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2144 J. Opt. Soc. Am. A/Vol. 27, No. 10 /October 2010 Natalie Baddour

Operational and convolution properties ofthree-dimensional Fourier transforms in spherical

polar coordinates

Natalie Baddour

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

Received July 12, 2010; accepted August 12, 2010;posted August 12, 2010 (Doc. ID 131538); published September 13, 2010

For functions that are best described with spherical coordinates, the three-dimensional Fourier transform canbe written in spherical coordinates as a combination of spherical Hankel transforms and spherical harmonicseries. However, to be as useful as its Cartesian counterpart, a spherical version of the Fourier operationaltoolset is required for the standard operations of shift, multiplication, convolution, etc. This paper derives thespherical version of the standard Fourier operation toolset. In particular, convolution in various forms is dis-cussed in detail as this has important consequences for filtering. It is shown that standard multiplication andconvolution rules do apply as long as the correct definition of convolution is applied. © 2010 Optical Society ofAmerica

OCIS codes: 070.6020, 070.4790, 350.6980, 100.6950.

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. INTRODUCTIONhe Fourier transform has proved to be a powerful tool inany diverse disciplines and indispensable to signal pro-

essing. One of its powerful features is it can easily be ex-ended to n dimensions. The strength of the Fourierransform is that it is accompanied by a toolset of opera-ional properties that simplify the calculation of moreomplicated transforms through the use of these standardules, turning otherwise complex calculations into thosehat can be done via a look-up table and a core set ofransforms. Specifically, the standard Fourier toolset con-ists of results for translation (spatial shift), multiplica-ion, and convolution, along with the basic transforms ofhe Dirac-delta function and complex exponential. Thisasic toolset of operational rules is well known for theegular Fourier transform in single and multiple dimen-ions [1,2]. The convolution rules are particularly impor-ant as they form the basis of most filtering operations.

As is also known, the Fourier transform in three di-ensions can be developed in terms of spherical polar co-

rdinates [3], most usefully when the function beingransformed has some underlying spherical symmetry.or example, this has seen application in the field of pho-oacoustics [4] and some attempts to translate ideas fromontinuous to discrete domain [5]. The primary inspira-ion behind this paper actually lies in the Fourier diffrac-ion theorem [6] of acoustic tomography, versions of whichlso appear in other imaging modalities [7]. The Fourieriffraction theorem relates the Fourier transform of theorward scattered acoustic field to the value of the Fourierransform of the object on a circular [two-dimensional2D)] or a spherical [three-dimensional (3D)] arc. The facthat the scattered field is related to the Fourier transformn an arc is one of the primary motivators to switch theormulation of tomographic problems to curvilinear coor-

1084-7529/10/102144-12/$15.00 © 2

inates. The development of this operational toolset washus motivated by the desire to write various (acoustic,hermal, and photoacoustic) tomographic problems in cur-ilinear coordinates, while no Fourier toolset to aid in thisormulation could be found in the literature.

A complete interpretation of the standard Fourier op-rational toolset in terms of 2D polar coordinates haseen developed in [8], and the equivalent toolset for theD transform in spherical coordinates is still missingrom the literature. Thus, the aim of this paper is to de-elop this missing toolset for the 3D Fourier transform inpherical coordinates. Some results are already known,ut the results on shift, multiplication, and in particularonvolution are incomplete. Therefore, we feel that it isorthwhile to present a complete, detailed, and unifiedccount of the 3D curvilinear toolset for archival pur-oses, describing the mathematical foundation underly-ng all results.

What is to the author’s knowledge of particular noveltyn this paper is the treatment of shift, multiplication, andonvolution. It is known that 3D Fourier transforms foradially (spherically) symmetric functions can be inter-reted in terms of a (zeroth order) spherical Hankelransform. It is also known that Hankel transforms do notave a multiplication/convolution rule, a rule which hasound so much use in the Cartesian version of the trans-orm. In this paper, the multiplication/convolution rule isreated in detail for the curvilinear version of the trans-orm, and in particular it is shown that the sphericalankel transform does obey a multiplication/convolution

ule—once the proper interpretation of convolution is ap-lied. This paper carefully considers the definition of con-olution and derives the correct interpretation of this inerms of the curvilinear coordinates so that the standardultiplication/convolution rule is again applicable. Con-

010 Optical Society of America

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Natalie Baddour Vol. 27, No. 10 /October 2010 /J. Opt. Soc. Am. A 2145

olution forms the basis of filtering, so developingonvolution/multiplication rules is akin to developing altering toolset for multidimensional curvilinear signals.The outline of the paper is as follows. Section 2 intro-

uces the mathematical language of the spherical Hankelransform and spherical harmonics, upon which the 3Dourier transform in spherical polar coordinates is intro-uced in Section 3. Sections 4 and 5 introduce the multi-imensional Dirac-delta function and the complex expo-ential, respectively, and their curvilinear transforms.ection 6 discusses the Fourier rules for multiplication,nd Section 7 introduces the rules for working with thehift operator. The combination of multiplication and shifts required in order to finally arrive at defining the con-olution rules, addressed in Section 8. Section 9 brieflyonsiders the special case of convolution of sphericallyymmetric functions. Section 10 addresses angular convo-utions (spherical and full) and shows that the results inhis paper match those of the seminal paper by Driscollnd Healy [9], when the same interpretation of convolu-ion is applied. Section 11 outlines the Parseval relation-hips while Section 12 finally concludes the paper. Table 1ummarizes the spherical polar Fourier “toolset.”

