Abstract

We consider how to interpret, filter, and cross-correlate complex-value color (hue and saturation) images by using a single discrete Fourier transform: the spatiochromatic discrete Fourier transform. The model defines new types of spatiochromatic oriented sinusoidal gratings, termed rainbow gratings, which encode the variation of color over space. We demonstrate how color-opponent detectors observed within the vertebrate visual system can be easily defined by linear filters within this representation. This model also allows us to filter and detect both spatial and chromatic patterns in images by using a single cross-correlation procedure. In doing so, we explore a new form of the Cauchy–Schwartz inequality applied to complex-valued scalar products. Results demonstrate the power of this form of spatiochromatic matched filtering in detecting signals embedded in such a significant amount of noise that they are not visible to the unaided human eye.

© 2000 Optical Society of America

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References

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  1. J. Davidoff, “Color perception,” in The Handbook of Brain Theory and Neural Networks, M. A. Arbib, ed. (MIT Press, Cambridge, Mass., 1995), pp. 210–215.
  2. T. Caelli, D. Reye, “On the classification of image regions by color, texture, and shape,” Pattern Recogn. 26, 461–470 (1993).
    [CrossRef]
  3. D. Slater, G. Healey, “The illuminant-invariant recognition of 3D objects using local color invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 206–210 (1996).
    [CrossRef]
  4. D. Carevic, T. Caelli, “Region-based coding of colour images using the Karhunen–Loève transform,” Graphical Models Image Process. 59, 27–38 (1997).
    [CrossRef]
  5. G. Healey, “Modeling color images for machine vision,” in Image Technology: Advances in Image Processing, Multimedia and Machine Vision, J. Sanz, ed. (Springer-Verlag, Berlin, 1995), pp. 129–146.
  6. M. Das, E. M. Riseman, B. A. Draper, “FOCUS: Searching for multi-colored objects in a diverse image,” in IEEE Conference on Computer Vision and Pattern Recognition 97 (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 756–761.
  7. J. Huang, S. R. Kumar, M. Mitra, W. Zhu, R. Zabih, “Image indexing using color correlograms,” in Computer Vision and Pattern Recognition 97 (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 762–768.
  8. A. Jain, G. Healey, “A multiscale representation including opponent color features for texture recognition,” IEEE Trans. Image Process. 7, 124–128 (1998).
    [CrossRef]
  9. B. Thai, G. Healey, “Modeling and classifying symmetries using a multiscale opponent color representation,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1224–1235 (1998).
    [CrossRef]
  10. E. Persoon, K. S. Fu, “Shape discrimination using Fourier descriptors,” IEEE Trans. Syst. Man Cybern. SMC-7, 170–179 (1977).
    [CrossRef]
  11. G. Bonmasser, E. Schwartz, “Lie groups, space-variant Fourier analysis and the exponential chirp transform,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 492–498.
  12. A. L. Thornton, S. J. Sangwine, “Colour object location using complex coding in the frequency domain,” in Proceedings of the 5th International Conference on Image Processing and Its Applications (Heriot-Watt University, Edinburgh, UK, 1995), pp. 820–824.
  13. R. Boylestad, Introductory Circuit Analysis, 6th ed. (Macmillan, New York, 1990).
  14. CIE, “Colorimetry,” (CIE: International Commission on Illumination, Vienna, 1986; 2nd ed. (1996).
  15. S. Marcelja, “Mathematical description of the responses of simple cortical cells,” J. Opt. Soc. Am. 70, 1297–1300 (1980).
    [CrossRef] [PubMed]
  16. J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 847–856 (1980).
    [CrossRef] [PubMed]
  17. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978).
  18. M. Livingstone, D. Hubel, “Anatomy and Physiology of Q colour system in the primate visual cortex,” J. Physiol. 4, 309–356 (1984).
  19. J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall Inc., Englewood Cliffs, N.J., 1990).

1998 (2)

A. Jain, G. Healey, “A multiscale representation including opponent color features for texture recognition,” IEEE Trans. Image Process. 7, 124–128 (1998).
[CrossRef]

B. Thai, G. Healey, “Modeling and classifying symmetries using a multiscale opponent color representation,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1224–1235 (1998).
[CrossRef]

1997 (1)

D. Carevic, T. Caelli, “Region-based coding of colour images using the Karhunen–Loève transform,” Graphical Models Image Process. 59, 27–38 (1997).
[CrossRef]

1996 (1)

D. Slater, G. Healey, “The illuminant-invariant recognition of 3D objects using local color invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 206–210 (1996).
[CrossRef]

1993 (1)

T. Caelli, D. Reye, “On the classification of image regions by color, texture, and shape,” Pattern Recogn. 26, 461–470 (1993).
[CrossRef]

1984 (1)

M. Livingstone, D. Hubel, “Anatomy and Physiology of Q colour system in the primate visual cortex,” J. Physiol. 4, 309–356 (1984).

