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Natalie Baddour

Operational and convolution properties of three-dimensional Fourier transforms in spherical polar coordinates Natalie Baddour Department of Mechanical Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, Ontario K1N 6N5, Canada (nbaddour@uottawa.ca) 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 can be written in spherical coordinates as a combination of spherical Hankel transforms and spherical harmonic series. However, to be as useful as its Cartesian counterpart, a spherical version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, etc. This paper derives the spherical version of the standard Fourier operation toolset. In particular, convolution in various forms is discussed in detail as this has important consequences for filtering. It is shown that standard multiplication and convolution rules do apply as long as the correct definition of convolution is applied. © 2010 Optical Society of America OCIS codes: 070.6020, 070.4790, 350.6980, 100.6950.

1. INTRODUCTION The Fourier transform has proved to be a powerful tool in many diverse disciplines and indispensable to signal processing. One of its powerful features is it can easily be extended to n dimensions. The strength of the Fourier transform is that it is accompanied by a toolset of operational properties that simplify the calculation of more complicated transforms through the use of these standard rules, turning otherwise complex calculations into those that can be done via a look-up table and a core set of transforms. Specifically, the standard Fourier toolset consists of results for translation (spatial shift), multiplication, and convolution, along with the basic transforms of the Dirac-delta function and complex exponential. This basic toolset of operational rules is well known for the regular Fourier transform in single and multiple dimensions [1,2]. The convolution rules are particularly important as they form the basis of most filtering operations. As is also known, the Fourier transform in three dimensions can be developed in terms of spherical polar coordinates [3], most usefully when the function being transformed has some underlying spherical symmetry. For example, this has seen application in the field of photoacoustics [4] and some attempts to translate ideas from continuous to discrete domain [5]. The primary inspiration behind this paper actually lies in the Fourier diffraction theorem [6] of acoustic tomography, versions of which also appear in other imaging modalities [7]. The Fourier diffraction theorem relates the Fourier transform of the forward scattered acoustic field to the value of the Fourier transform of the object on a circular [two-dimensional (2D)] or a spherical [three-dimensional (3D)] arc. The fact that the scattered field is related to the Fourier transform on an arc is one of the primary motivators to switch the formulation of tomographic problems to curvilinear coor1084-7529/10/102144-12/$15.00

dinates. The development of this operational toolset was thus motivated by the desire to write various (acoustic, thermal, and photoacoustic) tomographic problems in curvilinear coordinates, while no Fourier toolset to aid in this formulation could be found in the literature. A complete interpretation of the standard Fourier operational toolset in terms of 2D polar coordinates has been developed in [8], and the equivalent toolset for the 3D transform in spherical coordinates is still missing from the literature. Thus, the aim of this paper is to develop this missing toolset for the 3D Fourier transform in spherical coordinates. Some results are already known, but the results on shift, multiplication, and in particular convolution are incomplete. Therefore, we feel that it is worthwhile to present a complete, detailed, and unified account of the 3D curvilinear toolset for archival purposes, describing the mathematical foundation underlying all results. What is to the author’s knowledge of particular novelty in this paper is the treatment of shift, multiplication, and convolution. It is known that 3D Fourier transforms for radially (spherically) symmetric functions can be interpreted in terms of a (zeroth order) spherical Hankel transform. It is also known that Hankel transforms do not have a multiplication/convolution rule, a rule which has found so much use in the Cartesian version of the transform. In this paper, the multiplication/convolution rule is treated in detail for the curvilinear version of the transform, and in particular it is shown that the spherical Hankel transform does obey a multiplication/convolution rule—once the proper interpretation of convolution is applied. This paper carefully considers the definition of convolution and derives the correct interpretation of this in terms of the curvilinear coordinates so that the standard multiplication/convolution rule is again applicable. Con© 2010 Optical Society of America

