V. Sierra-Vázquez and I. Serrano-Pedraza Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. A 781
Application of Riesz transforms to the isotropicAM–PM decomposition of geometrical-optical
illusion images
Vicente Sierra-Vázquez1 and Ignacio Serrano-Pedraza2,*1Departamento de Psicología Básica I, Facultad de Psicología, Universidad Complutense de Madrid, Campus de
Somosaguas, 28223 Madrid, Spain2Institute of Neuroscience, Faculty of Medical Sciences, Newcastle University, Newcastle upon Tyne, NE2 4HH, UK
. INTRODUCTIONeometrical-optical illusions are visual phenomena inhich the perceived geometrical properties are distortedith respect to the corresponding properties of the physi-
al images that originate the illusions. Based on the kindf distortion they present, geometrical-optical illusionsave been classified as illusions of extent and illusions ofirection [1–3]. Illusions of extent are the optical illusionsn which length or size is misjudged. Standard versions ofome of them are shown in Fig. 1: the Müller-Lyer illusion4–6] [see Fig. 1(a)], in which a shaft with arrow-feathersooks longer than another one with arrow-heads, althoughoth shafts are in fact equal in length; the Ponzo illusion7] [see Fig. 1(b)], in which two horizontal lines of equalength appear unequal in length when they are enclosedetween convergent straight lines (the line closer to thepex looks longer than the one farther from it); the Del-oeuf illusion [8] [see Fig. 1(c)], in which an inner circleppears to be larger than an outer one of same size. Illu-ions of direction are those in which the percept presentsistortions in the orientation of simple line elements oristortions in the shape of figures. Some of them are: theoggendorff illusion (described by Zollner [9]) [see Fig.(d)], in which two oblique collinear lines do not appear toe so when interrupted by two vertical straight lines; theering variant [see Fig. 1(e)] of the Wundt-Hering illu-
ion [2,10,11], in which the two central straight and par-llel lines placed over a radial inducing pattern appear toe curved outward; and finally, the Ehrenstein illusion11] [see Fig. 1(f)], in which the shape of a square appears
istorted when placed over a radial pattern. We have de-cribed the commonest forms of these illusions (and, inome cases, not the configuration originally designed byheir authors); a number of versions (or forms [3]) of eachne of them, as well as other geometrical-optical illusions,an be seen in [2,3].
These visual phenomena are so surprising that a greatmount of research has been devoted to them (see [1–3]or historical references). Although illusions are interest-ng in themselves, since William James [12] the rationaleor the research on illusions has been the belief that theechanisms underlying them must also underlie veridi-
al perception, because “between normal perception andllusion…there is no break, the process being identicallyhe same in both” (see [12], p. 757; italics are of W. Jamesimself). Thus, a number of biological [13], psychophysi-al [13], and cognitive [14,15] explanations of these illu-ions have been given (see other references in [2,3]). Un-ortunately, most of them are ad hoc explanations and areot based on known properties of human visual process-
ng. Among the explanations that take into account theiological and psychophysical machinery, the most influ-ntial one so far in the field of study of spatial vision ishe Ginsburg explanation [13,16]. Briefly, this explana-ion proposed that the spatial information leading toeometrical-optical illusions (and other visual phenom-na) lies in the lower spatial frequency range of the imageourier spectra (i.e., in the coarse scales). Thus, the per-eptual illusion could be based on the results of a visualow-frequency band-pass filtering. Both optical blurring
010 Optical Society of America
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782 J. Opt. Soc. Am. A/Vol. 27, No. 4 /April 2010 V. Sierra-Vázquez and I. Serrano-Pedraza
3,17] and band-pass digital filtering of illusion images byhe 2D human contrast sensitivity function (CSF) [13,16],n addition to some experimental work [18,19], supporthe claim that geometrical-optical illusions may be due toisual selection of the lower spatial frequency componentshat are present in the Fourier spectra of the illusion im-ges. However, it has been shown that these illusions areaintained in a number of stimuli containing few or no
ow or medium spatial frequencies, for example, images ofome of these illusions composed of balanced dots [20] oralanced squares [21], optically [22] or digitally [23] high-ass-filtered [22] illusion images, and digitally altered23] versions of them, for all of which the first-order or lu-inance psychophysical channels sensitive to coarse
cales are inactive. So, as low spatial frequencies are notecessary for the perception of these illusory patterns,here is the spatial information about these illusions?The aforementioned Ginsburg explanation is based on
he image representation as a linear combination of or-hogonal and periodic elementary functions (i.e., Fourierynthesis) and on the biological spatial frequency filteringf visual stimuli (e.g., filtering by the CSF). However,here is a complementary way of thinking about the im-ge and early spatial visual processing. According to aingle-band modulation model [24–26], any narrow-band-ass image can be obtained over the spatial domain ashe modulation product of a smoothly varying two-imensional (2D) spatial envelope (in relation to themplitude-modulation, or AM component) and a fine spa-ial structure (spatial frequency-modulated 2D carrier, or
(a)Muller-Lyer
(b)Ponzo
(c)Delboeuf
(d)Poggendorff
(e)Hering
(f)Ehrenstein
ig. 1. Standard versions of geometrical-optical illusions: (a)üller-Lyer [4,5]. (b) Ponzo [7]. (c) Delboeuf [8]. (d) Poggendorff
gure after Robinson [2]. (e) Hering figure after Ehrenstein [11].f) Ehrenstein [11].
D FM carrier). Put in mathematical terms: Let a realarrow-band 2D signal f be the contrast function corre-ponding to a given physical image I (i.e., f�x��I�x� /Iave�−1, where Iave is the average luminance (orC component) and x= �x ,y� is the spatial coordinate vec-
or); then
f�x� = A�x�cos���x��, �1�
here the local amplitude A is the amplitude-modulationAM) component (its magnitude, �A��0, is called the sig-al envelope, and �A�2 is the local energy), the local phase, ��x�� �−� ,��, is the phase-modulation (PM) compo-ent, and cos���x�� is the 2D frequency-modulated (FM)arrier. The FM function, which provides the instanta-eous or emergent spatial frequencies [27], is just thehase gradient, ���x�= ����x� /�x ,���x� /�y�. The AM–PMecomposition (also called demodulation transform [24])f a given signal is the process of estimating its AM andM (alternatively, FM) components, i.e., A and � of Eq.
1). In the image processing field, envelope and PM car-ier have been interpreted as local contrast and imageexture, respectively [25]. It has also been suggested thatur perceptual organization of the visual stimulus corre-ponds in certain respects to its AM–PM decomposition24]. Separately, it has been found that the visual systemas both neurobiological [28,29] and psychophysicalecond-order machinery sensitive to (and able to recover)he envelope of periodic or noisy visual stimuli (for re-iews see [30] and Subsection 1.B in [31]). This evidenceuggests that the human visual system may be sensitiveo the image envelope and that therefore the envelope ofn image could have functional consequences in visualerception.The image model of Eq. (1) is inherently ambiguous be-
ause for a given image, there is an unlimited number ofunction pairs A and � that fulfill Eq. (1). Because there iso single solution for the AM–PM decomposition of aiven image, a number of demodulation algorithms (i.e.,ffective procedures to calculate the AM–PM decomposi-ion) have been proposed in the past. In Sierra-Vázqueznd Serrano-Pedraza [32] two well-known demodulationlgorithms (the DESA or Discrete Energy Separation Al-orithm [33] and the AMPM algorithm [24,34]) were ap-lied to compute the AM–PM decomposition of five ver-ions of the Müller–Lyer illusion. Results showed that themoothed envelope estimated by either one of the algo-ithms reveals physical distortions in the spatial proper-ies of Müller–Lyer illusion images that look like the per-eived illusion. In addition, an elementary, psychophys-cally based, filter-rectify-filter mechanism proposed forecond-order vision [35,36] was applied to the processingf the illusion images. Results showed that the outputs ofhe second-order mechanism had spatial properties simi-ar to the physical envelopes from the above-mentionedlgorithms. Unfortunately, both algorithms have someifficulties with estimating the AM and/or the PM compo-ents in non-periodic intrinsically 2D images because ofhe local nature of the operator involved in the DESA andhe anisotropic nature of the AMPM algorithm (see Dis-ussion and Appendix in [32]).