1980 (2)

S. Marcelja, “Mathematical description of the responses of simple cortical cells,” J. Opt. Soc. Am. 70, 1297–1300 (1980).
[CrossRef] [PubMed]

J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 847–856 (1980).
[CrossRef] [PubMed]

1977 (1)

E. Persoon, K. S. Fu, “Shape discrimination using Fourier descriptors,” IEEE Trans. Syst. Man Cybern. SMC-7, 170–179 (1977).
[CrossRef]

Bonmasser, G.

G. Bonmasser, E. Schwartz, “Lie groups, space-variant Fourier analysis and the exponential chirp transform,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 492–498.

Boylestad, R.

R. Boylestad, Introductory Circuit Analysis, 6th ed. (Macmillan, New York, 1990).

Caelli, T.

D. Carevic, T. Caelli, “Region-based coding of colour images using the Karhunen–Loève transform,” Graphical Models Image Process. 59, 27–38 (1997).
[CrossRef]

T. Caelli, D. Reye, “On the classification of image regions by color, texture, and shape,” Pattern Recogn. 26, 461–470 (1993).
[CrossRef]

Carevic, D.

D. Carevic, T. Caelli, “Region-based coding of colour images using the Karhunen–Loève transform,” Graphical Models Image Process. 59, 27–38 (1997).
[CrossRef]

Das, M.

M. Das, E. M. Riseman, B. A. Draper, “FOCUS: Searching for multi-colored objects in a diverse image,” in IEEE Conference on Computer Vision and Pattern Recognition 97 (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 756–761.

Daugman, J. G.

J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 847–856 (1980).
[CrossRef] [PubMed]

Davidoff, J.

J. Davidoff, “Color perception,” in The Handbook of Brain Theory and Neural Networks, M. A. Arbib, ed. (MIT Press, Cambridge, Mass., 1995), pp. 210–215.

Draper, B. A.

M. Das, E. M. Riseman, B. A. Draper, “FOCUS: Searching for multi-colored objects in a diverse image,” in IEEE Conference on Computer Vision and Pattern Recognition 97 (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 756–761.

Fu, K. S.

E. Persoon, K. S. Fu, “Shape discrimination using Fourier descriptors,” IEEE Trans. Syst. Man Cybern. SMC-7, 170–179 (1977).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978).

Healey, G.

A. Jain, G. Healey, “A multiscale representation including opponent color features for texture recognition,” IEEE Trans. Image Process. 7, 124–128 (1998).
[CrossRef]

B. Thai, G. Healey, “Modeling and classifying symmetries using a multiscale opponent color representation,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1224–1235 (1998).
[CrossRef]

D. Slater, G. Healey, “The illuminant-invariant recognition of 3D objects using local color invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 206–210 (1996).
[CrossRef]

G. Healey, “Modeling color images for machine vision,” in Image Technology: Advances in Image Processing, Multimedia and Machine Vision, J. Sanz, ed. (Springer-Verlag, Berlin, 1995), pp. 129–146.

Huang, J.

J. Huang, S. R. Kumar, M. Mitra, W. Zhu, R. Zabih, “Image indexing using color correlograms,” in Computer Vision and Pattern Recognition 97 (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 762–768.

Hubel, D.

M. Livingstone, D. Hubel, “Anatomy and Physiology of Q colour system in the primate visual cortex,” J. Physiol. 4, 309–356 (1984).

Jain, A.

A. Jain, G. Healey, “A multiscale representation including opponent color features for texture recognition,” IEEE Trans. Image Process. 7, 124–128 (1998).
[CrossRef]

Kumar, S. R.

J. Huang, S. R. Kumar, M. Mitra, W. Zhu, R. Zabih, “Image indexing using color correlograms,” in Computer Vision and Pattern Recognition 97 (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 762–768.

Lim, J. S.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall Inc., Englewood Cliffs, N.J., 1990).

Livingstone, M.

M. Livingstone, D. Hubel, “Anatomy and Physiology of Q colour system in the primate visual cortex,” J. Physiol. 4, 309–356 (1984).

Marcelja, S.

Mitra, M.

J. Huang, S. R. Kumar, M. Mitra, W. Zhu, R. Zabih, “Image indexing using color correlograms,” in Computer Vision and Pattern Recognition 97 (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 762–768.

Persoon, E.

E. Persoon, K. S. Fu, “Shape discrimination using Fourier descriptors,” IEEE Trans. Syst. Man Cybern. SMC-7, 170–179 (1977).
[CrossRef]

Reye, D.