Natalie Baddour

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

volution forms the basis of filtering, so developing convolution/multiplication rules is akin to developing a filtering toolset for multidimensional curvilinear signals. The outline of the paper is as follows. Section 2 introduces the mathematical language of the spherical Hankel transform and spherical harmonics, upon which the 3D Fourier transform in spherical polar coordinates is introduced in Section 3. Sections 4 and 5 introduce the multidimensional Dirac-delta function and the complex exponential, respectively, and their curvilinear transforms. Section 6 discusses the Fourier rules for multiplication, and Section 7 introduces the rules for working with the shift operator. The combination of multiplication and shift is required in order to finally arrive at defining the convolution rules, addressed in Section 8. Section 9 briefly considers the special case of convolution of spherically symmetric functions. Section 10 addresses angular convolutions (spherical and full) and shows that the results in this paper match those of the seminal paper by Driscoll and Healy [9], when the same interpretation of convolution is applied. Section 11 outlines the Parseval relationships while Section 12 finally concludes the paper. Table 1 summarizes the spherical polar Fourier “toolset.”

2. SPHERICAL HANKEL TRANSFORM AND SPHERICAL HARMONICS The spherical Bessel function can be defined from the spherical Bessel equation and admits several forms [10], one of which is from the half-integer order Bessel function jn共z兲 =

冑

2z

Jn+1/2共z兲.

共1兲

The spherical Bessel functions satisfy an orthogonality relationship. The spherical Hankel transform can then be

2145

defined as [1,10]

共兲 = S 兵f共r兲其 = F n n

冕

⬁

f共r兲jn共r兲r2dr.

共2兲

0

Note uses of the capital letter and of the hat to denote the spherical Hankel transform. Sn is used to specifically denote the spherical Hankel transform of order n. The inverse transform is given by 2 f共r兲 =

冕

⬁

共兲j 共r兲2d . F n n

共3兲

0

Spherical harmonics are the solution to the angular portion of Laplace’s equation in spherical polar coordinates and can be shown to be orthogonal. These spherical harmonics are given by [3]

冑

Ylm共, 兲 =

共2l + 1兲共l − m兲! 4共l + m兲!

Plm共cos 兲eim ,

共4兲

where Ylm is called a spherical harmonic function of degree l and order m, Plm is an associated Legendre function, 0 ⱕ ⱕ represents the colatitude, and 0 ⱕ ⱕ 2 represents the longitude. With the normalization given in Eq. (4), they are orthonormal so that

冕冕 2

0

0

YlmYl⬘ ⬘ sin dd = ␦ll⬘␦mm⬘ . m

共5兲

Here ␦ij is the Kronecker delta and the overbar indicates the complex conjugate. Different normalizations of the spherical harmonics are possible [11]. The spherical harmonics form a complete set of orthonormal functions and thus form a vector space. When restricted to the surface of a sphere, functions may be expanded on the sphere into a

Table 1. 3D Spherical Polar Fourier Toolseta flk共r兲

Flk共兲

F共 ជ兲

f共rជ 兲 = f共r , r , r兲 = 兺flk共r兲Ylk共r , r兲

兰02兰0f共rជ 兲Ylk共r , r兲d⍀r

4共−i兲l兰0⬁flk共r兲jl共r兲r2dr

F共 ជ 兲 = F共 , , 兲 = 兺Flk共兲Ylk共 , 兲

␦共rជ − rជ 0兲 = ␦共r − r0兲 / 共r sin r兲 . ␦ 共 r − 0兲 ␦ 共 r − 0兲

␦共r − r0兲 / r

4共−i兲

f共rជ 兲 L

L

2

eiជ 0·rជ

2

Ylk共r0 , r0兲

l

e−iជ ·rជ 0

jl共r0兲Ylk共r0 , r0兲

共2兲3 / 2 ␦共 − 0兲Ylk共0 , 0兲

共2兲3␦共 ជ − ជ 0兲

关4共−i兲ljl共r0兲Ylk共r0 , r0兲兴 ⴱ Flk共兲

e−iជ ·rជ 0F共 ជ兲

l+l⬘ k k−k hlk共r兲 = flk共r兲 ⴱ glk共r兲 ª 兺 fl ⬘ 兺 cl⬙共l , k , l⬘ , k⬘兲gl ⬘ ⬘ ⬙ L⬘ l⬙=兩l−l⬘兩

