Complex signal representation, Mandel’s theorem, andspiral phase quadrature transform

Kedar KhareGE Global Research, One Research Circle, Niskayuna, New York 12309, USA

*khare@ge.com

Received 3 January 2008; accepted 18 February 2008;posted 7 March 2008 (Doc. ID 91215); published 8 April 2008

Complex (or analytic) signal representation as introduced by Gabor plays an important role in opticalsignal processing and in coherence theory of optical fields. Several definitions for extending the notionof complex signal representation to two dimensions have appeared in the literature. These defini-tions differ in their choice of the quadrature transform for a two-dimensional signal. We study the pro-blem of determining the complex representation for two-dimensional real signals (or images) using aleast-square minimization framework first used by Mandel [J. Opt. Soc. Am. 57, 613 (1967)]. In parti-cular, we seek a suitable quadrature transform such that the resultant complex image has the least fluc-tuating envelope in an ensemble-averaged sense. It is observed that the spiral phase quadraturetransform for two-dimensional signals is a solution of this analysis. © 2008 Optical Society of America

OCIS codes: 000.3860, 070.0070, 200.3050.

1. Introduction

The concept of complex signal representation forone-dimensional real signals was introduced in com-munication theory by Gabor [1]. This representationis also referred to as the “analytic signal” to empha-size that it is a boundary value of a function thatis analytic in the lower half of the complex plane.Gabor’s complex signal representation is often astarting point in the study of problems related todemodulation of AM–FM signals, local frequencymethods in signal analysis, spectral analysis, inter-ferometry, radar, coherence theory of optical fields,etc. The widespread usage of the complex signalrepresentation for one-dimensional signals hasprompted several definitions in the literature thatextend this concept to two-dimensional signals.The analytic or complex signal corresponding to a

real image is typically obtained as a complex sumof the real image and its quadrature transform. Acommon theme in the extensions of the complexrepresentation from one to two dimensions is tochoose a two-dimensional equivalent of the one-

dimensional sgnðνÞ filter function in the Fourierdomain, where

sgnðνÞ ¼ 1; ν > 0 ¼ −1; ν < 0: ð1Þ

One then seeks a transform that approximatelyturns a function of the form cos½ψðx; yÞ� intosin½ψðx; yÞ� and vice versa with a sign change. Someof the choices for two-dimensional sgn filter have ahalf-plane or quadrant-based symmetry [2–4], withthe result that the corresponding complex imagesare dependent on the image orientation. Optical pro-cessing setups for obtaining quadrature transformsthat remove the angular anisotropy in the imageirradiance have also been proposed [5,6]. The spiralphase quadrature transform has received muchattention in the past few years after the detailedanalysis of its properties by Larkin et al. [7,8], whoalso provide an extensive literature survey on thesubject of multidimensional quadrature transforms.The spiral phase transform of an image is obtainedby multiplication of the Fourier transform of theimage with a spiral phase factor expðiϕÞ followedby the inverse Fourier transform operation, whereϕ is the polar angle in the Fourier domain. This

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process is equivalent to a two-dimensional convolu-tion of the image with a kernel i expðiθÞ=ð2πr2Þ,where ðr; θÞ are the polar coordinates in the imagedomain. The function expðiϕÞ is treated here as atwo-dimensional generalization of the sgn functionas it behaves similar to the one-dimensional sgnfunction along any line through the origin. Larkinet al. illustrated this transform (along with an addi-tional orientational phase factor) with demodulationof fingerprint and fringe patterns and also presenteda stationary phase analysis for this transform. Highcompression factors for demodulated fingerprintpatterns have been achieved recently by employingthe spiral phase transform [9]. In a separate ap-proach Servin et al. [10] proposed an n-dimensionalquadrature transform operator in the form of aninner product of two n-dimensional vectors fields.Their operator also reduces to the spiral phase trans-form in the two-dimensional case.The extension of the quadrature transform to the

