Abstract

A signal-processing algorithm has been developed where a filter function is extracted from degraded data through mathematical operations. The filter function can then be used to restore much of the degraded content of the data through use of a deconvolution algorithm. This process can be performed without prior knowledge of the detection system, a technique known as blind deconvolution. The extraction process, designated self-deconvolving data reconstruction algorithm, has been used successfully to restore digitized photographs, digitized acoustic waveforms, and other forms of data. The process is non-iterative, computationally efficient, and requires little user input. Implementation is straightforward, allowing inclusion into many types of signal-processing software and hardware. The novelty of the invention is the application of a power law and smoothing function to the degraded data in frequency space. Two methods for determining the value of the power law are discussed. The first method assumes the power law is frequency dependent. The function derived comparing the frequency spectrum of the degraded data with the spectrum of a signal with the desired frequency response. The second method assumes this function is a constant of frequency. This approach requires little knowledge of the original data or the degradation.

© 2002 Optical Society of America

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References

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  1. J. N. Caron, N. M. Namazi, R. L. Lucke, C. J. Rollins, P. R. Lynn, “Blind data restoration with an extracted filter function,” Opt. Lett. 26, 1164–1166 (2001).
    [CrossRef]
  2. J. N. Caron, U.S. Patent pending, anticipated acceptance date: August, 2003.
  3. A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  4. M. I. Sezan, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
    [CrossRef]
  5. B. Jähne, Digital Image Processing (Springer-Verlag, Berlin, 1997).
  6. P. M. Clarkson, H. Stark, eds., Signal Processing Methods for Audio, Images, and Telecommunications (Academic, San Diego, Calif., 1995).
  7. J. G. Proakis, D. G. Manolakis, Introduction to Signal Processing (Macmillan, New York, 1988).
  8. N. F. Law, D. T. Nguyen, “Multiple frame projection based blind deconvolution,” Electron. Lett. 31, 1733–1734 (1995).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. G. Arfken, “Mathematical Methods for Physicists,” Academic, San Diego, Calif., 1985.
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    [CrossRef]
  17. D. Slepian, “Linear least-squares filtering of distorted images,” J. Opt. Soc. Am. 57, 918–922 (1998).
    [CrossRef]
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    [CrossRef]
  19. Image courtesy of National Aeronautics and Space Administration/Jet Propulsion Laboratory/California Institute of Technology.
  20. Image courtesy of the Arizona Board of Regents and the Center for Image Processing in Education, Tucson, Ariz.
  21. J. N. Caron, Y. Yang, J. B. Mehl, K. V. Steiner, “Gas-coupled laser acoustic detection for ultrasound inspection of composite materials,” Mater. Eval. 58, 667–671 (2000).

2001 (1)

2000 (1)

J. N. Caron, Y. Yang, J. B. Mehl, K. V. Steiner, “Gas-coupled laser acoustic detection for ultrasound inspection of composite materials,” Mater. Eval. 58, 667–671 (2000).

1999 (2)

1998 (2)

1995 (1)

N. F. Law, D. T. Nguyen, “Multiple frame projection based blind deconvolution,” Electron. Lett. 31, 1733–1734 (1995).

1994 (1)

1993 (1)

O. Shalvi, E. Weinstein, “Super-exponential methods for blind deconvolution,” IEEE Trans. Info. Theory 39, 504–519 (1993).
[CrossRef]

1990 (1)

M. I. Sezan, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

1988 (1)

1967 (1)

Arfken, G.

G. Arfken, “Mathematical Methods for Physicists,” Academic, San Diego, Calif., 1985.

Ayers, G. R.

Barraza-Felix, S.

Bones, P. J.

Caron, J. N.

J. N. Caron, N. M. Namazi, R. L. Lucke, C. J. Rollins, P. R. Lynn, “Blind data restoration with an extracted filter function,” Opt. Lett. 26, 1164–1166 (2001).
[CrossRef]

J. N. Caron, Y. Yang, J. B. Mehl, K. V. Steiner, “Gas-coupled laser acoustic detection for ultrasound inspection of composite materials,” Mater. Eval. 58, 667–671 (2000).

J. N. Caron, U.S. Patent pending, anticipated acceptance date: August, 2003.

Dainty, J. C.

Frieden, B. R.

Helstrom, C. W.

Jähne, B.

B. Jähne, Digital Image Processing (Springer-Verlag, Berlin, 1997).

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Kopeika, N. S.

Lantzman, A.

Law, N. F.

N. F. Law, D. T. Nguyen, “Multiple frame projection based blind deconvolution,” Electron. Lett. 31, 1733–1734 (1995).

Lucke, R. L.

Lynn, P. R.

Manolakis, D. G.

J. G. Proakis, D. G. Manolakis, Introduction to Signal Processing (Macmillan, New York, 1988).

Mehl, J. B.

J. N. Caron, Y. Yang, J. B. Mehl, K. V. Steiner, “Gas-coupled laser acoustic detection for ultrasound inspection of composite materials,” Mater. Eval. 58, 667–671 (2000).

Milberg, R.

Mor, I.

Namazi, N. M.

Nguyen, D. T.