4共−i兲l兰0⬁hlk共r兲jl共r兲r2dr

G共 ជ 兲 ⴱ ⴱ ⴱ F共 ជ兲

f共r兲 ⴱ ⴱ ⴱ g共r兲

il / 22 兰0⬁Hlk共兲jl共r兲2d

Hlk共兲 = Glk共兲 ⴱ Flk共兲

G共 ជ 兲F共 ជ兲

f共rជ 兲ⴱ共,兲g共rជ 兲

hlk共r兲 = flk共r兲glk共r兲

4共−i兲l兰0⬁hlk共r兲jl共r兲r2dr

兺Hlk共兲Ylk共 , 兲

4共−i兲l兰0⬁hlk共r兲jl共r兲r2dr

兺Hlk共兲Ylk共 , 兲

f共rជ − rជ 0兲

f共r兲g共r兲

共f ⴱR g兲共r , r , r兲 a

4共i兲ljl共0r兲Ylk共0 , 0兲 ⬁ l+l⬘ l⬘ k 兺 兺 8共i兲l−l⬘Yl ⬘共r0 , r0兲 兺 共−i兲l⬙ · ⬘ l⬘=0 k⬘=−l⬘ l⬙=兩l−l⬘兩 k−k l,l cl⬙共l , k , l⬘ , k⬘兲兰0⬁fl ⬘共u兲Sl ⬘共u , r , r0兲u2du ⬙ ⬙

hlk共r兲 = flk共r兲gl0共r兲共−1兲k

冑 4 / 2l + 1

l ⬁ 兺k=−l 兵 其. Note that the following shorthand is used in the table: d⍀ = sin dd , 兺L兵 其 = 兺l=0

L

L

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Natalie Baddour

series approximation much like a Fourier series [2].

3. CONNECTION BETWEEN 3D FOURIER TRANSFORMS AND SPHERICAL HARMONICS WITH SPHERICAL HANKEL TRANSFORMS The 3D Fourier transform of f共rជ 兲 = f共x , y , z兲 is defined as F共 ជ 兲 = F共x, y, z兲 =

冕

⬁

f共rជ 兲e−i共ជ ·rជ 兲drជ .

共6兲

−⬁

The spatial position vector rជ can be rewritten in terms of spherical coordinates by specifying the position rជ in terms of 共r , r , r兲 via the usual coordinate transformation from Cartesian to spherical polar coordinates. Similarly, the frequency-space variables are also converted into their own set of polar coordinates so that position in frequency space ជ is given by 共 , , 兲. Hence, Eq. (6) becomes

Most importantly, it has been shown that the operation of taking the 3D Fourier transform is equivalent to (1) first finding a spherical harmonic expansion in the angular variables and (2) then finding the lth order spherical Hankel transform (of the radial variable to the spatial radial variable) of the 共l , k兲th coefficient in the spherical harmonic series. Since each of these operations involves integration over one variable only, the order of these operations is interchangeable. For the inverse 3D Fourier transform, we invoke the (coordinate-less) definition of the inverse transform, f共rជ 兲 =

=

冕 冕冕

0

0

e

= 4

f共r, r, r兲e−iជ ·rជ r2 sin rdrdrdr . 共7兲

兺 兺 共− i兲 j 共r兲Y 共 , 兲Y 共 , 兲. l

k l

l

r

r

k l

共8兲

l=0 k=−l

Due to the commutative property of the dot product, the complex conjugate in Eq. (8) may be taken on either the r or spherical harmonic functions. Similarly, any wellbehaved function f共rជ 兲 = f共r , r , r兲 can be expanded in terms of spherical harmonics so that ⬁

f共rជ 兲 = f共r, r, r兲 =

l

兺 兺 f 共r兲Y 共 , 兲, k l

k l

r

r

共9兲

l=0 k=−l

⬁

where

冕冕 0

F共, , 兲 =

f共r, r, r兲Ylk共r, r兲sin rdrdr .

