Abstract

When imaging is performed by using a coherent signal, the result is frequently a realization of the stochastic process known as speckle. The information sought from this process is often the mean value of its envelope or intensity at each point in the image plane. When only a single realization of the process is available, ergodicity is required within a sufficiently large region for accurate estimation of the mean. The identification of these regions is the segmentation problem that is addressed. The approach presented clips the speckle image at a constant threshold level and analyzes the resulting bilevel image based on the level-crossing statistics of the speckle process. An analysis of the level-crossing process leads to a decision rule for identifying or segmenting distinct regions of the image based on the sizes of the fades and the excursions in the clipped speckle. The measurement of these sizes is accomplished by using the morphological transformations of opening and closing. This new approach has been applied to computer-generated speckle images and may prove useful in laser, ultrasound, and radar imaging, in which speckle phenomena are manifest.

© 1991 Optical Society of America

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References

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  1. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
    [CrossRef]
  2. D. Middleton, An Introduction to Statistical Communications Theory (McGraw-Hill, New York, 1960).
  3. J. Marron, G. M. Morris, “Properties of clipped laser speckle,” in Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 39–44 (1985).
    [CrossRef]
  4. R. Barakat, “Clipped photon-counting covariance functions,” J. Opt. Soc. Am. A 5, 1248–1253 (1988).
    [CrossRef]
  5. S. O. Rice, “Distribution of the durations of fades in radio transmission: Gaussian noise model,” Bell Syst. Tech. J. 27, 581–635 (1958).
    [CrossRef]
  6. R. Barakat, “Level-crossing statistics of aperature integrated speckle,” J. Opt. Soc. Am. A 5, 1244–1247 (1988).
    [CrossRef]
  7. K. J. Parker, “Attenuation measurement uncertainties caused by speckle statistics,”J. Acoust. Soc. Am. 80, 727–734 (1986).
    [CrossRef] [PubMed]
  8. R. H. Sperry, “Segmentation of speckle images based on level crossing statistics,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1989).
  9. T. A. Tuthill, R. H. Sperry, K. J. Parker, “Deviations from Rayleigh statistics in ultrasound speckle,” Ultrasonic Imag. 10, 81–89 (1988).
  10. J. L. Melsa, D. L. Cohen, Decision and Estimation Theory (McGraw-Hill, New York, 1978).
  11. H. Cramer, M. R. Leadbetter, Stationary and Related Stochastic Processes (Wiley, New York, 1967).
  12. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1984).
  13. R. F. Wagner, M. F. Insana, D. G. Brown, “Statistical properties of radio-frequency and envelope-detected signals with applications to medical ultrasound,” J. Opt. Soc. Am. A 4, 910–922 (1987).
    [CrossRef] [PubMed]
  14. A. J. Rainal, “Duration of fades associated with radar clutter,” Bell Syst. Tech. J. 45, 1285–1298 (1966).
    [CrossRef]
  15. J. Serra, Image Analysis and Mathematical Morphology (Academic, London, 1982).
  16. K. A. Wear, R. L. Popp, “Methods for estimation of statistical properties of envelopes of ultrasonic echoes from myocardium,”IEEE Trans. Med. Imag. MI-6, 281–291 (1987).
    [CrossRef]

1988 (3)

1987 (2)

K. A. Wear, R. L. Popp, “Methods for estimation of statistical properties of envelopes of ultrasonic echoes from myocardium,”IEEE Trans. Med. Imag. MI-6, 281–291 (1987).
[CrossRef]

R. F. Wagner, M. F. Insana, D. G. Brown, “Statistical properties of radio-frequency and envelope-detected signals with applications to medical ultrasound,” J. Opt. Soc. Am. A 4, 910–922 (1987).
[CrossRef] [PubMed]

1986 (1)

K. J. Parker, “Attenuation measurement uncertainties caused by speckle statistics,”J. Acoust. Soc. Am. 80, 727–734 (1986).
[CrossRef] [PubMed]

1966 (1)

A. J. Rainal, “Duration of fades associated with radar clutter,” Bell Syst. Tech. J. 45, 1285–1298 (1966).
[CrossRef]

1958 (1)

S. O. Rice, “Distribution of the durations of fades in radio transmission: Gaussian noise model,” Bell Syst. Tech. J. 27, 581–635 (1958).
[CrossRef]

Barakat, R.