. SPHERICAL HANKEL TRANSFORM ANDPHERICAL HARMONICShe spherical Bessel function can be defined from thepherical Bessel equation and admits several forms [10],ne of which is from the half-integer order Bessel function

jn�z� =� �

2zJn+1/2�z�. �1�

he spherical Bessel functions satisfy an orthogonalityelationship. The spherical Hankel transform can then be

Table 1. 3D Spherica

�r�� flk�r�

�r��= f�r ,�r ,�r�=�L

flk�r�Yl

k��r ,�r� �02��0

�f�r��Ylk��r ,�r�d

�r� −r�0�=��r−r0� / �r2 sin �r� .��r−�0��r−�0�

��r−r0� / r2Ylk��r0

,�

i�� 0·r� 4��i�ljl��0r�Ylk��0

,

�r� −r�0� �l�=0

�

�k�=−l�

l�8�i�l−l�Yl�

k��r0,�r0

�l

cl��l ,k , l� ,k��0�fl�

k−k��u�Sl�l,l��u

�r�g�r� hlk�r�= fl

k�r��glk�r�ª�

L�fl�

k� �l�=l−l�

l+l�cl

�r�� g�r� il / 2�2�0�Hl

k��jl��r�

�r��,��g�r�� hlk�r�= fl

k�r�glk�r�

f �R g��r ,�r ,�r� hlk�r�= fl

k�r�gl0�r��−1�k�4�

aNote that the following shorthand is used in the table: d=sin �d�d� , � � �=
L l=0 k=−
efined as [1,10]

F�n�� = Sn�f�r�� =�0

�

f�r�jn��r�r2dr. �2�

ote uses of the capital letter and of the hat to denote thepherical Hankel transform. Sn is used to specifically de-ote the spherical Hankel transform of order n. The in-erse transform is given by

f�r� =2

��

0

�

F�n��jn��r��2d�. �3�

pherical harmonics are the solution to the angular por-ion of Laplace’s equation in spherical polar coordinatesnd can be shown to be orthogonal. These spherical har-onics are given by [3]

Ylm��,�� =��2l + 1��l − m�!

4��l + m�!Pl

m�cos ��eim�, �4�

here Ylm is called a spherical harmonic function of de-

ree l and order m, Plm is an associated Legendre func-

ion, 0�� represents the colatitude, and 0��2�epresents the longitude. With the normalization given inq. (4), they are orthonormal so that

�0

2��0

�

YlmYl�

m� sin �d�d� = �ll��mm�. �5�

ere �ij is the Kronecker delta and the overbar indicateshe complex conjugate. Different normalizations of thepherical harmonics are possible [11]. The spherical har-onics form a complete set of orthonormal functions and

hus form a vector space. When restricted to the surface ofsphere, functions may be expanded on the sphere into a

ar Fourier Toolseta

Flk�� F��

4��−i�l�0�fl

k�r�jl��r�r2dr F�� =F�� ,�� ,��=�

LFl

k��Ylk�� ,��

4��−i�ljl��r0�Ylk��r0

,�r0� e−i�� ·r�0

�2��3 / �2��−�0�Ylk��0

,��0� �2��3�� −�� 0�

i�l� ·

u2du

4��−i�ljl��r0�Ylk��r0

,�r0��Fl

k�� e−i�� ·r�0F��

� ,k��gl�k−k� 4��−i�l�0

�hlk�r�jl��r�r2dr G�� F��

Hlk��=Gl

k��Flk�� G�� F��

4��−i�l�0�hl

k�r�jl��r�r2dr �L

Hlk��Yl

k�� ,��

1 4��−i�l�0�hl

k�r�jl��r�r2dr �L

Hlk��Yl

k�� ,��

� �.

l Pol

r

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��0�

��=l−l�

l+l��−

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��l ,k , l

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eries approximation much like a Fourier series [2].

. CONNECTION BETWEEN 3D FOURIERRANSFORMS AND SPHERICALARMONICS WITH SPHERICAL HANKELRANSFORMShe 3D Fourier transform of f�r��= f�x ,y ,z� is defined as

F�� = F��x,�y,�z� =�−�

�

f�r��e−i�� ·r��dr� . �6�

he spatial position vector r� can be rewritten in terms ofpherical coordinates by specifying the position r� in termsf �r ,�r ,�r� via the usual coordinate transformation fromartesian to spherical polar coordinates. Similarly, the

requency-space variables are also converted into theirwn set of polar coordinates so that position in frequencypace �� is given by �� ,�� ,��. Hence, Eq. (6) becomes

F�� = F��,��,��

=�0

2��0

��0

�

f�r,�r,�r�e−i�� ·r�r2 sin �rdrd�rd�r. �7�

o proceed in terms of spherical harmonics, both the origi-al function f and the Fourier kernel are expanded inpherical harmonic series. First, the Fourier kernel isritten in terms of spherical harmonics as [3]

e−i�� ·r� = 4��l=0

�

�k=−l

l

�− i�ljl��r�Ylk��r,�r�Yl

k��,��. �8�

ue to the commutative property of the dot product, theomplex conjugate in Eq. (8) may be taken on either the rr � spherical harmonic functions. Similarly, any well-ehaved function f�r��= f�r ,�r ,�r� can be expanded inerms of spherical harmonics so that

f�r�� = f�r,�r,�r� = �l=0

�

�k=−l

l

flk�r�Yl

k��r,�r�, �9�

here

flk�r� =�

0

2��0

�

f�r,�r,�r�Ylk��r,�r�sin �rd�rd�r. �10�

hese spherical harmonic expansions in Eqs. (8) and (9)re now substituted into the definition of the Fourierransform (7), and using the orthonormality of the spheri-al harmonics, we get

F��,��,�� = �l=0

�

�k=−l

l

4��− i�l��0

�

flk�r�jl��r�r2dr Yl

k��,��

= �l=0

�

�k=−l

l

4��− i�lF� lk��Yl

k��,��, �11�

here the integration with respect to r within the curlyraces has been recognized as a spherical Hankel trans-orm. That is, F� l

k�� is the lth order spherical Hankelransform of fk�r� so that F�k��=S �fk�r� r→��.
l l l l
Most importantly, it has been shown that the operationf taking the 3D Fourier transform is equivalent to (1)rst finding a spherical harmonic expansion in the angu-

ar variables and (2) then finding the lth order sphericalankel transform (of the radial variable to the spatial ra-ial variable) of the �l ,k�th coefficient in the sphericalarmonic series. Since each of these operations involves

ntegration over one variable only, the order of these op-rations is interchangeable.

For the inverse 3D Fourier transform, we invoke thecoordinate-less) definition of the inverse transform,

f�r�� =1

�2��3�−�

�

F�� ei�� ·r��d�� , �12�

nd note that this may be written in spherical coordi-ates. The 3D Fourier transform is expanded in sphericalarmonics as

F�� = F��,��,�� = �l=0

�

�k=−l

l

Flk��Yl

k��,��, �13�

here

Flk�� =�

0

2��0

�

F��,��,��Ylk��,��sin ��d��d��.