The new approach used in this paper is as follows. Forgiven 1D real signal, it is known that its AM and PM
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V. Sierra-Vázquez and I. Serrano-Pedraza Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. A 783
omponents are, respectively, the instantaneous ampli-ude and the argument of its associated Gabor analyticignal [37–40]. Analytic signal is defined as the complex-alued function whose real part is the given original sig-al itself and whose imaginary part (the conjugate func-ion) is computed via the Hilbert transform of the originalignal. Indeed, the theoretical problem in the case of im-ges rests on the isotropic generalization of the Hilbertransform and the notion of analytic signal to multi-imensional signals because there is no unique counter-art for the Hilbert transform definition for multidimen-ional signals without implying a preferred direction39–41]. As Felsberg and Sommer [42] point out, we needo find an odd filter with an isotropic energy distribution,problem that does not have any solution in the fields of
eal or complex numbers. A solution (i.e., a multidimen-ional isotropic generalization of Hilbert transform), ishe Riesz transform, a vector-valued transform surpris-ngly well known to mathematicians since the 1950s inhe field of Calderón–Zygmund theory [43–47], introducednto signal theory in [48] and used for image processing42,49–58]. The conjugate function of an n-dimensionalignal is the n-dimensional vector-valued function whoseomponents are its Riesz transforms [45]; the monogenicignal [50], which generalizes the concept of Gabor’s ana-ytic signal to multidimensional functions, is the n+1ector-valued function generated by linearly combininghe original signal and its n Riesz transforms. (A brief butxcellent historical survey on the presence of the Rieszransform in harmonic analysis and its applications toeophysics and optics can be found in the Introduction of56]).
The aims of this work are to use the Riesz transformsnd the monogenic signal to perform isotropic AM–PM de-omposition of the representative versions of theeometrical-optical illusions mentioned above, to esti-ate the AM and PM components of these geometrical-
ptical-illusion images, and to determine which compo-ent conveys the perceived (illusory) distortions of these
llusions. This is a qualitative paper; a quantitative studyf the relationship between numerical outputs and illu-ion measurement data is beyond the scope of this paper.his paper is organized as follows. Because the Rieszransform and the monogenic signal are not widelynown in the vision community, in Section 2 we will pro-ide a short introduction to them, focusing on their appli-ation to image processing. The mathematical fundamen-als are partly based on the authoritative books of Stein46,47] on harmonic analysis and partly on the extensivend complete exposition about this issue that can beound in a set of papers from the Kiel University Cogni-ive Systems Group [42,49–55]. However, herein, to keepathematics under control, the material is presented
airly informally, as others have done [57,58] (i.e., withouthe aid of more complicated geometric algebra). In Section, we will describe the test illusion images and details ofhe numerical algorithm used to compute their Rieszransforms and monogenic signals. In Section 4, we willresent numerical results of single-band isotropicM–PM demodulation of these illusion images. Finally, inection 5, we discuss our results and draw some conclu-ions.
. FUNDAMENTALS: RIESZ TRANSFORMS,ONOGENIC SIGNAL OF A GIVEN
MAGE, AND ISOTROPIC ESTIMATION OFTS LOCAL AMPLITUDE, LOCALHASE, AND LOCAL ORIENTATION. Riesz Transformsor f�Lp�Rn�, 1�p��, the Riesz transform Rj of f, fRj
, isefined in [[46], Eq. (5)] as
fRj�x� � Rj�f��x� = lim
�→�
cn��y���
yj
�y�n+1 f�x − y�dy,
j = 1, . . . ,n, �2�
here cn= ���n+1� /2� /��n+1�/2 (cn is a constant depend-ng on the dimension n, and is the gamma function) andhere x= �x1 ,x2 , . . . ,xn� is the position vector, y�y1 ,y2 , . . . ,yn� is the vector that represents the dummyariable of integration, and �y� is its vector norm. Thus, Rjs defined by the convolution kernel [46]
Kj�x� = cn
x
�x�n+1 , j = 1, . . . ,n. �3�
he Riesz transform is an example of a singular integralperator with odd kernel [47]. Note that for every f :Rn
R there are n Riesz transforms. In [46] Proposition 2,p. 58), it is established that the Riesz transform is a mul-idimensional generalization of the Hilbert transform.he conjugate function of the n-dimensional real functionis just the vector-valued function or vector field, fR, de-ned in [45] by
fR�x� = �fR1�x�, . . . ,fRn
�x��. �4�
ote that the conjugate function of an n-dimensional sig-al contains n elements at each position x. Let us definehe multidimensional Fourier transform of a signal f, f, as
f�x� =�Rn
f�x�exp�− i2��,x��dx, �5�
here � ,�= �1 ,2 , . . . ,n�, is the vector of spatial frequen-ies, i is the square root of −1, and .,.� denotes the innerr scalar product of vectors. Then, the Fourier transformf the jth Riesz kernel, Kj, or Fourier multiplier, is [47]
Kj��� = − ij
���, j = 1, . . . ,n, �6�
nd by the convolution theorem [38], the Fourier trans-orm of the jth Riesz transform of f is [47]
Rj�r�∧��� � fRj��� = − i
j
���f���, j = 1, . . . ,n. �7�
he vector-valued linear filter characterized in the fre-uency domain by the vector field K of kernel transforms,
ˆ ���= �−i1 / ��� ,−i2 / ��� , . . . ,−in / ����, fulfills the require-ents cited in the Introduction for the appropriate filtere are looking for: it has isotropic energy distribution (in
act, it has unit length in any direction) and is odd withespect to the coordinate origin (i.e., K�−��=−K���). The
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784 J. Opt. Soc. Am. A/Vol. 27, No. 4 /April 2010 V. Sierra-Vázquez and I. Serrano-Pedraza
iesz operator commutes with translations and dilationsf Rn and is equivariant under rotations in Rn; i.e., theiesz operators transform in the same way as the compo-ents of a vector [46]. (See the properties of the Rieszransforms, presented in a comprehensive form, in [50], p.140, [51], p.79, and [57], p. 41).
. Riesz Transforms and Monogenic Signal of a Givenmage and Estimation of Its Local Amplitude,ocal Phase, and Local Orientationo apply the Riesz transforms to 2D signals, we slightlyhange the symbolic notation and the definitions. Assual, points in R2 will be notated as x= �x ,y� in the spa-ial domain, and u= �u ,v� in the Fourier domain. To keepefinitions parallel to definitions of Hilbert transform,ourier multiplier, and analytic signal currently used inhe literature [38,39] and, at same time, to retain the con-olution operation, we change the sign of the Riesz trans-orm [59]. It follows, according to Eq. (2) with a minusign, that the two Riesz transforms of a given image f, fR1
,nd fR2
, are
fR1�x� = p.v.�
R2−
x�
2��x��3f�x − x��dx�, �8�
nd
fR2�x� = p.v.�
R2−
y�
2��x��3f�x − x��dx�, �9�
here x�= �x� ,y�� ,p.v. stands for the Cauchy principalalue, and cn=1/ �2�� because n=2. Therefore, accordingo Eqs. (8), (9), and (6), the two Riesz kernels for 2D sig-als are K1�x�=−x / �2��x�3� and K2�x�=−y / �2��x�3�, re-pectively, in the spatial domain, and their Fourier trans-orms are K1�u�= iu / �u� and K2�u�= iv / �u�, respectively.Note that these latter functions are spherical harmonicsf first order. Also note that Riesz kernels and multipliersoincide with the Hilbert kernel and with the Fourierransform of the Hilbert kernel respectively when n=1).