共10兲

0

l

兺 兺 4共− i兲

l

l=0 k=−l ⬁

=

k l

k l

冕冕 2

Flk共兲 =

共13兲

0

F共, , 兲Ylk共, 兲sin dd .

0

共14兲 Here Flk共兲 denotes the 共l , k兲th coefficient in the expansion of F共 ជ 兲 and does not imply the Fourier transform of flk共r兲. This relation will be found. Substituting the expansion in Eq. (13), as well as that for the Fourier kernel from (8) into the definition of the 3D Fourier transform in (12) and using the orthonormality of the spherical harmonics yields f共rជ 兲 = f共r, r, r兲 ⬁

=

l

1

兺 兺 4

共i兲l

再冕 2

⬁

0

冎

Flk共兲jl共r兲2d Ylk共r, r兲,

These spherical harmonic expansions in Eqs. (8) and (9) are now substituted into the definition of the Fourier transform (7), and using the orthonormality of the spherical harmonics, we get ⬁

l

兺 兺 F 共兲Y 共 , 兲,

where

l=0 k=−l

flk共r兲 =

共12兲

−⬁

⬁

l

2

F共 ជ 兲ei共ជ ·rជ 兲d ជ,

l=0 k=−l

0

⬁

冕

F共 ជ 兲 = F共, , 兲 =

To proceed in terms of spherical harmonics, both the original function f and the Fourier kernel are expanded in spherical harmonic series. First, the Fourier kernel is written in terms of spherical harmonics as [3] −i ជ ·rជ

共2兲3

⬁

and note that this may be written in spherical coordinates. The 3D Fourier transform is expanded in spherical harmonics as

F共 ជ 兲 = F共, , 兲 2

1

再冕

冎

⬁

flk共r兲jl共r兲r2dr Ylk共, 兲

0

l

兺 兺 4共− i兲 F 共兲Y 共 , 兲, l

k l

k l

共11兲

l=0 k=−l

where the integration with respect to r within the curly braces has been recognized as a spherical Hankel trans k共兲 is the lth order spherical Hankel form. That is, F l k共兲 = S 兵fk共r兲 兩 r → 其. transform of flk共r兲 so that F l l l

共15兲

where the quantity in curly braces is the inverse spherical Hankel transform of Flk共兲. Comparing Eqs. (11) and (13), then

k共兲 = 4共− i兲lS 兵fk共r兲其, Flk共兲 = 4共− i兲lF l l l

共16兲

from which the reverse relationship can be obtained. Equation (16) and its inverse thus provide the means of obtaining the coefficients in the Fourier transform expansion from those of the spatial domain expansion, and vice versa. Namely, starting with the knowledge of flk共r兲 (the coefficients for f in the spatial domain), Eq. (16) indicates how to obtain Flk共兲 (the coefficients for the 3D Fourier transform F in the spatial frequency domain), and vice versa. With appropriate constants, the relationship is essentially one of a spherical Hankel transform. The expressions derived here contain slight corrections to those given in [3,10].

Natalie Baddour

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

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

4. DIRAC-DELTA FUNCTION AND ITS TRANSFORM The Dirac-delta function in 3D spherical polar coordinates can be written as f共rជ 兲 = ␦共rជ − rជ 0兲 =

6. MULTIPLICATION

1 r2 sin r

␦共r − r0兲␦共r − r0兲␦共r − r0兲. 共17兲

From the definition, the spherical harmonic expansion coefficients for the Dirac-delta function are given by flk共r兲 =