�14�

ere Flk�� denotes the �l ,k�th coefficient in the expansion

f F�� and does not imply the Fourier transform of flk�r�.

his relation will be found. Substituting the expansion inq. (13), as well as that for the Fourier kernel from (8)

nto the definition of the 3D Fourier transform in (12) andsing the orthonormality of the spherical harmonicsields

f�r�� = f�r,�r,�r�

= �l=0

�

�k=−l

l 1

4��i�l� 2

��

0

�

Flk��jl��r��2d� Yl

k��r,�r�,

�15�

here the quantity in curly braces is the inverse sphericalankel transform of Fl

k��. Comparing Eqs. (11) and (13),hen

Flk�� = 4��− i�lF� l

k�� = 4��− i�lSl�flk�r��, �16�

rom which the reverse relationship can be obtained.quation (16) and its inverse thus provide the means ofbtaining the coefficients in the Fourier transform expan-ion from those of the spatial domain expansion, and viceersa. Namely, starting with the knowledge of fl

k�r� (theoefficients for f in the spatial domain), Eq. (16) indicatesow to obtain Fl

k�� (the coefficients for the 3D Fourierransform F in the spatial frequency domain), and viceersa. With appropriate constants, the relationship is es-entially one of a spherical Hankel transform. The expres-ions derived here contain slight corrections to thoseiven in [3,10].

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. DIRAC-DELTA FUNCTION AND ITSRANSFORMhe Dirac-delta function in 3D spherical polar coordi-ates can be written as

f�r�� = ��r� − r�0� =1

r2 sin �r��r − r0��r − �r0

��r − �r0�.

�17�

rom the definition, the spherical harmonic expansion co-fficients for the Dirac-delta function are given by

flk�r� =

1

r2��r − r0�Ylk��r0

,�r0�, �18�

o that the full 3D transform itself becomes

F�� = F��,��,��

= �l=0

�

�k=−l

l

4��− i�ljl��r0�Ylk��r0

,�r0�Yl

k��,��

= e−i�� ·r�0. �19�

he Fourier transform of the Dirac-delta function is thexponential function—as expected and as shown in Eq.19). However, the toolbox development requires the coef-cients of the expansion of the Fourier transform, andhese are

Flk�� = 4��− i�ljl��r0�Yl

k��r0,�r0

�. �20�

he spherical harmonic coefficients of the function [Eq.18)] and transform [Eq. (20)] form a Fourier pair. This ishe second entry in Table 1, following the basic definitionsf the transform.

. COMPLEX EXPONENTIAL AND ITSRANSFORMrom Eq. (8), the spherical harmonic coefficients of theomplex exponential are given by

flk�r� = 4��i�ljl��0r�Yl

k��0,��0

�. �21�

sing the orthogonality of the spherical Bessel functions5], the coefficients of the transform can be calculatedrom Eq. (11):

F�� = �l=0

�

�k=−l

l

�4��2�

2�2�� − �0�Ylk��0

,��0�Yl

k��,��.

�22�

his gives

Flk�� = �2��3

1

�2�� − �0�Ylk��0

,��0�. �23�

e recognize from Eq. (18) that these are the coefficientsor the expansion of the Dirac-delta function so that Eq.22) gives the traditional Fourier transform of the com-lex exponential as the Dirac-delta function, as it should.he spherical harmonic coefficients of the function

Eq. (21)] and its transform [Eq. (23)] form a Fourierransform pair and are also included in Table 1.

. MULTIPLICATIONe consider the product of two functions h�r��= f�r��g�r��,here f�r��=�l=0

� �k=−ll fl

k�r�Ylk��r ,�r�; similarly for g, the co-

fficients flk�r� and gl

k�r� are given by Eq. (10), and we seeko find the equivalent coefficients hl

k�r� such that h�r��f�r��g�r��=�l=0

� �k=−ll hl

k�r�Ylk��r ,�r�. This shall be accom-

lished by finding the Fourier transform of h�r�� and usinghe expansions for f�r�� and g�r��, along with Eq. (8):

H�� =�−�

�

f�r��g�r��e−i�� ·r�dr�

=�0

��0

2��0

�

�L�

fl�k��r�Yl�

k��r,�r��L�


k��r,�r�

�L

4��− i�ljl��r�Ylk��r,�r�Yl

k��,��

sin �rd�rd�rr2dr, �24�

here the shorthand notation �L� �=�l=0� �k=−l

l � � has beensed. Performing the integration over the angular vari-bles requires the evaluation of the following integral:

�0

2��0

�

Yl�k��r,�r�Yl�

k��r,�r�Ylk��r,�r�sin �rd�rd�r.

�25�

n fact, it turns out that the integrals in Eq. (25) arenown as Slater coefficients which are defined as [12,13]

cl��l,k,l�,k�� =�0

2��0

�

Ylk��r,�r�Yl�

k��r,�r�Yl�k−k��r,�r�

sin �rd�rd�r, �26�

nd are nonzero only for l− l�� l�� l+ l�. Equation (25)an also be expressed in terms of Clebsh–Gordan coeffi-ients [14] or Wigner 3j-symbols [15]. In the following, thelater coefficients of Eq. (26) will be used since their defi-ition in terms of an integral is the most straightforward.sing this definition of Slater coefficients, Eq. (24) be-

omes

H�� = �l=0

�

�k=−l

l

4��− i�l��0

�

�l�=0

�

�k�=−l�

l�

fl�k��r�

�l�=l−l�

l+l�

cl��l,k,l�,k��gl�k−k��r�jl��r�r2dr Yl

k��,��.

�27�

omparing Eq. (27) with Eq. (11), the quantity in curlyraces is

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hlk�r� = �

l�=0

�

�k�=−l�

l�

fl�k��r� �

l�=l−l�

l+l�

cl��l,k,l�,k��gl�k−k��r�. �28�

ased on knowledge of Fourier transforms and that mul-iplication in one domain becomes a convolution in thether domain and following the methodology used for po-ar coordinates in [8], Eq. (28) is to be interpreted as theonvolution of the spherical harmonic series for f and g.learly, since the operation of multiplication commutes

hen Eq. (28) still holds with the roles of f and g reversed.n other words, we define the convolution of two harmoniceries for f and g as

flk�r� � gl

k�r� � �l�=0

�

�k�=−l�

l�

fl�k��r� �

l�=l−l�

l+l�

cl��l,k,l�,k��gl�k−k��r�,

�29�

nd the results of the preceding analysis demonstratehat

�fg�lk = fl

k � glk. �30�

efining the convolution of two spherical harmonic seriess given in Eqs. (29) and (30) yields the familiar resulthat the spherical harmonic transform of a product is theonvolution of their respective transforms. This main-ains the usual multiplication/convolution rule for stan-ard Fourier transforms to the spherical harmonic trans-orm (which is often called a spherical Fourier transform).nlike Cartesian or polar coordinates, definitions of con-olution for spherical harmonic series are not standard inhe literature. For example, in [9,16], although convolu-ions of spherical harmonic expansions are mentioned,either author offers a definition for the operation of theonvolution of two expansions.

