Abstract

Two new two dimensional (2-D) complex operators for estimating the energy and orientation of 2-D oriented patterns are proposed. The starting point for our work is a new 2-D extension of the Teager-Kaiser energy operator incorporating orientation estimation. The first new energy operator is based on partial derivatives and can be considered a local (point-based) estimator. Using a nonlocal (pseudo-differential) operator we derive a second and more general energy operator. A scale invariant nonlocal operator is derived from the recently proposed spiral phase quadrature (or Riesz) transform. The Teager-Kaiser energy operator and the phase congruency local energy are unified in a single equation for both 1-D and 2-D. Robust orientation estimation, important for isotropic demodulation of fringe patterns is demonstrated. Theoretical error analysis of the local operator is greatly simplified by a logarithmic formulation. Experimental results using the operators on noisy images are shown. In the presence of Gaussian additive noise both the local and nonlocal operators give improved performance when compared with a simple gradient based estimator.

©2005 Optical Society of America

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2005 (1)

M. Felsberg and E. Jonsson, “Energy tensors: Quadratic phase invariant image operators,” DAGM Symposium, Mustererkennung, Wien, (2005),

2004 (3)

M. Felsberg and G. H. Granlund, “Detection using Channel Clustering and the 2D Energy Tensor,” Pattern Recognition: 26th DAGM Symposium, Tübingen, Germany, (2004), pp. 103–110.

J. Bigun, T. Bigun, and K. Nilsson., “Recognition by symmetry derivatives and the generalized structure tensor,” IEEE Trans. Pattern Anal. Mach. Intell. 26, (2004).
[Crossref] [PubMed]

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, and N. I. Smith, et al., “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[Crossref] [PubMed]

2003 (3)

M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20, 925–934 (2003).
[Crossref]

B. Forster, T. Blu, and M. Unser., “A New Family of Complex Rotation-Covariant Multiresolution Bases in 2D,” Proceedings of the SPIE Conference on Mathematical Imaging: Wavelet Applications in Signal and Image Processing X, San Diego CA, USA, (2003), 475–479.

J. M. Huntley, “Fringe analysis today and tomorrow,” Speckle Metrology 2003, Trondheim, Norway, (2003), 167–174.

2002 (3)

M. Felsberg, “Low-Level Image Processing with the Structure Multivector,” PhD thesis. Christian-Albrechts-University of Kiel, 2002. http://www.isy.liu.se/~mfe/Diss.ps.gz

P. A. Fletcher and K. G. Larkin, “Direct Embedding and Detection of RST Invariant Watermarks,” IH2002, Fifth International Workshop on Information Hiding, Noordwijkerhout, The Netherlands, (2002), 129–144.

J. A. Quiroga, M. Servin, and F. Cuevas, “Modulo two pi fringe orientation angle estimation by phase unwrapping with a regularized phase tracking algorithm,” J. Opt. Soc. Am. A 19, 1524–1531 (2002).
[Crossref]

2001 (7)

P.-E. Danielsson, Q. Lin, and Q.-Z. Ye, “Efficient detection of second degree variations in 2D and 3D images,” Journal of Vis. Commun. Image Represent. 12, 255–305 (2001).
[Crossref]

J. P. Da Costa, F. Le Pouliquen, C. Germain, and P. Baylou, “New operators for optimized orientation estimation,” ICIP 2001, Thessaloniki, Greece, (2001).

K. G. Larkin, D. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns: I. General background to the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, pp.1862–1870 (2001).
[Crossref]

K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns: II. Stationary phase analysis of the spiral phase quadrature transform.,” J. Opt. Soc. Am. A 18, pp.1871–1881 (2001).
[Crossref]

K. G. Larkin, “Topics in Multi-dimensional Signal Demodulation,” PhD thesis. Dept. of Physical Optics, University of Sydney, 2001. http://setis.library.usyd.edu.au/~thesis/adt-NU/public/

K. G. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express 9, 236–253 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-5-236
[Crossref] [PubMed]

J. V. d. Weijer, L. J. v. Vliet, P. W. Verbeek, and M. v. Ginkel, “Curvature estimation in oriented patterns using curvilinear models applied to gradient vector fields,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 1035–1042 (2001).
[Crossref]

2000 (1)

M. Alonso and G. W. Forbes, “Measures of spread for periodic distributions and the associated uncertainty relations,” Am. J. Phys. 69, 340–347 (2000).

