operational and convolution properties of two-dimensional fourier transforms in polar coordinates

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1768 J. Opt. Soc. Am. A/Vol. 26, No. 8 /August 2009 Natalie Baddour

Operational and convolution propertiesof two-dimensional Fourier transforms

in polar coordinates

Natalie Baddour

Department of Mechanical Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa,Ontario, K1N 6N5, Canada ([email protected])

Received February 24, 2009; revised May 26, 2009; accepted June 11, 2009;posted June 12, 2009 (Doc. ID 108008); published July 10, 2009

For functions that are best described in terms of polar coordinates, the two-dimensional Fourier transform canbe written in terms of polar coordinates as a combination of Hankel transforms and Fourier series—even if thefunction does not possess circular symmetry. However, to be as useful as its Cartesian counterpart, a polarversion of the Fourier operational toolset is required for the standard operations of shift, multiplication, con-volution, etc. This paper derives the requisite polar version of the standard Fourier operations. In particular,convolution—two dimensional, circular, and radial one dimensional—is discussed in detail. It is shown thatstandard multiplication/convolution rules do apply as long as the correct definition of convolution is applied.© 2009 Optical Society of America

OCIS codes: 070.6020, 070.4790, 350.6980, 100.6950.

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. INTRODUCTIONhe Fourier transform needs no introduction, and itould be an understatement to say that it has proved toe invaluable in many diverse disciplines such as engi-eering, mathematics, physics, and chemistry. Its appli-ations are numerous and include a wide range of topicsuch as communications, optics, astronomy, geology, im-ge processing, and signal processing. It is known thathe Fourier transform can easily be extended to n dimen-ions. The strength of the Fourier transform is that it isccompanied by a toolset of operational properties thatimplify the calculation of more complicated transformshrough the use of these standard rules. Specifically, thetandard Fourier toolset consists of results for scaling,ranslation (spatial shift), multiplication, and convolu-ion, along with the basic transforms of the Dirac-deltaunction and complex exponential that are so essential tohe derivation of the shift, multiplication, and convolutionesults. This basic toolset of operational rules is wellnown for the Fourier transform in single and multiple-imensions [1,2].As is also known, the Fourier transform in two dimen-

ions can be developed in terms of polar coordinates [3]nstead of the usual Cartesian coordinates, most usefullyn the case where the function being transformed is natu-ally describable in polar coordinates. For example, thisas seen application in the field of photoacoustics [4], andttempts have been made to translate ideas from the con-inuous domain to the discrete domain by developing nu-erical algorithms for such calculations [5]. However, to

he best of the author’s knowledge, a complete interpreta-ion of the standard Fourier operational toolset in termsf polar coordinates is missing from the literature. Someesults are known, such as the Dirac-delta function inoth polar and spherical polar coordinates, but the results

1084-7529/09/081768-11/$15.00 © 2

n shift, multiplication, and in particular convolution, arencomplete.

This paper thus aims to develop the Fourier opera-ional toolset of Dirac-delta, exponential, spatial shift,ultiplication, and convolution for the two-dimensional

2D) Fourier transform in polar coordinates. Of particularovelty is the treatment of the shift, multiplication, andonvolution theorems, which can also be adapted for thepecial cases of circularly symmetric functions that haveo angular dependence. It is well known from the litera-ure that 2D Fourier transforms for radially symmetricunctions can be interpreted in terms of a (zeroth-order)ankel trasnform. It is also known that the Hankel

ransforms do not have a multiplication/convolution rule,rule that has found so much use in the Cartesian ver-

ion of the transform. In this paper, the multiplication/onvolution rule is treated in detail for the curvilinearersion of the transform, and in particular it is shownhat the zeroth-order Hankel transform does obey aultiplication/convolution rule once the proper interpre-

ation of convolution is applied. This paper carefully con-iders the definition of convolution and derives the correctnterpretation of it in terms of the curvilinear coordinateso that the standard multiplication/convolution rule willnce again be applicable.

The outline of the paper is as follows. For completeness,he Hankel transform and the interpretation of the 2Dourier transform in terms of a Hankel transform and aourier series are introduced in Sections 2 and 3. Sectionsand 5 treat the special functions of the Dirac-delta and

omplex exponential. Sections 6–8 address the multiplica-ion, spatial shift, and convolution operations. In particu-ar, the nature of the spatial shift and its role in the en-uing convolution theorem are discussed. Sections 9 and0 discuss the spatial shift and convolution operators

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Natalie Baddour Vol. 26, No. 8 /August 2009 /J. Opt. Soc. Am. A 1769

hen dealing with the special case of radially (circularly)ymmetric functions. Sections 11 and 12 discuss the spe-ial cases of angular or radial convolution only—that is,ot a full 2D but a special one-dimensional (1D) convolu-ion as restricted to convolving over only one of the vari-bles of the polar coordinates. In particular, it is shownhat while the angular convolution yields a simple convo-ution relationship, the radial-only convolution does not.ection 13 derives the Parseval relationships. The derivedperational toolset is summarized in a table at the end ofhe paper.