The monogenic signal of a given 2D signal f, fM, is de-ned as the linear combination of its own original signalnd its two corresponding Riesz transforms [50], i.e.,
fM�x� = − ifR1�x� − jfR2
�x� + f�x� �10�
Eq. (13) from [50] expanded and the real part moved tohe right], where i, j are the two imaginary units (or hy-ercomplex units) and �i , j ,1 is an orthonormal basis of3. Like the Riesz transform with respect to the Hilbert
ransform, the monogenic signal is a multidimensionalsotropic generalization of the Gabor analytic signal.roperties of monogenic signals as the isotropic generali-ation of analytic signals (quadrature, doubling energy ofhe original signal, possibility of absence of negative partf spectra, rotation invariance) can be seen in [50],. 3140, [51], pp. 79, 80, and [57] pp. 41, 42.The use of the monogenic signal as defined in Eq. (10)
equires complicated Clifford algebra [45]; so, as othersave done [57,58], we prefer to define the monogenic sig-al of an image by means of a vector-valued function orector field f , f :R2→R3 as
M M
fM�x� = �− fR�x�,f�x�� = �− fR1�x�,− fR2
�x�,f�x��. �11�
or our aims, the expression of monogenic signal compo-ents in polar (spherical) coordinates is more useful.rom Eq. (11), the radius or local amplitude of the mono-enic signal is the pointwise L2-norm,
�fM�x�� = �f2�x� + �fR�x��2, �12�
here �fR�x�� is the pointwise norm of (vector-valued) con-ugate function fR. The azimuth angle �M at position x is
�M�x� = arctanfR2
�x�
fR1�x�
, �M � �− �/2,�/2�. �13�
he elevation or zenith angle �M at position x is �fR�x��:
�M�x� = arg�f�x� + i�fR�x���, �M � �0,��. �14�
Later on, in the Method section (Section 3), we computehe local-phase angle in a way that the interval will be−� ,��). We can see now that the local amplitude of theonogenic signal is the spatial envelope of the original
eal image (and the squared modulus, �fM�x��2, is the localnergy), the azimuth angle is the local orientation, andhe zenith angle the local (scalar) 1D phase. Note that lo-al orientation and 1D local phase can be combined in theo-called local phase-vector r (or monogenic phase)57,60],
r�x� =fR�x�
�fR�x��arctan �fR�x��
f�x� � = ��M�x��
�sgn��M�x��cos��M�x��,sgn��M�x��sin��M�x���,
�15�
here ��M�x��, the absolute value of local (scalar) phase athe spatial location x, is the vector length and ��x�, theocal orientation at the spatial location x, is its argument.nstead of what is established in the seminal paper ofelsberg and Sommer [50] in which monogenic phase isefined as an operator (rotation vector) in Eq. (15), theonogenic phase vector points in the direction of the local
rientation and not orthogonally to it. (Later on, this wille illustrated in Fig. 4). It follows from Eqs. (11), (12), and14) that the expression for a 2D contrast function f inerms of the modulation product from the local amplitudend the local scalar phase of its monogenic signal is
f�x� = �fM�x��cos��M�x��, 0 � �fM�x�� � 1, �16�
here the AM component [i.e., A of Eq. (1)] is just the lo-al amplitude of the corresponding monogenic signaliven by Eq. (12), and the PM component (argument of co-ine term) [i.e., � of Eq. (1)] is the 1D local phase given byq. (14), and the cosine term in Eq. (16) is the PM modu-
ated carrier wave. Because the AM component of the mo-ogenic signal is a non-negative value and is equal to thenvelope, hereafter, the terms AM component, envelope,nd local amplitude will be used interchangeably. (Notehat the reconstruction formula of Eq. (16) is formallydentical to Eq. (24) in [52] with s0=0). Finally, the exactepresentation of a narrow-band image I with the localmplitude normalized to unity and Iave as DC components
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V. Sierra-Vázquez and I. Serrano-Pedraza Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. A 785
I�x� = Iave�1 + �fM�x��cos��M�x��, �17�
hich is a solution of the isotropic AM–PM decompositionf a given real image. (To summarize, Table 1 shows thearallelisms between the analytic signal of a 1D signalnd the monogenic signal of a 2D signal).Since Felsberg and Sommer’s paper [50], deeper com-
rehension has been achieved in the interpretation of po-ar components of the monogenic signal of a real imageeyond the direct meaning as local amplitude, localhase, and local orientation, respectively, of the given im-ge. It must be clear that different polar components rep-esent mutually independent properties of the image.hus, the monogenic signal orthogonally divides the im-ge spatial information into energetic information (localmplitude), structural information (local phase), mainlyymmetries, and, additionally, geometric information (lo-al orientation) [50,51]. These components allow us toompute the location, type, and orientation of intrinsicallyD (i1D) [51] classic visual features (bright and dark lineseven symmetries), and left and right edges (odd symme-
Table 1. Conceptual Parallelisms betw
1D Signal [37]
Concept Formula
ssociated Analytichyper)complex signal fA�x�= f�x�− ifH�xignal fA
ransform Hilberttransfrom fH�x�=p .v . �K�x�fH
onjugateignal
fH
ernel(s) foronvolution(s) K�x�=−
1�x
ourierultiplier(s) K�u�= i
u�u�
ocalmplitude
Envelopeof signal
�fA�x��=�f2�x�+ fH2
rgument (1D)enith (2D)
Localphase
��x�=arg�f�x�+ if
zimuth (2D)
M–FMepresentation
f�x�= �fA�x��cos���
aSee in the Method section the computation of the local-phase angle.
ries) considered as a natural basis of image processingoth in biological [61] and psychophysical [62] research).he local amplitude is a measure of the presence of a fea-
ure at a specific location [63].In particular, a local maximum of the local amplitude
t a given position indicates the presence of a 1D featuren that spatial location, whereas the local phase value in-icates the class of discrete feature (±�, bright or darkines and ±� /2, right or left edges), and the local orienta-ion indicates the main orientation of the feature. In theq. (13) used here, local orientation is taken in the direc-
ion of the luminance modulation of the feature, i.e.,ointing in the direction of highest luminance variation ofhe i1D feature and perpendicular to its constant lumi-ance elongation. Therefore, we define a single vertical
ine or edge as having an orientation of 0° and a singleorizontal line or edge an orientation of 90°. More impor-antly, because the Riesz transform is a non-local opera-or, the local amplitude, interpreted as the image enve-ope, carries the global spatial structure of the image (i.e.,he structure of the image as a whole), whereas the local
786 J. Opt. Soc. Am. A/Vol. 27, No. 4 /April 2010 V. Sierra-Vázquez and I. Serrano-Pedraza
hase and local orientation pick up the fine spatial struc-ure of the image. In this vein, it has been shown that,ue to the low-pass nature of the local amplitude of theonogenic signal, it conveys holistic spatial information
bout the original image, information that is relevant toatch the global spatial structure of a variety of texturednd non-textured natural as well as synthetic images64].