N. F. Law, D. T. Nguyen, “Multiple frame projection based blind deconvolution,” Electron. Lett. 31, 1733–1734 (1995).

Proakis, J. G.

J. G. Proakis, D. G. Manolakis, Introduction to Signal Processing (Macmillan, New York, 1988).

Rollins, C. J.

Satherly, B. L.

Sezan, M. I.

M. I. Sezan, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

Shalvi, O.

O. Shalvi, E. Weinstein, “Super-exponential methods for blind deconvolution,” IEEE Trans. Info. Theory 39, 504–519 (1993).
[CrossRef]

Slepian, D.

Steiner, K. V.

J. N. Caron, Y. Yang, J. B. Mehl, K. V. Steiner, “Gas-coupled laser acoustic detection for ultrasound inspection of composite materials,” Mater. Eval. 58, 667–671 (2000).

Weinstein, E.

O. Shalvi, E. Weinstein, “Super-exponential methods for blind deconvolution,” IEEE Trans. Info. Theory 39, 504–519 (1993).
[CrossRef]

Wiener, N.

N. Wiener, The Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications (Wiley, New York, 1949).

Yang, Y.

J. N. Caron, Y. Yang, J. B. Mehl, K. V. Steiner, “Gas-coupled laser acoustic detection for ultrasound inspection of composite materials,” Mater. Eval. 58, 667–671 (2000).

Yitzhaky, Y.

Yohaev, S.

Appl. Opt. (3)

Electron. Lett. (1)

N. F. Law, D. T. Nguyen, “Multiple frame projection based blind deconvolution,” Electron. Lett. 31, 1733–1734 (1995).

IEEE Trans. Info. Theory (1)

O. Shalvi, E. Weinstein, “Super-exponential methods for blind deconvolution,” IEEE Trans. Info. Theory 39, 504–519 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Mater. Eval. (1)

J. N. Caron, Y. Yang, J. B. Mehl, K. V. Steiner, “Gas-coupled laser acoustic detection for ultrasound inspection of composite materials,” Mater. Eval. 58, 667–671 (2000).

Opt. Eng. (1)

M. I. Sezan, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

Opt. Lett. (2)

Other (9)

J. N. Caron, U.S. Patent pending, anticipated acceptance date: August, 2003.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

B. Jähne, Digital Image Processing (Springer-Verlag, Berlin, 1997).

P. M. Clarkson, H. Stark, eds., Signal Processing Methods for Audio, Images, and Telecommunications (Academic, San Diego, Calif., 1995).

J. G. Proakis, D. G. Manolakis, Introduction to Signal Processing (Macmillan, New York, 1988).

G. Arfken, “Mathematical Methods for Physicists,” Academic, San Diego, Calif., 1985.

N. Wiener, The Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications (Wiley, New York, 1949).

Image courtesy of National Aeronautics and Space Administration/Jet Propulsion Laboratory/California Institute of Technology.

Image courtesy of the Arizona Board of Regents and the Center for Image Processing in Education, Tucson, Ariz.

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Figures (7)

Fig. 1
Fig. 1

(a) Truth image of a bowl of fruit taken with a digital camera; (b) convolution of the image with a point spread function.

Fig. 2
Fig. 2

Line plot of the frequency spectrum of the four images. The y axis is a logarithmic plot of the frequency response amplitude normalized to unity. The second y axis is a plot of the alpha function. Notice how the value can be approximated by 0.28 for all but the lowest frequencies.

Fig. 3
Fig. 3

(a) Restoration of the image using α = 0.28 and a pseudo-inverse filter; (b) restoration of the image by using a frequency dependent α(u) and a pseudo-inverse filter.

Fig. 4
Fig. 4

This image (a) was taken by the space probe Galileo of the surface of Io.19 Although the image already exhibits a fair amount of high resolution, application of the frequency-dependent restoration process brings out more detail (b).

Fig. 5
Fig. 5

This image (a) is an x-ray of a muscovy duck.20 The deconvolution (b) was restored with a frequency dependent α(u). The white line indicates the path of the line plot shown in Fig. 6.

Fig. 6
Fig. 6

This is a line plot taken across the wing as shown in Fig. 5. The restored image shows improved resolution when the algorithm is applied.

Fig. 7
Fig. 7

(a) Lower plot (labelled Original Data) is an ultrasound waveform that traveled through a centimeter of air. In this frequency range, the higher frequencies are strongly attenuated by air.21 The frequency dependent alpha function was determined using Eq. (9), producing an appropriate transfer function. The data was deconvolved with the extracted filter function. The restored data (labelled) exhibits several peaks that were not apparent in the original data. An arbitrary offset was added to the data for demonstrative purposes. The frequency spectra are shown on (b). Note the improved higher-frequency sensitivity between 1 and 4 MHz.

Equations (9)

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

gx=fx * dx+wx,
Gu=FuDu+Wu,
Fu=GuD*u|Du|2+K,
Du=KGS|Gu-Wu|αu,
Du=S|Gu|S|Fu|,
S|Gu|S|Fu|KGS|Gu|KFS|Fu|KGS|Gu|αu.
KFS|Fu|KFS|Fu|.
αuLnKGS|Gu|-LnKFS|Fu|LnKGS|Gu|.
Du=KGS|Gu|αu,

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