Brown, D. G.

Cohen, D. L.

J. L. Melsa, D. L. Cohen, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Cramer, H.

H. Cramer, M. R. Leadbetter, Stationary and Related Stochastic Processes (Wiley, New York, 1967).

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

Insana, M. F.

Leadbetter, M. R.

H. Cramer, M. R. Leadbetter, Stationary and Related Stochastic Processes (Wiley, New York, 1967).

Marron, J.

J. Marron, G. M. Morris, “Properties of clipped laser speckle,” in Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 39–44 (1985).
[CrossRef]

Melsa, J. L.

J. L. Melsa, D. L. Cohen, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Middleton, D.

D. Middleton, An Introduction to Statistical Communications Theory (McGraw-Hill, New York, 1960).

Morris, G. M.

J. Marron, G. M. Morris, “Properties of clipped laser speckle,” in Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 39–44 (1985).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1984).

Parker, K. J.

T. A. Tuthill, R. H. Sperry, K. J. Parker, “Deviations from Rayleigh statistics in ultrasound speckle,” Ultrasonic Imag. 10, 81–89 (1988).

K. J. Parker, “Attenuation measurement uncertainties caused by speckle statistics,”J. Acoust. Soc. Am. 80, 727–734 (1986).
[CrossRef] [PubMed]

Popp, R. L.

K. A. Wear, R. L. Popp, “Methods for estimation of statistical properties of envelopes of ultrasonic echoes from myocardium,”IEEE Trans. Med. Imag. MI-6, 281–291 (1987).
[CrossRef]

Rainal, A. J.

A. J. Rainal, “Duration of fades associated with radar clutter,” Bell Syst. Tech. J. 45, 1285–1298 (1966).
[CrossRef]

Rice, S. O.

S. O. Rice, “Distribution of the durations of fades in radio transmission: Gaussian noise model,” Bell Syst. Tech. J. 27, 581–635 (1958).
[CrossRef]

Serra, J.

J. Serra, Image Analysis and Mathematical Morphology (Academic, London, 1982).

Sperry, R. H.

T. A. Tuthill, R. H. Sperry, K. J. Parker, “Deviations from Rayleigh statistics in ultrasound speckle,” Ultrasonic Imag. 10, 81–89 (1988).

R. H. Sperry, “Segmentation of speckle images based on level crossing statistics,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1989).

Tuthill, T. A.

T. A. Tuthill, R. H. Sperry, K. J. Parker, “Deviations from Rayleigh statistics in ultrasound speckle,” Ultrasonic Imag. 10, 81–89 (1988).

Wagner, R. F.

Wear, K. A.

K. A. Wear, R. L. Popp, “Methods for estimation of statistical properties of envelopes of ultrasonic echoes from myocardium,”IEEE Trans. Med. Imag. MI-6, 281–291 (1987).
[CrossRef]

Bell Syst. Tech. J. (2)

S. O. Rice, “Distribution of the durations of fades in radio transmission: Gaussian noise model,” Bell Syst. Tech. J. 27, 581–635 (1958).
[CrossRef]

A. J. Rainal, “Duration of fades associated with radar clutter,” Bell Syst. Tech. J. 45, 1285–1298 (1966).
[CrossRef]

IEEE Trans. Med. Imag. (1)

K. A. Wear, R. L. Popp, “Methods for estimation of statistical properties of envelopes of ultrasonic echoes from myocardium,”IEEE Trans. Med. Imag. MI-6, 281–291 (1987).
[CrossRef]

J. Acoust. Soc. Am. (1)

K. J. Parker, “Attenuation measurement uncertainties caused by speckle statistics,”J. Acoust. Soc. Am. 80, 727–734 (1986).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

Ultrasonic Imag. (1)

T. A. Tuthill, R. H. Sperry, K. J. Parker, “Deviations from Rayleigh statistics in ultrasound speckle,” Ultrasonic Imag. 10, 81–89 (1988).

