Abstract

An efficient and accurate algorithm for determining the magnitude of noise as a function of signal in an arbitrary digital image is presented and demonstrated. The algorithm is robust and largely independent of the form of the image, returning the noise function with subcount error across the full dynamic range of a synthetic test image where noise of a known form has been added. The noise performance of a CCD under different image recording and processing conditions is examined using the algorithm. The effect of different noise functions on pattern-matching measurements of electronic structure by quantitative convergent beam electron diffraction is investigated.

© 2012 Optical Society of America

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References

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  1. J. M. Zuo, M. Kim, M. O’Keeffe, and J. C. H. Spence, Nature 401, 49 (1999).
    [CrossRef]
  2. J. M. Zuo, Rep. Prog. Phys. 67, 2053 (2004).
    [CrossRef]
  3. P. N. H. Nakashima, A. E. Smith, J. Etheridge, and B. C. Muddle, Science 331, 1583 (2011).
    [CrossRef]
  4. J. C. H. Spence, Acta Crystallogr. Sect. A 49, 231 (1993).
    [CrossRef]
  5. M. Saunders, P. A. Midgley, T. D. Walsh, E. S. K. Menon, A. G. Fox, and R. Vincent, Scanning Microsc. 11, 23 (1997).
  6. J. M. Zuo, Ultramicroscopy 66, 21 (1996).
    [CrossRef]
  7. P. N. H. Nakashima and A. W. S. Johnson, Ultramicroscopy 94, 135 (2003).
    [CrossRef]
  8. P. N. H. Nakashima and B. C. Muddle, Phys. Rev. B 81, 115135 (2010).
    [CrossRef]

2011 (1)

P. N. H. Nakashima, A. E. Smith, J. Etheridge, and B. C. Muddle, Science 331, 1583 (2011).
[CrossRef]

2010 (1)

P. N. H. Nakashima and B. C. Muddle, Phys. Rev. B 81, 115135 (2010).
[CrossRef]

2004 (1)

J. M. Zuo, Rep. Prog. Phys. 67, 2053 (2004).
[CrossRef]

2003 (1)

P. N. H. Nakashima and A. W. S. Johnson, Ultramicroscopy 94, 135 (2003).
[CrossRef]

1999 (1)

J. M. Zuo, M. Kim, M. O’Keeffe, and J. C. H. Spence, Nature 401, 49 (1999).
[CrossRef]

1997 (1)

M. Saunders, P. A. Midgley, T. D. Walsh, E. S. K. Menon, A. G. Fox, and R. Vincent, Scanning Microsc. 11, 23 (1997).

1996 (1)

J. M. Zuo, Ultramicroscopy 66, 21 (1996).
[CrossRef]

1993 (1)

J. C. H. Spence, Acta Crystallogr. Sect. A 49, 231 (1993).
[CrossRef]

Etheridge, J.

P. N. H. Nakashima, A. E. Smith, J. Etheridge, and B. C. Muddle, Science 331, 1583 (2011).
[CrossRef]

Fox, A. G.

M. Saunders, P. A. Midgley, T. D. Walsh, E. S. K. Menon, A. G. Fox, and R. Vincent, Scanning Microsc. 11, 23 (1997).

Johnson, A. W. S.

P. N. H. Nakashima and A. W. S. Johnson, Ultramicroscopy 94, 135 (2003).
[CrossRef]

Kim, M.

J. M. Zuo, M. Kim, M. O’Keeffe, and J. C. H. Spence, Nature 401, 49 (1999).
[CrossRef]

Menon, E. S. K.

M. Saunders, P. A. Midgley, T. D. Walsh, E. S. K. Menon, A. G. Fox, and R. Vincent, Scanning Microsc. 11, 23 (1997).

Midgley, P. A.

M. Saunders, P. A. Midgley, T. D. Walsh, E. S. K. Menon, A. G. Fox, and R. Vincent, Scanning Microsc. 11, 23 (1997).

Muddle, B. C.

P. N. H. Nakashima, A. E. Smith, J. Etheridge, and B. C. Muddle, Science 331, 1583 (2011).
[CrossRef]

P. N. H. Nakashima and B. C. Muddle, Phys. Rev. B 81, 115135 (2010).
[CrossRef]

Nakashima, P. N. H.

P. N. H. Nakashima, A. E. Smith, J. Etheridge, and B. C. Muddle, Science 331, 1583 (2011).
[CrossRef]

P. N. H. Nakashima and B. C. Muddle, Phys. Rev. B 81, 115135 (2010).
[CrossRef]

P. N. H. Nakashima and A. W. S. Johnson, Ultramicroscopy 94, 135 (2003).
[CrossRef]

O’Keeffe, M.

