Abstract

The problem of detecting objects in noisy backgrounds is addressed. We derive detection filters by training a linear classifier, using features obtained from subimages corresponding to circular channels in the Fourier domain. The classifier weights approach the prewhitening matched filter when the classifier is trained for the detection of known objects in stationary noise. A simple form of rotation invariance is attained for considerably less computation than by the direct application of multiple matched filters. The method is demonstrated for the task of detecting simulated tumors in simulated nuclear medical images.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. N. Strickland, IEEE Trans. Med. Imag. 13, 491 (1994).
    [CrossRef]
  2. H. Hotelling, Ann. Math. Stat. 2, 360 (1931).
    [CrossRef]
  3. A. B. Watson, Comput. Vision Graphics Image Process. 39, 311 (1987).
    [CrossRef]
  4. A. Cohen, I. Daubechies, and J. C. Feauveau, (AT&T Bell Laboratories, Murray Hill, N.J., 1990).
  5. J. P. Rolland and H. H. Barrett, J. Opt. Soc. Am. A 9, 649 (1992).
    [CrossRef] [PubMed]
  6. W. E. Smith and H. H. Barrett, J. Opt. Soc. Am. A 3, 717 (1986).
    [CrossRef]
  7. K. J. Myers and H. H. Barrett, J. Opt. Soc. Am. A 4, 2447 (1987).
    [CrossRef] [PubMed]
  8. J. Yao and H. H. Barrett, Proc. SPIE 1768, 161 (1992).
    [CrossRef]
  9. C. Goresnic and S. R. Rotman, Graph. Models Image Process. 54, 329 (1992).
    [CrossRef]
  10. R. D. Fiete and H. H. Barrett, Opt. Lett. 12, 643 (1987).
    [CrossRef] [PubMed]
  11. B. V. K. Vijaya Kumar, Appl. Opt. 31, 4773 (1992).
    [CrossRef] [PubMed]

1994 (1)

R. N. Strickland, IEEE Trans. Med. Imag. 13, 491 (1994).
[CrossRef]

1992 (4)

J. Yao and H. H. Barrett, Proc. SPIE 1768, 161 (1992).
[CrossRef]

C. Goresnic and S. R. Rotman, Graph. Models Image Process. 54, 329 (1992).
[CrossRef]

J. P. Rolland and H. H. Barrett, J. Opt. Soc. Am. A 9, 649 (1992).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, Appl. Opt. 31, 4773 (1992).
[CrossRef] [PubMed]

1987 (3)

1986 (1)

1931 (1)

H. Hotelling, Ann. Math. Stat. 2, 360 (1931).
[CrossRef]

Barrett, H. H.

Cohen, A.

A. Cohen, I. Daubechies, and J. C. Feauveau, (AT&T Bell Laboratories, Murray Hill, N.J., 1990).

Daubechies, I.

A. Cohen, I. Daubechies, and J. C. Feauveau, (AT&T Bell Laboratories, Murray Hill, N.J., 1990).

Feauveau, J. C.

A. Cohen, I. Daubechies, and J. C. Feauveau, (AT&T Bell Laboratories, Murray Hill, N.J., 1990).

Fiete, R. D.

Goresnic, C.

C. Goresnic and S. R. Rotman, Graph. Models Image Process. 54, 329 (1992).
[CrossRef]

Hotelling, H.

H. Hotelling, Ann. Math. Stat. 2, 360 (1931).
[CrossRef]

Myers, K. J.

Rolland, J. P.

Rotman, S. R.

C. Goresnic and S. R. Rotman, Graph. Models Image Process. 54, 329 (1992).
[CrossRef]

Smith, W. E.

Strickland, R. N.

R. N. Strickland, IEEE Trans. Med. Imag. 13, 491 (1994).
[CrossRef]

Vijaya Kumar, B. V. K.

Watson, A. B.

A. B. Watson, Comput. Vision Graphics Image Process. 39, 311 (1987).
[CrossRef]

Yao, J.

J. Yao and H. H. Barrett, Proc. SPIE 1768, 161 (1992).
[CrossRef]

Ann. Math. Stat. (1)

H. Hotelling, Ann. Math. Stat. 2, 360 (1931).
[CrossRef]

Appl. Opt. (1)

Comput. Vision Graphics Image Process. (1)

A. B. Watson, Comput. Vision Graphics Image Process. 39, 311 (1987).
[CrossRef]

Graph. Models Image Process. (1)

C. Goresnic and S. R. Rotman, Graph. Models Image Process. 54, 329 (1992).
[CrossRef]

IEEE Trans. Med. Imag. (1)

R. N. Strickland, IEEE Trans. Med. Imag. 13, 491 (1994).
[CrossRef]

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

Opt. Lett. (1)

Proc. SPIE (1)

J. Yao and H. H. Barrett, Proc. SPIE 1768, 161 (1992).
[CrossRef]

Other (1)

A. Cohen, I. Daubechies, and J. C. Feauveau, (AT&T Bell Laboratories, Murray Hill, N.J., 1990).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Two-dimensional spatial frequency channel con-figuration with eight uniformly spaced circular channels (or bands) plus one high-frequency residual band. Further decomposition of the circular bands into oriented subbands, indicated by the intersecting dashed lines, is required for the discrimination of noncircularly symmetric objects.

Fig. 2
Fig. 2

Detectability da for several channel configurations. The task in this case is the detection of two-dimensional circularly symmetric Gaussian blobs in simulated anatomical background noise. (See, e.g., Fig. 4 below.) I, Four-octave wavelet transform, using the biorthogonal B-spline wavelet in Ref. 4. II, Four-octave circular channel configuration with cutoff frequencies 0.05625, 0.1125, 0.225, 0.45, and 0.9. III, Eight circular channels uniformly spaced over 0.0–0.2. The range 0.2–1.0 is integrated to give a ninth feature. (See Fig. 1.) IV, 40 circular channels uniformly spaced over 0–1. V, Prewhitening matched filter. Filter III performs almost as well as the prewhitening matched filter (V) and the densely sampled filter of IV. (All frequencies are normalized by half of the sampling frequency. Error bars represent ±1σ of experimental variation.)

Fig. 3
Fig. 3

Comparison of the prewhitening matched filter (solid curve) and the optimum nine-element optimum weight vector [w in Eq. (4)] of the channel configuration shown in Fig. 1. The matched filter response is the one-dimensional profile of a circularly symmetric two-dimensional function. Each weight is plotted at the center frequency of the channel that it represents. The error bars represent ±1σ measured over 20 different training data sets.

Fig. 4
Fig. 4

Example of tumor detection in 128×128 pixel simulated anatomical background image. Left, the simulated Gaussian tumor (σ=4 pixels) is added to Gaussian-filtered white noise; the sum is then convolved with a Gaussian to simulate the point-spread function of a typical pinhole imaging system. Poisson noise is then added. (Details in Ref. 5.) Right, test statistic λx,y computed with Eq. (9) using the weights shown in Fig. 3.

Equations (10)

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

HMFωx,ωy=F*ωx,ωySnnωx,ωyexp-jωxx0+ωyy0,
g=f+n,
λ=wTv,
wT=v1-v0TS-1.
S=½v0-v0v0-v0T+½v1-v1v1-v1T
wT=FTKn-1,
limN Kn=diagSnn0,Snn1,,SnnN-1,
da=λ1-λ0½σ12+σ021/2,
λx,y=wTvx,y,
λx,y=maxwvx,y.

Metrics