. HANKEL TRANSFORMhe nth-order Hankel transform is defined by the integral

6]

F̂n�� = Hn�f�r�� =�0

�

f�r�Jn��r�rdr, �1�

here Jn�z� is the nth-order Bessel function and the over-at indicates a Hankel transform as shown in Eq. (1).ere, n may be an arbitrary real or complex number.owever, an integral transform needs to be invertible in

rder to be useful, and this restricts the allowable valuesf n. If n is real and n�1/2, the transform is self-eciprocating and the inversion formula is given by

f�r� =�0

�

Fn��Jn��r��d �. �2�

he inversion formula for the Hankel transform followsmmediately from Hankel’s repeated integral, whichtates that under suitable boundary conditions and sub-ect to the condition that �0

�f�r��rdr is absolutely conver-ent, then for n�1/2

�0

�

sds�0

�

f�r�Jn�sr�Jn�su�rdr =1

2�f�u + � + f�u − ��. �3�

he most important cases correspond to n=0 or n=1. Theankel transform exists only if the Dirichlet condition is

atisfied, that is, if the following integral exists:

0��r1/2f�r��dr. The Hankel transform is particularly usefulor problems involving cylindrical symmetry.

. CONNECTION BETWEEN THE 2DOURIER TRANSFORM AND THE HANKELRANSFORMhe 2D Fourier transform of a function f�x ,y� is defined as

F�� = F��x,�y� =�−�

� �−�

�

f�x,y�e−j��xx+�yy�dxdy. �4�

he inverse Fourier transform is given by

f�r�� = f�x,y� =1

�2��2�−�

� �−�

�

F��x,�y�ej�� ·r�d�xd�y, �5�

here the shorthand notation of �� = ��x ,�y�, r� = �x ,y� haseen used.

Polar coordinates can be introduced as x=r cos �, yr sin � and similarly in the spatial frequency domain asx=� cos � �y=� sin �. It then follows that the 2D Fourier

ransform can be written as

F��,�� =�0

��−�

�

f�r,��e−ir� cos��−��rdrd�. �6�

hus, in terms of polar coordinates, the Fourier transformperation transforms the spatial position radius andngle �r ,�� to the frequency radius and angle �� ,��. Thesual polar coordinate relationships apply in each domaino that r2=x2+y2, �=arctan�y /x� and �2=�x

2+�y2, �

arctan��y /�x�. Using r� to represent �r ,�� in physical po-ar coordinates and �� to denote the frequency vector �� ,��n frequency polar coordinates, the following expansionsre valid [3]:

ei�� ·r� = n=−�

�

inJn��r�ein� e−in�, �7�

e−i�� ·r� = n=−�

�

i−nJn��r�e−in� ein�. �8�

hese expansions can be used to convert the 2D Fourierransform into polar coordinates.

. Radially Symmetric Functionsf it is assumed that f is radially symmetric, then it can beritten as function of r only and can thus be taken out of

he integration over the angular coordinate so that Eq. (6)ecomes

F��,�� =�0

�

rf�r�dr�−�

�

e−ir � cos��−��d�. �9�

sing the integral definition of the zero-order Besselunction,

J0�x� =1

2��

−�

�

e−ix cos��−��d� =1

2��

−�

�

e−ix cos �d�, �10�

q. (9) can be written as

F�� = F2Df�r�� = 2��0

�

f�r�J0��r�rdr, �11�

hich can be recognized as 2� times the Hankel trans-orm of order zero. Thus the special case of the 2D Fourierransform of a radially symmetric function is the same ashe Hankel transform of that function:

F�� = F2Df�r�� = 2�H0f�r��. �12�

ith reference to Eq. (12), f�r� and F�� are functions withadial syummetry in a 2D regime, F2D·� is an operator inhe 2D regime, while H0·� is an operator in 1D regime.

. Nonradially Symmetric Functionshen the function f�r ,�� is not radially symmetric and isfunction of both r and �, the preceding result can be gen-

ralized. Since f�r ,�� depends on �, it can be expandednto a Fourier series,

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f�r�� = f�r,�� = n=−�

�

fn�r�ejn�, �13�

here

fn�r� =1

2��

0

2�

f�r,��e−jn� d�. �14�

his transform is well suited to functions that are sepa-able in r and �. This case is extensively treated in Good-an’s widely used text on Fourier optics [7].Similarly, the 2D Fourier transform F�� ,�� can also be

xpanded into its own Fourier series so that

F�� = F��,�� = n=−�

�

Fn��ejn� �15�

nd

Fn�� =1

2��

0

2�

F��,��e−jn�d�. �16�

ote that Fn�� is not the Fourier transform of fn�r�. Inact, it is the relationship between fn�r� and Fn�� that weeek to define.

. Forward Transformxpansions (8) and (13) are substituted into the definitionf the forward Fourier transform to give

F�� =�−�

�

f�r��e−j�� ·r� dr�

=�0

��0

2�

m=−�

�

fm�r�ejm� n=−�

�

i−nJn��r�e−in� ein�d� rdr

= n=−�

�

2�i−nein��0

�

fn�r�Jn��r�rdr. �17�

his follows since

�0

2�

ein�d� = 2�n0 = �2� if n = 0

0 otherwise , �18�

here nm denotes the kronecker delta function.By comparing Eqs. (15) and (17), we obtain the expres-

ion for Fn��, the Fourier coefficients of the Fourier do-ain expansion. Using Eq. (1), this can also be inter-

reted in terms of a Hankel transform as

Fn�� = 2�i−n�0

�

fn�r�Jn��r�rdr = 2�i−nHnfn�r��. �19�

. Inverse Transformhe corresponding 2D inverse Fourier transform is writ-en as

f�r�� =1

�2��2�0

��0

2�

F�� ej�� ·r�d� �d�. �20�

sing Eq. (15) along with the expansion (7) yields

f�r�� = n=−�

� 1

2�in ein��

0

�

Fn��Jn��r��d � �21�

o that

fn�r� =in

2��

0

�

Fn��Jn��r��d � =in

2�HnFn��. �22�

hus, it can be observed that the nth term in the Fouriereries for the original function will Hankel transform intohe nth term of the Fourier series of the Fourier trans-orm function. However, it is an nth-order Hankel trans-orm for the nth term, so all the terms are not equiva-ently transformed. Furthermore, we recall that theeneral 2D Fourier transform for radially symmetricunctions was shown to be equivalent to the zeroth-orderankel transform. Therefore the mapping from fn�r� ton��, which is an nth-order Hankel transform, is not aD Fourier transform.Most importantly, it has been shown that the operation