two-dimensional case by generalizing the sgnðνÞfunction and requiring approximate transform rela-tions between sines and cosines is usually justifiedby analogy with the one-dimensional case. In thispaper we take a different approach to the problemof defining the complex signal representation intwo dimensions by using a least-square minimiza-tion framework in early work by Mandel [11,12].Mandel’s result may be stated as follows. Considera one-dimensional real signal xðtÞ that is a realiza-tion of a stationary random process. We relate thesignal xðtÞ to its quadrature yðtÞ by means of a con-volution kernel kðtÞ,

yðtÞ ¼Z

∞

−∞

dt0kðt − t0Þxðt0Þ; ð2Þ

and define a complex representation correspondingto xðtÞ as

zðtÞ ¼ 12½xðtÞ þ iyðtÞ� ¼ zoðtÞ expði2πνotÞ: ð3Þ

Here νo is the carrier frequency of the signal and zoðtÞrepresents its complex envelope. Mandel showedthat the choice KðνÞ ¼ FTfkðtÞg ¼ −isgnðνÞ corre-sponding to the Hilbert transform is a solution ofa minimization problem—this choice minimizesthe fluctuation in the complex envelope as measuredby the ensemble average:

�����dzoðtÞdt

����2�:

The complex (or analytic) signal thus provides thesmoothest envelope-carrier type representation inthe sense of Mandel’s theorem. The above one-dimensional definition for analytic signal is not di-rectly applicable to truly two-dimensional signalsthat may potentially possess an envelope-carriertype representation, e.g., closed circular fringes,

fingerprint patterns, and digitally recorded holo-grams. We believe that Mandel’s approach providesan independent criterion that is suitable for extend-ing the notion of complex signal representation tothe multidimensional case. In what follows we firstformulate a two-dimensional version of this minimi-zation problem and then seek an appropriate quad-rature transform as its solution. We show thatthe spiral phase quadrature transform is a solutionof this analysis. Further we also suggest a general-ization of the spiral phase transform to the three-dimensional case.

2. Complex signal representation in two dimensions

Consider a two-dimensional real signal gðr; θÞ thatis a realization of a wide-sense homogeneous (or sta-tionary) random field [13]. We express the signal asa Fourier expansion:

gðr; θÞ ¼Z Z

ρdρdϕGðρ;ϕÞei2πrρ cosðθ−ϕÞ: ð4Þ

The Fourier expansion above will be assumed toexist, in the sense of generalized function theory.The terminology “wide-sense homogeneous randomfield” [13] above implies that the mean hgðrÞi is a con-stant and the autocorrelation hgðrÞgðr0Þi is only afunction of the difference vector ðr − r0Þ as denotedby Cðr − r0Þ. We will further assume that the meanhgðrÞi is equal to zero. The spectrum Sðρ;ϕÞ ofgðr; θÞ can be expressed as follows:

hG�ðρÞGðρ0Þi ¼Z Z

d2rd2r0hgðrÞgðr0Þi

× exp½i2πðr · ρ − r0 · ρ0Þ�

¼Z Z

d2rd2r0Cðr − r0Þ

× exp½i2πððr − r0Þ · ρþ r0 · ðρ − ρ0ÞÞ�¼ SðρÞδðρ − ρ0Þ; ð5Þ

where, in the last step above we define the spec-trum SðρÞ through the relation (Wiener—Khintchinetheorem):

SðρÞ ¼Z

d2uCðuÞ expði2πu · ρÞ: ð6Þ

Equation (5) may be explicitly written in polar coor-dinates ðρ;ϕÞ as

hG�ðρ;ϕÞGðρ0;ϕ0Þi ¼ Sðρ;ϕÞ 1ρ δðρ − ρ0Þδðϕ − ϕ0Þ: ð7Þ

The expression for the two-dimensional deltafunction,

δðρ − ρ0Þ ¼ 1ρ δðρ − ρ0Þδðϕ − ϕ0Þ; ð8Þ

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is not valid at zero frequency ρ ¼ 0, where the polarangle ϕ is undefined. This does not cause any trou-ble for us since we have assumed gðr; θÞ to be a zeromean signal. The spectrum Sðρ;ϕÞ of a real sig-nal gðr; θÞ is also expected to have an inversionsymmetry:

Sðρ;ϕÞ ¼ Sðρ;ϕþ πÞ: ð9Þ

In analogy with the one-dimensional case, we de-fine the quadrature transform of gðr; θÞ through itsconvolution with a kernel kðr; θÞ. Denoting the two-dimensional Fourier transform of kðr; θÞ by Kðρ;ϕÞ,we define the complex signal corresponding togðr; θÞ as

gzðr; θÞ ¼12

Z ZρdρdϕGðρ;ϕÞ½1

þ iKðρ;ϕÞ�ei2πrρ cosðθ−ϕÞ: ð10Þ

Here Kðρ;ϕÞ is the filter function that we want to de-termine. In the one-dimensional case, the sgnðνÞfunction retains the amplitude of all the Fouriercomponents. We will stick to this property in thehigher dimensional extensions, so that Kðρ;ϕÞ maybe assumed to be of unit amplitude given by

Kðρ;ϕÞ ¼ exp½iαðρ;ϕÞ�; ð11Þ

where αðρ;ϕÞ is some real function of ðρ;ϕÞ. Further,we expect this new definition in two dimensions toreduce to the more familiar one-dimensional caseif we suppress one of the dimensions. We thereforeput another condition on Kðρ;ϕÞ:

Kðρ;ϕÞ ¼ −Kðρ;ϕþ πÞ: ð12Þ

In other words Kðρ;ϕÞ is multiplied by −1 along anyradial line as one crosses the origin.We now define the complex envelope aðr; θÞ of the

analytic signal gzðr; θÞ in Eq. (10) as

aðr; θÞ ¼ 12

Z ZρdρdϕGðρ;ϕÞ½1

þ iKðρ;ϕÞ�ei2πrðρ−ρoÞ cosðθ−ϕÞ: ð13Þ

Here ρo is the carrier frequency of the signal. To get asimple understanding of ρo, consider a pure carriersignal Joð2πrρoÞ, which is an inverse Fourier trans-form of a delta ring: Gðρ;ϕÞ ¼ δðρ − ρoÞ=ð2πÞ. The cor-responding complex envelope,

aðr; θÞ ¼ ρo4π

Zdϕ½1þ iKðρo;ϕÞ� ¼ constant; ð14Þ

conforms with our notion of an envelope. A moreclear understanding of ρo will be provided in the dis-cussion later in this section. We will now consider the

fluctuation of the complex envelope aðr; θÞ. The par-tial derivatives of aðr; θÞ are

∂aðr; θÞ∂r

¼ iπZ Z

ρdρdϕGðρ;ϕÞ

× ½1þ iKðρ;ϕÞ�ei2πrðρ−ρoÞ cosðθ−ϕÞðρ − ρoÞ× cosðθ − ϕÞ; ð15Þ

1r∂aðr; θÞ

∂θ ¼ −iπZ Z

ρdρdϕGðρ;ϕÞ

× ½1þ iKðρ;ϕÞ�ei2πrðρ−ρoÞ cosðθ−ϕÞðρ − ρoÞ× sinðθ − ϕÞ: ð16Þ

From the definition of the complex signal gzðr; θÞ inEq. (10), we note that the Fourier transform of gzðr; θÞis given by

Gzðρ;ϕÞ ¼12Gðρ;ϕÞ½1þ iKðρ;ϕÞ�: ð17Þ

We can write an equation similar to Eq. (7) forGzðρ;ϕÞ and relate the spectrum Sðρ;ϕÞ of gðr; θÞ tothe spectrum Szðρ;ϕÞ of gzðr; θÞ as

Szðρ;ϕÞ ¼14Sðρ;ϕÞj1þ iKðρ;ϕÞj2: ð18Þ

With the help of Eqs. (7), (15), (16), and (18), we nowcalculate the fluctuation of the complex envelope:

�����∂aðr; θÞ∂r

����2þ 1

r2

����∂aðr; θÞ∂θ

����2�

¼ π2Z Z

ρdρdϕðρ − ρoÞ2Sðρ;ϕÞj1þ iKðρ;ϕÞj2

¼ 4π2Z Z

ρdρdϕðρ − ρoÞ2Szðρ;ϕÞ: ð19Þ

The fluctuation of the complex envelope is thus seento be proportional to the mean-square deviation ofthe spectrum Szðρ;ϕÞ of the analytic signal gzðr; θÞ,from the carrier frequency ρo. Our goal now is to findan appropriate filter function Kðρ;ϕÞ that will mini-mize this mean-square deviation. It is clear that themean-square deviation will be a minimum when itis calculated with respect to the mean frequency ρodefined as

