Abstract

The existence of a special second-order mechanism in the human visual system, able to demodulate the envelope of visual stimuli, suggests that spatial information contained in the image envelope may be perceptually relevant. The Riesz transform, a natural isotropic extension of the Hilbert transform to multidimensional signals, was used here to demodulate band-pass filtered images of well-known visual illusions of length, size, direction, and shape. We show that the local amplitude of the monogenic signal or envelope of each illusion image conveys second-order information related to image holistic spatial structure, whereas the local phase component conveys information about the spatial features. Further low-pass filtering of the illusion image envelopes creates physical distortions that correspond to the subjective distortions perceived in the illusory images. Therefore the envelope seems to be the image component that physically carries the spatial information about these illusions. This result contradicts the popular belief that the relevant spatial information to perceive geometrical-optical illusions is conveyed only by the lower spatial frequencies present in their Fourier spectrum.

© 2010 Optical Society of America

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2007 (1)

V. Sierra-Vázquez and I. Serrano-Pedraza, “Single-band amplitude demodulation of Müller-Lyer illusion images,” Spanish J. Psychol. 10, 3-19 (2007).

2005 (2)

K. G. Larkin, “Uniform estimation of orientation using local and nonlocal 2-D energy operators,” Opt. Express 13, 8097-8121 (2005).
[CrossRef] [PubMed]

G. S. Hesse and M. A. Georgeson, “Edges and bars: where do people see features in 1-D images?” Vision Res. 45, 507-525 (2005).
[CrossRef]

2004 (2)

M. Felbsberg and G. Sommer, “The monogenic scale-space: A unifying approach to phase-based image processing in scale-space,” J. Math. Imaging Vision 21, 5-26 (2004).
[CrossRef]

A. P. Johnson and C. L. Baker, “First- and second-order information in natural images: a filter-based approach to image statistics,” J. Opt. Soc. Am. A 21, 913-925 (2004).
[CrossRef]

2003 (1)

A. J. Schofield and M. A. Georgeson, “Sensitivity to contrast modulation: the spatial frequency dependence of second-order vision,” Vision Res. 43, 243-259 (2003).
[CrossRef] [PubMed]

2001 (4)

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

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136-3144 (2001).
[CrossRef]

A. Oliva and A. Torralba, “Modeling the shape of the scene: a holistic representation of the spatial envelope,” Int. J. Comput. Vis. 42, 145-175 (2001).
[CrossRef]

C. Chubb, L. Olzak, and A. Derrington, “Second-order processes in vision: introduction,” J. Opt. Soc. Am. A 18, 2175-2178 (2001).
[CrossRef]

2000 (2)

B. C. Skottun, “Amplitude and phase in the Müller-Lyer illusion,” Perception 29, 201-209 (2000).
[CrossRef] [PubMed]

J. A. Solomon, “Channel selection with non-white-noise mask,” J. Opt. Soc. Am. A 17, 986-993 (2000).
[CrossRef]

1999 (2)

R. Müller and J. Marquard, “Die Hilbertransformation und ihre Verallgemeinerung in Optik und Bildverarbeitung,” Optik (Stuttgart) 110, 99-109 (1999).

I. Mareschal and C. L. Baker, “Cortical processing of second-order motion,” 16, 527-540 (1999).

1995 (2)

1993 (1)

Y-X. Zhou and C. L. Baker, “A processing stream in mammalian visual cortex neurons for non-Fourier responses,” Science 261, 98-101 (1993).
[CrossRef] [PubMed]

1992 (1)

A. C. Bovik, N. Gopal, T. Emmoth, and A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691-712 (1992).
[CrossRef]

1991 (1)

M. A. García-Pérez, “Visual phenomena without low spatial frequencies: a closer look,” Vision Res. 31, 1647-1653 (1991).
[CrossRef] [PubMed]

1988 (2)

J. R. Bergen and E. H. Adelson, “Early vision and texture perception,” Nature 333, 363-364 (1988).
[CrossRef] [PubMed]

M. C. Morrone and D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London, Ser. B 235, 221-245 (1988).
[CrossRef]

1986 (1)

M. Carrasco, J. G. Figueroa, and J. D. Willen, “A test of the spatial-frequency explanation of the Müller-Lyer illusion,” Perception 15, 553-562 (1986).
[CrossRef] [PubMed]

1984 (1)

C. R. Carlson, J. R. Moeller, and C. H. Anderson, “Visual illusions without low spatial frequencies,” Vision Res. 24, 1407-1413 (1984).
[CrossRef] [PubMed]

1979 (1)

A. P. Ginsburg and P. W. Evans, “Predicting visual illusions from filtered images based upon biological data,” J. Opt. Soc. Am. 69, 1443 (1979).

1975 (1)

D. H. Kelly, “Spatial frequency selectivity in the retina,” Vision Res. 15, 665-672 (1975).
[CrossRef] [PubMed]

1969 (1)

S. Coren, “The influence of optical aberrations on the magnitude of the Poggendorf illusion,” Percept. Psychophys. 6, 185-186 (1969).
[CrossRef]

1962 (1)

D. H. Hubel and T. N. Wiesel, “Receptive fields, binocular interaction, and functional architecture in the cat's visual cortex,” J. Physiol. (London) 160, 106-154 (1962).

1956 (1)

A. P. Calderón and A. Zygmund, “On singular integrals,” Am. J. Math. 78, 289-309 (1956).
[CrossRef]

1953 (1)

J. Horváth, “Sur les functions conjugées à plusieurs variables,” Indagationes Matematicae 15, 17-29 (1953).

1952 (1)

A. P. Calderón and A. Zygmund, “On the existence of certain singular integrals,” Acta Math. 88, 85-139 (1952).
[CrossRef]

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. (part III) 93, 429-457 (1946).

1928 (1)

M. Ponzo, “Urteilstäuschungen über Mengen,” Arch. Ges. Psychol. 65, 129-162 (1928).

1925 (1)

W. Ehrenstein, “Versuche über die Beziehungen zwischen Bewegungs- und Gestaltwahrnehmung,” Z. Psychol. 95, 305-352 (1925).