1 r2

␦共r − r0兲Ylk共r0, r0兲,

共18兲

We consider the product of two functions h共rជ 兲 = f共rជ 兲g共rជ 兲, ⬁ l 兺k=−l flk共r兲Ylk共r , r兲; similarly for g, the cowhere f共rជ 兲 = 兺l=0 efficients flk共r兲 and glk共r兲 are given by Eq. (10), and we seek to find the equivalent coefficients hlk共r兲 such that h共rជ 兲 ⬁ l 兺k=−l hlk共r兲Ylk共r , r兲. This shall be accom= f共rជ 兲g共rជ 兲 = 兺l=0 plished by finding the Fourier transform of h共rជ 兲 and using the expansions for f共rជ 兲 and g共rជ 兲, along with Eq. (8):

so that the full 3D transform itself becomes

=

l

l=0 k=−l

=e

−i ជ ·rជ 0

0

4共− i兲ljl共r0兲Ylk共r0, r0兲Ylk共, 兲

Flk共兲 = 4共− i兲ljl共r0兲Ylk共r0, r0兲.

共20兲

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

5. COMPLEX EXPONENTIAL AND ITS TRANSFORM From Eq. (8), the spherical harmonic coefficients of the complex exponential are given by flk共r兲 = 4共i兲ljl共0r兲Ylk共0, 0兲.

共21兲

Using the orthogonality of the spherical Bessel functions [5], the coefficients of the transform can be calculated from Eq. (11): ⬁

⫻

共19兲

.

The Fourier transform of the Dirac-delta function is the exponential function—as expected and as shown in Eq. (19). However, the toolbox development requires the coefficients of the expansion of the Fourier transform, and these are

F共 ជ兲 =

l

兺 兺 共4兲 2 ␦共 − 兲Y 共 2

2

⬁

=

兺兺

f共rជ 兲g共rជ 兲e−iជ ·rជ drជ

−⬁

F共 ជ 兲 = F共, , 兲 ⬁

冕 冕冕 冕兺 ⬁

H共 ជ兲 =

0

2

l=0 k=−l

k l

k 0, 0兲Yl 共, 兲.

= 共2兲

1

2

␦共 −

0兲Ylk共0, 0兲.

共23兲

We recognize from Eq. (18) that these are the coefficients for the expansion of the Dirac-delta function so that Eq. (22) gives the traditional Fourier transform of the complex exponential as the Dirac-delta function, as it should. The spherical harmonic coefficients of the function

L⬘

0

k

兺 f ⬙⬙共r兲Y ⬙⬙共 , 兲 L⬙

k l

k l

r

r

兺 4共− i兲 j 共r兲Y 共 , 兲Y 共 , 兲 l

k l

l

r

r

k l

共24兲

⬁ l where the shorthand notation 兺L兵 其 = 兺l=0 兺k=−l 兵 其 has been used. Performing the integration over the angular variables requires the evaluation of the following integral:

冕冕 2

0

0

Yl⬘⬘共r, r兲Yl⬙⬙共r, r兲Ylk共r, r兲sin rdrdr . k

k

共25兲 In fact, it turns out that the integrals in Eq. (25) are known as Slater coefficients which are defined as [12,13] cl⬙共l,k,l⬘,k⬘兲 =

冕冕 2

0

0

k k−k Ylk共r, r兲Yl⬘⬘共r, r兲Yl⬙ ⬘共r, r兲

⫻sin rdrdr ,

共26兲

and are nonzero only for 兩l − l⬘兩 ⱕ l⬙ ⱕ l + l⬘. Equation (25) can also be expressed in terms of Clebsh–Gordan coefficients [14] or Wigner 3j-symbols [15]. In the following, the Slater coefficients of Eq. (26) will be used since their definition in terms of an integral is the most straightforward. Using this definition of Slater coefficients, Eq. (24) becomes ⬁

l

兺兺

4共− i兲l

l=0 k=−l

3

k

⫻sin rdrdrr2dr,

H共 ជ兲 = Flk共兲

fl⬘⬘共r兲Yl⬘⬘共r, r兲

L

共22兲 This gives

0

l+l⬘

⫻

兺

l⬙=兩l−l⬘兩

再冕

⬁

⬁

l⬘

兺 兺

0 l⬘=0 k⬘=−l⬘

fl⬘⬘共r兲 k

冎

k−k cl⬙共l,k,l⬘,k⬘兲gl⬙ ⬘共r兲jl共r兲r2dr Ylk共, 兲.