. SPATIAL SHIFThe expression for a Fourier series shifted in space is

ound by finding the inverse Fourier transform of thexponential-weighted transform. In other words, f�r� −r�0�s defined from

f�r� − r�0� = F−1�e−i�� ·r�0F�� . �31�

he reason for defining the shifted function from (31) ishat we have already found the expansion for the complexxponential as well as the rules for product of two expan-ions. It is not sufficient to find any expression for thepatial shift, but rather we seek the expression that is (i)n the form of a spherical harmonic series and (ii) in termsf the unshifted coefficients of the original function, ashis builds the rule for what to do to the coefficients when

shift is desired. Thus, by building on the previouslyound results, the relevant spatial shift result can beound in the desired form.

Using the inverse Fourier transform in Eq. (12), alongith expansions (8) and (11), then the desired quantity isiven by

f�r� − r�0� =1

�2��3�0

��0

2��0

�

�l�=0

�

�k�=−l�

l�

4��− i�l�

��0

�

fl�k��u�jl��u�u2du Yl�

k��,��

�l�=0

�

�k�=−l�

l�

4��− i�l�jl��r0�Yl�k��r0

,�r0�Yl�

k��,��

�l=0

�

�k=−l

l

4��i�ljl��r�Ylk��,��Yl

k��r,�r��2

sin ��d��d��d�. �32�

ntegration over the angular variables is nonzero only if�=k−k� and l− l�� l�� l+ l� since the same Slater inte-ral as given in Eq. (25) appears, and thus the discussionubsequent to Eq. (25) still applies. Integration over thengular variables will again lead to the Slater coeffi-ients. The integration over the radial variable yields ashift” type operator featuring a triple Bessel producthich is defined as

Sl�l,l��u,r,r0� =�

0

�

jl��u�jl��r0�jl��r��2d�. �33�

ith the shift operator and Slater coefficients, Eq. (32)ecomes

f�r� − r�0� = �l=0

�

�k=−l

l

8�i�lYlk��r,�r��

l�=0

�

�k�=−l�

l�

�− i�l�Yl�k��r0

,�r0�

�l�=l−l�

l+l�

�− i�l�cl��l,k,l�,k��

�0

�

fl�k−k��u�Sl�

l,l��u,r,r0�u2du. �34�

ence, the coefficients for the shifted function are

f�r� − r�0��lk = �

l�=0

�

�k�=−l�

l�

8�i�l−l�Yl�k��r0

,�r0�

�l�=l−l�

l+l�

�− i�l�cl��l,k,l�,k��

�0

�


l,l��u,r,r0�u2du. �35�

he preceding equation thus defines the rule for findinghe coefficients of the shifted function f�r� −r�0��l

k, once theoefficients of the original function f�r�l

k are known. Thiss the shift rule while remaining in the spatial domain.

The “shift operator” in Eq. (33) is non-trivial and haseen the topic of several papers, e.g., [17–19]. The mostmportant observation regarding these results is that

l�l,l��u ,r ,r0� is zero unless �u ,r ,r0� can form a triangle or aegenerate triangle [that is to say �u ,r ,r0� are collinear].

hat is, Sl,l��u ,r ,r � is zero unless r−r �u�r+r , which
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hould be geometrically intuitive. If �u ,r ,r0� cannot formtriangle, the proposed geometry is not physically realiz-

ble. Mathematics (and intuition) indicates that the shiftperator evaluates to zero.

Corresponding coefficients of the 3D transform of thehifted function can also be found. That is, if we define�r��= f�r� −r�0�, then H�� =e−i�� ·r�0F�� , and Hl

k�� areought. Knowing that multiplication in the Fourier do-ain implies convolution of the coefficients, with the con-

olution operation defined in Eq. (29), and also knowinghe coefficients for the complex exponential from (21),hen

Hlk�� = 4��− i�ljl��r0�Yl

k��r0,�r0

�� Flk��, �36�

r more explicitly

Hlk�� = �

l�=0

�

�k�=−l�

l�

4��− i�l�jl��r0�Yl�k��r0

,�r0�

�l�=l−l�

l+l�

cl��l,k,l�,k��Fl�k−k��, �37�

s per the definition of series convolution in Eq. (29).

. CONVOLUTIONhe 3D convolution of two functions is defined by

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h�r�� = f�r�� g�r�� =�−�

�

g�r�0�f�r� − r�0�dr�0. �38�

he triple-star notation of �� is used to emphasize thathis is a 3D convolution and to distinguish it from one-imensional (1D) and 2D convolutions, and it follows theonvention in [8].

Using the spherical harmonic expansion for g and thehifted version of f given by Eq. (34), Eq. (38) is

h�r�� =�0

��0

2��0

�

�l�=0

�

�k�=−l�

l�

gl�k��r0�Yl�

k��r0,�r0

�

�l=0

�

�k=−l

l


l�=0

�

�k�=−l�

l�


,�r0�

�l�=l−l�

l+l�

�− i�l�cl��l,k,l�,k��

�0

�


l,l��u,r,r0�u2du

sin �r0d�r0

d�r0r0

2dr0. �39�

ntegrating over the angular variables and using the or-hogonality of the spherical harmonics gives l�= l� and

=k . This and the definition of the shift operator give
� �
h�r�� = �l=0

�

�k=−l

l


l�=0

�

�k�=−l�

l�

�− i�l� �l�=�l−l��

l+l�

�− i�l�cl��l,k,l�,k��0

��0

�

gl�k��r0�jl��r0�r0

2dr0

=il�

4�Gl�

k��

�0

�

fl�k−k��u�jl��u�u2du

=il�

4�Fl�

k−k��

jl��r��2d�.