1896 (1)

F. C. Müller-Lyer, “Zur Lehre von den optischen Täuschungen. Über Kontrast und Konfluxion,” Z. Psychol. 9, 1-16 (1896).

1892 (2)

F. Brentano, “Über ein optisches Paradoxon,” Z. Psychol. 3, 349-358 (1892).

J. L. R. Delboeuf, “Sur une nouvelle illusion d'optique,” Bull. Acad. Roy. Belg. 24, 545-558 (1892).

1889 (1)

F. C. Müller-Lyer, “Optische Urtheilstäuschungen,” Archiv. Anat. Phys., Phys. Abthlg. Suppl. 2, 263-270 (1889).

1860 (1)

F. Zöllner, “Über eine neue Art von Pseudoskopie und ihre Beziehungen zu den von Plateau und Oppel beschriebenen Bewegungsphäenomenen,” Ann. Phys. 186, 500-525 (1860).

Adelson, E. H.

J. R. Bergen and E. H. Adelson, “Early vision and texture perception,” Nature 333, 363-364 (1988).
[CrossRef] [PubMed]

Anderson, C. H.

C. R. Carlson, J. R. Moeller, and C. H. Anderson, “Visual illusions without low spatial frequencies,” Vision Res. 24, 1407-1413 (1984).
[CrossRef] [PubMed]

Baker, C. L.

A. P. Johnson and C. L. Baker, “First- and second-order information in natural images: a filter-based approach to image statistics,” J. Opt. Soc. Am. A 21, 913-925 (2004).
[CrossRef]

I. Mareschal and C. L. Baker, “Cortical processing of second-order motion,” 16, 527-540 (1999).

Y-X. Zhou and C. L. Baker, “A processing stream in mammalian visual cortex neurons for non-Fourier responses,” Science 261, 98-101 (1993).
[CrossRef] [PubMed]

Bergen, J. R.

J. R. Bergen and E. H. Adelson, “Early vision and texture perception,” Nature 333, 363-364 (1988).
[CrossRef] [PubMed]

Blake, R.

R. Sekuler and R. Blake, Perception, 3rd ed. (McGraw-Hill, 1994).
[PubMed]

Bone, D. J.

Boring, E. G.

E. G. Boring, Sensation and Perception in the History of Experimental Psychology (Appleton-Century-Crofts, 1942).

Bovik, A. C.

P. Maragos and A. C. Bovik, “Image demodulation using multidimensional energy separation,” J. Opt. Soc. Am. A 12, 1867-1876 (1995).
[CrossRef]

A. C. Bovik, N. Gopal, T. Emmoth, and A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691-712 (1992).
[CrossRef]

J. P. Havlicek and A. C. Bovik, “Image modulation models,” in Handbook of Image and Video Processing, A.C.Bovik, ed. (Academic, 2000), pp. 313-324.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1978).

Brady, M.

M. Brady, Advanced Transform Methods (Department of Engineering Science, Oxford University, 2006). Available at http://www.robots.ox.ac.uk/~jmb/transformlectures.html.

Brentano, F.

F. Brentano, “Über ein optisches Paradoxon,” Z. Psychol. 3, 349-358 (1892).

Bülow, T.

T. Bülow and G. Sommer, “A novel approach to the 2D analytic signal,” in 8th Conference on Computer Analysis of Images and Patterns, Ljubljana, F.Solina and A.Leonardis, eds., LNCS Vol. 1689 (Springer-Verlag, 1999), pp. 25-32.
[CrossRef]

T. Bülow, D. Pallek, and G. Sommer, “Riesz transform for the isotropic estimation of the local phase moiré interferograms,” in 22nd DAGM Symposium für Mustererkennung, Kiel (2000), G.Sommer, N.Krüger, and C.Perwass, eds. (Springer-Verlag, 2000), pp. 333-340.

Burr, D. C.

M. C. Morrone and D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London, Ser. B 235, 221-245 (1988).
[CrossRef]

Calderón, A. P.

A. P. Calderón and A. Zygmund, “On singular integrals,” Am. J. Math. 78, 289-309 (1956).
[CrossRef]

A. P. Calderón and A. Zygmund, “On the existence of certain singular integrals,” Acta Math. 88, 85-139 (1952).
[CrossRef]

Carlson, C. R.

C. R. Carlson, J. R. Moeller, and C. H. Anderson, “Visual illusions without low spatial frequencies,” Vision Res. 24, 1407-1413 (1984).
[CrossRef] [PubMed]

Carrasco, M.

M. Carrasco, J. G. Figueroa, and J. D. Willen, “A test of the spatial-frequency explanation of the Müller-Lyer illusion,” Perception 15, 553-562 (1986).
[CrossRef] [PubMed]

Chubb, C.

C. Chubb, L. Olzak, and A. Derrington, “Second-order processes in vision: introduction,” J. Opt. Soc. Am. A 18, 2175-2178 (2001).
[CrossRef]

C. Chubb and M. S. Landy, “Orthogonal distribution analysis: a new approach to the study of texture perception,” in Computational Models of Visual Processing, M.S.Landy and J.A.Movshon, eds. (MIT Press, 1991), pp. 291-301.

Coren, S.

S. Coren, “The influence of optical aberrations on the magnitude of the Poggendorf illusion,” Percept. Psychophys. 6, 185-186 (1969).
[CrossRef]

S. Coren and J. S. Girgus, “Visual illusions,” in Handbook of Sensory Physiololgy. Vol. VIII. Perception, R.Held, H.W.Leibowitz, and H.-L.Teuber, eds. (Springer-Verlag, 1978), pp. 551-568.

S. Coren and J. S. Girgus, Seeing Is Deceiving. The Psychology of Visual Illusions (Lawrence Erlbaum, 1978).

Daugman, J. G.

J. G. Daugman and C. J. Downing, “Demodulation, predictive coding, and spatial vision,” J. Opt. Soc. Am. A 12, 641-660 (1995).
[CrossRef]

J. G. Daugman and C. J. Downing, “Demodulation: a new approach to texture vision,” in Spatial Vision in Human and Robots, L.H.Harris and M.Jenkin, eds. (Cambridge U. Press, 1993), pp. 63-87.

Delboeuf, J. L. R.

J. L. R. Delboeuf, “Sur une nouvelle illusion d'optique,” Bull. Acad. Roy. Belg. 24, 545-558 (1892).

Derrington, A.

Downing, C. J.

J. G. Daugman and C. J. Downing, “Demodulation, predictive coding, and spatial vision,” J. Opt. Soc. Am. A 12, 641-660 (1995).
[CrossRef]

J. G. Daugman and C. J. Downing, “Demodulation: a new approach to texture vision,” in Spatial Vision in Human and Robots, L.H.Harris and M.Jenkin, eds. (Cambridge U. Press, 1993), pp. 63-87.