ρo ¼R R

ρdρdϕρSzðρ;ϕÞR RρdρdϕSzðρ;ϕÞ

: ð20Þ

We can therefore write the minimization problem as

minimizeZ Z

ρdρdϕðρ − ρoÞ2Szðρ;ϕÞ; ð21Þ

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with the constraint:

Z Zρdρdϕðρ − ρoÞSzðρ;ϕÞ ¼ 0: ð22Þ

3. Solution of the minimization problem

We have assumed Kðρ;ϕÞ to be of the formexp½iαðρ;ϕÞ� [see Eq. (11)], so that, j1þ iKðρ;ϕÞj2 ¼2½1 − sin αðρ;ϕÞ� and the minimization problem maybe restated as

minimizeZ Z

ρdρdϕðρ − ρoÞ2Sðρ;ϕÞ½1 − sin αðρ;ϕÞ�;ð23Þ

with the constraint

Z Zρdρdϕðρ − ρoÞSðρ;ϕÞ½1 − sin αðρ;ϕÞ� ¼ 0: ð24Þ

Assuming that the solution of the minimization pro-blem is αðρ;ϕÞ, if we vary the solution to a neighbor-ing function αðρ;ϕÞ þ ϵηðρ;ϕÞ, where ϵ is a smallpositive number, the integral in Eq. (23) should re-main unchanged to the first order in ϵ. In perfomingthis variation the condition (12) must be satisfied,so that

exp½iαðρ;ϕÞ þ iϵηðρ;ϕÞ� ¼ − exp½iαðρ;ϕþ πÞþ iϵηðρ;ϕþ πÞ�: ð25Þ

Since the above condition is already satisfied whenϵ ¼ 0, we will assume ηðρ;ϕÞ to obey the relation:

ηðρ;ϕÞ ¼ ηðρ;ϕþ πÞ: ð26Þ

Apart from this condition the function ηðρ;ϕÞ is com-pletely arbitrary. This variation in αðρ;ϕÞ alsochanges the mean frequency to ðρo − δρoÞ accordingto Eq. (24), with δρo proportional to ϵ. Making thesesubstitutions in Eq. (23) we collect the terms that arefirst order in ϵ and equate them to zero. Thus,

2δρoZ Z

ρdρdϕðρ − ρoÞSðρ;ϕÞ

× ½1 − sin αðρ;ϕÞ� − ϵZ Z

ρdρdϕðρ − ρoÞ2

× Sðρ;ϕÞηðρ;ϕÞ cos½αðρ;ϕÞ� ¼ 0: ð27Þ

The first term in the above equation vanishes accord-ing to the constraint (24). The integration over ϕ inthe second term above can be rewritten in view of thesymmetry properties (9) and (26) of Sðρ;ϕÞ andηðρ;ϕÞ, respectively, to get

Z π

ϕ¼0

Zρdρdϕðρ − ρoÞ2Sðρ;ϕÞηðρ;ϕÞfcos½αðρ;ϕÞ�

þ cos½αðρ;ϕþ πÞ�g¼ 0: ð28Þ

In order for the above integral to vanish identicallyfor arbitrary functions Sðρ;ϕÞ and ηðρ;ϕÞ, we musthave

cos½αðρ;ϕÞ� þ cos½αðρ;ϕþ πÞ� ¼ 0: ð29Þ

The desired filter function Kðρ;ϕÞ ¼ exp½iαðρ;ϕÞ�must obey this cosine condition. Note that the choicecorresponding to the spiral phase transform,

