Abstract

A new approach for characterizing man-made and natural objects from two-dimensional data has been developed. The approach is based on the use of a fractal signature, which is a measure of local fractal dimension as a function of scale and space. Estimates of the fractal signature are obtained by using computationally efficient morphological filters. Consistency of the fractal signature and the effects of edges and anomalies are examined by using real and synthetic data. Preliminary results using optical and infrared data demonstrate the potential of using fractal signatures for object characterization and discrimination.

© 1990 Optical Society of America

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  1. B. L. Bullock, “Finding structure in outdoors scenes,” in Pattern Recognition and Artificial Intelligence, C. H. Chen, ed. (North-Holland, Amsterdam, 1976), pp. 61–85.
  2. R. Nevatia, K. E. Price, “Locating structure in aerial images,” in Digital Image Analysis, Vol. 2 of Digital Image Processing and Analysis, R. Chllapa, A. A. Sawchuck, eds. (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 247–255.
  3. A. D. Gross, A. Rosenfeld, “Multiresolution object detection and delineation,” Comput. Vision Graphics Image Process. 39, 102–115 (1987).
    [CrossRef]
  4. T. F. Quatieri, “Object detection by two dimensional linear prediction,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 108–111.
  5. A. P. Pentland, “Fractal based description of natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661–674 (1984).
    [CrossRef]
  6. T. Lundahl, W. Ohley, S. K. Kay, R. Siffert, “Fractional Brownian motion: a maximum likelihood estimator and its application to image texture,” IEEE Trans. Med. Imaging MI-5, 152–161 (1986).
    [CrossRef]
  7. M. J. Carlotto, M. C. Stein, “Detecting man-made change in imagery,” in Intelligent Robots and Computer Vision: Seventh in a Series, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1002, 6–11 (1985).
  8. S. Peleg, J. Naor, R. Hartely, D. Avnir, “Multiple resolution texture analysis and classification,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 518–523 (1984).
    [CrossRef]
  9. B. B. Mandelbort, The Fractal Geometry of Nature (Freeman, New York, 1983).
  10. F. Hausdorff, “Dimension and Ausseres Mass,” Math. Annalen 79, 157 (1919).
    [CrossRef]
  11. P. Kube, A. P. Pentland, “On imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704–707 (1988).
    [CrossRef]
  12. A. N. Kolmogorov, “A new invariant for transitive dynamical systems,” Dokl. Akad. Nauk SSSR 119, 861–864 (1958).
  13. B. B. Mandelbrot, B. J. Van Ness, “Fractional Brownian motion, fractional noises and applications,” SIAM Rev. 10, 442–438 (1968).
    [CrossRef]
  14. J. Serra, Image Analysis and Mathematical Morphology (Academic, London, 1982).
  15. F. Arduini, C. Dambra, S. Dellepiane, S. B. Serpico, G. Vernazza, R. Viviani, ”Fractal dimension by adaptive mask selection,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1116–1119 (1988).
  16. M. Nagao, T. Matsuyama, A Structural Analysis of Complex Aerial Photographs (Plenum, New York, 1980).
    [CrossRef]

1988 (1)

P. Kube, A. P. Pentland, “On imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704–707 (1988).
[CrossRef]

1987 (1)

A. D. Gross, A. Rosenfeld, “Multiresolution object detection and delineation,” Comput. Vision Graphics Image Process. 39, 102–115 (1987).
[CrossRef]

1986 (1)

T. Lundahl, W. Ohley, S. K. Kay, R. Siffert, “Fractional Brownian motion: a maximum likelihood estimator and its application to image texture,” IEEE Trans. Med. Imaging MI-5, 152–161 (1986).
[CrossRef]

1984 (2)

S. Peleg, J. Naor, R. Hartely, D. Avnir, “Multiple resolution texture analysis and classification,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 518–523 (1984).
[CrossRef]

A. P. Pentland, “Fractal based description of natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661–674 (1984).
[CrossRef]

1968 (1)

B. B. Mandelbrot, B. J. Van Ness, “Fractional Brownian motion, fractional noises and applications,” SIAM Rev. 10, 442–438 (1968).
[CrossRef]

1958 (1)

A. N. Kolmogorov, “A new invariant for transitive dynamical systems,” Dokl. Akad. Nauk SSSR 119, 861–864 (1958).

1919 (1)

F. Hausdorff, “Dimension and Ausseres Mass,” Math. Annalen 79, 157 (1919).
[CrossRef]

Arduini, F.