Ehrenstein, W.

W. Ehrenstein, “Versuche über die Beziehungen zwischen Bewegungs- und Gestaltwahrnehmung,” Z. Psychol. 95, 305-352 (1925).

Emmoth, T.

A. C. Bovik, N. Gopal, T. Emmoth, and A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691-712 (1992).
[CrossRef]

Evans, P. W.

A. P. Ginsburg and P. W. Evans, “Predicting visual illusions from filtered images based upon biological data,” J. Opt. Soc. Am. 69, 1443 (1979).

Felbsberg, M.

M. Felbsberg and G. Sommer, “The monogenic scale-space: A unifying approach to phase-based image processing in scale-space,” J. Math. Imaging Vision 21, 5-26 (2004).
[CrossRef]

Felsberg, M.

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136-3144 (2001).
[CrossRef]

M. Felsberg and G. Sommer, “The multidimensional isotropic generalization of quadrature filters in geometric algebra,” in Lecture Notes in Computer Science, Vol 1888, G.Sommer and Y.Zeevi, eds. (Springer-Verlag, 2000), pp. 175-185.
[CrossRef]

M. Felsberg and G. Sommer, “Structure multivector for local analysis of images,” in Multi-image Analysis, Proceedings of Daugstuhl Workshop on Theoretical Foundations of Computer Vision, LNCS 2032, R.Klette, T.Huang, and G.Gimel'farb, eds. (Springer-Verlag, 2001), pp. 93-104.

M. Felsberg and U. Köthe, “GET: The connection between monogenic scale-space and Gaussian derivatives,” in Scale-Space and PDE Methods in Computer Vision, Proceedings of Scale-Space 2005, LNCS Vol. 3459, R.Kimmel, N.Sochen, J.Weickert, eds. (Springer-Verlag, 2005), pp. 192-203.
[CrossRef]

M. Felsberg, “Low-level image processing with the structure multivector,” Ph.D. thesis, Tech. Rep. #0203 (Institute of Computer Science and Applied Mathematics, Christian-Albrechts Universität, Kiel, Germany, 2002), available at http://www.informatik.uni-kiel.de/reports/2002/0203.html.

M. Felsberg, “Disparity from monogenic phase,” in DAGM 2002, LNCS Vol. 2449, L.Van Gool, ed. (Springer-Verlag, Berlin, 2002), pp. 248-256.

M. Felsberg, “Optical flow estimation from monogenic phase,” in Complex Motion, First International Workshop, IWCM 2004, LNCS 34517, B.Jähne, R.Mester, E.Barth, and H.Scharr, eds. (Springer-Verlag, 2007), pp. 1-13.

M. Felsberg and G. Sommer, “A new extension of linear signal processing for estimating local properties and detecting features,” in 22nd Symposium für Mustererkennung, DAGM, Kiel, G.Sommer, N.Krüger, and C.Perwass, eds. (Springer-Verlag, 2000), pp. 195-202.

Figueroa, J. G.

M. Carrasco, J. G. Figueroa, and J. D. Willen, “A test of the spatial-frequency explanation of the Müller-Lyer illusion,” Perception 15, 553-562 (1986).
[CrossRef] [PubMed]

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. (part III) 93, 429-457 (1946).

García-Pérez, M. A.

M. A. García-Pérez, “Visual phenomena without low spatial frequencies: a closer look,” Vision Res. 31, 1647-1653 (1991).
[CrossRef] [PubMed]

Georgeson, M. A.

G. S. Hesse and M. A. Georgeson, “Edges and bars: where do people see features in 1-D images?” Vision Res. 45, 507-525 (2005).
[CrossRef]

A. J. Schofield and M. A. Georgeson, “Sensitivity to contrast modulation: the spatial frequency dependence of second-order vision,” Vision Res. 43, 243-259 (2003).
[CrossRef] [PubMed]

Ginsburg, A. P.

A. P. Ginsburg and P. W. Evans, “Predicting visual illusions from filtered images based upon biological data,” J. Opt. Soc. Am. 69, 1443 (1979).

A. P. Ginsburg, Visual Information Processing Based on Spatial Filters Constrained by Biological Data. Tech. Rep. AMRL-TR-78-129 (Aerospace Medical Research Laboratory, Wright Patterson Air Force Base, Dayton, Ohio, USA, 1978).

A. P. Ginsburg, “Spatial filtering and visual form perception,” in Handbook of Perception and Human Performance. Vol. II, K.R.Boff, L.Kaufman, and J.P.Thomas, eds. (Wiley, 1986), Chap. 34.

Girgus, J. S.

S. Coren and J. S. Girgus, Seeing Is Deceiving. The Psychology of Visual Illusions (Lawrence Erlbaum, 1978).

S. Coren and J. S. Girgus, “Visual illusions,” in Handbook of Sensory Physiololgy. Vol. VIII. Perception, R.Held, H.W.Leibowitz, and H.-L.Teuber, eds. (Springer-Verlag, 1978), pp. 551-568.

Goldstein, E. B.

E. B. Goldstein, Sensation and Perception, 2nd ed. (Wadsworth, 1984).

Gopal, N.

A. C. Bovik, N. Gopal, T. Emmoth, and A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691-712 (1992).
[CrossRef]

Grandlund, G. H.

G. H. Grandlund and H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, 1995).

Gregory, R.

R. Gregory, Eye and Brain. The Psychology of Seeing, 4th ed. (Oxford U. Press, 1994).

Hahn, S. L.

S. L. Hahn, Hilbert Transform in Signal Processing (Artech House, 1996).

Havlicek, J. P.

J. P. Havlicek, “AM-FM image models,” Ph.D. dissertation (The University of Texas at Austin, 1996).

J. P. Havlicek and A. C. Bovik, “Image modulation models,” in Handbook of Image and Video Processing, A.C.Bovik, ed. (Academic, 2000), pp. 313-324.

Hering, E.

Robinson quotes that Hering's illusion appeared in E. Hering, Beitrage zur Psychology Vol. 1 (Engelman, Leipzig, 1891). Unfortunately, we could not obtain the original source.

Hesse, G. S.

G. S. Hesse and M. A. Georgeson, “Edges and bars: where do people see features in 1-D images?” Vision Res. 45, 507-525 (2005).
[CrossRef]

Horváth, J.

J. Horváth, “Sur les functions conjugées à plusieurs variables,” Indagationes Matematicae 15, 17-29 (1953).

Hubel, D. H.