Other (8)

J. L. Melsa, D. L. Cohen, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

H. Cramer, M. R. Leadbetter, Stationary and Related Stochastic Processes (Wiley, New York, 1967).

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1984).

R. H. Sperry, “Segmentation of speckle images based on level crossing statistics,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1989).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

D. Middleton, An Introduction to Statistical Communications Theory (McGraw-Hill, New York, 1960).

J. Marron, G. M. Morris, “Properties of clipped laser speckle,” in Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 39–44 (1985).
[CrossRef]

J. Serra, Image Analysis and Mathematical Morphology (Academic, London, 1982).

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Figures (10)

Fig. 1
Fig. 1

Example of the segmentation process. (a) Synthetically generated speckle pattern for which the mean of the exterior is 1.0 and the mean of the interior is 3.0. (b) Result of clipping (a) at the maximum-likelihood decision point. (c) Closing of (b) with a circular structuring element of 15 pixels in diameter. (d) Opening of (c) by a circular structuring element of 15 pixels in diameter.

Fig. 2
Fig. 2

Mean lengths of fades (dashed curve), the mean length of excursions (solid curve), and the mean distance between fades (excursions) for the envelope of a speckle process.

Fig. 3
Fig. 3

Log-likelihood ratio of an excursion of normalized length x1 using the large ratio of means approximation for ratios of 5.0 (solid curve), 10.0 (short-dashed curve), 100.0 (long dashed curve), 1000.0 (dotted-dashed curve).

Fig. 4
Fig. 4

Log-likelihood ratio for a fade of normalized length x2 using the large ratios of means approximation for ratios 5.0 (solid curve), 10.0 (short-dashed curve), 100.0 (long dashed curve), 1000.0 (dotted–dashed curve).

Fig. 5
Fig. 5

Segmentation of a 256 × 256-pixel synthetic speckle image with a circle of 128 pixels in diameter. The top row is the image displayed with a logarithmic palette. The center row is the result of thresholding at the maximum-likelihood decision point. The bottom row displays the result of the segmentation algorithm with known means. The columns from left to right have means of (a) 10–100, (b) 20–100, (c) 30–100, (d) 40–100.

Fig. 6
Fig. 6

Same as Fig. 5 for known means with ratios of (a) 50:100, (b) 150:100, (c) 200:100, (d) 250:100.

Fig. 7
Fig. 7

Same as Fig. 5 for known means with ratios of (a) 300:100, (b) 350:100, (c) 500:100, (d) 1000:100.

Fig. 8
Fig. 8

Circles of mean 300 in a background of 100 with diameters of (a) 32 pixels, (b) 64 pixels, (c) 182 pixels, (d) 256 pixels. The threshold was set to the maximum likelihood by using the known parameters.

Fig. 9
Fig. 9

Same as Fig. 5 for unknown means with ratios of (a) 50:100, (b) 150:100, (c) 200:100, (d) 250:100.

Fig. 10
Fig. 10

Same as Fig. 5 for unknown means with ratios of (a) 300:100, (b) 350:100, (c) 500:100, (d) 1000:100.

Tables (4)

Tables Icon

Table 1 Statistics Resulting from the Segmentation of the Images in Fig. 5a

Tables Icon

Table 2 Statistics Resulting from the Segmentation of the Images in Fig. 6a

Tables Icon

Table 3 Statistics Resulting from the Segmentation of the Images in Fig. 7a

Tables Icon

Table 4 Statistics Resulting from the Segmentation of the Images in Fig. 8a

Equations (44)