�40�

s indicated in Eq. (40), the integrals in the last term cane recognized as the definition of the spherical harmonicoefficients of the Fourier transform of G and H. Thus,he preceding equation can be written in the form of Eq.15) as

h�r�� = �l=0

�

�k=−l

l �i�l

4�� 2

��

0

�

Hlk��jl��r��2d� Yl

k��r,�r�

= �l=0

�

�k=−l

l

hlk�r�Yl

k��r,�r�, �41�

here

Hlk�� = �

l�=0

�

�k�=−l�

l�

Gl�k��

l�=l−l�

l+l�

cl��l,k,l�,k��Fl�k−k��

= Glk�� Fl

k��. �42�

hus, the convolution of two functions in space is the con-olution of the spherical harmonic transform coefficients

f their 3D Fourier transform. From the section on theultiplicative property, it is known that convolving the

oefficients is equivalent to multiplication of the functionshemselves; or, in other words, the result that Hl

k��Gl

k��Flk�� is equivalent to stating that H��

G�� F�� , as would be expected. This is, of course, a wellnown Fourier result and serves to confirm the accuracyf the development. However, for the purposes of this de-elopment, the main result we seek is that of Eq. (41),hich gives the values of hl

k�r� in terms of glk�r� and fl

k�r�or rather the spherical harmonic transform of those) andssentially defines the convolution operation for functionsiven as a spherical harmonic series. This equation de-nes the convolution operation so that to find the convo-

ution of two functions given in spherical harmonic seriesorm, one must first find the �l ,k�th coefficient of the 3Dourier transform of each function, namely, Gl

k�� and

lk��, and then subsequently convolve the resulting seriess per Eq. (42) to get Hk��. The final step is then to in-erse spherical Hankel transform the result to finally ob-ain hk�r�. It is pointed out that convolving the two func-
l

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ions f�r�� and g�r�� is not equivalent to convolving theireries—the convolution of the series was shown previ-usly to be equivalent to the multiplication of the func-ions themselves.

. CONVOLUTION OF SPHERICALLYYMMETRIC FUNCTIONSpherical harmonic series expansions of spherically sym-etric functions include the (0,0) term only. The 3D con-

olution of the two spherically symmetric functions is de-ned as usual as

h�r�� = f�r�� g�r�� =�−�

�

g�r�0�f�r� − r�0�dr�0, �43�

here it is noted that the integration is over all r�0, whichncludes all possible values of radial and angular vari-bles. For spherically symmetric functions, Eq. (43)hould be properly interpreted as a 3D convolution alongith Eq. (34) for the shifted function. In other words, Eq.

43) is not to be interpreted as

�0

�

g�r0�f�r − r0�dr0, �44�

hich would be a 1D convolution in the radial variable.sing the fact that since the functions are spherically

ymmetric only the l=k=0 term is retained in the seriesnd that Y0

0�� ,��=1/�4�, then the convolution of twopherically symmetric functions becomes

h�r�� = f�r�� g�r�� =�−�

�

g�r0�81

�4�

�l�=0

�

�k�=−l�

l�


,�r0� �l�=−l�

l�

�− i�l�cl��0,0,l�,k��

�0

�

fl�−k��u�Sl�

0,l��u,r,r0�u2dudr�0. �45�

ince l�= l� and then l�=k�=0, because the sphericallyymmetric function only has the (0,0) entry, given by Eq.10) as f0

0�r�= f�r��4�, Eq. (45) simplifies to

h�r�� =�0

��0

2��0

�

g�r0�81

�4�

1

�4�c0�0,0,0,0�

�0

�

f00�u�S0

0,0�u,r,r0�u2du sin �r0d�r0

d�r0r0

2dr0.

�46�

rom Eq. (26), c0�0,0,0,0�=1/�4� so that integrationver the angular variables yields

h�r�� = f�r� � � � g�r�

=�0

�

g�r0�8�0

�

f�u�S00,0�u,r,r0�u2dur0

2dr0, �47�

here the triple star emphasizes that this is a 3D convo-ution. Equation (47) can be thought of as

f�r�� g�r�� = f�r� � � � g�r� =�0

�

g�r0��3D�r − r0�r0dr0,

�48�

ith

�3D�r − r0� =�0

�

f�u�8S00,0�u,r,r0�u2du

=�0

2��0

�

f�r� − r�0�sin �r0d�r0

d�r0. �49�

hus the correct process of a full 3D convolution is thathe function f is shifted from r� to r�0 (destroying sphericalymmetry), and then the resulting shifted function is inte-rated over all angular variables. The unshifted functionis still spherically symmetric and unaffected by the in-

egration over angular variables so that the final result isiven by Eq. (48). While it is tempting to write Eq. (44) for3D convolution, it is in fact incorrect. The correct ver-

ion is given by Eq. (48).The 3D convolution of two radially symmetric functions

ields another radially symmetric function; and moreover,y using the proper definition of a 3D convolution insteadf the tempting definition of a 1D convolution, the well-nown relationship between convolutions and multiplica-ions in space and frequency domains is preserved,amely,

h�r� = f�r� � � � g�r� ⇒ H�� = F��G��. �50�

he key point to this relationship is the proper definitionf the convolution as a 3D convolution.

0. ANGULAR CONVOLUTIONSor any two functions f�r��= f�r ,� ,�� and g�r��=g�r ,� ,�� an-ular convolutions can also be defined as was done in twoimensions in [8]. These shall be discussed next.

. Rotation Formulas for Spherical Harmonicso consider angular convolutions, the rotation of a spheri-al harmonic needs to be considered. These formulas cane found in the literature, for example, [20,21], and theost relevant are presented here without proofs.We write a single-rotation operator as Rz��, with the

ubscript denoting the axis about which to take a rotationnd the argument denoting the angle of rotation. Hencez�� defines a rotation about the z axis by an angle �. Aeneral 3D rotation operator can written as R ,�,�Rz��Ry� �Rz�� and is essentially the standard Euler-ngle rotations about axes z, y, and z by angles �, , and. The reader is reminded that the order in which rota-ions are performed does matter. The rotation formula forpherical harmonics is given by [20]

Ylm�R ,�,��,�� = �

m�=−l

l

Dmm�l � ,�,��Yl

m��,��. �51�

t is important to note that the m indices are mixed—apherical harmonic after rotation must be expressed as aombination of other spherical harmonics with different

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indices. However, the l indices are not mixed; rotationsf spherical harmonics of order l are composed entirely ofther spherical harmonics with order l. For a given order, Dl is a matrix that tells how a spherical harmonic trans-orms under rotation, i.e., how to rewrite a rotated spheri-al harmonic as a linear combination of spherical har-onics of the same order. Analytical forms for Dl can be

ound in [21]. In particular, since R ,�,�=Rz��Ry� �Rz��ependences on � and � are simple since they are rota-ions about the z axis, so

Dmm�l � ,�,�� = dmm�

l � �eim�eim��, �52�

here dl is a matrix that defines how a spherical har-onic transforms under rotation about the y axis. It can

e shown [20] that dl satisfies

dmm�l � � =�

0

2��0

�

Ylm�Ry� ��,��Yl

m��,��sin��d�d�.