D. H. Hubel and T. N. Wiesel, “Receptive fields, binocular interaction, and functional architecture in the cat's visual cortex,” J. Physiol. (London) 160, 106-154 (1962).

James, W.

W. James, The Principles of Psychology, Vol. I (Harvard U. Press, 1981).

Johnson, A. P.

Kelly, D. H.

D. H. Kelly, “Spatial frequency selectivity in the retina,” Vision Res. 15, 665-672 (1975).
[CrossRef] [PubMed]

Knutsson, H.

G. H. Grandlund and H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, 1995).

Köthe, U.

M. Felsberg and U. Köthe, “GET: The connection between monogenic scale-space and Gaussian derivatives,” in Scale-Space and PDE Methods in Computer Vision, Proceedings of Scale-Space 2005, LNCS Vol. 3459, R.Kimmel, N.Sochen, J.Weickert, eds. (Springer-Verlag, 2005), pp. 192-203.
[CrossRef]

Landy, M. S.

C. Chubb and M. S. Landy, “Orthogonal distribution analysis: a new approach to the study of texture perception,” in Computational Models of Visual Processing, M.S.Landy and J.A.Movshon, eds. (MIT Press, 1991), pp. 291-301.

Larkin, K. G.

Maragos, P.

Mareschal, I.

I. Mareschal and C. L. Baker, “Cortical processing of second-order motion,” 16, 527-540 (1999).

Marquard, J.

R. Müller and J. Marquard, “Die Hilbertransformation und ihre Verallgemeinerung in Optik und Bildverarbeitung,” Optik (Stuttgart) 110, 99-109 (1999).

Moeller, J. R.

C. R. Carlson, J. R. Moeller, and C. H. Anderson, “Visual illusions without low spatial frequencies,” Vision Res. 24, 1407-1413 (1984).
[CrossRef] [PubMed]

Morrone, M. C.

M. C. Morrone and D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London, Ser. B 235, 221-245 (1988).
[CrossRef]

Müller, R.

R. Müller and J. Marquard, “Die Hilbertransformation und ihre Verallgemeinerung in Optik und Bildverarbeitung,” Optik (Stuttgart) 110, 99-109 (1999).

Müller-Lyer, F. C.

F. C. Müller-Lyer, “Zur Lehre von den optischen Täuschungen. Über Kontrast und Konfluxion,” Z. Psychol. 9, 1-16 (1896).

F. C. Müller-Lyer, “Optische Urtheilstäuschungen,” Archiv. Anat. Phys., Phys. Abthlg. Suppl. 2, 263-270 (1889).

Oldfield, M. A.

Oliva, A.

A. Oliva and A. Torralba, “Modeling the shape of the scene: a holistic representation of the spatial envelope,” Int. J. Comput. Vis. 42, 145-175 (2001).
[CrossRef]

Olzak, L.

Pallek, D.

T. Bülow, D. Pallek, and G. Sommer, “Riesz transform for the isotropic estimation of the local phase moiré interferograms,” in 22nd DAGM Symposium für Mustererkennung, Kiel (2000), G.Sommer, N.Krüger, and C.Perwass, eds. (Springer-Verlag, 2000), pp. 333-340.

Papoulis, A.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, 1962).

Ponzo, M.

M. Ponzo, “Urteilstäuschungen über Mengen,” Arch. Ges. Psychol. 65, 129-162 (1928).

Restrepo, A.

A. C. Bovik, N. Gopal, T. Emmoth, and A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691-712 (1992).
[CrossRef]

Robinson, J. O.

J. O. Robinson, The Psychology of Visual Illusion (Hutchinson U. Library, 1972).

Rock, I.

I. Rock, “The description and analysis of object and event perception,” in Handbook of Perception and Human Performance. Vol. II, K.R.Boff, L.Kaufman, and J.P.Thomas, eds. (Wiley, 1986), Chap. 33.

San José Estépar, R.

R. San José Estépar, “Local structure tensor for multidimensional signal processing. Applications to medical image analysis,” Ph.D. thesis (Escuela Técnica Superior de Ingenieros de Telecommunicación, Universidad de Valladolid, Valladolid, Spain, 2005). Available athttp://lmi.bwh.harvard.edu/papers/pdfs/2005/san-joseThesis05.pdf.

Schofield, A. J.

A. J. Schofield and M. A. Georgeson, “Sensitivity to contrast modulation: the spatial frequency dependence of second-order vision,” Vision Res. 43, 243-259 (2003).
[CrossRef] [PubMed]

Sekuler, R.

R. Sekuler and R. Blake, Perception, 3rd ed. (McGraw-Hill, 1994).
[PubMed]

Serrano-Pedraza, I.

V. Sierra-Vázquez and I. Serrano-Pedraza, “Single-band amplitude demodulation of Müller-Lyer illusion images,” Spanish J. Psychol. 10, 3-19 (2007).

Sierra-Vázquez, V.

V. Sierra-Vázquez and I. Serrano-Pedraza, “Single-band amplitude demodulation of Müller-Lyer illusion images,” Spanish J. Psychol. 10, 3-19 (2007).

V. Sierra-Vázquez, “Riez transform and the image spatial global structure,” in Proceedings of The First Iberian Conference on Perception (2005), pp. 17-18.

Skottun, B. C.

B. C. Skottun, “Amplitude and phase in the Müller-Lyer illusion,” Perception 29, 201-209 (2000).
[CrossRef] [PubMed]

Solomon, J. A.

Sommer, G.

M. Felbsberg and G. Sommer, “The monogenic scale-space: A unifying approach to phase-based image processing in scale-space,” J. Math. Imaging Vision 21, 5-26 (2004).
[CrossRef]

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136-3144 (2001).
[CrossRef]

M. Felsberg and G. Sommer, “Structure multivector for local analysis of images,” in Multi-image Analysis, Proceedings of Daugstuhl Workshop on Theoretical Foundations of Computer Vision, LNCS 2032, R.Klette, T.Huang, and G.Gimel'farb, eds. (Springer-Verlag, 2001), pp. 93-104.

M. Felsberg and G. Sommer, “The multidimensional isotropic generalization of quadrature filters in geometric algebra,” in Lecture Notes in Computer Science, Vol 1888, G.Sommer and Y.Zeevi, eds. (Springer-Verlag, 2000), pp. 175-185.
[CrossRef]

T. Bülow, D. Pallek, and G. Sommer, “Riesz transform for the isotropic estimation of the local phase moiré interferograms,” in 22nd DAGM Symposium für Mustererkennung, Kiel (2000), G.Sommer, N.Krüger, and C.Perwass, eds. (Springer-Verlag, 2000), pp. 333-340.