f taking the 2D Fourier transform of a function is equiva-ent to first (1) finding its Fourier series expansion in thengular variable and then (2) finding the nth-order Han-el transform (of the spatial radial variable to the spatialrequency radial variable) of the nth coefficient in theourier series and appropriately scaling the result.learly, for functions with cylindrical-type symmetries

hat are naturally described by cylindrical coordinates,he operation of taking a three-dimensional (3D) Fourierransform will be equivalent to (a) a regular 1D Fourierransform in the z coordinate, then (2) a Fourier series ex-ansion in the angular variable, and then (3) nth-orderankel transform (of the radial variable to the spatial ra-ial variable) of the nth coefficient in the Fourier series. Ithould further be noted that since each of these opera-ions involves integration over one variable only with thethers being considered parameters vis-à-vis the integra-ion, the order in which these operations are performed isnterchangeable.

. THE DIRAC-DELTA FUNCTION AND ITSRANSFORMhe unit-mass Dirac-delta function in 2D polar coordi-ates is defined as

f�r�� = �r� − r�0� =1

r�r − r0�� − �0�. �23�

o find the Fourier transform, the Fourier series expan-ion is required, followed by a Hankel transform, as pre-iously discussed. Thus for the Dirac-delta function, theourier series expansions terms are

fn�r� =1

2��

0

2� 1

r�r − r0�� − �0�e−in� d�

=1

2�r�r − r0�e−in�0, �24�

nd then the full transform is given by

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F�� = n=−�

�

Fn��ejn� = n=−�

�

2�i−n ein��0

�

fn�r�Jn��r�rdr

= n=−�

�

2�i−nein��0

� �r − r0�

2�re−in�0 Jn��r�rdr

= n=−�

�

i−nJn��r0�e−in�0 ein� = e−i�� ·r�0, �25�

here Eq. (25) is the 2D linear exponential phase func-ion. It can be seen that the Fourier transform of theirac delta function is the exponential function, as woulde expected from the results in Cartesian coordinates.ore importantly, and the main result that we seek, is

hat the coefficients of the Fourier expansions of theransform are given by

Fn�� = i−nJn��r0�e−in�0. �26�

. Dirac-Delta Function at the Originf r�0 is at the origin, then it is multiply covered by the an-ular variable, and the Dirac-delta function in two dimen-ions is given by

�r�� =1

2�r�r�, �27�

here the difference in notation between �r�� and �r� ismphasized. The notation �r�� is used to represent theirac-delta function in the appropriate multidimensional

oordinate system, whose actual form may vary. The no-ation �r� denotes the standard 1D scalar version of theirac-delta function that is most familiar. Either Eq. (11)r Eq. (26) can be used to calculate the Fourier transform,s they both yield the correct transform for the Dirac-elta function at the origin, which is 1.

. Ring-Delta Functionn often-used function is the Ring-delta function given by

1

2�r�r − r0�. �28�

he function given in Eq. (28) is nonzero only on the ringf radius r0. The 2D Fourier transform of the Ring-delta isost easily found from Eq. (11) and is given by

F�� = 2��0

�� 1

2�r�r − r0� J0��r�rdr = J0��r0�. �29�

. COMPLEX EXPONENTIAL AND ITSRANSFORMrom Eq. (7), the 2D complex exponential function can beritten in polar coordinates as

f�r�� = ei�� 0·r� = n=−�

�

inJn��0r�e−in�0 ein�, �30�

o the Fourier coefficients can be directly seen from Eq.30) or can be found from the formula by

fn�r� =1

2��

0

2�

m=−�

�

imJm��0r�e−im�0 eim� e−in�d�

= inJn��0r�e−in�0. �31�

he Fourier transform is given by

F�� = n=−�

�


�

fn�r�Jn��r�rdr

= n=−�

�


�

inJn��0r�e−in�0�Jn��r�rdr

= n=−�

�

2� e−in�01

�� − �0�ein�, �32�

here the last line follows from the orthogonality of theessel functions [8]. This gives

Fn�� = 2� e−in�01

�� − �0�. �33�

t is also noted that the closure of the complex exponen-ials (or, equivalently, finding the Fourier series of the 1Dirac-delta function) gives

�� − �0� = n=−�

� 1

2�e−in�0 ein�, �34�

o Eq. (32) actually gives the traditional Fourier trans-orm of the complex exponential as it should:

F�� = �2��21

�� − �0�� − �0� = �2��2�� − �� 0�. �35�

. Special Cases a special case of this, the Fourier transform of f�r��=1

an be computed by subsituting �0=0 into the above for-ulas. This gives

F�� = �2��21

�� = �2��2�� , �36�

r, alternatively, in series form the Fourier transform of�r��=1 is given by

F�� =2�

��

n=−�

�

ein�. �37�

. MULTIPLICATIONe consider the product of two functions h�r��= f�r��g�r��,here f�r��=n=−�

� fn�r�ein� and g�r��=n=−�� gn�r�ein�. The

ourier series coefficients fn�r� and gn�r� are given by Eq.14), and we seek to find the equivalent coefficients hn�r�f h�r��= f�r��g�r��=n=−�

� hn�r�ein�. This is accomplished bynding the Fourier transform of h�r�� and using the expan-ions f�r��=n=−�

� fn�r�ein� and g�r��=n=−�� gn�r�ein� along

ith Eq. (8) for the polar form of the 2D complex exponen-ial:

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H�� =�−�

�

f�r��g�r��e−i�� ·r�dr�

=�0

��0

2�

n=−�

�

fn�r�ein� m=−�

�

gm�r�eim�

k=−�

�

i−kJk��r�e−ik� eik�d� rdr. �38�

erforming the integration over the angular variableields

H�� = k=−�

�

2�i−keik��0

�

m=−�

�

fk−m�r�gm�r�Jk��r�rdr

= k=−�

�

Hk��eik�, �39�

here

Hk�� = 2�i−k�0

�

m=−�

�

fk−m�r�gm�r�Jk��r�rdr. �40�

owever, it is known from Eq. (19) that

Hk�� = 2�i−k�0

�

hk�r�Jk��r�rdr; �41�

ence it follows that

hk�r� = m=−�

�

fk−m�r�gm�r�, �42�

hich is in fact the convolution of the Fourier series of f�r��nd g�r��. In other words, we have that

�fg�k = fk � gk, �43�

ith the convolution of two series defined as

�fk � gk��r� � m=−�

�

fk−m�r�gm�r�. �44�

he definition of the discrete convolution of two seriesiven in Eq. (44) is the same as the standard definitioniven in the literature [9].

. SPATIAL SHIFThe correct expression for a Fourier series shifted inpace is obtained by finding the inverse Fourier transformf the complex exponential-weighted transform. In otherords, f�r� −r�0� is defined from

f�r� − r�0� = F−1e−i�� ·r�0 F�� . �45�

he shifted function is defined according to Eq. (45) sincee have already found the expansion for the complex ex-onential, and the rules for finding the product of two ex-ansions was found in the previous section. It is not suf-cient to find any expression for the spatial shift, butather the expression that is sought is one that is (i) inhe form of a Fourier series and (ii) in terms of the un-hifted coefficients of the original function, as this buildshe rule for what to do to the coefficients if a shift is de-ired. Thus, by building on the previously found results,he relevant spatial shift result can be found in the de-ired form.

Using the definition of the inverse Fourier transformiven in Eq. (20), along with the expansions in Eqs. (7),8), and (15), the desired quantity is then given by

f�r� − r�0� =1

�2��2�0

��0

2�

m=−�

�

i−mJm��r0�e−im�0 eim�

n=−�

�

Fn��ejn� k=−�

�

ikJk��r�eik� e−ik� d� �d �.

�46�

erforming the integration over � yields a nonzero valuenly if k−n−m=0, so the preceding equation simplifies to

f�r� − r�0� =1

2� n=−�

�

k=−�

�

in e−i�k−n��0 eik�

�0

�

Fn��Jk−n��r0�Jk��r��d �. �47�

t can be observed from Eq. (47) that we obtain the Fou-ier coefficients of the shifted function as

�f�r� − r�0��k�r� =1

2� n=−�

�

in e−i�k−n��0

�0

�

Fn��Jk−n��r0�Jk��r��d �, �48�

hich gives the Fourier coefficients of the shifted functionn r space in terms of the unshifted frequency-space coef-cients Fn��. If the value of the shifted coefficeints areesired in terms of the original unshifted coefficients fn�r�,here f�r��=n=−�

� fn�r�ejn�, then the definition of theourier-space coefficients from Eq. (19) can be invoked;

Fn�� = 2�i−n�0

�

fn�r�Jn��r�rdr = 2�i−nHnfn�r��, �49�

nd can be substituted into Eq. (48) so that

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f�r� − r�0� = k=−�

�

eik� n=−�

�

e−i�k−n��0�0

�

fn�u�Snk�u,r,r0�udu,

�50�

here Snk�u ,r ,r0� is defined as a shift-type operator given

y the integral of the triple-Bessel product:

Snk�u,r,r0� =�

0

�

Jn��u�Jk−n��r0�Jk��r��d �. �51�

hus, the Fourier coefficients of the shifted function inerms of the unshifted function coefficients are given by

�f�r� − r�0��k = n=−�

�

e−i�k−n��0�0

�

fn�u�Snk�u,r,r0�udu. �52�

quation (52) describes the shift operation in terms ofourier polar coordinates.In the special case that r�0=0, then the definition of the

hift operator along with the fact that Jn�0�=n0 implieshat in this case the shift operator becomes

Snk�u,r,0� =�

0

�

nkJn��u�Jn��r��d � =1

u�u − r�nk

�53�

o that Eq. (52) returns the correct unshifted value of fn,s it should.The corresponding coefficients of the 2D Fourier trans-

orm can be found. If we define h�r��= f�r� −r�0�, then H�� e−i�� ·r�0F�� , and the Fourier coefficients Hn�� are sought.hus H�� is defined as a product of the complex exponen-ial and F�� , and it was previously shown that multipli-ation in the Fourier series domain implies convolution ofhe coefficients of the respective series, with the convolu-ion operation defined in Eq. (42). Since the coefficientsor the complex exponential are given in Eq. (8), it followshat

Hk�� = �i−kJk��r0�e−ik�0� � Fk�� 54�

r, explicitly,

Hk�� = m=−�

�

i−mJm��r0�e−im�0 Fk−m�� 55�

s per the definition of the convolution operation for twoourier series coefficients.

