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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Acoustics, Speech, Signal Process. 1991 (1)

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.

Am. J. Phys. (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).

Ann. Hum. Genet. ,Lond. (1)

R. Penrose, �??The topology of ridge systems,�?? Ann. Hum. Genet.,Lond. 42, 435-444 (1979).
[CrossRef]

Appl. Opt. (5)

CVGIP (1)

M. Kass, and A. Witkin, �??Analyzing oriented patterns,�?? CVGIP 37, 362-385 (1987).

CVGIP-Image Understanding (1)

C. F. Shu, and R. C. Jain, �??Direct Estimation and Error Analysis For Oriented Patterns,�?? CVGIP-Image Understanding 58, 383-398 (1993).
[CrossRef]

DAGM Symposium, 2005 (1)

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

Electron. Lett. (1)

K. G. Larkin, �??Efficient Demodulator for Bandpass Sampled AM Signals,�?? Electron. Lett. 32, 101-102 (1996).
[CrossRef]

Fifth Int'l Wkshp Information Hiding2002 (1)

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.

Fringe 2001 (1)

K. G. Larkin, �??Natural demodulation of 2D fringe patterns,�?? Fringe'01 - The Fourth International Workshop on Automatic Processing of Fringe Patterns, Bremen, Germany, (2001), Elsevier, The Data Science Library, eds W. Juptner and W. Osten, ISBN : 2-84299-318-7

ICASSP '94 (1)

H. Knutsson, and M. Andersson, �??Robust N-Dimensional Orientation Estimation using Quadrature Filters and Tensor Whitening,�?? ICASSP '94, Adelaide, Australia, (1994) <a href="http://www.cvl.isy.liu.se/ScOut/Publications/PaperInfo/ka94.html">http://www.cvl.isy.liu.se/ScOut/Publications/PaperInfo/ka94.html</a>

ICIP 2001 (1)

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

IEEE Conf. Image Processing 1997 (1)

J. P. Havlicek, J. W. Havlicek, and A. C. Bovik, �??The Analytic Image,�?? IEEE International Conference on Image Processing, Santa Barbara, California, (1997), 446-449.

IEEE Int. Conf. ASSP 1991 (1)

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.

IEEE Trans. Commun. (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]

IEEE Trans. Image Process. (1)

G. Krieger, and C. Zetzche, �??Nonlinear image operators for evaluation of local intrinsic dimension,�?? IEEE Trans. Image Process. 5, 1026-1042 (1996).
[CrossRef] [PubMed]

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

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]

L. Hong, Y. F. Wan, and A. Jain, �??Fingerprint Image Enhancement - Algorithm and Performance Evaluation,�?? IEEE Trans. Pattern Anal. Mach. Intell. 20, 777-789 (1998).
[CrossRef]

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]

IEEE Trans. Sig. Process. (3)

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]

A. C. Bovik, and P. Maragos, �??Conditions for positivity of an energy operator,�?? IEEE Trans. Sig. Process. 42, 469-471 (1994).
[CrossRef]

IEEE Trans. Signal Process. (2)

C. Bovik, J. Havlicek, M. Desai, and D. Harding, �??Limits on discrete modulated signals,�?? IEEE Trans. Signal Process. 45, 867-879 (1997).
[CrossRef]

R. Hamila, J. Astola, M. A. Cheikh, M. Gabbouj, et al., �??Teager energy and the ambiguity function,�?? IEEE Trans. Signal Process. 47, 260-262 (1999).
[CrossRef]

Image Understanding and Man Machine inte (1)

J. G. Daugman, �??Image analysis and compact coding by oriented Gabor primitives,�?? Image Understanding and Man Machine interface, 19-30, (1987).

Image Vision Comput. (1)

B. Rieger, and Lucas J. van Vliet, �??A systematic approach to nD orientation representation,�?? Image Vision Comput. 22, 453-459 (2004).
[CrossRef]

J. Microsc. (1)

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, et al., �??Linear phase imaging using differential interference contrast microscopy,�?? J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (8)

Journal of the IEE (1)

D. Gabor, �??Theory of communications,�?? Journal of the IEE, 93, 429-457 (1947).

Journal of Vis. Commun. Image Represent. (1)

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]

Multidimens. Syst. Signal Process. (1)

J. P. Havlicek, D. S. Harding, and A. C. Bovik, �??Multicomponent Multidimensional Signals,�?? Multidimens. Syst. Signal Process. 9, 391-398 (1998).
[CrossRef]

NATO Advanced Study Institute (1)

H. M. Teager, and S. M. Teager, �??Evidence for nonlinear sound production mechanisms in the vocal tract,�?? Speech production and speech modelling, ed. Hardcastle, W. J., and Marchal, A. (France: NATO Advanced Study Institute, Series D, 1989) pp. 55.

Nonlinear Image Processing (1)

S. Thurnhofer, �??Two-dimensional Teager filters,�?? Nonlinear Image Processing, ed. Mitra, S. K., and Sicuranza, G. L. (San Diego: Academic Press, 2001) 167-202.
[CrossRef]

Opt. Express (1)

Opt. Soc. Am. A (1)

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]

Optik (1)

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

Pattern Rec. 26th DAGM Symposium 2004 (1)

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.

Pattern Recogn. Lett. (1)

S. Venkatesh, and R. Owens, �??On the classification of image features,�?? Pattern Recogn. Lett. 11, 339-349 (1990).
[CrossRef]

Proc IEEE Acoust. Speech, Signal 1990 (1)

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.

Proc IEEE Int. Conf. ASSP 1992 (1)

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.

Proc. of the Royal Society of London (1)

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]

Proc. SPIE (1)

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.

Scale Space and PDE Methods in Computer (1)

M. Felsberg, and U. Köthe, �??Get: The connection between monogenic scale-space and Gaussian derivatives,�?? Scale Space and PDE Methods in Computer Vision, ed. R. Kimmel, Sochen, N., and J. Weickert. LNCS, Springer, 2005) 3459: 192-203.
[CrossRef]

Signal Process. (1)

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]

Soc. Am. A (1)

P. Maragos, and A. C. Bovik, �??Image Demodulation Using Multidimensional Energy Separation�?? J. Opt. Soc. Am. A 12, pp.1867-1876 (1995).
[CrossRef]

Speckle Metrology 2003 (1)

J. M. Huntley, �??Fringe analysis today and tomorrow,�?? Speckle Metrology 2003, Trondheim, Norway, (2003), 167-174.

SPIE Conf. on Math. Imaging 2003 (1)

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.

SPIE Visual Communications 1992 (1)

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Figures (9)

Fig. 1.
Fig. 1.

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

Fig. 2.
Fig. 2.

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

Fig. 3.
Fig. 3.

Gradient based magnitude and orientation estimate

Fig. 4.
Fig. 4.

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

Fig. 5.
Fig. 5.

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

Fig. 6.
Fig. 6.

10dB test pattern

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

Fig. 8.
Fig. 8.

NIST digitized fingerprint image

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.

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.

Equations (47)

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f x y = b x y cos [ ψ x y ] .
f A x y = b x y exp [ 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 ) .
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 ( ) 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 ( ) G = exp ( i [ ϕ + π / 2 ] ) G , u = q cos ϕ , v = q sin ϕ .
i exp ( ) FT 1 2 π r 2 exp ( ) .
{ D m f s = b 0 exp ( ) 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 ) .
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 ( ) G .
( D α f s ) 2 f s . D α 2 f s = ( b 0 q α ) 2 exp ( 2 ) .
D general FT R ( q ) exp ( ) .
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 ) 2 i . D g { exp ( 2 ) } = D g { β } .

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