In the following section, we will apply the Riesz trans-orms and monogenic signal for isotropic single-bandM–PM decomposition of the geometrical-optical illusion
mages shown in Fig. 1.
. METHOD. Input Images
nput images used here (see Fig. 1) were the standardonfigurations of the six illusions as they appear in au-horitative works on geometrical-optical illusions [2,3]nd perceptual organization [15] and are widely extendedn popular textbooks on perception [65,66]. All input im-ges had 512 512 pixels, two gray levels, and a constantine width of six pixels. Details on original bibliographicources, description of the relevant parameters of illusionmages, and modifications of the original figures appear inable 2. Before processing, all images were transformed
nto a contrast function with a zero DC component, thenadded with 1 s images up to 1024 1024 pixels to avoidraparound effects.
Table 2. Relevant Parameter Description of GeomDifferences from the
llusion Original sourceValues of relevinput images
üller-Lyer Müller-Lyer, 1889 [4]Figs. 2(b) and 2(f)
Length of shafRatio of length0.410Angle of arrow
onzo Ponzo, 1982 [7]Fig. 6
Length of horizHorizontal pos295Angle betweenlines: 45°
elboeuf Delboeuf, 1892 [8]Fig. 25
Fixed circle raInner circle raOuter circle ra
oggendorf Robinson, 1972 [2]Fig. 3.37
Separation of vOrientation of120°
ering Ehrenstein, 1925 [11]Fig. 40
Separation of vAngle betweendivergent linesNumber of div
hrenstein Ehrestein, 1925 [11]Fig. 43
Length of squaAngle betweendivergent linesNumber of div
. Procedure for Computing Single-Band AM–PMecomposition and the Monogenic Signal of a Given
magehese illusion images (like other synthetic and natural
mages) have broadband Fourier spectra, so the imageodel described in Eq. (1) does not fit them particularlyell. Thus, it is necessary to pre-filter the image with a
hosen band-pass filter before applying the Riesz trans-orms. Specifically, single-band AM–PM demodulation ofgiven image is the process of computing the AM and PM
omponents of a band-pass version of the image or,quivalently, at a chosen spatial scale. There are twoays of computing the monogenic signal. One way is first
o filter the image with a chosen band-pass filter and theno apply the Riesz transforms. The other way, based onhe associative property of convolution operation, is to ap-ly the Riesz transform to the chosen band-pass filter andhen to filter the image with these new filters. This latterrocedure generates a triplet of so-called sphericaluadrature filters (SQFs) (defined in [53]): the even parts the isotropic band-pass basis filter, with hBP as theoint-spread function (PSF), and the odd part (its conju-ate filter) is jointly composed of the two Riesz transformsf the basis filter, h1 ,h2. SQFs generated by Riesz trans-orms are a multidimensional generalization of the 1Duadrature filter generated by Hilbert transform. There-ore, in the spatial domain, the components h1 ,h2 ofhe (vector-valued) conjugate filter, h, are h1�x�K1�x��hBP�x� and h2�x�=K2�x��hBP�x�, with K1 ,K2 de-
al-Optical Illusions Used as Input Images and theinal Illusion Images
rameters of Main differences from theoriginal figure
pixelsrow and shaft:
Original figures are vertical.One part is above the other part.
bars: 97 pixelst pixel 189 and
o convergent
Original figure is horizontal.Vertical bars are made ofdiscrete points.
4 pixels4 pixels4 pixels
Original figure has smallerrelative distance between innerand outer circles.
l lines: 96 pixelse collinear lines:
Orientation of oblique collinearlines: 60°
l lines: 120 pixelsccessive
lines: 10
Original figure is horizontal.Angles between divergent linesdecrease with orientation.Number of divergent lines: 8
: 76 pixelsccessive
lines: 11
Original figure has larger lengthof square sides.Smaller angle betweendivergent lines.Number of divergent lines: 17
etricOrig
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miiwpsalwuNnf[pnagtdmt
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V. Sierra-Vázquez and I. Serrano-Pedraza Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. A 787
ned above, where the symbol � denotes convolution of 2Dunctions. Hence, according to Eq. (11), the expression forhe monogenic signal of a band-pass filtered image will be
The properties of the Riesz transform impose condi-ions on the choice of the PSF of the band-pass filter, hBP57]: it has to be symmetric to preserve the odd characterf Riesz transform kernels and with null DC component.abor sine functions are odd; popular Gabor cosine func-
ions, though symmetric, are not DC-free functions (inact, the value of the DC component depends on its shapeatio). Here, as a band-pass filter, we chose a log-normalunction in the Fourier domain for mathematical, compu-ational, and empirical reasons. Mathematically, it canave arbitrary bandwidth without introducing a DC com-onent; computationally, it is easy to implement numeri-ally in the Fourier domain; and finally, it is reminiscentf a 1D modulation transfer function of visual first-orderisual channels [67], and, with suitable parameter values,t is similar to the shape of the spatial CSF [68]. Thus, theasis (real) function is the radial log-normal function de-ned in the Fourier domain as
hBP�u� = �exp�−ln2��u�/�0�
2�2 � ⇔ �u� � 0
0 ⇔ �u� = 0� , �19�
here �u� , �u�=+�u2+v2�1/2 is the radius; �0, in cycles permage width (c/iw), is the peak radial spatial frequency;nd �, ��0, is a constant indicating the spatial spread ofhe filter and is proportional to the relative radial band-idth, in octaves (full width at half-height), Boct, �
��ln 2/2�2�Boct=0.2943Boct. Parameter values for theognormal filter used in Figs. 5–7 below are �0=32 c/ iwnd Boct=2.5 octaves (one octave for Hering illusion). Nottempt has been made to find the scales and spectralpread, i.e., the pair of �0 and Boct, for which the physicalistortions will correspond to known experimental data.hus, according to Eq. (7) and sign modification, the Fou-ier transform of even imaginary filters are
h1�u� = �iu
�u�exp�−
ln2��u�/�0�
2�2 � ⇔ �u� � 0
0 ⇔ �u� = 0� , �20�
nd
h2�u� = �iv
�u�exp�−
ln2��u�/�0�
2�2 � ⇔ �u� � 0
0 ⇔ �u� = 0� . �21�
nce the triplet of SQFs in the Fourier domain is gener-ted, we compute the Euclidean components of the mono-enic signal fR1
, fR2, fBP by means of inverse FT. From
hese components, the local amplitude, local orientation,nd local phase are easily computed from Eqs. (12)–(14)espectively. To rule out local amplitudes, local phases,nd local orientations without meaning, we set to zero thealues of f , f , f lower than max�f �x�� /100 (this has
R1 R2 BP M
inimal consequences on the reconstruction of fBP fromts computed local amplitude and local phase). For visual-zation only, the envelopes (whether smoothed or not)ere normalized to unity. The local orientation is com-uted as �M�x�=atan���h2� f��x�� / ��h1� f��x���. For theake of clarity, we will display the local orientation valuest locations in which the local amplitude has sufficientlyarge values and the Riesz transforms are not zero; other-ise, white is used. According to Eq. (14), local phase val-es should be computed as �M�x�=atan2��fR�x�� , fBP�x��.evertheless, and because �fR�x�� is always a non-egative value and fBP�x� is in R, still using the atan2unction, local phase is wrapped between 0 and � as in[57], Fig 2.11 (a)]. Therefore, we would have ��x�=0 inositive even symmetries or light lines and ��x�=� inegative even symmetries or dark lines, but ��x�=� /2 forll odd symmetries, meaning that we could not distin-uish between light-to-dark and dark-to-light edges. Al-hough in this paper we are not concerned with featureetection, we want our analysis to preserve feature infor-ation. For this purpose, we used the following equation
o compute scalar local phase:
��x� = atan2�sign��h1 � f��x���fR�x��,fBP�x��
55,63]. The algorithm used to compute the local phase is
NR, NC are the row and column dimensionsDO I=1, NRDO J=1, NC
x is the Riesz transform fR1 along x of band-passed imageBP
x=fR1 (i, j)y is the Riesz transform fR2 along y of band-passed image
o �� �−� ,��.In order to study the spatial frequency content of enve-
opes and PM carriers obtained from monogenic signals,e compute their distributions of energy as a function of
he radial spatial frequency. To avoid jumps due to theiscrete nature of our computations and Fourier trans-orm, each amplitude spectrum was weighted with aarrow-band Gaussian ring function successively cen-
tarEf
EErmbs
aa1ptwacoNB
gde
Swc(ds(2sprsNi
e3ttoetpactlt
FLafhnrs(wi8
788 J. Opt. Soc. Am. A/Vol. 27, No. 4 /April 2010 V. Sierra-Vázquez and I. Serrano-Pedraza
ered at the radial spatial frequency. Therefore, the aver-ged energy, N, in decibels (dB) referred to as Emax (dBe Emax), is defined as N=10 log�E��0� /Emax�, where¯ ��0� is the averaged, weighted energy centered at spatialrequency �0, E��0�=E��0� / �2��0�, Emax=max
�0
�E��0��, and
��0� is the weighted energy in a band centered at �0,��0�=�0
��02��F�� ,���2H�� ,� ;�0�d�d�, where F is the Fou-
ier transform both of envelope and carrier, and H is theodulation transfer function (MTF) of the weighting
and-pass Gaussian filter centered at �0 and with fixedtandard deviation (s.d.) of 2 c/ iw.