F. Arduini, C. Dambra, S. Dellepiane, S. B. Serpico, G. Vernazza, R. Viviani, ”Fractal dimension by adaptive mask selection,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1116–1119 (1988).

Avnir, D.

S. Peleg, J. Naor, R. Hartely, D. Avnir, “Multiple resolution texture analysis and classification,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 518–523 (1984).
[CrossRef]

Bullock, B. L.

B. L. Bullock, “Finding structure in outdoors scenes,” in Pattern Recognition and Artificial Intelligence, C. H. Chen, ed. (North-Holland, Amsterdam, 1976), pp. 61–85.

Carlotto, M. J.

M. J. Carlotto, M. C. Stein, “Detecting man-made change in imagery,” in Intelligent Robots and Computer Vision: Seventh in a Series, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1002, 6–11 (1985).

Dambra, C.

F. Arduini, C. Dambra, S. Dellepiane, S. B. Serpico, G. Vernazza, R. Viviani, ”Fractal dimension by adaptive mask selection,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1116–1119 (1988).

Dellepiane, S.

F. Arduini, C. Dambra, S. Dellepiane, S. B. Serpico, G. Vernazza, R. Viviani, ”Fractal dimension by adaptive mask selection,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1116–1119 (1988).

Gross, A. D.

A. D. Gross, A. Rosenfeld, “Multiresolution object detection and delineation,” Comput. Vision Graphics Image Process. 39, 102–115 (1987).
[CrossRef]

Hartely, R.

S. Peleg, J. Naor, R. Hartely, D. Avnir, “Multiple resolution texture analysis and classification,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 518–523 (1984).
[CrossRef]

Hausdorff, F.

F. Hausdorff, “Dimension and Ausseres Mass,” Math. Annalen 79, 157 (1919).
[CrossRef]

Kay, S. K.

T. Lundahl, W. Ohley, S. K. Kay, R. Siffert, “Fractional Brownian motion: a maximum likelihood estimator and its application to image texture,” IEEE Trans. Med. Imaging MI-5, 152–161 (1986).
[CrossRef]

Kolmogorov, A. N.

A. N. Kolmogorov, “A new invariant for transitive dynamical systems,” Dokl. Akad. Nauk SSSR 119, 861–864 (1958).

Kube, P.

P. Kube, A. P. Pentland, “On imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704–707 (1988).
[CrossRef]

Lundahl, T.

T. Lundahl, W. Ohley, S. K. Kay, R. Siffert, “Fractional Brownian motion: a maximum likelihood estimator and its application to image texture,” IEEE Trans. Med. Imaging MI-5, 152–161 (1986).
[CrossRef]

Mandelbort, B. B.

B. B. Mandelbort, The Fractal Geometry of Nature (Freeman, New York, 1983).

Mandelbrot, B. B.

B. B. Mandelbrot, B. J. Van Ness, “Fractional Brownian motion, fractional noises and applications,” SIAM Rev. 10, 442–438 (1968).
[CrossRef]

Matsuyama, T.

M. Nagao, T. Matsuyama, A Structural Analysis of Complex Aerial Photographs (Plenum, New York, 1980).
[CrossRef]

Nagao, M.

M. Nagao, T. Matsuyama, A Structural Analysis of Complex Aerial Photographs (Plenum, New York, 1980).
[CrossRef]

Naor, J.