M. Felsberg and G. Sommer, “A new extension of linear signal processing for estimating local properties and detecting features,” in 22nd Symposium für Mustererkennung, DAGM, Kiel, G.Sommer, N.Krüger, and C.Perwass, eds. (Springer-Verlag, 2000), pp. 195-202.

T. Bülow and G. Sommer, “A novel approach to the 2D analytic signal,” in 8th Conference on Computer Analysis of Images and Patterns, Ljubljana, F.Solina and A.Leonardis, eds., LNCS Vol. 1689 (Springer-Verlag, 1999), pp. 25-32.
[CrossRef]

Stein, E. M.

E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton U. Press, 1970).

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton U. Press, 1971).

Torralba, A.

A. Oliva and A. Torralba, “Modeling the shape of the scene: a holistic representation of the spatial envelope,” Int. J. Comput. Vis. 42, 145-175 (2001).
[CrossRef]

Weiss, G.

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton U. Press, 1971).

Wiesel, T. N.

D. H. Hubel and T. N. Wiesel, “Receptive fields, binocular interaction, and functional architecture in the cat's visual cortex,” J. Physiol. (London) 160, 106-154 (1962).

Willen, J. D.

M. Carrasco, J. G. Figueroa, and J. D. Willen, “A test of the spatial-frequency explanation of the Müller-Lyer illusion,” Perception 15, 553-562 (1986).
[CrossRef] [PubMed]

Zhou, Y-X.

Y-X. Zhou and C. L. Baker, “A processing stream in mammalian visual cortex neurons for non-Fourier responses,” Science 261, 98-101 (1993).
[CrossRef] [PubMed]

Zöllner, F.

F. Zöllner, “Über eine neue Art von Pseudoskopie und ihre Beziehungen zu den von Plateau und Oppel beschriebenen Bewegungsphäenomenen,” Ann. Phys. 186, 500-525 (1860).

Zygmund, A.

A. P. Calderón and A. Zygmund, “On singular integrals,” Am. J. Math. 78, 289-309 (1956).
[CrossRef]

A. P. Calderón and A. Zygmund, “On the existence of certain singular integrals,” Acta Math. 88, 85-139 (1952).
[CrossRef]

Acta Math. (1)

A. P. Calderón and A. Zygmund, “On the existence of certain singular integrals,” Acta Math. 88, 85-139 (1952).
[CrossRef]

Am. J. Math. (1)

A. P. Calderón and A. Zygmund, “On singular integrals,” Am. J. Math. 78, 289-309 (1956).
[CrossRef]

Ann. Phys. (1)

F. Zöllner, “Über eine neue Art von Pseudoskopie und ihre Beziehungen zu den von Plateau und Oppel beschriebenen Bewegungsphäenomenen,” Ann. Phys. 186, 500-525 (1860).

Arch. Ges. Psychol. (1)

M. Ponzo, “Urteilstäuschungen über Mengen,” Arch. Ges. Psychol. 65, 129-162 (1928).

Archiv. Anat. Phys., Phys. Abthlg. Suppl. (1)

F. C. Müller-Lyer, “Optische Urtheilstäuschungen,” Archiv. Anat. Phys., Phys. Abthlg. Suppl. 2, 263-270 (1889).

Bull. Acad. Roy. Belg. (1)

J. L. R. Delboeuf, “Sur une nouvelle illusion d'optique,” Bull. Acad. Roy. Belg. 24, 545-558 (1892).

IEEE Trans. Inf. Theory (1)

A. C. Bovik, N. Gopal, T. Emmoth, and A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691-712 (1992).
[CrossRef]

IEEE Trans. Signal Process. (1)

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136-3144 (2001).
[CrossRef]

Indagationes Matematicae (1)

J. Horváth, “Sur les functions conjugées à plusieurs variables,” Indagationes Matematicae 15, 17-29 (1953).

Int. J. Comput. Vis. (1)

A. Oliva and A. Torralba, “Modeling the shape of the scene: a holistic representation of the spatial envelope,” Int. J. Comput. Vis. 42, 145-175 (2001).
[CrossRef]

J. Inst. Electr. Eng. (part III) (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. (part III) 93, 429-457 (1946).

J. Math. Imaging Vision (1)

M. Felbsberg and G. Sommer, “The monogenic scale-space: A unifying approach to phase-based image processing in scale-space,” J. Math. Imaging Vision 21, 5-26 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

A. P. Ginsburg and P. W. Evans, “Predicting visual illusions from filtered images based upon biological data,” J. Opt. Soc. Am. 69, 1443 (1979).

J. Opt. Soc. Am. A (6)

J. Physiol. (London) (1)

D. H. Hubel and T. N. Wiesel, “Receptive fields, binocular interaction, and functional architecture in the cat's visual cortex,” J. Physiol. (London) 160, 106-154 (1962).

Nature (1)

J. R. Bergen and E. H. Adelson, “Early vision and texture perception,” Nature 333, 363-364 (1988).
[CrossRef] [PubMed]

Opt. Express (1)

Optik (Stuttgart) (1)

R. Müller and J. Marquard, “Die Hilbertransformation und ihre Verallgemeinerung in Optik und Bildverarbeitung,” Optik (Stuttgart) 110, 99-109 (1999).

Percept. Psychophys. (1)

S. Coren, “The influence of optical aberrations on the magnitude of the Poggendorf illusion,” Percept. Psychophys. 6, 185-186 (1969).
[CrossRef]

Perception (2)

B. C. Skottun, “Amplitude and phase in the Müller-Lyer illusion,” Perception 29, 201-209 (2000).
[CrossRef] [PubMed]

M. Carrasco, J. G. Figueroa, and J. D. Willen, “A test of the spatial-frequency explanation of the Müller-Lyer illusion,” Perception 15, 553-562 (1986).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. B (1)

M. C. Morrone and D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London, Ser. B 235, 221-245 (1988).
[CrossRef]

Science (1)

Y-X. Zhou and C. L. Baker, “A processing stream in mammalian visual cortex neurons for non-Fourier responses,” Science 261, 98-101 (1993).
[CrossRef] [PubMed]

Spanish J. Psychol. (1)

V. Sierra-Vázquez and I. Serrano-Pedraza, “Single-band amplitude demodulation of Müller-Lyer illusion images,” Spanish J. Psychol. 10, 3-19 (2007).