. Alternative Interpretation of the Expressionshe preceding expressions permit a great deal of simpli-cation by using the result derived in [10] that the triple-roduct integral given by

�0

�

Jn1�k1��Jn2

�k2��Jn3�k3��d � �56�

s nonzero only if n1+n2+n3=0. This is a restrictive con-traint and thus permits the simplification of the shift op-rator so that Sn

k�u ,r ,r0� is nonzero only for k=0, whichives

Snk�u,r,r0� = Sn

0�u,r,r0�

=�0

�

Jn��u�J−n��r0�J0��r��d �, r0 � 0.

�57�

t is emphasized that the result Snk�u ,r ,r0�=Sn

0�u ,r ,r0� isrue only for r0�0; otherwise, Sn

k�u ,r ,0�= 1u�u−r�nk, as

reviously discussed. This result for the shift operator im-lies that Eq. (50) becomes

f�r� − r�0� = n=−�

�

ein�0�0

�

fn�u�Sn0�u,r,r0�udu. �58�

Equation (58) essentially points out that shifting theunction to a new point involves a translation only andot a rotation. The translation does not materially affecthe angular portion of the expansion, and it can thus bes easily expanded in terms of the angular coordinatesith ein�0 as in terms of ein�. The translation portion of the

hift is handled with the integration with the shift opera-or. Thus, from Eq. (58), we see that the coefficients in thexpansion for the shifted function are

�f�r� − r�0��n =�0

�

fn�u�Sn0�u,r,r0�udu. �59�

quation (59) becomes

�f�r� − r�0��n =�0

��0

�

fn�u�Jn��u�uduJ−n��r0�J0��r��d �

=�− i�n

2��

0

�

Fn��Jn��r0�J0��r��d �, �60�

here the result J−n�x�= �−1�nJn�x� has been used, alongith the definition of Fn��. Equation (60) is identical toq. (47) with the simplification k=0 that became appar-nt only from the simplification of the shift operator.hile this simplification benefits the expression for the

hifted function, there are benefits to not simplifying toooon, as shall be seen in the following section on convolu-ions.

To find the simplified version of the forward transformf h�r��= f�r� −r�0�, we invoke the definition to find

Hn�� = 2�i−n�0

�

hn�r�Jn��r�rdr = 2�i−n�0

�

��− i�n

2��

0

�

Fn�u�Jn�ur0�J0�ur�udu�Jn��r�rdr.

�61�

o evaluate this integral, the integration over r needs toe performed first, and in fact, this can be found in closedorm [3] as the nth-order Hankel transform of J0�ur�. De-ote this integral with K so that we have

Kn�u,�� =�0

�

J0�ur�Jn��r�rdr. �62�

iven this definition, the expression for H �� becomes
n

8T

Tt1

s

wo

W(

T

wt

Ilmh

atow(dFttkn


Hn�� = �− 1�n�0

�

Fn�u�Jn�ur0�Kn�u,��udu. �63�

. CONVOLUTIONhe 2D convolution of two functions is defined by

h�r�� = f�r�� g�r�� =�−�

�

g�r�0�f�r� − r�0�dr�0. �64�

he double-asterisk notation of �� is used to emphasizehat this is a 2D convolution and to distinguish it from a
D convolution. S
k k

tsntvSf

9SFtFvcfb

w

Itotoodtt

Using the Fourier expansions for g and the shifted ver-ion of f given by Eq. (50), Eq. (64) becomes

h�r�� =�0

��0

2�

m=−�

�

gm�r0�eim�0 k=−�

�

eik� n=−�

�

e−i�k−n��0

�0

�

fn�u�Snk�u,r,r0�udud�0r0dr0, �65�

hich can be simplified by performing the integrationver the angular variable and by using the definition ofk
n�u ,r ,r0� to give
�66�

ith the simplifications in the brackets as shown in Eq.66), this becomes

h�r�� =1

2� k=−�

�

ik eik��0

�� n=−�

�

Gk−n��Fn�� Jk��r��d �.

�67�

his can be written in the form

h�r�� =1

2� k=−�

�

ik eik��0

�

Hk��Jk��r��d � = k=−�

�

hk�r�eik�,

�68�

here the coefficients Hk�� are clearly the convolution ofhe two series given by

Hk�� = n=−�

�

Gk−n��Fn��. �69�

n the results of Section 7, it was shown that the convo-ution of two sets of Fourier coefficients is equivalent to

ultiplication of the functions so that Gk�Fk= �GF�k;ence it follows from Eq. (69) that

H�� = G�� F�� , �70�

s would be expected from the standard results of Fourierheory, and it serves to confirm the accuracy of the devel-pment. However, the main result we seek is Eq. (67),hich gives the values of hk�r� in terms of gk�r� and fk�r�

or rather the Hankel transform of those) and essentiallyefines the convolution operation for functions given inourier series forms. This equation defines the convolu-ion operation so that to find the convolution of two func-ions given in Fourier series form, one must first find theth coefficient of the Fourier transform of each function,amely, G �� and F ��, and then subsequently convolve

he resulting series as per Eq. (69) to get Hk��. The finaltep is then to inverse Hankel transform the result to fi-ally obtain hk�r�. It is important to note that convolvinghe two functions f�r�� and g�r�� is not equivalent to con-olving their series: this in fact was shown previously inection 6 to be equivalent to the multiplication of the

unctions themselves.