Relative low-spatial-frequency content of each envelopend PM carrier were selected by smoothing them by mean
Gaussian low-pass filter with an s.d. of 9 c/ iw and6 c/ iw, respectively (later on, the justification of thesearameter values based on the computed radial distribu-ion of energies will be provided). Custom software wasritten in FORTRAN 77 using NAG routines C06GCFnd C06FUF for FFT [69]. Software was calibrated byomparing our Riesz transform estimates with the resultsbtained in [48,50,55,57,58] with the same test images.umerical computations were run on an SGI Altix 3700x2 Server System.Figure 2 shows the steps taken to compute the mono-
enic signal corresponding to a band-pass-filtered stan-ard Müller-Lyer configuration, and the physical pres-nce of the illusion in the smoothed envelope.
PSFof SQF
Euclidean componenof monogenic signalof BP filtered input im
(a)
(b) (c)
Input image
f
1 256 512Spatial position (pixel)
512
256
1
Spa
tialp
ositi
on(p
ixel
)
hBP^
h1^
h2^
hBP
* h1
*
h2
*
h(x)
x
y
fBP
fR1
fR2
0.00
0.72
1.43
ig. 2. Computation of the Riesz transforms and polar componyer illusion image. (a) The input image is (b) convolved with thes 3D plots). The basis function in the 2D Fourier domain, hBP,requency of 8 c/ iw and relative bandwidth (full width at half-hei
1 and h2 and its corresponding Fourier transforms (displayed aary parts of the Fourier transforms because their real parts vaesult in (c) the Euclidean components of the monogenic signalpectively, for band-pass (BP) filtered input image and its two Rspherical) coordinates and displayed as gray maps, with the intehich the local amplitude has a sufficiently large value. (e) Next
mage of the local amplitude, the smoothed envelope (displayedc/ iw) is shown. (f) The envelope profiles along the horizontal s
imultaneous convolutions of the input image [Fig. 2(a)]ith the triplet of SQF [Fig. 2(b)] result in the three Eu-
lidean components of the monogenic signal [Fig. 2(c)].Note the opposite symmetries in the spatial and Fourieromains for the imaginary filters). Then the SQF re-ponses to the image input are transformed into polarspherical) components displayed as gray images [Fig.(d)]. Next to local phase, the corresponding PM carrier ishown [Fig. 2(e)]. The smoothed envelope (displayed as aerspective plot) is shown under the raw envelope. On theight-bottom part of the panel, the profiles of themoothed envelope along the shafts are shown [Fig. 2(f)].ote that these profiles present different lengths accord-
ng to the perceived illusion.Figure 3(a) presents the relationship between Euclid-
an and polar (spherical) coordinates. Figures 3(b) and(c) depict, in more understandable circular configura-ions, the meaning of the local phase and the local orien-ation, respectively. Image patches and intensity profilesf i1D features are shown with the local phase circle; ori-nted dark lines (just for convenience) are depicted withhe local orientation circle. Figure 4 shows the monogenichase of the top left of the Muller–Lyer image displayeds a vector field overlaid with local amplitude and edge lo-ation. According to Eq. (15), note that at edge location,he local monogenic phase vector points in the direction ofocal orientation (i.e., along the direction perpendicular tohe elongation of local structure rather than parallel to it
Polar components of monogenic signalof BP filtered input image
⎫ ⎥ ⎥⎥ ⎬ ⎥ ⎥ ⎥ ⎭
(d)
(e)
−−π2
0
−π2
Local orientation
θ
−π
0
π
Local phase(wrapped)
ϕ PM carrier
-1
0
1
mplitudeelope)| fM |
Envelope profilesalong the shafts
(f)
0 128 256 384 512Spatial position (pixel)
0
0.5
1
Nor
mal
ized
ampl
itude
passering
monogenic signal corresponding to a band-pass-filtered Müller-f the triplet of spherical SQFs, labeled hBP, h1, and h2 (displayedyed as an intensity image, is a radial log-normal function [peak2.5 octaves]. To understand the (odd) symmetry of Riesz kernelssity images) h1 and h2, note that these functions are the imagi-r all spatial frequencies. Convolutions of input image and SQF
ponding to the Müller-Lyer figure. Labels fBP, fR1, fR2
stand, re-ansforms. (d) Outputs of the filters were transformed into polarodes put aside. Local orientation is displayed only at locations inlocal phase, the PM carrier (i.e., cos���x��) is shown. Under theplot) result of an isotropic Gaussian low-pass filtering (s.d. ofhe low-pass-filtered envelope was normalized to unity.
ts
age
Local a(env
lowfilt
ents ofPSF odisplaght) ofs intennish focorresiesz trnsity cto theas 3D
hafts. T
am
CIltsssptPDblbltPaTfptatt
4Ftinmulbav
ibgAv
f
Ftpoo
Fspeampi(rtEtAtf
V. Sierra-Vázquez and I. Serrano-Pedraza Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. A 789
s in the Fig. 4 of [50] because of the different definition ofonogenic phase).
. Presence of Illusionsn this study, the presence of physical distortions in enve-opes of illusions of extent (length and size) was qualita-ively determined by signaling the spatial span betweenpatial locations of absolute maxima in the respectivemoothed envelopes at particular locations and along cho-en directions. Thus, we compute the absolute maxima ofrofiles along the horizontal shafts in the input image inhe Müller-Lyer illusion, along the horizontal bars in theonzo illusion, and along the horizontal diameter in theelboeuf illusion. In order to separate the effects from theackground, the presence of distortions in envelopes of il-usions of direction (orientation and shape) were signaledy thresholding the total smoothed envelopes and by se-ecting the pixels that correspond to the pertinent fea-ures of illusion images: vertical and oblique lines in theoggendorf illusion, vertical bars in the Hering illusion,nd the sides of square in the Ehrenstein illusion.hreshold values are 0.5 for Poggendorff illusion and 0.75
or Hering and Ehrenstein illusions. The same generalrocedure was used with smoothed PM carriers, althoughhe spatial span was computed between spatial location ofbsolute or relative minima for the extent illusions and ahreshold value equal to zero was used for thresholdinghe smoothed carriers of the direction illusions.