Vision Res. (5)

C. R. Carlson, J. R. Moeller, and C. H. Anderson, “Visual illusions without low spatial frequencies,” Vision Res. 24, 1407-1413 (1984).
[CrossRef] [PubMed]

M. A. García-Pérez, “Visual phenomena without low spatial frequencies: a closer look,” Vision Res. 31, 1647-1653 (1991).
[CrossRef] [PubMed]

D. H. Kelly, “Spatial frequency selectivity in the retina,” Vision Res. 15, 665-672 (1975).
[CrossRef] [PubMed]

G. S. Hesse and M. A. Georgeson, “Edges and bars: where do people see features in 1-D images?” Vision Res. 45, 507-525 (2005).
[CrossRef]

A. J. Schofield and M. A. Georgeson, “Sensitivity to contrast modulation: the spatial frequency dependence of second-order vision,” Vision Res. 43, 243-259 (2003).
[CrossRef] [PubMed]

Z. Psychol. (3)

W. Ehrenstein, “Versuche über die Beziehungen zwischen Bewegungs- und Gestaltwahrnehmung,” Z. Psychol. 95, 305-352 (1925).

F. C. Müller-Lyer, “Zur Lehre von den optischen Täuschungen. Über Kontrast und Konfluxion,” Z. Psychol. 9, 1-16 (1896).

F. Brentano, “Über ein optisches Paradoxon,” Z. Psychol. 3, 349-358 (1892).

Other (37)

E. G. Boring, Sensation and Perception in the History of Experimental Psychology (Appleton-Century-Crofts, 1942).

J. O. Robinson, The Psychology of Visual Illusion (Hutchinson U. Library, 1972).

S. Coren and J. S. Girgus, Seeing Is Deceiving. The Psychology of Visual Illusions (Lawrence Erlbaum, 1978).

Robinson quotes that Hering's illusion appeared in E. Hering, Beitrage zur Psychology Vol. 1 (Engelman, Leipzig, 1891). Unfortunately, we could not obtain the original source.

W. James, The Principles of Psychology, Vol. I (Harvard U. Press, 1981).

A. P. Ginsburg, Visual Information Processing Based on Spatial Filters Constrained by Biological Data. Tech. Rep. AMRL-TR-78-129 (Aerospace Medical Research Laboratory, Wright Patterson Air Force Base, Dayton, Ohio, USA, 1978).

I. Rock, “The description and analysis of object and event perception,” in Handbook of Perception and Human Performance. Vol. II, K.R.Boff, L.Kaufman, and J.P.Thomas, eds. (Wiley, 1986), Chap. 33.

R. Gregory, Eye and Brain. The Psychology of Seeing, 4th ed. (Oxford U. Press, 1994).

A. P. Ginsburg, “Spatial filtering and visual form perception,” in Handbook of Perception and Human Performance. Vol. II, K.R.Boff, L.Kaufman, and J.P.Thomas, eds. (Wiley, 1986), Chap. 34.

S. Coren and J. S. Girgus, “Visual illusions,” in Handbook of Sensory Physiololgy. Vol. VIII. Perception, R.Held, H.W.Leibowitz, and H.-L.Teuber, eds. (Springer-Verlag, 1978), pp. 551-568.

I. Mareschal and C. L. Baker, “Cortical processing of second-order motion,” 16, 527-540 (1999).

J. P. Havlicek, “AM-FM image models,” Ph.D. dissertation (The University of Texas at Austin, 1996).

J. P. Havlicek and A. C. Bovik, “Image modulation models,” in Handbook of Image and Video Processing, A.C.Bovik, ed. (Academic, 2000), pp. 313-324.

C. Chubb and M. S. Landy, “Orthogonal distribution analysis: a new approach to the study of texture perception,” in Computational Models of Visual Processing, M.S.Landy and J.A.Movshon, eds. (MIT Press, 1991), pp. 291-301.

J. G. Daugman and C. J. Downing, “Demodulation: a new approach to texture vision,” in Spatial Vision in Human and Robots, L.H.Harris and M.Jenkin, eds. (Cambridge U. Press, 1993), pp. 63-87.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1978).

G. H. Grandlund and H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, 1995).

S. L. Hahn, Hilbert Transform in Signal Processing (Artech House, 1996).

T. Bülow and G. Sommer, “A novel approach to the 2D analytic signal,” in 8th Conference on Computer Analysis of Images and Patterns, Ljubljana, F.Solina and A.Leonardis, eds., LNCS Vol. 1689 (Springer-Verlag, 1999), pp. 25-32.
[CrossRef]

M. Felsberg and G. Sommer, “A new extension of linear signal processing for estimating local properties and detecting features,” in 22nd Symposium für Mustererkennung, DAGM, Kiel, G.Sommer, N.Krüger, and C.Perwass, eds. (Springer-Verlag, 2000), pp. 195-202.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, 1962).

NAG, The NAG FORTRAN Library Manual, Mark 15 (The Numerical Algorithms Group Ltd, Oxford, 1991).

M. Felsberg and U. Köthe, “GET: The connection between monogenic scale-space and Gaussian derivatives,” in Scale-Space and PDE Methods in Computer Vision, Proceedings of Scale-Space 2005, LNCS Vol. 3459, R.Kimmel, N.Sochen, J.Weickert, eds. (Springer-Verlag, 2005), pp. 192-203.
[CrossRef]

V. Sierra-Vázquez, “Riez transform and the image spatial global structure,” in Proceedings of The First Iberian Conference on Perception (2005), pp. 17-18.

E. B. Goldstein, Sensation and Perception, 2nd ed. (Wadsworth, 1984).

R. Sekuler and R. Blake, Perception, 3rd ed. (McGraw-Hill, 1994).
[PubMed]

T. Bülow, D. Pallek, and G. Sommer, “Riesz transform for the isotropic estimation of the local phase moiré interferograms,” in 22nd DAGM Symposium für Mustererkennung, Kiel (2000), G.Sommer, N.Krüger, and C.Perwass, eds. (Springer-Verlag, 2000), pp. 333-340.

M. Felsberg, “Low-level image processing with the structure multivector,” Ph.D. thesis, Tech. Rep. #0203 (Institute of Computer Science and Applied Mathematics, Christian-Albrechts Universität, Kiel, Germany, 2002), available at http://www.informatik.uni-kiel.de/reports/2002/0203.html.