. SPATIAL SHIFT OF RADIALLYYMMETRIC FUNCTIONSrom the previous discussion on radially symmetric func-

ions embodied in Eq. (12) and the definition of the 2Dourier transform as given in Eqs. (13) and (14), it is ob-ious that the Fourier series for a radially symmetric fun-ion and its transform include only the n=0 term. Thus,or a radially symmetric function, the shifted function cane written from Eq. (47) as

f�r� − r�0� =1

2� k=−�

�

e−ik�0 eik��0

�

F0��Jk��r0�Jk��r��d �,

�71�

here

F0�� = 2��0

�

f�u�J0��u�udu. �72�

t is clear from the previous two equations that the fac-ors of 2� in the numerator and denominator will cancelut; however, they have been retained in order to main-ain the formulas that have been developed in the previ-us sections. Although f�r� is radially symmetric and hasnly the n=0 term in its Fourier series, f�r� −r�0� is not ra-ially symmetric and so needs the full complement of en-ries in its Fourier series. It is equally important to notehat this is the case even if the new “center” r to which
�0

tac

I

Tftfait−t=sfit

da

wg

ariwdsFosiFfpfocrcan2o

1STbtt

wwvp

a

wsv

w

setpb

w

Twcf


he function f has been shifted is located on the radialxis so that �0=0. If this is the case, then Eq. (71) be-omes

f�r� − r�0� = k=−�

�

eik��0

�

f�u��0

�

J0��u�Jk��r0�Jk��r��d �udu.

�73�

n general, Eq. (71) can be written using Eq. (51) as

f�r� − r�0� = k=−�

�

eik��−�0��0

�

f�u�S0k�u,r,r0�udu. �74�

hus, we have the result that a function of r becomes aunction of r and � only once it has been shifted away fromhe origin or, in other words, that a radially symmetricunction is no longer radially symmetric once shiftedway from the origin. This may generate some confusionn that f�r� −r�0� is the shift of the radially symmetric f�r� sohat it is now centered at r�0. This is not the same as f�rr0�, which is a radially symmetric function. To illustrate

hese points, consider the Gaussian “dome” of f�r�e−�r2/4�, f�r−4�=e−��r−4�2/4�, and f�r� −r�0� as f�r�=e−�r2/4�

hifted so that is it centered at r�0= �r=4,�=0�. Clearly, therst two of these will be radially symmetric, while thehird is not.

From Eq. (55), the Fourier transform of the shifted ra-ially symmetric function is also not radially symmetricnd is given by

F�f�r� − r�0�� = k=−�

�

i−kJk��r0�e−ik�0 F0��eik�, �75�

hich, using the expansion for the complex exponentialiven in Eq. (8), becomes

F�f�r� − r�0�� = F0�� k=−�

�

i−kJk��r0�e−ik�0 eik� = F0��e−i�� ·r�0

= F0��e−ir0� cos��−�0�, �76�

s would be expected from the standard Fourier theoryesult for a shifted function. As previously mentioned, it isntuitively obvious from a 2D polar-coordinate view of theorld that a radially symmetric function is no longer ra-ially symmetric once shifted away from the origin. Thus,ince the function is no longer radially symmetric, the 2Dourier transform is no longer equivalent to a zeroth-rder Hankel transform as was the case for the radiallyymmetric function. In fact, Eq. (76) clearly shows that its the e−i�� ·r�0 term that destroys the symmetry of the 2Dourier transform. In particular, e−i�� ·r�0=e−ir0� cos��−�0� is a

unction of magnitude 1 but with a nonradially symmetrichase. Continuing with this argument, if Hankel trans-orms are considered to be merely transforming a functionf a variable r to another function of a variable �, withoutonsideration of its role in the 2D perspective, it would beeasonable to look for a Hankel analog of the shift andonvolution theorems of standard Fourier theory. Clearly,

1D shift/convolution rule for Hankel transforms doesot exist, and it is only in considering the matter from theD perspective that the reason one cannot exist becomesbvious.

0. CONVOLUTION OF TWO RADIALLYYMMETRIC FUNCTIONSwo radially symmetric functions are functions for whichoth Fourier series expansions include only the n=0erm. The convolution of the two radially symmetric func-ions is defined as

h�r�� = f�r�� g�r�� =�−�

�

g�r�0�f�r� − r�0�dr�0, �77�

here it is emphasized that the integration is over all r�0,hich includes all possible values of radial and angularariables. In other words, Eq. (77) must be properly inter-reted as a 2D convolution along with Eq. (71) as

h�r�� = f�r�� g�r�� =�0

��0

2�

g�r0� k=−�

�

e−ik�0 eik�

�0

�

f�u�S0k�u,r,r0�udu d�0r0dr0 �78�

nd not interpreted as

�0

�

g�r0�f�r − r0�dr0, �79�

hich would be a 1D convolution. Equation (78) can beimplified by performing the integration over the angularariable, which is nonzero only if k=0, so that

f�r�� g�r�� = f�r� � � g�r�

=�0

�

g�r0��0

�

f�u�S00�u,r,r0�udu r0dr0,

�80�

ith

S00�u,r,r0� =�

0

�

J0��u�J0��r0�J0��r��d �, �81�

o that Eq. (80) is hardly recognizable as a convolution op-ration. The double � notation has been used to highlighthe fact that a 2D convolution is being taken. In fact, theroper, correct definition of the convolution, Eq. (78), cane interpreted from Eq. (80) as

f�r�� g�r�� = f�r� � � g�r� =�0

�

g�r0��r − r0�r0dr0,

�82�

here

��r − r0� =�0

�

f�u�S00�u,r,r0�udu =�

0

2�

f�r� − r�0�d�0.

�83�

hus, we observe that the definition of convolution thate are tempted to write, as given by Eq. (79), is almost

orrect. What needed to be done was to shift the functionfrom r to r (thus destroying radial symmetry) and
� �0

tgstg

s

IzGoitsostdp

To

1Fonam

Nle

T

IlFei(

1St

Sgto

Ur�

L

w�b


hen to integrate the resulting shifted function over all an-ular variables. The unshifted function g is still radiallyymmetric and thus is not affected by the integration overhe angular variable, and the final result is the formiven in Eq. (82).