J. M. Zuo, M. Kim, M. O’Keeffe, and J. C. H. Spence, Nature 401, 49 (1999).
[CrossRef]

Saunders, M.

M. Saunders, P. A. Midgley, T. D. Walsh, E. S. K. Menon, A. G. Fox, and R. Vincent, Scanning Microsc. 11, 23 (1997).

Smith, A. E.

P. N. H. Nakashima, A. E. Smith, J. Etheridge, and B. C. Muddle, Science 331, 1583 (2011).
[CrossRef]

Spence, J. C. H.

J. M. Zuo, M. Kim, M. O’Keeffe, and J. C. H. Spence, Nature 401, 49 (1999).
[CrossRef]

J. C. H. Spence, Acta Crystallogr. Sect. A 49, 231 (1993).
[CrossRef]

Vincent, R.

M. Saunders, P. A. Midgley, T. D. Walsh, E. S. K. Menon, A. G. Fox, and R. Vincent, Scanning Microsc. 11, 23 (1997).

Walsh, T. D.

M. Saunders, P. A. Midgley, T. D. Walsh, E. S. K. Menon, A. G. Fox, and R. Vincent, Scanning Microsc. 11, 23 (1997).

Zuo, J. M.

J. M. Zuo, Rep. Prog. Phys. 67, 2053 (2004).
[CrossRef]

J. M. Zuo, M. Kim, M. O’Keeffe, and J. C. H. Spence, Nature 401, 49 (1999).
[CrossRef]

J. M. Zuo, Ultramicroscopy 66, 21 (1996).
[CrossRef]

Acta Crystallogr. Sect. A (1)

J. C. H. Spence, Acta Crystallogr. Sect. A 49, 231 (1993).
[CrossRef]

Nature (1)

J. M. Zuo, M. Kim, M. O’Keeffe, and J. C. H. Spence, Nature 401, 49 (1999).
[CrossRef]

Phys. Rev. B (1)

P. N. H. Nakashima and B. C. Muddle, Phys. Rev. B 81, 115135 (2010).
[CrossRef]

Rep. Prog. Phys. (1)

J. M. Zuo, Rep. Prog. Phys. 67, 2053 (2004).
[CrossRef]

Scanning Microsc. (1)

M. Saunders, P. A. Midgley, T. D. Walsh, E. S. K. Menon, A. G. Fox, and R. Vincent, Scanning Microsc. 11, 23 (1997).

Science (1)

P. N. H. Nakashima, A. E. Smith, J. Etheridge, and B. C. Muddle, Science 331, 1583 (2011).
[CrossRef]

Ultramicroscopy (2)

J. M. Zuo, Ultramicroscopy 66, 21 (1996).
[CrossRef]

P. N. H. Nakashima and A. W. S. Johnson, Ultramicroscopy 94, 135 (2003).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Noise characterization algorithm is summarized by the flow diagram with more detail shown above it: (a) Noisy input image. (b) Short-range interpolation point spread function (PSF). (c) Intensity in the example pixel marked is plotted (black diamonds) after each of four convolutions with the interpolation PSF. A cubic fit (red line) to points 1 through 4 is used to extrapolate to “zeroth” smoothing order (circled). Doing this for all pixels in (a) gives the (d) “zeroth” image. (e) Noise image = (a)–(d). Slicing (d) into intensity intervals allows σ from (e) to be plotted against I for each slice in (d), giving (f). Fitting Eq. (2) determines the noise function [red, top left in (f)], which closely agrees with the known noise function coefficients, a, b, and c.

Fig. 2.
Fig. 2.

Noise analysis applying the present algorithm to a (a)–(d) CBED pattern is compared to independent noise analysis with uniform illumination recorded over a (e)–(h) range of exposure times with the same CCD. The image recording and processing details are: (b) and (f) single-frame recordings, (c) and (g) 25-frame summed recordings, and (d) and (h) after deconvolution of the instrumental PSF of the CCD detector from the 25-frame summed images. All data were collected with a Gatan Ultrascan 1000 CCD camera (2048×2048 pixels) operating at 25°C to minimize dark noise.

Fig. 3.
Fig. 3.

Refinement trajectories of structure factors F111, F113, and F004, refined from pattern matching of Fig. 2(a), summed over 25 frames and corrected for the detector PSF. The first pattern-matching refinement uses the noise function σ=1.40×102I+5.59I+722, determined as per Fig. 2(d) (solid lines). The second uses σ=I in Eq. (1) (dotted lines). The refined structure factors are plotted as a fraction of previous QCBED determinations [3], which were averaged over a large number of independent measurements.

Equations (2)

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χ2=1i=1nwi.i=1nwi.(IiexptIitheor)2σi2.
σ=a.I+b.I+c.

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