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I u ( x ) = { 1 , I ( x ) u 0 , I ( x ) < u .
p V ( v ) = { v ψ exp ( - v 2 2 ψ ) , v > 0 0 , otherwise
v d i d e [ 2 ln ( ψ i ψ e ) 1 ψ i - 1 ψ e ] 1 / 2 ,
C u = f ξ ( u ) ξ ( t ) ξ ( t ) = u ,
C u 2 = D u = U u .
x u = μ - 1 { P [ ξ ( 0 ) - v 0 ( ) > u ] } ,
x u = μ - 1 { P [ ξ ( 0 ) - w 0 ( ) < u ] } ,
ξ ( t ) = - exp ( j t λ ) d ζ ( λ ) .
λ n = - λ n d F ( λ ) ,
r ( t ) = - f ( λ ) exp ( j t λ ) d λ .
C u = ( 2 Δ π ) 1 / 2 u exp ( - u 2 2 ) .
x u = ( 2 π Δ ) 1 / 2 exp ( u 2 / 2 ) - 1 u
x u = ( 2 π Δ ) 1 / 2 1 u .
u u ( λ 0 ) 1 / 2 , Δ λ 0 λ 2 - λ 1 2 λ 0 2 .
f ( τ , u ) = exp ( - τ ) ,
f ( ζ , u 0 ) = exp ( - ζ ) .
f ( τ , u 0 ) = 2 π z 2 exp ( - z ) [ I 0 ( z ) - ( 1 + 1 2 z ) I 1 ( z ) ] ,
f ( ζ , u ) = π 2 ζ exp ( - π 4 ζ 2 ) .
p ( z m 1 ) p ( z m 2 ) d 1 d 2 p ( m 2 ) p ( m 1 ) ,
lim u 2 0 f ( x ) = 1 x ¯ 2 exp ( - x / x ¯ 2 ) ,
lim u 1 f ( x ) = π 2 x x ¯ 1 2 exp [ - π 4 ( x x ¯ 1 ) 2 ] ,
Λ u ( x ) = [ 1 x ¯ 2 exp ( - x x ¯ 2 ) ] / { π x 2 x ¯ 1 2 exp [ - π 4 ( x x ¯ 1 ) 2 ] } .
l u ( x ) = π 4 x 1 2 - x 1 r - ln ( x 1 ) - ln ( π 2 r ) ,
l u ( x ) d 2 d 1 0.
P ( z m 1 ) P ( z m 2 ) d 1 d 2 P ( m 2 ) P ( m 1 ) ,
Λ u ( x ) d 1 d 2 Λ ( u ) ,
lim u 2 0 f ( x ) = 1 x ¯ 2 2 π z 2 exp ( - z ) [ I 0 ( z ) - ( 1 + 1 2 z ) I 1 ( z ) ] ,
lim u 1 f ( x ) = 1 x ¯ 1 exp ( - x x ¯ 1 ) ,
lim u 2 0 x ¯ 2 = ( 2 π Δ ) 1 / 2 u 2 2 + O ( u 2 3 )
lim u 2 0 f ( x ) = 1 x ¯ 2 6 π x ¯ 2 4 x 4 exp ( - 2 x ¯ 2 2 π x 2 ) .
Λ u ( x ) = [ 1 x ¯ 2 6 π x ¯ 2 4 x 4 exp ( - 2 x ¯ 2 2 π x 2 ) ] / [ 1 x ¯ 1 exp ( - x x ¯ 1 ) ] .
l u ( x ) = ln ( 6 x ¯ 1 π x ¯ 2 ) - 4 ln ( x 2 ) - 2 π x 2 2 + x x ¯ 1 .
x ¯ 1 = ( 2 π Δ ) 1 / 2 { exp [ 2 ln ( r ) 1 - r ] - 1 } / [ 2 ln ( r ) 1 - r ] 1 / 2 .
A x = A + x = { y + x y A } .
c A = { c · x x A } .
A ˇ = - A = { - x x A } .
A B = { x + y x A , y B } .
A B = y B A y = x A B x .
A B = y B A y ,
A B = ( A B ˇ ) B
A B = x B y B A x - y ,
A B = ( A B ˇ ) B
A B = x B y B A x - y ,
[ J 1 ( x ) 2 x ] ,

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