S. Peleg, J. Naor, R. Hartely, D. Avnir, “Multiple resolution texture analysis and classification,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 518–523 (1984).
[CrossRef]

Nevatia, R.

R. Nevatia, K. E. Price, “Locating structure in aerial images,” in Digital Image Analysis, Vol. 2 of Digital Image Processing and Analysis, R. Chllapa, A. A. Sawchuck, eds. (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 247–255.

Ohley, W.

T. Lundahl, W. Ohley, S. K. Kay, R. Siffert, “Fractional Brownian motion: a maximum likelihood estimator and its application to image texture,” IEEE Trans. Med. Imaging MI-5, 152–161 (1986).
[CrossRef]

Peleg, S.

S. Peleg, J. Naor, R. Hartely, D. Avnir, “Multiple resolution texture analysis and classification,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 518–523 (1984).
[CrossRef]

Pentland, A. P.

P. Kube, A. P. Pentland, “On imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704–707 (1988).
[CrossRef]

A. P. Pentland, “Fractal based description of natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661–674 (1984).
[CrossRef]

Price, K. E.

R. Nevatia, K. E. Price, “Locating structure in aerial images,” in Digital Image Analysis, Vol. 2 of Digital Image Processing and Analysis, R. Chllapa, A. A. Sawchuck, eds. (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 247–255.

Quatieri, T. F.

T. F. Quatieri, “Object detection by two dimensional linear prediction,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 108–111.

Rosenfeld, A.

A. D. Gross, A. Rosenfeld, “Multiresolution object detection and delineation,” Comput. Vision Graphics Image Process. 39, 102–115 (1987).
[CrossRef]

Serpico, S. B.

F. Arduini, C. Dambra, S. Dellepiane, S. B. Serpico, G. Vernazza, R. Viviani, ”Fractal dimension by adaptive mask selection,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1116–1119 (1988).

Serra, J.

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

Siffert, R.

T. Lundahl, W. Ohley, S. K. Kay, R. Siffert, “Fractional Brownian motion: a maximum likelihood estimator and its application to image texture,” IEEE Trans. Med. Imaging MI-5, 152–161 (1986).
[CrossRef]

Stein, M. C.

M. J. Carlotto, M. C. Stein, “Detecting man-made change in imagery,” in Intelligent Robots and Computer Vision: Seventh in a Series, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1002, 6–11 (1985).

Van Ness, B. J.

B. B. Mandelbrot, B. J. Van Ness, “Fractional Brownian motion, fractional noises and applications,” SIAM Rev. 10, 442–438 (1968).
[CrossRef]

Vernazza, G.

F. Arduini, C. Dambra, S. Dellepiane, S. B. Serpico, G. Vernazza, R. Viviani, ”Fractal dimension by adaptive mask selection,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1116–1119 (1988).

Viviani, R.

F. Arduini, C. Dambra, S. Dellepiane, S. B. Serpico, G. Vernazza, R. Viviani, ”Fractal dimension by adaptive mask selection,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1116–1119 (1988).

Comput. Vision Graphics Image Process. (1)

A. D. Gross, A. Rosenfeld, “Multiresolution object detection and delineation,” Comput. Vision Graphics Image Process. 39, 102–115 (1987).
[CrossRef]

Dokl. Akad. Nauk SSSR (1)

A. N. Kolmogorov, “A new invariant for transitive dynamical systems,” Dokl. Akad. Nauk SSSR 119, 861–864 (1958).

IEEE Trans. Med. Imaging (1)

T. Lundahl, W. Ohley, S. K. Kay, R. Siffert, “Fractional Brownian motion: a maximum likelihood estimator and its application to image texture,” IEEE Trans. Med. Imaging MI-5, 152–161 (1986).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

S. Peleg, J. Naor, R. Hartely, D. Avnir, “Multiple resolution texture analysis and classification,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 518–523 (1984).
[CrossRef]