. RESULTSigure 5 shows, from left to right and for each illusion,he input image [column (a)], the band-pass filtered inputmage (i.e., fBP or third Euclidean component of the mo-ogenic signal) [column (b)], the polar component of theonogenic signal of the band-pass-filtered image [col-
mns (c), (d), (e)], and the PM carrier (i.e., the cosine ofocal phase) [column (f)]. Illusions can still be seen in theand-passed images of column (b), even though these im-ges lack low spatial frequencies. In all cases, the raw en-elopes of the psychophysically band-pass-filtered input
(a)
R1(x)
Riesz axis 1
fR2(x)
Riesz axis 2
f (x)
Real signal axis
fR(x)θ(x)
| fR (x)|
fM(x)
ϕ(x)
|f M(x)
|
ig. 3. Graphic code to show the meaning of polar (spherical) chat the real component of the monogenic signal is the third coohase. Only patches of local spatial structures and intensity profir peak, right edge, dark line or pit, and left edge) are depicted. (f phase) in four orientations. Dark bars were chosen for clarity.
mages [Fig. 5(c)] do not contain any physical distortionut seem to present low-pass characteristics related tolobal spatial structure of the image at the scale of filter.careful inspection of local phases and orientations re-
eals the type i1D features and their orientations in each
-90 °
-45°
0°
45°
90°
0 rad
−π/2 rad
π/2 rad
−π/2 rad
d
−π/2 rad−π/2 rad
(b) (c)ents of Figs. 2, 5, and 7. (a) Polar (spherical) coordinates. (Notee). Meaning of symbols in the text. (b) Circle code for the localresponding to i1D discrete features (counter clockwise, light linele code for the local orientation and an ideal dark line (i.e., � rad
Light barRight edgeDark barLeft edge
ig. 4. Monogenic phase from local phase and orientationhown in Fig. 2 is displayed as a vector field atop the local am-litude and edges. For clarity, only the monogenic phase and thenvelope of the top left quadrant of the Müller-Lyer input imagere depicted. Local amplitude is displayed as a gray intensityap (dark is higher amplitude). Overlaid monogenic phase is dis-
layed as a vector field in which the absolute local (scalar) phases proportional to the length of vector, and the local orientationaffected by the local phase sign) is represented by the angle ofotation. Arrows are displayed in a square grid and only at loca-ions in which local amplitude has a sufficiently large intensity.dge locations were found with a trivial algorithm that indicates
he locations in which local phase is equal to or near � /2 or −� /2.straight segment perpendicular to edge orientation was drawn
o indicate the edge location. The inset shows the code for theour i1D features (length of arrows indicates the feature type).
±π ra
omponrdinatles corc) Circ
otttncewtamng
acftt
rteacir
FIiHFft
790 J. Opt. Soc. Am. A/Vol. 27, No. 4 /April 2010 V. Sierra-Vázquez and I. Serrano-Pedraza
f them. Also, note how accurately the type and orienta-ion of edges and lines are encoded by the local phase andhe local orientations, respectively. The isotropic nature ofhe Riesz transforms (after filtering by the isotropic log-ormal filter) is shown in the panels of local orientationorresponding to the Hering and Ehrenstein illusions andspecially in the local orientation of the Delboeuf illusion,hich replicates almost exactly the circle code for orien-
ation of Fig. 3(c). Each PM carrier conveys informationbout the fine structure of the input image (in fact, iterely replicates, in an almost posterized way, the origi-
al input image of the first column, but surrounded byray or black continuous ribbons) and does not present
(a)Input image
(b)BP filtered
input image
Polar
⎫
(c)Envelope
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Muller-Lyer
Ponzo
Delboeuf
Poggendorff
Hering
Ehrenstein
ig. 5. Polar (spherical) components of monogenic signals of thenput images. (b) Band-pass (BP) filtered input images (i.e., fBPsotropic MTF whose radial profile was a lognormal function (peering and 2.5 for the rest). (c) Envelopes displayed as an intensior clarity, local orientation is displayed in white at locations in w
orms are zero. (e) Local phases. (f) PM carriers from local phasation, and PM carrier are depicted at the bottom of each column
ny physical distortion correlating to the respective per-eived illusion. In the following, we will study the spatialrequency content of envelopes and PM carriers by findingheir distribution of energy as a function of the radial spa-ial frequency.
Figure 6 shows the averaged weighted energy N, in dBe Emax, as a function of the radial spatial frequency forhe envelope [Fig. 6(a)] and the PM carrier [Fig. 6(b)] ofach image illusion. Except for the Delboeuf illusion, theveraged energy of each envelope decreases monotoni-ally with the radial spatial frequency, but in all illusions,t becomes negligibly small (around −30 dB re Emax) foradial spatial frequency larger than 32 c/ iw, showing the
ents of monogenic signaliltered input image
⎥ ⎬ ⎥ ⎥ ⎥ ⎭
(d)al orientation
(e)Local phase
(f)PM carrier
512
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−π 0 π -1 0 1
trical-optical illusion images of Fig. 1 and their PM carriers. (a)rd Euclidean coordinate of monogenic signal) obtained with antial frequency of 32 c/ iw and 1 octave of relative bandwidth forge (all envelopes are normalized to unity). (d) Local orientations.he normalized envelope is lower than 0.01 and both Riesz trans-cos���x��. Intensity codes for envelope, local phase, local orien-
componof BP f
⎥ ⎥
Loc
512 1
512 1Spati
1 −−π2
geomeor thi
ak spaty imahich t
es, i.e.,.
lpb(p
[tfpdcEssiFnb
tcwa=FmApstshm
us[rfe(lcltveiaaacpsuiicmssstsrIr
FsrEesTfitttd((c
V. Sierra-Vázquez and I. Serrano-Pedraza Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. A 791
ow-pass nature of envelopes. The inset at the bottom leftart of Fig. 6(a) depicts, for comparison, the radial distri-utions of energy of low-pass (solid curves) and band-passdashed curves) signals. In contrast to the example low-ass signal, the energy distribution of each PM carrier
(a)
(b)
1 16 32 48 64
Radial spatial frequency (c/image width)
-60 -60
-40 -40
-20 -20
0 0
Ave
rage
den
ergy
(dB
reE
max
)
MT
Fradialam
plitude(dB
reA
max)
1 16 32 48 64
Radial spatial frequency (c/image width)
-60 -60
-40 -40
-20 -20
0 0
Ave
rage
den
ergy
(dB
reE
max
)
MT
Fradialam
plitude(dB
reA
max)
Muller-LyerPonzoDelboeufMTF radial profile of low-pass filter
PoggendorffHeringEhrenstein
ig. 6. Distribution of averaged, weighted energy with radialpatial frequency up to 64 c/ iw for (a) envelopes and (b) PM car-iers from the respective monogenic signal of each illusion image.ach panel shows for each illusion image the averaged, weightednergy N, in dB referred (re) to Emax, as a function of the radialpatial frequency �0, in c/iw. (See text for computation details).he thickest curves are the MTF radial profiles of the low-passlters, in dB re Amax �Amax=1�. Arrows on the x axis indicatehe spatial frequency values at which MTF radial profiles takehe amplitudes indicated by arrows in the y axis. The insets inhe bottom left part of each panel depict, for comparison, the ra-ial distributions of (a) a monotonic-energy decreasing signalsolid curve) and a purely band-pass (dashed curve) signal andb) a mixed-energy decreasing and band-pass signal (dashedurve).