1999 (3)

P. Kovesi, “Image Features From Phase Congruency,” Videre: A Journal of Computer Vision Research, MIT Press 1, (1999). http://mitpress.mit.edu/e-journals/Videre/001/v13.html

R. Hamila, J. Astola, M. A. Cheikh, and M. Gabbouj, et al., “Teager energy and the ambiguity function,” IEEE Trans. Signal Process. 47, 260–262 (1999).
[Crossref]

X. Zhou, J. P. Baird, and J. F. Arnold, “Fringe-Orientation Estimation by use of a Gaussian Gradient Filter and Neighboring-Direction Averaging,” Appl. Opt. 38, 795–804 (1999).
[Crossref]

1998 (5)

1997 (2)

A. C. Bovik, J. Havlicek, M. Desai, and D. Harding, “Limits on discrete modulated signals,” IEEE Trans. Signal Process.  45, 867–879 (1997).
[Crossref]

B. Jahne, Practical handbook on Image processing for Scientific applications, CRC Press, Boca Raton, Florida, 1997.

1996 (5)

G. Krieger and C. Zetzche, “Nonlinear image operators for evaluation of local intrinsic dimension,” IEEE Trans. Image Process. 5, 1026–1042 (1996).
[Crossref] [PubMed]

J. P. Havlicek, “AM-FM Image models,” PhD thesis. University of Texas, 1996. http://hotnsour.ou.edu/joebob/PdfPubs/JPHavlicekDiss.pdf

B. Strobel, “Processing of Interferometric Phase Maps As Complex-Valued Phasor Images,” Appl. Opt. 35, 2192–2198 (1996).
[Crossref] [PubMed]

K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
[Crossref]

K. G. Larkin, “Efficient Demodulator for Bandpass Sampled AM Signals,” Electron. Lett. 32, 101–102 (1996).
[Crossref]

1995 (3)

1994 (4)

H. Knutsson and M. Andersson, “Robust N-Dimensional Orientation Estimation using Quadrature Filters and Tensor Whitening,” ICASSP ′94, Adelaide, Australia, (1994). http://www.cvl.isy.liu.se/ScOut/Publications/PaperInfo/ka94.html

Q. Yu and K. Andresen, “Fringe-orientation maps and fringe skeleton extraction by the two-dimensional derivative-sign binary-fringe method,” Appl. Opt. 33, 6873–6878 (1994).
[Crossref] [PubMed]

A. C. Bovik and P. Maragos, “Conditions for positivity of an energy operator,” IEEE Trans. Sig. Process. 42, 469–471 (1994).
[Crossref]

A. Potamianos and P. Maragos, “A Comparison of the Energy Operator and the Hilbert Transform Approach to Signal and Speech Demodulation,” Signal Process. 37, 95–120 (1994).
[Crossref]

1993 (4)

K. Andresen and Q. Yu, “Robust Phase Unwrapping By Spin Filtering Combined With a Phase Direction Map,” Optik 94, pp.145–149 (1993).

C. F. Shu and R. C. Jain, “Direct Estimation and Error Analysis For Oriented Patterns,” CVGIP-Image Understanding 58, 383–398 (1993).
[Crossref]

P. Maragos, J. F. Kaiser, and T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Sig. Process.  41, 1532–1550 (1993).
[Crossref]

A. C. Bovik, P. Maragos, and T. F. Quatieri, “AM-FM energy detection and separation in noise using multiband energy operators,” IEEE Trans. Sig. Process. 41, 3245–3265 (1993).
[Crossref]

1992 (2)

P. Maragos, A. C. Bovik, and T. F. Quatieri, “A multidimensional energy operator for image processing,” SPIE Conference on Visual Communications and Image Processing, Boston, MA, (1992), pp. 177–186.

P. Maragos, T. F. Quatieri, and J. F. Kaiser, “On separating amplitude from frequency modulations using energy operators,” Proc IEEE Int. Conf. ASSP, San Francisco, CA, (1992), 1–4.