A. P. Pentland, “Fractal based description of natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661–674 (1984).
[CrossRef]

P. Kube, A. P. Pentland, “On imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704–707 (1988).
[CrossRef]

Math. Annalen (1)

F. Hausdorff, “Dimension and Ausseres Mass,” Math. Annalen 79, 157 (1919).
[CrossRef]

SIAM Rev. (1)

B. B. Mandelbrot, B. J. Van Ness, “Fractional Brownian motion, fractional noises and applications,” SIAM Rev. 10, 442–438 (1968).
[CrossRef]

Other (8)

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

F. Arduini, C. Dambra, S. Dellepiane, S. B. Serpico, G. Vernazza, R. Viviani, ”Fractal dimension by adaptive mask selection,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1116–1119 (1988).

M. Nagao, T. Matsuyama, A Structural Analysis of Complex Aerial Photographs (Plenum, New York, 1980).
[CrossRef]

B. B. Mandelbort, The Fractal Geometry of Nature (Freeman, New York, 1983).

M. J. Carlotto, M. C. Stein, “Detecting man-made change in imagery,” in Intelligent Robots and Computer Vision: Seventh in a Series, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1002, 6–11 (1985).

T. F. Quatieri, “Object detection by two dimensional linear prediction,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 108–111.

B. L. Bullock, “Finding structure in outdoors scenes,” in Pattern Recognition and Artificial Intelligence, C. H. Chen, ed. (North-Holland, Amsterdam, 1976), pp. 61–85.

R. Nevatia, K. E. Price, “Locating structure in aerial images,” in Digital Image Analysis, Vol. 2 of Digital Image Processing and Analysis, R. Chllapa, A. A. Sawchuck, eds. (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 247–255.

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Figures (17)

Fig. 1
Fig. 1

Construction of an area by the upper and lower functions of the covering-blanket method.

Fig. 2
Fig. 2

(a) Sample of a tree of size 32 × 32 and (b) fractal signature computed according to the area- and the volume-differences methods.

Fig. 3
Fig. 3

Kernel used to generate lower and upper surfaces for a two (spatial)-dimensional signal.

Fig. 4
Fig. 4

Construction of the upper and lower functions (surfaces) as a function of scale: (a) original signal, (b) scale 1, (c) scale 2, (d) scale 3.

Fig. 5
Fig. 5

Computing the fractal signature with morphological filters.

Fig. 6
Fig. 6

Four images of two classes of cloud: (a) class 1(a) image; (b) class 1(b) image; (c) class 2(a) image; (d) class 2(b) image.

Fig. 7
Fig. 7

Fractal signatures of the four images of the two classes of cloud shown in Fig. 6.

Fig. 8
Fig. 8

Trees scenery of size 64 × 64.

Fig. 9
Fig. 9

Consistency of the fractal signature for (a) class 1(b) cloud image, (b) class 2(b) cloud image and its right and left halves, and (c) class 2(b) cloud image and its top and bottom halves for the tree example of Fig. 8 (d).

Fig. 10
Fig. 10

Synthetic images of size 32 × 32 square (a), step edge (b), and their corresponding fractal signatures (c).

Fig. 11
Fig. 11

Fractal signature is used at large scale to highlight objects of high contrast and of the same scale: (a) original; (b) pixels of fractal dimension greater than 2.8 at scale 20 are highlighted.

Fig. 12
Fig. 12

Fractal signatures of anomalies: (a) an anomaly, (b) a thick anomaly, (c) their corresponding signatures, (d) an isolated tree on a flat background, (e) its corresponding signature.

Fig. 13
Fig. 13

Hypothetical example of signatures of two classes. Scales 4 and 5 have values of the fractal dimension that are compact within classes and are apart for the two different classes.

Fig. 14
Fig. 14

Samples of three classes of scenery: (a) vehicle and tanks, (b) trees, (c) background.