Fig. 6(b)], though decreasing, does not decrease mono-onically. Thus it is neither a low-pass nor a band-passunction but a mixed one with a monotonically decreasingatch followed by a (narrow-band) main bump and otherecreasing secondary ones. Cutoff frequency for the de-reasing patch ranges from 9 to 11 c/ iw at −20 dB remax (Hering and Ehrenstein extrapolated), and the low-
patial frequency main bump peaks at 19 c/ iw for Ehren-tein, 29 c/ iw for Hering, and 20 c/ iw for the rest of thellusions. For comparison, the inset at the bottom left ofig. 6(b) depicts the radial distribution of energy of a sig-al with one decreasing patch and one energy-decreasingump (dashed curve).Next, we selected the spatial information conveyed by
he relative low spatial frequencies of envelopes and PMarriers by means of isotropic low-pass Gaussian filtersith an s.d. of 9 c/ iw and 16 c/ iw, respectively. Radialmplitude profiles of filter MTFs (in dB re Amax, Amax1) are indicated by the thickest curves in both panels ofig. 6. The MTF of the filter for envelopes covers theeaningful part of envelope spectra (it drops to −55 dB remax at 32 c/ iw). The MTF of filter for PM carriersasses, almost undistorted, the decreasing patch of theirpectra (it falls to −2 dB re Amax at 11 c/ iw) at the sameime that it passes, though attenuated, the main low-patial-frequency band-pass bump (it drops to half-eight, i.e., −6 dB re Amax, at 19 c/ iw) to maintain theeaning of carrier concept.Figure 7 shows the images of smoothed envelopes [col-
mn (b)], measures of illusory effects taken frommoothed envelopes [column (c)], smoothed PM carrierscolumn (d)], and measures taken from smoothed PM car-iers [column (e)]. Original illusion images are also shownor comparison [column (a)]. The smoothed envelope ofach illusion shows differences from the raw envelopei.e., without filtering) that reveal physical distortions ofength, size, direction, or shape correlating with the per-eived (illusory) distortions in the input image and corre-ating with the conscious perceived illusion. In column (c)he presence of distortions is shown in the smoothed en-elope intensity profiles at specific locations (illusions ofxtent) or in the total envelope (illusions of direction). Forllusions of extent, panels depict the envelope profileslong each horizontal shaft for the Müller–Lyer figure,long the horizontal bars for the Ponzo figure, and thelong the horizontal diameter for the Delboeuf illusion. Asan be seen in the Müller–Lyer case, distances betweeneaks in the smoothed envelope are longer for the upperhaft than for the lower one. For the Ponzo illusion, thepper line is longer than the lower line. For the Delboeuf
llusion, the distance between the maxima of the envelopes greater for the right-hand circle than for the left-handircle. For illusion of direction, note that the centers ofass of the oblique lines in the Poggendorff illusion
moothed envelope are not collinear. In the Hering illu-ion, by thresholding the smoothed envelope, the envelopehows the curvature of the intermediate parts of the ver-ical bars. Finally, the illusory distortion of horizontalides of the square clearly appears physically in the Eh-enstein illusion by thresholding the smoothed envelope.n contrast to the smoothed envelopes, smoothed PM car-iers of Fig. 7(d) closely replicate the unfiltered PM carri-
ecst
5Tao
bbctefiittw
Fesl((fibiipat
792 J. Opt. Soc. Am. A/Vol. 27, No. 4 /April 2010 V. Sierra-Vázquez and I. Serrano-Pedraza
rs [cf. Fig. 5(d)] and do not show any physical distortionsorrelating with the respective perceived illusions, ashown by the envelope profiles of extent illusions andhresholded images of direction illusions [Fig. 7(e)].
. DISCUSSIONhis paper presents numerical results about amplitudend phase demodulation of a sample of geometrical-ptical illusions. We have shown that envelopes obtained
(a)Input image
(b)Low-pass filtered
envelopeMeas
LP
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Muller-Lyer
Ponzo
Delboeuf
Poggendorff
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ig. 7. Physical presence of illusory distortions in the smoothenvelopes displayed as intensity images, obtained by filtering themoothed envelopes are normalized to unity. (c) 1D profiles (exteopes. For illusions of extent, thick curves indicate 1D envelopPonzo), and left horizontal diameter (Delboeuf), respectively; thMüller-Lyer), lower bar (Ponzo), and right horizontal diameter (Dles indicate the physical length between absolute maxima. Distoy thresholding the smoothed envelopes (threshold values are 0.llusions). Intensities larger than the respective threshold are disan filter (s.d. of 16 c/ iw). (e) 1D profiles (extent illusions) and 2Drofiles is the same as in (c). Straight segments drawn over profind Ponzo) or relative (Delboeuf) minima. Amplitude threshold vahreshold value are displayed in black.
y means of Riesz transforms and monogenic signal ofand-passed images of these illusions present low-passharacteristics related to the global spatial structure ofhe original images. In addition, the local phase and ori-ntation components of monogenic signals pick up thene spatial structure, such as the symmetries related to
1D spatial features and their orientations. A low-pass fil-ering of these envelopes reveals physical distortions inhe directions of perceived illusions. Because of this fact,e can conclude that, contrary to widespread belief, spa-
t ofe
(d)Low-pass filtered
PM carrier
(e)Measurement of
LP PM carrier
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lope. (a) Input images (depicted for comparison). (b) Smoothedpes with an isotropic Gaussian low-pass filter (s.d. of 9 c/ iw). Allsions) and 2D thresholds (direction illusions) of smoothed enve-les along the upper horizontal shafts (Müller-Lyer), upper barves indicate envelope profiles along the lower horizontal shaft
uf), respectively. Straight segments over smoothed envelope pro-in smoothed envelopes of direction illusions have been enhancedthe Poggendorff illusion and 0.75 for the Hering and Erhensteinin black. (d) PM carriers filtered by an isotropic low-pass Gauss-olds (direction illusions) of smoothed PM carriers. Meaning of 1Dillusion of extent show the spans between absolute (Müller-Lyerillusions of direction is equal to zero. Amplitudes lower than the
(c)uremenenvelop
256
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V. Sierra-Vázquez and I. Serrano-Pedraza Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. A 793
ial information about these illusions does not reside inhe lower spatial frequencies of the image spectrum but inhe local amplitude or envelope result of its isotropic (i.e.,iesz-made) AM–PM decomposition.