1991 (3)

T.-H. Yu and S. K. Mitra, “A novel nonlinear filter for image enhancement,” Image Processing algorithms and Techniques II, Proc. SPIE 1452 ,(1991), pp. 303–309.

S. K. Mitra, H. Li, I-S. Lin, and T-H. Yu, “A new class of nonlinear filters for image enhancement,” Int. Conf. Acoustics, Speech, and Signal Processing, Toronto, Canada, (1991), pp. 2525–2528.

P. Maragos, T. F. Quatieri, and J. F. Kaiser, “Speech nonlinearities, modulations, and energy operators,” Proc IEEE Int. Conf. ASSP, Toronto, Canada, (1991), 421–424.

1990 (2)

J. F. Kaiser, “On a simple algorithm to calculate the 'energy' of a signal,” Proc IEEE Int. Conf. Acoust. Speech, Signal Processing, Albuquerque, NM, (1990), pp. 381–384.

S. Venkatesh and R. Owens, “On the classification of image features,” Pattern Recogn. Lett. 11, 339–349 (1990).
[Crossref]

1988 (2)

J. J. Koenderink and W. Richards, “Two-dimensional curvature operators,” J. Opt. Soc. Am. A 5, 1136–1141 (1988).
[Crossref]

M. C. Morrone and D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proceedings of the Royal Society of London, B 235, 221–245 (1988).
[Crossref]

1987 (2)

M. Kass and A. Witkin, “Analyzing oriented patterns,” CVGIP 37, 362–385 (1987).

J. G. Daugman, “Image analysis and compact coding by oriented Gabor primitives,” Image Understanding and Man Machine interface, 19–30, (1987).

1985 (1)

1983 (1)

H. Knutsson, R. Wilson, and G. H. Granlund, “Anisotropic Non-Stationary Image Estimation and its Applications ┅ Part I: Restoration of Noisy Images,” IEEE Trans. Commun. 31, 388╌397 (1983).
[Crossref]

1982 (1)

H. Knutsson, “Filtering and Reconstruction in Image Processing,” PhD. Linkoping University, 1982.

1979 (1)

R. Penrose, “The topology of ridge systems,” Ann. Hum. Genet.,Lond. 42, 435–444 (1979).
[Crossref]

1947 (1)

D. Gabor, “Theory of communications,” Journal of the IEE,  93, 429–457 (1947).

Adelson, E. H.

Alonso, M.

M. Alonso and G. W. Forbes, “Measures of spread for periodic distributions and the associated uncertainty relations,” Am. J. Phys. 69, 340–347 (2000).

Andersson, M.

H. Knutsson and M. Andersson, “Robust N-Dimensional Orientation Estimation using Quadrature Filters and Tensor Whitening,” ICASSP ′94, Adelaide, Australia, (1994). http://www.cvl.isy.liu.se/ScOut/Publications/PaperInfo/ka94.html

Andresen, K.

Q. Yu and K. Andresen, “Fringe-orientation maps and fringe skeleton extraction by the two-dimensional derivative-sign binary-fringe method,” Appl. Opt. 33, 6873–6878 (1994).
[Crossref] [PubMed]

K. Andresen and Q. Yu, “Robust Phase Unwrapping By Spin Filtering Combined With a Phase Direction Map,” Optik 94, pp.145–149 (1993).

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, and N. I. Smith, et al., “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[Crossref] [PubMed]

Arnold, J. F.

Astola, J.

R. Hamila, J. Astola, M. A. Cheikh, and M. Gabbouj, et al., “Teager energy and the ambiguity function,” IEEE Trans. Signal Process. 47, 260–262 (1999).
[Crossref]

Baird, J. P.

Baylou, P.