E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton U. Press, 1970).

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton U. Press, 1971).

R. San José Estépar, “Local structure tensor for multidimensional signal processing. Applications to medical image analysis,” Ph.D. thesis (Escuela Técnica Superior de Ingenieros de Telecommunicación, Universidad de Valladolid, Valladolid, Spain, 2005). Available athttp://lmi.bwh.harvard.edu/papers/pdfs/2005/san-joseThesis05.pdf.

M. Brady, Advanced Transform Methods (Department of Engineering Science, Oxford University, 2006). Available at http://www.robots.ox.ac.uk/~jmb/transformlectures.html.

In the literature, both the definition of the Hilbert and the Riesz transforms and the computation of their Fourier transforms (or multipliers) create an awkward situation, caused by the selection of the sign of convolution kernel in Eq. and the definition of the direct Fourier transform. This is not a trivial issue because it has consequences for the computation of the true phase of odd symmetries. As the four possibilities exist in the literature, the definition of the j Riesz transform as convolution with the kernel Kj(x)=−cnxj|x|−n−1, j=1,...,n, adopted in Eqs. is consistent with the definition of the Hilbert transform in and accords with the definition adopted by some authors in harmonic analysis , Eq. (4.1), and image processing but differs by a minus sign from those used in harmonic analysis , signal theory , and image processing . This decision also affects the definitions of analytic signal and monogenic signal. If the definition of the direct Fourier transform contains a minus sign in the argument of the complex exponential [i.e., Eq. ], then the FT of the Riesz kernel j is K̂j(ξ)=iξj/|ξ|, j=1,...,n, and it follows from this that K̂1(u)=iu/|u| and K̂2(u)=iv/|u| for a two-dimensional signal.

M. Felsberg, “Disparity from monogenic phase,” in DAGM 2002, LNCS Vol. 2449, L.Van Gool, ed. (Springer-Verlag, Berlin, 2002), pp. 248-256.

M. Felsberg and G. Sommer, “The multidimensional isotropic generalization of quadrature filters in geometric algebra,” in Lecture Notes in Computer Science, Vol 1888, G.Sommer and Y.Zeevi, eds. (Springer-Verlag, 2000), pp. 175-185.
[CrossRef]

M. Felsberg and G. Sommer, “Structure multivector for local analysis of images,” in Multi-image Analysis, Proceedings of Daugstuhl Workshop on Theoretical Foundations of Computer Vision, LNCS 2032, R.Klette, T.Huang, and G.Gimel'farb, eds. (Springer-Verlag, 2001), pp. 93-104.

M. Felsberg, “Optical flow estimation from monogenic phase,” in Complex Motion, First International Workshop, IWCM 2004, LNCS 34517, B.Jähne, R.Mester, E.Barth, and H.Scharr, eds. (Springer-Verlag, 2007), pp. 1-13.

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Figures (7)

Fig. 1
Fig. 1

Standard versions of geometrical-optical illusions: (a) Müller-Lyer [4, 5]. (b) Ponzo [7]. (c) Delboeuf [8]. (d) Poggendorff figure after Robinson [2]. (e) Hering figure after Ehrenstein [11]. (f) Ehrenstein [11].

Fig. 2
Fig. 2

Computation of the Riesz transforms and polar components of monogenic signal corresponding to a band-pass-filtered Müller-Lyer illusion image. (a) The input image is (b) convolved with the PSF of the triplet of spherical SQFs, labeled h BP , h 1 , and h 2 (displayed as 3D plots). The basis function in the 2D Fourier domain, h ̂ BP , displayed as an intensity image, is a radial log-normal function [peak frequency of 8 c iw and relative bandwidth (full width at half-height) of 2.5 octaves]. To understand the (odd) symmetry of Riesz kernels h 1 and h 2 and its corresponding Fourier transforms (displayed as intensity images) h ̂ 1 and h ̂ 2 , note that these functions are the imaginary parts of the Fourier transforms because their real parts vanish for all spatial frequencies. Convolutions of input image and SQF result in (c) the Euclidean components of the monogenic signal corresponding to the Müller-Lyer figure. Labels f BP , f R 1 , f R 2 stand, respectively, for band-pass (BP) filtered input image and its two Riesz transforms. (d) Outputs of the filters were transformed into polar (spherical) coordinates and displayed as gray maps, with the intensity codes put aside. Local orientation is displayed only at locations in which the local amplitude has a sufficiently large value. (e) Next to the local phase, the PM carrier (i.e., cos [ φ ( x ) ] ) is shown. Under the image of the local amplitude, the smoothed envelope (displayed as 3D plot) result of an isotropic Gaussian low-pass filtering (s.d. of 8 c iw ) is shown. (f) The envelope profiles along the horizontal shafts. The low-pass-filtered envelope was normalized to unity.

Fig. 3
Fig. 3

Graphic code to show the meaning of polar (spherical) components of Figs. 2, 5, 7. (a) Polar (spherical) coordinates. (Note that the real component of the monogenic signal is the third coordinate). Meaning of symbols in the text. (b) Circle code for the local phase. Only patches of local spatial structures and intensity profiles corresponding to i1D discrete features (counter clockwise, light line or peak, right edge, dark line or pit, and left edge) are depicted. (c) Circle code for the local orientation and an ideal dark line (i.e., π rad of phase) in four orientations. Dark bars were chosen for clarity.

Fig. 4
Fig. 4

Monogenic phase from local phase and orientation shown in Fig. 2 is displayed as a vector field atop the local amplitude and edges. For clarity, only the monogenic phase and the envelope of the top left quadrant of the Müller-Lyer input image are depicted. Local amplitude is displayed as a gray intensity map (dark is higher amplitude). Overlaid monogenic phase is displayed as a vector field in which the absolute local (scalar) phase is proportional to the length of vector, and the local orientation (affected by the local phase sign) is represented by the angle of rotation. Arrows are displayed in a square grid and only at locations in which local amplitude has a sufficiently large intensity. Edge locations were found with a trivial algorithm that indicates the locations in which local phase is equal to or near π 2 or π 2 . A straight segment perpendicular to edge orientation was drawn to indicate the edge location. The inset shows the code for the four i1D features (length of arrows indicates the feature type).