From Eqs. (68) and (69), it follows that

h�r� = f�r� � � g�r� =1

2��

0

�

G0��F0��J0��r��d �, �84�

ince the interpretation of Eq. (69) in this case is

Hk�� = n=−�

�

Gk−n��Fn�� = �G0��F0�� k = 0

0 otherwise .

�85�

n the previous two equations, G0�� and F0�� are theeroth-order coefficients in the expansions for F�� and�� themselves. Equation (85) can also be obtained by

bserving that the integration over the angular variablen Eq. (78) forces k to be zero. Thus, the 2D convolution ofwo radially symmetric functions yields another radiallyymmetric function, as can be seen from Eq. (84). More-ver, by using the proper definition of a 2D convolution in-tead of using the tempting definition of a 1D convolution,he well-known relationship between convolutions in oneomain leading to multiplication in the other domain isreserved, namely, that

h�r�� = f�r�� g�r�� = f�r� � � g�r� ⇒ H�� = F��G��.

�86�

he key point to this relationship is the proper definitionf the convolution as a 2D convolution.

1. CIRCULAR (ANGULAR) CONVOLUTIONor two functions f�r��= f�r ,�� and g�r��=g�r ,��, the notionf a circular or angular convolution can be defined. This isot the 2D convolution as previously discussed but ratherconvolution over the angular variable only so that itay be defined as

f�r��g�r�� =1

2��

0

2�

f�r,�0�g�r,� − �0�d�0. �87�

ote the notation of �� used to denote the angular convo-ution. A Fourier relationship can be defined for this op-ration of angular convolution:

h�r,�� = f�r��g�r�� =1

2��

0

2� � n=−�

�

fn�r�ejn�0��

m=−�

�

gm�r�ejm��−�0��d�0

= n=−�

�

m=−�

�

fn�r�gm�r�ejm�1

2��

0

2�

ejn�0 e−jm�0 d�0.

�88�

he value of the integral in the preceding equation is 2�, and the following simplification results:
mn
f�r��g�r�� = m=−�

�

fn�r�gn�r�ejn�. �89�

n other words, the nth coefficient of the angular convo-ution of two functions is simply the product of the nthourier coefficient of each of the two functions. Math-matically, if h�r ,��= f�r��g�r��, then hn�r�= fn�r�gn�r�. Thiss also the same result as obtained with Fourier series of1D) periodic functions.

2. RADIAL CONVOLUTIONimilarly to the definition of angular or circular convolu-ion, we can define the notion of a radial convolution as

h�r�� = f�r��rg�r�� =�0

�

g�r0,��f�r − r0,��r0dr0. �90�

uch convolutions are less often seen than their 2D or an-ular counterparts; however, they are included herein forhe sake of completeness. Using Eq. (50) for the radially-nly shifted function yields

h�r,�� =�0

� � m=−�

�

gm�r0�eim��

k=−�

�

eik� n=−�

� �0

�

fn�u�Snk�u,r,r0�udu�r0dr0.

�91�

sing the result for the product of two series so that theesulting Fourier coefficients are convolved as fk�gkm=−�

� fm�r�gk−m�r� gives

h�r,�� = k=−�

�

eik��0

�

m=−�

�

gk−m�r0�

n=−�

� �0

�

fn�u�Snm�u,r,r0�udu r0dr0

= k=−�

�

eik��0

�

m=−�

�

gk−m�r0�

n=−�

� in

2��

0

�

Fn��Jm−n��r0�Jm��r��d � r0dr0.

�92�

et us define

Gk−mm−n�� =�

0

�

gk−m�r0�Jm−n��r0�r0dr0, �93�

hich is the �m–n�th-order Hankel transform of thek–m�th Fourier coefficient. With this definition, Eq. (92)ecomes

TftaodEftp

1A“qt

I=

wwbewraEt

w

Ea

Fl0

Fsg

Is


h�r,�� = k=−�

�

eik� m=−�

�

n=−�

� in

2��

0

�

Gk−mm−n��Fn��Jm��r��d �.

�94�

his is the closest we can come to a convolution theoremor the radial convolution. Without the shift and integra-ion over the angular portion of the function, the result iswkward because the order of the Hankel transform ofne of the functions does not always correspond to the or-er of the Fourier coefficient as seen by the definition inq. (93). The angular shift and integration portion of the

ull 2D convolution effectively washes out all these crosserms to yield the nice result of the full 2D convolution, asreviously discussed.

3. PARSEVAL RELATIONSHIPSParseval relantionship is important, as it deals with the

power” of a signal or function in the spatial and fre-uency domains. As previously mentioned, it is knownhat we can write

f�r�� = f�r,�� = n=−�

�

fn�r�ejn�. �95�

t is noted that in polar coordinates if r� = �r ,��, then −r��r ,�+��. Hence

wi

C

f�− r�� = f�r,� + �� = n=−�

�

fn�r�ejn��+�� = n=−�

�

fn�r��− 1�ne−jn�

= n=−�

�

f−n�r��− 1�nejn�, �96�

here the overbar indicates a complex conjugate. Thus ife define a function g�r��= f�−r��, its coefficients are giveny gn�r�= f−n�r��−1�n. To derive the Parseval relation, wevaluate that convolution of two functions f�r�� and g�r��ith g�r��= f�−r�� and where the convolution is evaluated at