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Figures (9)

Fig. 1.
Fig. 1. (a) Sample fringe pattern for analysis and (b) its Fourier magnitude

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Fig. 2.
Fig. 2. (a) Ideal 2-D energy operator magnitude and (b) ideal orientation phase for test pattern

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Fig. 3.
Fig. 3. Gradient based magnitude and orientation estimate

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Fig. 4.
Fig. 4. (a) Differential 2-D energy operator magnitude (b) differential 2-D energy operator orientation phase

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Fig. 5.
Fig. 5. (a) Spiral-phase 2-D energy operator magnitude and (b) Spiral-phase 2-D energy operator phase

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Fig. 6.
Fig. 6. 10dB test pattern

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Fig. 7.
Fig. 7. Estimates from 10dB test pattern: (a) and (b) magnitude and phase of the gradient estimator, (c) and (d) magnitude and phase of the differential energy operator, and (e) and (f) magnitude and phase of the spiral phase (Riesz) estimator

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Fig. 8.
Fig. 8. NIST digitized fingerprint image

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Fig. 9.
Fig. 9. Fingerprint magnitude and orientation phase estimates. Note that the phase is displayed as a pseudo-color map between -π and +π, with the color scale shown (blue is zero). (a) and (b) are the magnitude and phase of the gradient squared operator, (c) and (d) are the magnitude and phase of the differential (local) energy operator, and (e) and (f) are the magnitude and phase of the spiral phase (nonlocal) energy operator.

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Tables (1)

Tables Icon

Table 1. Standard deviation of the 2β orientation error for various estimators and SNR ratios. Note that β errors are half the double –angle errors and are given in brackets.

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Equations (47)

