tichonov, V. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).
  2. C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2002).
    [CrossRef]
  3. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 2002).
  4. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  5. R. G. Lane, “Methods for maximum-likelihood deconvolution,” J. Opt. Soc. Am. A 13, 1992–1998 (1996).
    [CrossRef]
  6. S. Prasad, “Statistical-information-based performance criteria for Richardson-Lucy image deblurring,” J. Opt. Soc. Am. A 19, 1286–1296 (2002).
    [CrossRef]
  7. S. F. Gull, G. J. Daniel, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686–690 (1978).
    [CrossRef]
  8. J. Skilling, “Probabilistic data analysis: an introductory guide,” J. Microsc. (Oxford) 190, 28–36 (1998).
    [CrossRef]
  9. H. Graafsma, R. Y. de Vries, “Deconvolution of the two-dimensional point-spread function of area detectors using the maximum-entropy algorithm,” J. Appl. Crystallogr. 32, 683–691 (1999).
    [CrossRef]
  10. R. C. Puetter, “Pixons and Bayesian image reconstruction,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 112–131 (1994).
    [CrossRef]
  11. For more detailed information on the Pixon method, see http://www.pixon.com .
  12. T. E. Gureyev, R. Evans, “Tomography of objects with a priori known internal geometry,” Inverse Probl. 14, 1469–1480 (1998).
    [CrossRef]

2002 (1)

1999 (1)

H. Graafsma, R. Y. de Vries, “Deconvolution of the two-dimensional point-spread function of area detectors using the maximum-entropy algorithm,” J. Appl. Crystallogr. 32, 683–691 (1999).
[CrossRef]

1998 (2)

T. E. Gureyev, R. Evans, “Tomography of objects with a priori known internal geometry,” Inverse Probl. 14, 1469–1480 (1998).
[CrossRef]

J. Skilling, “Probabilistic data analysis: an introductory guide,” J. Microsc. (Oxford) 190, 28–36 (1998).
[CrossRef]

1996 (1)

1978 (1)

S. F. Gull, G. J. Daniel, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686–690 (1978).
[CrossRef]

1972 (1)

Arsenin, V.

A. N. Tichonov, V. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Daniel, G. J.

S. F. Gull, G. J. Daniel, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686–690 (1978).
[CrossRef]

de Vries, R. Y.

H. Graafsma, R. Y. de Vries, “Deconvolution of the two-dimensional point-spread function of area detectors using the maximum-entropy algorithm,” J. Appl. Crystallogr. 32, 683–691 (1999).
[CrossRef]

Evans, R.

T. E. Gureyev, R. Evans, “Tomography of objects with a priori known internal geometry,” Inverse Probl. 14, 1469–1480 (1998).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 2002).

Graafsma, H.

H. Graafsma, R. Y. de Vries, “Deconvolution of the two-dimensional point-spread function of area detectors using the maximum-entropy algorithm,” J. Appl. Crystallogr. 32, 683–691 (1999).
[CrossRef]

Gull, S. F.

S. F. Gull, G. J. Daniel, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686–690 (1978).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, R. Evans, “Tomography of objects with a priori known internal geometry,” Inverse Probl. 14, 1469–1480 (1998).
[CrossRef]

Lane, R. G.

Prasad, S.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 2002).

Puetter, R. C.

R. C. Puetter, “Pixons and Bayesian image reconstruction,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 112–131 (1994).
[CrossRef]

Richardson, W. H.

Skilling, J.

J. Skilling, “Probabilistic data analysis: an introductory guide,” J. Microsc. (Oxford) 190, 28–36 (1998).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 2002).

Tichonov, A. N.

A. N. Tichonov, V. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 2002).

Vogel, C. R.

C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2002).
[CrossRef]

Inverse Probl. (1)

T. E. Gureyev, R. Evans, “Tomography of objects with a priori known internal geometry,” Inverse Probl. 14, 1469–1480 (1998).
[CrossRef]

J. Appl. Crystallogr. (1)

H. Graafsma, R. Y. de Vries, “Deconvolution of the two-dimensional point-spread function of area detectors using the maximum-entropy algorithm,” J. Appl. Crystallogr. 32, 683–691 (1999).
[CrossRef]

J. Microsc. (Oxford) (1)

J. Skilling, “Probabilistic data analysis: an introductory guide,” J. Microsc. (Oxford) 190, 28–36 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nature (London) (1)

S. F. Gull, G. J. Daniel, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686–690 (1978).
[CrossRef]

Other (5)

R. C. Puetter, “Pixons and Bayesian image reconstruction,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 112–131 (1994).
[CrossRef]

For more detailed information on the Pixon method, see http://www.pixon.com .

A. N. Tichonov, V. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2002).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 2002).

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Figures (7)

Fig. 1
Fig. 1

Lena image with 512 × 512 pixels used in the deconvolution tests.

Fig. 2
Fig. 2

Image from Fig. 1 convolved with a 9 × 5 pixel-wide Gaussian PSF (σ x = 4.5 pixels and σ y = 2.5 pixels) and with 1% Poisson noise.

Fig. 3
Fig. 3

Iterative Taylor deconvolution of Fig. 2 using Eqs. (15) with the maximum derivative order equal to 2 and with a Gaussian low-pass 2 × 2 pixel filter used to calculate the regularized partial derivatives.

Fig. 4
Fig. 4

Iterative Taylor deconvolution of Fig. 2 using a 15 × 9 pixel-wide rectangular PSF.

Fig. 5
Fig. 5

Wiener deconvolution of Fig. 2 using a 2-D version of Eq. (2) with a 15 × 9 pixel-wide rectangular PSF.

Fig. 6
Fig. 6

Result of the iterative Taylor deconvolution of the image as in Fig. 2 but with a 5-pixel-wide strip along the edges masked with a constant value using Eqs. (15) with the maximum derivative order equal to 2 and with a Gaussian low-pass 2 × 2 pixel filter used to calculate the regularized partial derivatives.

Fig. 7
Fig. 7

Result of the Wiener deconvolution of the image as in Fig. 2 but with a 5-pixel-wide strip along the edges masked with a constant value by use of a 2-D version of Eq. (2).

Tables (1)

Tables Icon

Table 1 Accuracy of Deconvolution of the Lena Image from Fig. 1 with Various Amounts of Poisson Noise by the Wiener, Richardson-Lucy, Maximum Entropy, and Iterative Taylor Methods

Equations (17)

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Dx= Ix-xPxdx,
ÎWξ=DˆNξPˆ*ξ/|Pˆξ|2+α,
Dx=k=0 akdkIx/dxk,
ak=-1kk!  xkPxdx.
Ix=n=0 bndnDx/dxn
b0=1/a0, bn=-b0k=0n-1 bkan-k, n=1, 2, 3,.
|a0|>0, |ak|Cγk, k=1, 2,
|dnDx/dxn|C1δn, δ<1+C/|a0|γ-1, n=1, 2, 3,
IxDx/a0-a2Dx/a02,
Ix=k=01/k!-σ2/2kd2kDx/dx2k.
Ix, y=k=0l=0 bklxkylDx, y.
bkl=δk0δl0-m=0k-1n=0l-1 ak-ml-nbmn-m=0k-1 ak-m0bml-n=0l-1 a0l-nbkn/a00,
amn=-1n+mm!n!  xmynPx, ydxdy
Ix, yb0DNx, y+b22DNx, y,
I0=ADN, In=In-1+ADN-In-1*P, n=1, 2,,
In=k=0n BkADN, Bf=f-Af*P.
m,n |IWα * Pxm, yn-DNxm, yn|2=MNσ2DN2,

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