� =0� implying r=0, �=0. In other words, we evaluate �f�g�s given by Eq. (64) at r� =0� . This is done by making use ofq. (66), along with J−n�x�= �−1�nJn�x� and Jn�0�=n0 so

hat

�−�

�

g�− r�0�f�r�0�dr�0 = 2��0

�� n=−�

� �0

�

g−n�r0�J−n��r0�r0dr0

�0

�

fn�u�Jn��u�udu �d �, �97�

hich, with the given choice of function for g, becomes

�98�

quation (98) furnishes the desired Parseval relationships

�−�

�

�f�r��2dr� =1

2� n=−�

� �0

�

�Fn��2�d �. �99�

ollowing this same procedure and evaluating the convo-ution of any two well-behaved functions f�r�� and g�−r�� at� yields the generalized Parseval relationship:

�−�

�

g�r��f�r��dr� =1

2� n=−�

� �0

�

Gn��Fn��d �. �100�

urthermore, from the result on multiplication and ob-erving from Eq. (96) that the coefficients of g�r�� are−n�r�, we have that

f�r��g�r�� = k=−�

�

m=−�

�

fm�r�g−�k−m��r�eik�. �101�

ntegrating both sides of the previous equation over allpace gives

�0

�

f�r��g�r��dr� = 2� m=−�

� �0

�

fm�r�gm�r�rdr, �102�

hich gives another version of the Parseval relationshipn Eq. (100) as

n=−�

� �0

�

fn�r�gn�r�rdr =1

�2��2 n=−�

� �0

�

Fn��Gn��d �.

�103�

learly, it then follows from the preceding equation that

n=−�

� �0

�

�fn�r��2rdr =1

�2��2 n=−�

� �0

�

�Fn��2�d �.

�104�

1IcaFcuowcosfsmc

ATeart

R

1

f


4. SUMMARY AND CONCLUSIONSn summary, this paper has considered the polar-oordinate version of the standard 2D Fourier transformnd derived the operational toolset required for standardourier operations. As previously noted, the polar-oordinate version of the 2D Fourier transform is mostseful for functions that are naturally described in termsf polar coordinates. Additionally, Parseval relationshipsere also derived. The results of the paper are concisely

ollected in Table 1. Of particular interest are the resultsn convolution and spatial shift. With the application of apecial result on the integral of a triple product of Besselunctions, the expression for spatial shift allowed for con-iderable simplification. Notably, standard convolution/ultiplication rules do apply for 2D convolution and 1D

ircular convolution but not for 1D radial convolution.

CKNOWLEDGMENTShis work was financially supported by the Natural Sci-nces and Engineering Research Council of Canada. Theuthor also thanks one of the reviewers, whose thorougheview and helpful comments contributed to improving

Table 1. Summary of Fourier Transf

f�r�� fn�r�

�r ,��=n=−�� fn�r�ejn� 1

2��0

2�f�r ,��e−jn� d�

�r� −r�0�=�r−r0�

r��−�0�

�r−r0�2�r

e−in�0

ei�� 0·r� inJn��0r�e−in�0

f�r� −r�0� m=−�� e−i�n−m��0�0

�fm�u�Smn �u ,r ,r0�udu

f�r� −r�0�(alternate)

�−i�n

2��0

�Fn��Jn��r0�J0��r��d �

h�r��g�r�� fn�r�=hn�r��gn�r�=m=−�� hn−m�r�gm�r�

h�r�� g�r�� in

2��0

�Fn��Jn��r��d�

his paper.

EFERENCES1. R. Bracewell, The Fourier Transform and Its Applications

(McGraw-Hill, 1999).2. K. Howell, “Fourier transforms,” in The Transforms and

Applications Handbook, 2nd ed., A. D. Poularikas, ed. (CRCPress, 2000), pp. 2.1–2.159.

3. G. Chirikjian and A. Kyatkin, Engineering Applications ofNoncommutative Harmonic Analysis: With Emphasis onRotation and Motion Groups (CRC Press, 2001).

4. Y. Xu, M. Xu, and L. V. Wang, “Exact frequency-domainreconstruction for thermoacoustic tomography—II:Cylindrical geometry,” IEEE Trans. Med. Imaging 21,829–833 (2002).

5. A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M.Israeli, “Fast and accurate polar Fourier transform,” Appl.Comput. Harmonic Anal. 21, 145–167 (2006).

6. R. Piessens, “The Hankel transform,” in The Transformsand Applications Handbook, 2nd ed., A. D. Poularikas, ed.(CRC Press, 2000), pp. 9.1–9.30.

7. J. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2004).

8. G. Arfken and H. Weber, Mathematical Methods forPhysicists (Elsevier, 2005).

9. A. Oppenheim and R. Schafer, Discrete-time SignalProcessing (Prentice-Hall, 1989).

0. A. D. Jackson and L. C. Maximon, “Integrals of products ofBessel functions,” SIAM J. Math. Anal. 3, 446–460 (1972).

Relationships in Polar Coordinates

Fn�� F��

2�i−n�0�fn�r�Jn��r�rdr F�� ,��=n=−�

� Fn��ejn�

i−nJn��r0�e−in�0 e−i�� ·r�0

2� e−in�01�

��−�0��2��2�� −�� 0�

�i−nJn��r0�e−in�0��Fn��=m=−�

� i−mJm��r0�e−im�0 Fn−m��e−i�� ·r�0 F��

�−1�n�0�Fn�u�Jn�ur0�Kn�u ,��udu e−i�� ·r�0 F��

2�i−n�0�fn�r�Jn��r�rdr G�� H��

n��=Hn��Gn��=m=−�� Gn−m��Hm�� G�� H��

orm

F

operational and convolution properties of two-dimensional fourier transforms in polar coordinates

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