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f x y = b x y cos [ ψ x y ] .
f A x y = b x y exp [ iψ x y ] .
β x y = Arg ( ∂ ψ ∂ x + i ∂ ψ ∂ y ) .
b x y cos [ ψ x y ] ≡ b x y cos [ − ψ x y ] .
β est = atan ( f 01 f 10 ) = atan ( b ψ 01 sin ( ψ ) − b 01 cos ( ψ ) b ψ 10 sin ( ψ ) − b 10 cos ( ψ ) )
β est = atan ( ∓ b 01 ∓ b 10 ) .
E { g ( x ) } = ( dg dx ) 2 − g . ( d 2 g d x 2 ) .
E { g ( x ) } ≈ [ − ωb sin ( ωx ) ] 2 − b cos ( ωx ) [ − ω 2 b cos ( ωx ) ] = ω 2 b 2 .
∣ ∇ { g ( x ) } ∣ 2 ≈ ( ωb ) 2 sin 2 ( ωx ) .
E { g ( x ) } = D { g } . D { g } − g . D { D { g } } ≡ ( D { g } ) 2 − g . D 2 { g } .
D g = ∂ ∂ x + i ∂ ∂ y ,
E { f x y } = [ ( ∂ ∂ x + i ∂ ∂ y ) f x y ] 2 − f x y ( ∂ ∂ x + i ∂ ∂ y ) ( ∂ ∂ x + i ∂ ∂ y ) f x y .
E { f x y } = [ ( ∂ f ∂ x ) 2 − ( ∂ f ∂ y ) 2 − f ( ∂ 2 f ∂ x 2 − ∂ 2 f ∂ y 2 ) ] + i [ 2 ( ∂ f ∂ x ) ( ∂ f ∂ y ) − f ( ∂ 2 f ∂ x ∂ y + ∂ 2 f ∂ y ∂ x ) ] .
f simple x y = f s = b 0 cos [ ψ s ] = b 0 cos [ 2 π ( u 0 x + v 0 y ) + ψ 00 ] ,
where { u 0 = q 0 cos β , v 0 = q 0 sin β } .
{ ( D g f s ) 2 = ( − 2 π b 0 ( u 0 + i v 0 ) sin [ ψ S ] ) 2 f s D g 2 f s = − ( 2 π b 0 ( u 0 + i v 0 ) ) 2 cos 2 [ ψ S ] .
E { f } = ( D f s ) 2 − f . D 2 f s = ( 2 π b 0 ) 2 ( u 0 + i v 0 ) 2 ( sin 2 [ ψ S ] + cos 2 [ ψ S ] ) .
( D g f s ) 2 − f s . D g 2 f s = ( 2 π b 0 q 0 ) 2 exp ( 2 iβ ) .
E { f R + i f I } = define ( D g f R ) 2 − f R . D g 2 f R + ( D g f I ) 2 − f I . D g 2 f I .
E { f ( x ) } = ∣ df dx ∣ 2 − 1 2 ( f * d 2 f d x 2 + f d 2 f * d x 2 )
RHS ≡ ( d f R dx ) 2 − f R d 2 f R d x 2 + ( d f I dx ) 2 − f I d 2 f I d x 2 .
− d 2 d x 2 log g = ( dg dx ) 2 − g . d 2 g d x 2 g 2 = − E { g } g 2 .
{ ∇ 2 log g = 0 ⇒ log g is harmonic ∇ 2 log g < 0 ⇒ subharmonicity ⇒ E { g } g 2 > 0 ⇒ E { g } > 0 .
− 1 2 g 2 D 2 { log g 2 } = ( D { g } ) 2 − g . D 2 { g } = E { g } .
E { g } = [ ∇ g ] [ ∇ g ] T − g [ H g ] = − 1 2 g 2 [ H log ( g 2 ) ] .
G u v = F { g x y } = ∫ − ∞ + ∞ ∫ − ∞ + ∞ g x y exp [ − 2 πi ( ux + vy ) ] dxdy ,
F { D { g } } = 2 πi ( u + iv ) G = 2 πiq exp ( iϕ ) G , q 2 = u 2 + v 2 , tan ϕ = v / u .
{ F { D m { g } } = i ( u + iv ) ∣ u + iv ∣ G ≡ i sgn ( u + iv ) G = i exp ( iϕ ) G = exp ( i [ ϕ + π / 2 ] ) G , u = q cos ϕ , v = q sin ϕ .
i exp ( iϕ ) ↔ FT 1 2 π r 2 exp ( iθ ) .
{ D m f s = − b 0 exp ( iβ ) sin [ ψ S ] D m 2 f s = − b 0 exp ( i 2 β ) cos [ ψ S ] .
( D m f s ) 2 − f s . D m 2 f s = ( b 0 ) 2 exp ( 2 iβ ) .
g 2 + g ̂ 2 = − g g ̂ ̂ + g ̂ 2 ≡ ( S { g } ) 2 − g . S 2 { g } .
F { D α { g } } = ( u + iv ) ∣ u + iv ∣ 1 − α G = q α exp ( iϕ ) G .
( D α f s ) 2 − f s . D α 2 f s = ( b 0 q α ) 2 exp ( 2 iβ ) .
D general ↔ FT ∣ R ( q ) ∣ exp ( iϕ ) .
f 2 ( m ) − f ( m + 1 ) f ( m − 1 ) .
[ f ( m + 1 ) − f ( m − 1 ) ] 2 − [ f ( m + 2 ) − f ( m ) ] [ f ( m ) − f ( m − 2 ) ] .
{ D 1 { } ↔ FT ( u + iv ) M 1 u v exp [ i P 1 u v ] D 2 { } ↔ FT ( u + iv ) 2 M 2 u v exp [ i P 2 u v ] .
( D 1 f s ) 2 − f . D 2 f s = ( 2 π b 0 ) 2 ( u 0 + i v 0 ) 2 [ M 1 2 exp ( 2 i P 1 ) sin 2 ψ + M 2 exp ( i P 2 ) cos 2 ψ ] .
{ ( u + iv ) M 0 ↔ FT D 1 { } ( u + iv ) 2 M 0 2 ↔ FT D 2 { } .
exp ( i ψ test ) = exp ( 64 i [ cos ( 2 πx / 256 ) + cos ( 2 πx / 256 ) ] ) .
f test x y = int ( 127.5 [ 1 + cos ( 64 i [ cos ( 2 πx / 256 ) + cos ( 2 πx / 256 ) ] ) ] ) .
β test x y = Arg [ ∂ ψ test ∂ x + i ∂ ψ test ∂ y ] = arctan [ − π 2 sin ( 2 πy 256 ) / − π 2 sin ( 2 πx 256 ) ] .
f x y = e ρ x y cos [ ψ x y ] .
− 1 2 ( f ) 2 D g 2 log ( f ) 2 = e 2 ρ ( D g ψ ) 2 { 1 − cos 2 ψ ( D g ψ ) 2 D g 2 ρ + sin ψ cos ψ ( D g ψ ) 2 D g 2 ψ } .
SNR = 20 log 10 [ SDEV ( pattern ) ) SDEV ( noise ) ] .
exp ( − 2 iβ ) 2 i . D g { exp ( 2 iβ ) } = D g { β } .

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