. Riesz Transforms and Monogenic Signal as aathematical Tool to Factor a Given Image in Envelope
nd Carrierven though we are not concerned here with comparing
he performance of different multidimensional demodula-ion algorithms, it is clear that the Riesz transform andhe monogenic signal overcome the problems with therocedures used in [32] and reviewed in the Introduction.he monogenic signal is not the only procedure for isotro-ic AM–PM decomposition. Other Riesz-based isotropicemodulation algorithms [56,70] might be used to com-ute the envelope. In contrast to global directional de-odulation procedures (such as the directional Hilbert
ransform or Daugman’s AMPM algorithm), there are nopurious elongations in any direction; and unlike local op-rators such as DESA, local phase has realistic values inny location with no singularities at the input image dis-ontinuities. Although the Riesz transform is a typicalon-local operator, values of local phase and orientationaithfully represent the type and orientation of local linesnd edges. The performance of the Riesz transform doesot depend on the chosen band-pass basis filters if theyeet the requirements reported above; actually, we ob-
ained similar results with isotropic band-pass filter basisunctions other than the log-normal function, assuminghey have the same values for peak frequency and band-idth. On the other hand, visualization of the monogenic
ignal components of an image is not as complicated asome authors have suggested [56]. It is true that the re-ulting monogenic signal at every location is a three-omponent vector, implying that three separate imagesre required to display it. However, in fact, as shown inig. 4, the monogenic signal can be displayed in one single
mage, plotting the local amplitude as an intensity mapverlaid with local phase and orientation, both displayedointly as a 2D vector field according to Eq. (15).
So far, the Riesz transforms and the monogenic signals instruments for computing the envelope and PM car-ier of an image are not in common use within the visionommunity. However, some ad hoc attempts have beenade in both the image processing [71] and spatial vision
iterature [72] to use or “extend” to 2D images the con-epts of envelope and local energy. “Spatial envelope” ishe term coined by Oliva and Torralba [71] to model thehape of a scene with a set of perceptual dimensions. It isvery confusing term because it is hardly related to the
efinition of envelope from Hilbert and Riesz transforms.n [72] the pair of 2D oriented even (cosine) and odd (sine)lters used to compute “energy maps” simply generalizehe psychophysically based kernel used in [62] to 2D sig-als. “Energy” is then defined as the pointwise quadraticum of the pair of filtered outputs at every scale and ori-ntation; the energy map is obtained as the sum of the en-rgy over different scales and orientations. Image enve-ope and local energy computed from Riesz transformsan be used as a baseline against which to compare thesed hoc “envelope,” “second-order information,” or “energy”
btained from biologically or psychophysically based fil-ers.
. Image Envelope and Geometrical-Optical Illusionshere is both theoretical and experimental evidence that
ow spatial frequencies of input images are not necessaryo perceive these illusions [as the reader can see forimself/herself in Fig. 5(b)]. Low-spatial-frequency fil-ered images present physical distortions with respect tohe original images; but low-spatial-frequency filteringannot explain the existence of these illusions becauseistortions are also perceived in other spatial frequencyands. Hence, there must be some further component,ommon to all spatial frequency bands, that carries thehysical distortions correlative to the illusions. Thus, in-ormation about illusions will reside (also for low-spatial-requency filtered images) in the envelope, in the PM car-ier, or in both. Band-pass filtering implicitly introducesn amplitude modulation in broadband images (as anyass filter does [73]), including phase-only images23,32]), but not the physical distortions (see Fig. 5(b)).he envelope of these filtered images [Fig. 5(c)] presents
ow-pass characteristics in such a way that a furthermoothing with a low-pass filter introduces the illusionFig. 7(c)] similarly to the low-pass filtering of a broad-and luminance image. Actually, the procedure used heres a translation of Ginsburg’s original idea from the first-rder front-end mechanisms to second-order mechanisms.ere, the low-pass filtering is applied to the envelope of
he image instead to the input image directly. This fur-her low-pass filtering introduces physical distortions inhe spatial and in the intensity properties of the envelopehat look like the perceived illusion. On the other hand,fter low-pass filtering that maintains the image spatialharacteristics, PM carriers do not present any physicalistortion [Figs. 7(d) and 7(e)], though they convey thepatial structure of input images. These results suggesthat the spatial information about illusions is physicallyresent in the envelope of the filtered band-pass imagend not in the PM carrier, so the energy information inhese illusions is more decisive than the structural or geo-etrical information. That may not hold in general: for
ther tasks, such as discrimination or recognition tasks,he PM carrier might possibly play a more decisive struc-ural role than the envelope.
In this study, we computed the monogenic signals for aiven scale, although a scale can be varied continuouslyr discretely to obtain a representation in the so-calledonogenic scale-space [52]; so the scale is a parameter
hat we have to deal with. Because in this work we com-uted the envelopes from monogenic signals at a chosen,xed scale, it is pertinent to ask whether the envelope iscale-dependent in these images. Monogenic scale-spaceepresentation or multiband decomposition, channelinghe image into a bunch of SQFs parameterized by the ra-ial spatial frequency, is beyond the scope of this paper.evertheless, we also computed the monogenic signals of
nput images at scales centered at 8, 16, and 64 c/ iw.ith adequate parameter values, the results (not shown)
ndicate that there are also distortions in their envelopes.hus, spatial information about these illusions does noteem to be confined to a single spatial scale.
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794 J. Opt. Soc. Am. A/Vol. 27, No. 4 /April 2010 V. Sierra-Vázquez and I. Serrano-Pedraza
The result that the envelope physically carries theeometrical-optical illusions is very robust. A similar re-ult is obtained not only in different scales but also withhe same versions of these illusions, varying the relation-hips between their parts (at least for the Müller-Lyer il-usion, the magnitude of the illusion measured in themoothed envelope quantitatively follows the magnitudeariations of the perceived illusion) and with other ver-ions of illusions (such as the Müller-Lyer configurationsn [2,32]). (Results not shown here).
It is possible that there is no single cause for these il-usions and that many levels of processing contribute tohe final distorted percept [3], but one thing we haveearned is that a single image transformation is enough toxplain the physical distortions that are perceived in allhe geometrical-optical illusions studied here. Strangely,arly phenomenological explanations of some of these il-usions were confirmed by the smoothed envelopes ob-ained from the illusion monogenic signals shown in thisork. Müller-Lyer himself wrote about his own illusion
hat “The lines are judged to differ in length because theudgment not only takes the lines themselves into consid-ration, but, also, unintentionally, some part of the spacen either side” [4], (p. 266). Delboeuf wrote that “The is-ue is that the eye, dealing with measuring the diameter,s as it is constrained within the edges of the small centralircle, whereas on the other hand, when aiming at mea-uring the other diameter, it goes so to speak, beyond itsxtremities, attracted by the outer circle” [8], (pp. 553,54). That is exactly what the Riesz transforms show inhe envelopes of these illusions [as can be seen in the 3Durface of Fig. 2 and in the corresponding image enve-opes of Fig. 7(b)]. The human visual machinery is able toemodulate visual stimuli to produce an internal repre-entation that recovers the envelope of second-order (es-ecially contrast modulated) visual stimuli [28,29,74].ere we have demonstrated, with a procedure that is in-ependent of any second-order visual model, that the nu-erically processed envelopes of these geometrical-optical
llusions present physical distortions depending on the il-usions. Therefore, if the inner visual representations ofnput images of Fig. 1 were based on the spatial charac-eristics of their AM–PM decomposition, then our con-cious perceptual experience would reflect the (illusory)hysical distortions commonly labeled as geometrical-ptical illusions. Geometrical-optical illusions would beatural by-products of the same spatial second-order vi-ual process involved in envelope recovery.
CKNOWLEDGMENTShis research was supported in part by grant BSO2002-0192 from the Ministry of Science and TechnologySpain). The numerical computations were carried out athe Computer Centre of UCM, Spain. We thank Dr. Luis. López-Bascuas and Dr. Jenny C. A. Read for valuableomments on this manuscript. We also thank the two re-iewers for their helpful comments and suggestions. Pre-iminary results were reported in abstract form at the7th European Conference on Visual Perception held in
udapest, August 22–26, 2004, and at the Internationalongress of Mathematicians held in Madrid, August 22–0, 2006.
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