Fig. 5
Fig. 5

Polar (spherical) components of monogenic signals of the geometrical-optical illusion images of Fig. 1 and their PM carriers. (a) Input images. (b) Band-pass (BP) filtered input images (i.e., f BP or third Euclidean coordinate of monogenic signal) obtained with an isotropic MTF whose radial profile was a lognormal function (peak spatial frequency of 32 c iw and 1 octave of relative bandwidth for Hering and 2.5 for the rest). (c) Envelopes displayed as an intensity image (all envelopes are normalized to unity). (d) Local orientations. For clarity, local orientation is displayed in white at locations in which the normalized envelope is lower than 0.01 and both Riesz transforms are zero. (e) Local phases. (f) PM carriers from local phases, i.e., cos [ φ ( x ) ] . Intensity codes for envelope, local phase, local orientation, and PM carrier are depicted at the bottom of each column.

Fig. 6
Fig. 6

Distribution of averaged, weighted energy with radial spatial frequency up to 64 c iw for (a) envelopes and (b) PM carriers from the respective monogenic signal of each illusion image. Each panel shows for each illusion image the averaged, weighted energy N, in dB referred (re) to E max , as a function of the radial spatial frequency ρ 0 , in c/iw. (See text for computation details). The thickest curves are the MTF radial profiles of the low-pass filters, in dB re A max ( A max = 1 ) . Arrows on the x axis indicate the spatial frequency values at which MTF radial profiles take the amplitudes indicated by arrows in the y axis. The insets in the bottom left part of each panel depict, for comparison, the radial distributions of (a) a monotonic-energy decreasing signal (solid curve) and a purely band-pass (dashed curve) signal and (b) a mixed-energy decreasing and band-pass signal (dashed curve).

Fig. 7
Fig. 7

Physical presence of illusory distortions in the smoothed envelope. (a) Input images (depicted for comparison). (b) Smoothed envelopes displayed as intensity images, obtained by filtering the envelopes with an isotropic Gaussian low-pass filter (s.d. of 9 c iw ). All smoothed envelopes are normalized to unity. (c) 1D profiles (extent illusions) and 2D thresholds (direction illusions) of smoothed envelopes. For illusions of extent, thick curves indicate 1D envelope profiles along the upper horizontal shafts (Müller-Lyer), upper bar (Ponzo), and left horizontal diameter (Delboeuf), respectively; thin curves indicate envelope profiles along the lower horizontal shaft (Müller-Lyer), lower bar (Ponzo), and right horizontal diameter (Delboeuf), respectively. Straight segments over smoothed envelope profiles indicate the physical length between absolute maxima. Distortions in smoothed envelopes of direction illusions have been enhanced by thresholding the smoothed envelopes (threshold values are 0.50 for the Poggendorff illusion and 0.75 for the Hering and Erhenstein illusions). Intensities larger than the respective threshold are displayed in black. (d) PM carriers filtered by an isotropic low-pass Gaussian filter (s.d. of 16 c iw ). (e) 1D profiles (extent illusions) and 2D thresholds (direction illusions) of smoothed PM carriers. Meaning of 1D profiles is the same as in (c). Straight segments drawn over profiles of illusion of extent show the spans between absolute (Müller-Lyer and Ponzo) or relative (Delboeuf) minima. Amplitude threshold value for illusions of direction is equal to zero. Amplitudes lower than the threshold value are displayed in black.

Tables (2)

Tables Icon

Table 1 Conceptual Parallelisms between Analytic Signal and Monogenic Signal

Tables Icon

Table 2 Relevant Parameter Description of Geometrical-Optical Illusions Used as Input Images and the Differences from the Original Illusion Images

Equations (23)

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f ( x ) = A ( x ) cos [ ϕ ( x ) ] ,
f R j ( x ) R j ( f ) ( x ) = lim ϵ c n | y | > ϵ y j | y | n + 1 f ( x y ) d y ,
j = 1 , , n ,
K j ( x ) = c n x | x | n + 1 , j = 1 , , n .
f R ( x ) = ( f R 1 ( x ) , , f R n ( x ) ) .
f ̂ ( x ) = R n f ( x ) exp ( i 2 π ξ , x ) d x ,
K ̂ j ( ξ ) = i ξ j | ξ | , j = 1 , , n ,
R j ( r ) ( ξ ) f ̂ R j ( ξ ) = i ξ j | ξ | f ̂ ( ξ ) , j = 1 , , n .
f R 1 ( x ) = p.v. R 2 x 2 π | x | 3 f ( x x ) d x ,
f R 2 ( x ) = p.v. R 2 y 2 π | x | 3 f ( x x ) d x ,
f M ( x ) = i f R 1 ( x ) j f R 2 ( x ) + f ( x )
f M ( x ) = ( f R ( x ) , f ( x ) ) = ( f R 1 ( x ) , f R 2 ( x ) , f ( x ) ) .
| f M ( x ) | = f 2 ( x ) + | f R ( x ) | 2 ,
θ M ( x ) = arctan f R 2 ( x ) f R 1 ( x ) , θ M ( π 2 , π 2 ] .
φ M ( x ) = arg [ f ( x ) + i | f R ( x ) | ] , φ M [ 0 , π ] .
r ( x ) = f R ( x ) | f R ( x ) | arctan ( | f R ( x ) | f ( x ) ) = | φ M ( x ) | [ sgn ( φ M ( x ) ) cos ( θ M ( x ) ) , sgn ( φ M ( x ) ) sin ( θ M ( x ) ) ] ,
f ( x ) = | f M ( x ) | cos [ φ M ( x ) ] , 0 | f M ( x ) | 1 ,
I ( x ) = I ave { 1 + | f M ( x ) | cos [ φ M ( x ) ] } ,
f M ( x ) = ( f R 1 ( x ) , f R 2 ( x ) , f BP ( x ) ) = ( ( h 1 f ) ( x ) , ( h 2 f ) ( x ) , ( h BP f ) ( x ) ) .
h ̂ BP ( u ) = { exp [ ln 2 ( | u | ρ 0 ) 2 α 2 ] | u | 0 0 | u | = 0 } ,
h ̂ 1 ( u ) = { i u | u | exp [ ln 2 ( | u | ρ 0 ) 2 α 2 ] | u | 0 0 | u | = 0 } ,
h ̂ 2 ( u ) = { i v | u | exp [ ln 2 ( | u | ρ 0 ) 2 α 2 ] | u | 0 0 | u | = 0 } .
φ ( x ) = atan 2 [ sign ( ( h 1 f ) ( x ) ) | f R ( x ) | , f BP ( x ) ]

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