T. Caelli, D. Reye, “On the classification of image regions by color, texture, and shape,” Pattern Recogn. 26, 461–470 (1993).
[CrossRef]

Riseman, E. M.

M. Das, E. M. Riseman, B. A. Draper, “FOCUS: Searching for multi-colored objects in a diverse image,” in IEEE Conference on Computer Vision and Pattern Recognition 97 (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 756–761.

Sangwine, S. J.

A. L. Thornton, S. J. Sangwine, “Colour object location using complex coding in the frequency domain,” in Proceedings of the 5th International Conference on Image Processing and Its Applications (Heriot-Watt University, Edinburgh, UK, 1995), pp. 820–824.

Schwartz, E.

G. Bonmasser, E. Schwartz, “Lie groups, space-variant Fourier analysis and the exponential chirp transform,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 492–498.

Slater, D.

D. Slater, G. Healey, “The illuminant-invariant recognition of 3D objects using local color invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 206–210 (1996).
[CrossRef]

Thai, B.

B. Thai, G. Healey, “Modeling and classifying symmetries using a multiscale opponent color representation,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1224–1235 (1998).
[CrossRef]

Thornton, A. L.

A. L. Thornton, S. J. Sangwine, “Colour object location using complex coding in the frequency domain,” in Proceedings of the 5th International Conference on Image Processing and Its Applications (Heriot-Watt University, Edinburgh, UK, 1995), pp. 820–824.

Zabih, R.

J. Huang, S. R. Kumar, M. Mitra, W. Zhu, R. Zabih, “Image indexing using color correlograms,” in Computer Vision and Pattern Recognition 97 (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 762–768.

Zhu, W.

J. Huang, S. R. Kumar, M. Mitra, W. Zhu, R. Zabih, “Image indexing using color correlograms,” in Computer Vision and Pattern Recognition 97 (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 762–768.

Graphical Models Image Process. (1)

D. Carevic, T. Caelli, “Region-based coding of colour images using the Karhunen–Loève transform,” Graphical Models Image Process. 59, 27–38 (1997).
[CrossRef]

IEEE Trans. Image Process. (1)

A. Jain, G. Healey, “A multiscale representation including opponent color features for texture recognition,” IEEE Trans. Image Process. 7, 124–128 (1998).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

B. Thai, G. Healey, “Modeling and classifying symmetries using a multiscale opponent color representation,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1224–1235 (1998).
[CrossRef]

D. Slater, G. Healey, “The illuminant-invariant recognition of 3D objects using local color invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 206–210 (1996).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

E. Persoon, K. S. Fu, “Shape discrimination using Fourier descriptors,” IEEE Trans. Syst. Man Cybern. SMC-7, 170–179 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Physiol. (1)

M. Livingstone, D. Hubel, “Anatomy and Physiology of Q colour system in the primate visual cortex,” J. Physiol. 4, 309–356 (1984).

Pattern Recogn. (1)

T. Caelli, D. Reye, “On the classification of image regions by color, texture, and shape,” Pattern Recogn. 26, 461–470 (1993).
[CrossRef]

Vision Res. (1)

J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 847–856 (1980).
[CrossRef] [PubMed]

Other (10)

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978).

J. Davidoff, “Color perception,” in The Handbook of Brain Theory and Neural Networks, M. A. Arbib, ed. (MIT Press, Cambridge, Mass., 1995), pp. 210–215.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall Inc., Englewood Cliffs, N.J., 1990).

G. Bonmasser, E. Schwartz, “Lie groups, space-variant Fourier analysis and the exponential chirp transform,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 492–498.

A. L. Thornton, S. J. Sangwine, “Colour object location using complex coding in the frequency domain,” in Proceedings of the 5th International Conference on Image Processing and Its Applications (Heriot-Watt University, Edinburgh, UK, 1995), pp. 820–824.

R. Boylestad, Introductory Circuit Analysis, 6th ed. (Macmillan, New York, 1990).

CIE, “Colorimetry,” (CIE: International Commission on Illumination, Vienna, 1986; 2nd ed. (1996).

G. Healey, “Modeling color images for machine vision,” in Image Technology: Advances in Image Processing, Multimedia and Machine Vision, J. Sanz, ed. (Springer-Verlag, Berlin, 1995), pp. 129–146.

M. Das, E. M. Riseman, B. A. Draper, “FOCUS: Searching for multi-colored objects in a diverse image,” in IEEE Conference on Computer Vision and Pattern Recognition 97 (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 756–761.

J. Huang, S. R. Kumar, M. Mitra, W. Zhu, R. Zabih, “Image indexing using color correlograms,” in Computer Vision and Pattern Recognition 97 (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 762–768.

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Figures (9)

Fig. 1
Fig. 1

Phasor diagram representation for a complex-input DFT showing the two component phasors with different magnitudes and phases. The resultant path is shown on the right. (a) Real input, (b) complex input: same magnitude, different phase, (c) complex input, different magnitude and phase.