�53�

losed form formulas for dmm�l � � can be found in [21].

In order to work on a sphere, to be able to “sweep out”n entire sphere using only rotations of the north pole at0,0,1), it suffices to use rotations about the y and z axes.ence, for rotation on the sphere, the rotation operatorecomes

R ,� = R ,�,0 = Rz��Ry� �, �54�

nd the representation matrices are

Dmm�l � ,�� = Dmm�

l � ,�,0� = dmm�l � �eim�. �55�

hese representation matrices satisfy orthogonality con-itions

��=0

2� � =0

�

�Dmnl � ,��Dm�n

l� � ,��sin d d�

=4�

2l + 1�ll��mm�. �56�

ote that there is no orthogonality over the index n sincehe integration is over two rotations � ,�� instead of theull Eulerian angles � ,� ,��. Using the orthogonality ofim� over the interval 0,2��, Eq. (56) in terms of the

mm�l � � is

� =0

�

�dmnl � ��dmn

l� � ��sin d =2

2l + 1�ll�. �57�

. Convolution on the Spherepherical harmonics and the accompanying sphericalarmonic expansions of a function can be considered to bespherical Fourier transform as opposed to a full-fledged

D Fourier transform, which would require an additionalpherical Hankel transform of order n. These sphericalourier transforms have recently found many applica-ions in computer graphics and computer vision. Specifi-ally, they have been used to efficiently represent the bi-irectional reflection distribution function (BRDF) ofaterials [22] and to improve the modeling of the lighting

f scenes for computer graphics [16,20,22]. In computerraphics, the interaction between the incident illumina-ion and the BRDF is a basic building block in most ren-ering algorithms. In computer vision, it is often desiredo undo the effects of the reflection operator: to invert thenteraction between the BRDF and lighting. In otherords, it is desired to perform inverse rendering—the es-

imation of material and lighting properties from realhotographs. Ramamoorthi and Hanrahan presented aathematical theory of reflection for general complex

ighting environments and arbitrary BRDFs [20]. Theyormalized the notion of convolution as it applied to reflec-ion and showed the operation of reflection to be a (spheri-al) convolution [20] over angular coordinates. Thus, theathematical operations involved are those of convolu-

ion and de-convolution of spherical harmonic expansions.ephrasing these last results in terms of the terminologysed here, the “convolution” which these computer visionesearchers refer to is the equivalent of a convolution overnly the angular variables in 3D spherical polar coordi-ates. This angular convolution over the latitudinal and

ongitudinal angles but not over the radial variable im-lies that this is a convolution on the sphere, and not aull 3D convolution.

Based on these previous works, we define the angularonvolution on the sphere as

h�r�� = f�r��,��g�r�� =1

�2��2�0

2��0

�

f�r,�0,�0�

g�r,� − �0,� − �0�sin �0d�0d�0. �58�

he subscript on � serves to denote the type of convolu-ion so that there is no confusion with single-variable orull multidimensional convolutions.

The notation for g�r ,�−�0 ,�−�0� is that of a shift inerms of the angular variables, which is in effect a rota-ion of the function g�r ,� ,�� by �−�0 ,−�0�. To evaluate thentegral in Eq. (58), the integration is to be performedver the variables with the zero subscript ��0 ,�0�, and it isesired to have the final result in terms of the un-ubscripted variables �r ,� ,��. Therefore, it is desirable toxpand the shifted function g�r ,�−�0 ,�−�0� in terms ofnshifted spherical harmonics. In other words, althoughhe natural expansion for the rotated g�r ,�−�0 ,�−�0� isiven in terms of the rotated spherical harmonics as

g�r,� − �0,� − �0� = �l�=0

�

�k�=−l�

l�

gl�k��r�Yl�

k�� − �0,� − �0�,

�59�

here gl�k��r� is found from Eq. (10), the expansion that we

eally need is that of the rotated (“shifted”) function�r ,�−�0 ,�−�0� in un-rotated spherical harmonics:

g�r,� − �0,� − �0� = �l=0

�

�k=−l

l

�lk�r�Yl

k��,��. �60�

he coefficients �lk�r� can be found in the normal way by

he usual definition (10) along with the function whose co-fficients are sought [Eq. (59)], that is,

Ahrum

Hgn

ID

a

Wt

TDc�

CtcmefE

T

G

Tstt


�lk�r� =�

0

2��0

��l�=0

�

�k�=−l�

l�

gl�k��r�Yl�

k�� − �0,� − �0� Yl

k��,��sin �d�d�

= �l�=0

�

�k�=−l�

l�

gl�k��r�

�02��

0

�

Yl�k�� − �0,� − �0�Yl

k��,��sin �d�d�.

�61�

s previously mentioned, Yl�k��−�0 ,�−�0� is a spherical

armonic that has been rotated by �−�0 ,−�0� so that theotation formula for spherical harmonics [Eq. (51)] can besed. Using the rotation formula for the spherical har-onics, the integral in Eq. (61) can be written as

�0

2��0

�

Yl�k�� − �0,� − �0�Yl


=�0

2��0

�

�k�=−l�

l�

Dk�k�l� �− �0,− �0�

Yl�k��,��Yl

k��,��sin �d�d�. �62�

owever, the orthogonality of the spherical harmonicsives that the integral on the right hand side of Eq. (62) isonzero only for l= l� and k=k� so that the result becomes

�0

2��0

�

Yl�k�� − �0,� − �0�Yl


= �ll�Dk�kl �− �0,− �0�. �63�

n fact, Eq. (63) could be taken as the definition of

k�kl �−�0 ,−�0�. Given this result, Eq. (61) becomes

�lk�r� = �

l�=0

�

�k�=−l�

l�

gl�k��r��ll�Dk�k

l �− �0,− �0�

= �k�=−l

l

glk��r�Dk�k

l �− �0,− �0�, �64�

nd the sought expansion is finally given as

g�r,� − �0,� − �0�

= �l=0

�

�k=−l

l � �k�=−l

l

glk��r�Dk�k

l �− �0,− �0� Ylk��,��.