αðρ;ϕÞ ¼ ϕþ constant; ð30Þ

satisfies this condition everywhere except at ρ ¼ 0,where the phase ϕ is not defined. Once again thisshould not be a problem since we have assumedour signal gðr; θÞ to have zero mean. The constantterm above may be related to the overall fringe orien-tation factor in Larkin et al.’s work [7,9]. Expandingthe variation of the integral in (23) to second order inϵ one can further confirm that the choice of αðρ;ϕÞ inEq. (30) gives the minimum of the integral. For thereference of the reader, we also quote the conditionobtained by Mandel [11] for the one-dimensionalcase:

cos½αðνÞ� ¼ 0; ν > 0; with αð−νÞ ¼ −αðνÞ; ð31Þ

which leads to

KðνÞ ¼ exp½iαðνÞ� ¼ −isgnðνÞ: ð32Þ

We note that the generalizations of the sgn func-tion using the half-plane or the quadrant-based de-finitions do not satisfy the cosine condition (29)along the discontinuity lines and hence are not thesolutions of the constrained minimization problemin Eqs. (23) and (24). This is in agreement withthe observation by Larkin et al. [7] that the spiralphase quadrature transform tends to give complexrepresentations with smoother amplitude and phaseas compared with those using half-plane generaliza-tion of the sgn function.

We note that the spiral phase transform does notexactly convert a signal of the form cos½ψðx; yÞ� tosin½ψðx; yÞ� and vice versa; however, it satisfies thecosine condition (29). The minimization of fluc-tuation in the envelope of the complex signal maytherefore be treated as a more general criterion fordefining multidimensional quadrature transforms.As already mentioned in Ref. [7], the spiral phaseoperator provides exact quadrature relationshipbetween signals of the form JoðλrÞ and J1ðλrÞ for apositive constant λ. The Bessel functions Jo and J1

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are thus seen to take on the roles of cosines and sines,respectively, when the signal is radially symmetric.The least-square minimization problem can also

be formulated in higher dimensions to obtain cosineconditions similar to Eq. (29). For example, in three-dimensional polar coordinates ðρ; θ;ϕÞ in the Fourierdomain, one may formulate a similar constrainedminimization problem as above and obtain the cosinecondition as

cos½αðρ; θ;ϕÞ� þ cos½αðρ; π − θ;ϕþ πÞ� ¼ 0: ð33ÞThe transformations θ → π − θ and ϕ → ϕþ π appearin the above relation due to the inversion symmetryof the three-dimensional spectrum of the correspond-ing real signal. One possible solution satisfying thiscondition may be written as

αðρ; θ;ϕÞ ¼ ϕþ cos2θ þ constant; ð34Þwhich reduces to the two-dimensional case when θ ¼π=2. The corresponding filter function Kðρ; θ;ϕÞ ¼exp½iαðρ; θ;ϕÞ� is also seen to gain a multiplicativefactor of (−1) along a line through the origin atρ ¼ 0. In this case, we note that the filter func-tion has a line discontinuity along the z-axis in theFourier domain where the angle ϕ is not defined.A half-space or octant-based definition of sgn filteron the other hand would result in discontinuityplanes. We will not pursue this topic further in thispaper.We conclude this paper by noting that the defini-

tion of the multidimensional quadrature transformwas obtained with the simplifying assumption thatthe signal is a realization of a homogeneous randomfield. Alternate definitions for quadrature transformmay be deemed suitable, if the signals of interest donot have this property. The general idea of minimiz-ing the fluctuation in the complex envelope may,however, be employed even in such cases.

4. Summary

In summary, the problem of defining the complex sig-nal representation for two-dimensional real imageshas been studied by extending the least-square mini-mization approach due to Mandel. A suitable quad-rature transform was sought such that it minimizedthe fluctuation in the envelope of the complex imagein an ensemble-averaged sense. The spiral phasequadrature transform, which has received much at-

tention in the literature for the past few years, wasshown to be a solution of this minimization problem.The cosine condition (29) also clarifies the prior ob-servation in the literature that the spiral phase op-erator tends to give smoother representations ascompared with those using other generalizations ofsgn filter having line discontinuities. We suggest thatthis constrained minimization approachmay be trea-ted as a suitable method for defining multidimen-sional quadrature transforms.

The author would like to thank Prof. N. George andProf. E. Wolf for introducing him to the concept ofanalytic signal representation in their respectivegraduate level courses at University of Rochester.

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