Fig. 2
Fig. 2

(a)–(c): Spatiochromatic frequency gratings defined at 2 cycles/image in x and y directions and circular color modulation paths in color space. The gratings differ only in the starting phase angles. (d)–(g) Identical spectral frequencies with linear paths in color space [see Fig. 1(b)] differing only in starting (phase) values. (h) Isotropic Gaussian-modulated spatiochromatic grating having red-center green-surround receptive field profiles. (i) Anisotropic version of (h).

Fig. 3
Fig. 3

Intensity and chromatic-based filter operations. (a)–(f ), Complex filtering: (a) input image with four colors at two orientations, (b) real low-pass filtering, (c) real high-pass filtering; (d) input image as in (a), (e) imaginary low-pass filtering, (f ) imaginary high-pass filtering. (g)–(n) Convolution-based filtering with oriented opponent filters: (g) input image as in (a), (h) vertically oriented red-center green-surround PSF; (i) convolution results: real components, and ( j) convolution results: imaginary components; (k) input image as in (a), (l) diagonally oriented red-center green-surround PSF; (m) convolution results: real components, (n) convolution results: imaginary components.

Fig. 4
Fig. 4

Correlation operations represented with phasor A and conjugate phasor B*. u and v correspond to real and imaginary axes. (a) Positively correlated, (b) negatively correlated, (c) positive conjugate (in the dual opponent-color dimension) correlation and (d) negative conjugate correlation.

Fig. 5
Fig. 5

Cross-correlation with the Mondrian test image. (a) Input image, (b) small section of input image, (c) pseudocolored cross-correlation result, (d) pseudocolored indicator for correlation (similarity) results with labels referring to Fig. 4.

Fig. 6
Fig. 6

(a) Similarity (real) and (b) conjugate similarity (imaginary) outputs for the cross-correlation shown in Fig. 5. (c) Similarity and conjugate similarity responses for a cross-section cutting horizontally through the center of the template position in the cross-correlation output images above.

Fig. 7
Fig. 7

(a) Input image containing a composite of four sections of different Australian bank notes, (b) template for cross-correlating with the image, (c) pseudocolored result [see Fig. 5(d) for color code] of cross-correlation, (d)–(f ) as in (a)–(c) with sufficient additive Gaussian noise to render the image structures invisible.

Fig. 8
Fig. 8

(a) Similarity (real) and (b) conjugate similarity (imaginary) outputs for the bank-note cross-correlation, without additive noise [Fig. 7(a)]. (c) Similarity and conjugate similarity responses for a cross-section cutting through the template position in the image.

Fig. 9
Fig. 9

(a) Similarity (real) and (b) conjugate similarity (imaginary) outputs for the bank-note cross-correlation, with additive noise [Fig. 7(d)]. (c) Similarity and conjugate similarity responses for a cross-section cutting through the template position in the image.

Tables (1)

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Table 1 Fourier Transform Symmetry Properties

Equations (21)

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

F(P, Q)=p=N/2+1N/2q=-N/2+1N/2f(p, q)×exp[-j2π(pP+qQ)/N],
U(P, Q)+jV(P, Q)
=p=-N/2+1N/2q=-N/2+1N/2[u(p, q)+jv(p, q)]×exp[-2πj(Pp+Qq)/N].
 u(p, q)+jv(p, q)=P=-N/2+1N/2Q=-N/2+1N/2×[U(P, Q)+jV(P, Q)]×exp[2πj(Pp+Qq)/N].
H(P, Q)=AH(P, Q)exp[-jθH(P, Q)].
G(P, Q)=AF(P, Q) AH(P, Q)exp[-jθF(P, Q)],
G(P, Q)=AF(P, Q)exp{-j[θF(P, Q)+θH(P, Q)]},
G(P, Q)=AF(P, Q) AH(P, Q)exp{-j[θF(P, Q)+θH(P, Q)]}.
G(P, Q)=AF(P, Q) AH(P, Q)exp-jθF(P, Q)+π2,
G(P, Q)=-j{AF(P, Q) AH(P, Q)exp[-jθF(P, Q)]}.
AB*=(a+jb)(c-jd)
=(ac+bd)+j(bc-ad),
AB cos θAB=AB*sin θAB*=(ac+bd),
AB sin θAB=AB*cos θAB*=(bc-ad),
θ=tan-1(AB*cos θAB*/AB*sin θAB*),
similarity(realcomponent):AB*sin θAB*,
conjugatesimilarity(imaginarycomponent):
AB*cos θAB*.
k=1nakbk2k=1nak2k=1nbk2,
|AB|  |A||B|
|A×B*|  |A||B*|

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