�65�

e are now in a position to compute the convolution onhe sphere, as defined by Eq. (58):

f�r��,��g�r�� =1

�2��2�0

2��0

�

f�r,�0,�0�

g�r,� − �0,� − �0�sin �0d�0d�0

=1

�2��2�0

2��0

�

�l�=0

�

�k�=−l�

l�


k��0,�0�

��l=0

�

�k=−l

l

�k�=−l

l

glk��r�Dk�k

l �− �0,− �0�Ylk��,��

sin �0d�0d�0. �66�

o evaluate the integration in Eq. (66), the definition of

k�kl �−�0 ,−�0� is used, and it is observed that Yl

k�� ,�� canome outside the integral. Hence, the integration over−�0 ,−�0� reduces to evaluating

�0

2��0

�

Yl�k��0,�0�Dk�k

l �− �0,− �0�sin �0d�0d�0

=�0

2��0

�

Yl�k��0,�0��

0

2��0

�

Ylk�� − �0,�

− �0�Ylk��,��sin �d�d� sin �0d�0d�0. �67�

hanging the order of integration immediately yields thathe integral is nonzero only for l�= l, as previously dis-ussed. Furthermore, with the form of the spherical har-onics from Eq. (4) and their dependence on the complex

xponential, the integration over � and �0 can be per-ormed. Specifically, the � dependence of the integrand inq. (67) is given by

exp�ik��0�exp�ik�� − �0��exp�− ik��

= exp�i�k� − k��0�exp�i�k� − k��. �68�

hus, when Eq. (68) is integrated over all angles:

�0

2��0

2�

exp�i�k� − k��0�exp�i�k� − k��d�0d�

= �2��2�k�k��k�k. �69�

iven these simplifications, Eq. (66) becomes

f�r��,��g�r�� =1

�2��2 �l�=0

�

�k�=−l�

l�

fl�k��r��

l=0

�

�k=−l

l

�k�=−l

l

glk��r�

�2��2�ll��k�k��k�kYlk��,��

= �l=0

�

�k=−l

l

flk�r�gl

k�r�Ylk��,��. �70�

his is in fact a spectacularly useful result and is thepherical harmonic convolution theorem. What it says ishat if we define the spherical convolution of two func-ions as Eq. (58), then

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h�r�� = f�r��,��g�r�� = �l=0

�

�k=−l

l

hlk�r�Yl

k��,��

= �l=0

�

�k=−l

l

flk�r�gl

k�r�Ylk��,��, �71�

r, in other words, the standard convolution result ap-lies:

hlk�r� = fl

k�r�glk�r�. �72�

t cannot be emphasized enough that these “standard con-olution results,” by which we imply the equivalence ofonvolution in one domain to multiplication in the otheromain, rely on the correct definition of a convolution. Inarticular, the result in Eq. (72) depends on the definitionf convolution in Eq. (58).

. Full Rotational Convolutionquation (72) states that convolution on the sphere isquivalent to multiplication in the spherical Fourier do-ain; however this is not the same result as that obtained

n the seminal paper by Driscoll and Healey [9]. The rea-on for this is that the convolution defined in the preced-ng section, namely, Eq. (58), is not the same as the con-olution defined in [9], which is in fact a full rotationalonvolution, over all possible rotations in three dimen-ions. For each 3D rotation, R=R ,�,�, Driscoll and Healeyefined a rotation operator ��R� on functions on thephere:

��R�f�� = f�R−1�. �73�

n Eq. (73), f�� is a function on the sphere so that it is aunction of = �� ,��, and R−1 is the inverse rotation op-rator. Equation (73) indicates that the operator acts onhe function on the sphere f�� by rotating the angles ��y the rotation R−1.Driscoll and Healey [9] stated that any function on the

phere g may be used to define a convolution operator.his is accomplished by employing it as a weighting factor

or the rotation operators. They defined the operation ofeft convolution by the function g as

�f �R g��r,� =� f�r,R��R�dRg�r,�

=� f�r,R��g�r,R−1�dR, �74�

here is any point on the sphere, � is the north pole0,0,1), and the integration is taken over all 3D rotations, which implies integrating over all possible Eulerngles � ,� ,��. There are several important observationso be made regarding the definition of convolution in Eq.74). First, convolution in Eq. (74) does not necessarilyommute so that f �R g�g �R f. This follows upon closerxamination of the way the convolution operation in Eq.74) is defined. Second, in Eq. (74), the function f is evalu-ted at the rotation of the north pole by R, whereas theunction g itself is rotated by R−1. Thus, in this definitionf convolution, the way the two functions are treated isot the same. Further, although this was already men-

ioned, it bears repeating that the integrations in Eq. (74)re over all possible 3D rotations � ,� ,��, which is aarger set of rotations than those given in the angularonvolution of Eq. (58). The rotations in Eq. (58) are overhe longitudinal and latitudinal angles only, and not theull set of Euler angles as implied in Eq. (74).

Equation (74) can be interpreted in terms of the math-matics developed in this paper. To evaluate the integraln Eq. (74), we first need to write an expression for�r ,R��. In f�r ,R��, the north pole (0,0,1) is rotated by

,�,�, and since it is only a vector it is thus unaffected byhe final rotation � and is only a function of � ,��. Inther words, f�r ,R�� can be written as

f�r,R�� = f�r, ,�� = �l�=0

�

�k�=−l�

l�


k�� ,��. �75�

n the other hand, for the second term in the integral inq. (74), the whole function g�r ,� is rotated to g�r ,R−1�o that the rotation operator R ,�,� affects the whole func-ion, and more importantly in this case all three rotations ,� ,�� affect the final result. Using the rotation resultsiven in Eq. (65) and the simplification given in Eq. (52),e write

g�r,R−1� = g�r,R ,�,�−1 ��,��

= �l=0

�

�k=−l

l

�k�=−l

l

glk��r�Dk�k

l �− ,− �,− ��Ylk��,��

= �l=0

�

�k=−l

l

�k�=−l

l

glk��r�e−ik��Dk�k

l �− ,− ��Ylk��,��.