Fig. 15
Fig. 15

Fractal signatures of the three classes in Fig. 14.

Fig. 16
Fig. 16

Single-scale target discrimination: (a) original; (b) pixels of a fractal dimension of less than 2.2 at scale 2 are highlighted.

Fig. 17
Fig. 17

Single-scale target discrimination: (a) original image; (b) every pixel of (a) is highlighted if its fractal dimension at scale 4 is greater than 2.4.

Equations (24)

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M ( ɛ ) = K ɛ d D ,
M ( 2 ɛ ) = K ( 2 ɛ ) d D .
M ( 2 ɛ ) M ( ɛ ) = 2 d D
U ( i , j , 0 ) = L ( i , j , 0 ) = g ( i , j ) ,
U ( i , j , ɛ + 1 ) = max { U ( i , j , ɛ ) + 1 , max k , m η [ U ( k , m , ɛ ) ] } ,
L ( i , j , ɛ + 1 ) = min { L ( i , j , ɛ ) 1 , min k , m η [ L ( k , m , ɛ ) ] } ,
η = { ( k , m ) distance [ ( k , m ) , ( i , j ) ] 1 } .
A ( ɛ ) = i , j R U ( i , j , ɛ ) L ( i , j , ɛ ) 2 ɛ = V ( ɛ ) 2 ɛ .
A 1 ( ɛ ) = V ( ɛ ) V ( ɛ 1 ) 2 .
V ( ɛ ) = 2 ɛ A ( ɛ ) .
V ( ɛ ) = 2 K ɛ 3 D .
A 1 ( ɛ ) 0.5 V ( ɛ ) ɛ = ( 3 D ) K ɛ 2 D .
D ( i , j ) = ɛ C ɛ F ɛ ( i , j ) ɛ C ɛ ,
C ɛ = log ( ɛ ) log ( ɛ 1 ) log 2 .
F ɛ ( i , j ) = log A ( i , j , ɛ ) log A ( i , j , ɛ 1 ) log ɛ log ( ɛ 1 ) .
A ( i , j , ɛ ) A ( i , j , ɛ 1 ) = K ɛ ( 2 D ) K ( ɛ 1 ) ( 2 D ) = ( ɛ ɛ 1 ) 2 D .
log [ A ( i , j , ɛ ) A ( i , j , ɛ 1 ) ] = ( 2 D ) log ( ɛ ɛ 1 ) ,
log A ( i , j , ɛ ) log A ( i , j , ɛ 1 ) log ( ɛ ) log ( ɛ 1 ) = 2 D = F ɛ ( i , j ) ,
C ɛ F ɛ ( i , j ) = { log [ A ( ɛ ) ] log [ A ( ɛ 1 ) ] } log 2 .
ɛ C ɛ F ɛ ( i , j , ɛ ) = { log [ A ( i , j , ɛ ) ] log [ A ( i , j , ɛ 1 ) ] + log [ A ( i , j , ɛ 1 ) ] log [ A ( i , j , ɛ 2 ) ] + log [ A ( i , j , ɛ 2 ) log [ A ( i , j , 1 ) ] } / log 2 = { log [ A ( i , j , ɛ ) ] log [ A ( , j , 1 ) ] } / log 2 ,
ɛ C ɛ = [ log ( ɛ ) log ( ɛ 1 ) + log ( ɛ 1 ) log ( ɛ 2 ) + log ( ɛ 2 ) + log ( 1 ) ] / log 2 = log ( ɛ ) log ( 1 ) log 2 .
Erosion : ( g k ) ( x , y ) = min { g ( x + n , y + m ) k ( n , m ) } n , m Ros ( k ) ,
Dilation : ( g k ) ( x , y ) = max { g ( x + n , y + m ) + k ( n , m ) } n , m , Ros ( k ) ,
A ( k , l , ɛ ) = i = k = w k + w j = l w l + w U( i , j , ɛ ) L ( i , j , ɛ ) 2 ɛ .

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