�76�

he benefit of writing g�r ,R−1� in the form of the previ-us equation is that it is identical to the formulation ofhe convolution on the sphere, with the exception of thedditional e−ik�� term due to the full 3D rotation. Thus,ith expressions for the two functions, we may proceed tovaluating the integral defining convolution in Eq. (74) toive

�f �R g��r,� =� f�r,R��g�r,R−1�dR

=�0

2��0

��0

2�

�l�=0

�

�k�=−l�

l�


k�� ,��

�l=0

�

�k=−l

l

�k�=−l

l

glk��r�e−ik��Dk�k

l �− ,

− ��Ylk��,��d� sin d d�. �77�

he first integral over the rotation � is zero unless k�=0.his integration over � was not present in the convolutionn the sphere. Since the two functions in this full rota-ional convolution are not treated equally, the angle �nly appears in the expansion for the function g�r ,R−1�,nd after integrating over all � this forces k�=0. We areow left with an integral to evaluate that is very similaro Eq. (66), with the exception that since k =0, we need to
�

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Toowt

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valuate D0kl �− ,−��. An expression for this is derived

nd proved in the appendix of [20] as

D0kl �− ,− �� =� 4�

2l + 1Yl

k� ,��e−ik�. �78�

his in fact simplifies to

D0kl �− ,− �� =� 4�

2l + 1Yl

k� ,��e−ik�

= �− 1�k� 4�

2l + 1Yl

k� ,��. �79�

ith this considerable simplification, the integral that re-ains to be evaluated is given by

�f �R g��r,� =�0

2��0

�

�l�=0

�

�k�=−l�

l�


k�� ,��l=0

�

�k=−l

l

gl0�r�

�− 1�k� 4�

2l + 1Yl

k� ,��sin d d�Ylk��,��.

�80�

he orthonormality of the spherical harmonics guaran-ees that this integration over the angles and � givesk�k�l�l so that

�f �R g��r,� = �l=0

�

�k=−l

l

flk�r�gl

0�r��− 1�k� 4�

2l + 1Yl

k��,��.

�81�

his is precisely the result derived in [9] and, as previ-usly noted, differs from the result given for convolutionver the sphere as defined above in Eq. (58). In otherords, Eq. (81) states that the full 3D rotational convolu-

ion gives

�f �R g�lk = fl

k�r�gl0�r��− 1�k� 4�

2l + 1, �82�

hereas the spherical convolution of Eq. (58) yields

�f��,��g�lk = fl

k�r�glk�r�. �83�

1. PARSEVAL RELATIONSHIPSParseval relationship is important as it deals with the

power” of a signal or function in the spatial and fre-uency domains. In spherical polar coordinates if= �r ,� ,��, then −r� = �r ,�−� ,�+�� and Yl

k��r ,�r��−1�kYl

−k��r ,�r�. Hence, it can be shown that, for any f,

f�− r�� = �l=0

�

�k=−l

l

flk�r��− 1�kYl

−k��,�� = �l=0

�

�k=−l

l

fl−k�r�

�− 1�−kYlk��,��. �84�

onvolution of any two functions is given by Eq. (40). Theonvolution of the two functions evaluated at r=0 implieshat l=k=0 since jl�0�=�l0. Values of the Slater coeffi-ients at l=k=0 become

cl��0,0,l�,k�� =1

�4��

0

2��0

�

Yl�k��r,�r�Yl�

−k��r,�r�

sin �rd�rd�r

=1

�4��− 1�−k��l�l�. �85�

sing these simplifications, Eq. (40) for any two functionss

f � gr�=0� =�−�

�

g�r�0�f�− r�0�dr�0 =8

�4��l�=0

�

�k�=−l�

l�

�− 1�l��− 1�−k�

1

�4��

0

��0

�

gl�k��r0�jl��r0�r0

2dr0

�0

�

fl�−k��u�jl��u�u2du�2d�. �86�

n particular, if the preceding convolution is performed forny function g�r�� and another function f�−r��, using the re-ult of Eq. (84) the preceding equation gives one form of aeneralized Parseval relationship as

�−�

�

g�r��f�r��dr� =1

�2��3�l=0

�

�k=−l

l �0

�

Glk��Fl

k��2d�. �87�

he integral on the left hand side of the preceding equa-ion can be expressed in terms of the spherical harmonicxpansion. To do this, use the multiplication property inq. (28) to write

f�r��g�r�� = �l=0

�

�k=−l

l ��l�=0

�

�k�=−l�

l�

fl�k��r�

�l�=l−l�

l+l�

cl��l,k,l�,k��gl�k−k��r� Yl

k��r,�r�.

�88�

rom Eq. (84), coefficients of the expansion for any func-ion f�r�� are given by fl

−k�r��−1�k. Integrating both sides ofq. (88) over all space requires the integration of

lk��r ,�r� over all angles, which is nonzero only for l=k0, which yields �4�. As previously calculated, the valuef the required Slater coefficient is cl��0,0, l� ,k��1/�4��−1�−k��l�l�. All these simplifications togetherombine to yield

�−�

�

g�r��f�r��dr� = �l=0

�

�k=−l

l �0

�

glk�r�fl

k�r�r2dr. �89�

ombining the results of Eqs. (87) and (89) gives anotherersion of the generalized Parseval relationship as

�l=0

�

�k=−l

l �0

�

glk�r�fl

k�r�r2dr =1

�2��3�l=0

�

�k=−l

l �0

�

Glk��Fl

k��2d�.

�90�

learly, it then follows from the preceding equation that

1IptsrrATasctcacdcc

R

1

11

11

1

1

1

1

1

2

2

2


�l=0

�

�k=−l

l �0

�

flk�r�2r2dr =

1

�2��3�l=0

�

�k=−l

l �0

�

Flk��2�2d�.

�91�

2. SUMMARY AND CONCLUSIONSn summary, this paper has considered the sphericalolar-coordinate version of the standard 3D Fourierransform and derived the operational toolset required fortandard Fourier operations. This version of the 3D Fou-ier transform is most useful for functions that are natu-ally described in terms of spherical polar coordinates.dditionally, Parseval relationships were also derived.he results of the paper are concisely collected in Table 1t the end of the paper. Of particular interest are the re-ults on convolution and spatial shift. Notably, standardonvolution/multiplication rules do apply for 3D convolu-ion, and in particular special results apply for the specialases of angular-only convolutions. The results forngular-only convolutions bear special mention as spheri-al convolutions and full rotational convolutions admitifferent results, and it is important to define the type ofonvolution required in order to apply the correctonvolution/multiplication rule.

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3d fourier transforms

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