共27兲

Comparing Eq. (27) with Eq. (11), the quantity in curly braces is

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J. Opt. Soc. Am. A / Vol. 27, No. 10 / October 2010 ⬁

hlk共r兲 =

l⬘

兺 兺

l⬘=0 k⬘=−l⬘

fl⬘⬘共r兲 k

l+l⬘

兺

l⬙=兩l−l⬘兩

Natalie Baddour

k−k cl⬙共l,k,l⬘,k⬘兲gl⬙ ⬘共r兲. 共28兲

⬁

ⴱ

glk共r兲

⬅

l⬘

兺 兺

l⬘=0 k⬘=−l⬘

k fl⬘⬘共r兲

共2兲3

0

⫻

k

l⬘

⬁

兺 兺 ⬁

k

4共− i兲l⬘jl⬘共r0兲Yl⬘⬘共r0, r0兲Yl⬘⬘共, 兲 k

l⬘=0 k⬘=−l⬘

⫻

4共− i兲l⬙

0 l⬙=0 k⬙=−l⬙

0

fl⬙⬙共u兲jl⬙共u兲u2du Yl⬙⬙共, 兲

0

⫻

l⬙

⬁

2

⬁

⬁

Based on knowledge of Fourier transforms and that multiplication in one domain becomes a convolution in the other domain and following the methodology used for polar coordinates in [8], Eq. (28) is to be interpreted as the convolution of the spherical harmonic series for f and g. Clearly, since the operation of multiplication commutes then Eq. (28) still holds with the roles of f and g reversed. In other words, we define the convolution of two harmonic series for f and g as

flk共r兲

冕冕 冕兺兺 再冕 冎 1

f共rជ − rជ 0兲 =

k

l

兺 兺 4共i兲 j 共r兲Y 共 , 兲Y 共 , 兲 l

k l

l

k l

r

r

2

l=0 k=−l

⫻sin ddd .

l+l⬘

兺

c

l⬙=兩l−l⬘兩

l⬙

k−k 共l,k,l⬘,k⬘兲gl⬙ ⬘共r兲,

共29兲 and the results of the preceding analysis demonstrate that 共fg兲lk = flk ⴱ glk .

共30兲

Defining the convolution of two spherical harmonic series as given in Eqs. (29) and (30) yields the familiar result that the spherical harmonic transform of a product is the convolution of their respective transforms. This maintains the usual multiplication/convolution rule for standard Fourier transforms to the spherical harmonic transform (which is often called a spherical Fourier transform). Unlike Cartesian or polar coordinates, definitions of convolution for spherical harmonic series are not standard in the literature. For example, in [9,16], although convolutions of spherical harmonic expansions are mentioned, neither author offers a definition for the operation of the convolution of two expansions.

Integration over the angular variables is nonzero only if k⬙ = k − k⬘ and 兩l − l⬘兩 ⱕ l⬙ ⱕ l + l⬘ since the same Slater integral as given in Eq. (25) appears, and thus the discussion subsequent to Eq. (25) still applies. Integration over the angular variables will again lead to the Slater coefficients. The integration over the radial variable yields a “shift” type operator featuring a triple Bessel product which is defined as

冕

Sl⬙ ⬘共u,r,r0兲 = l,l

⬁

0

jl⬙共u兲jl⬘共r0兲jl共r兲2d .

共33兲

With the shift operator and Slater coefficients, Eq. (32) becomes ⬁

f共rជ − rជ 0兲 =

⬁

l

兺 兺 8共i兲

l

Ylk共r, r兲

l+l⬘

⫻

兺

l⬘

兺 兺

l⬘=0 k⬘=−l⬘

l=0 k=−l

共− i兲l⬘Yl⬘⬘共r0, r0兲 k

共− i兲l⬙cl⬙共l,k,l⬘,k⬘兲

l⬙=兩l−l⬘兩

⫻

冕

⬁

0

7. SPATIAL SHIFT

共32兲

k−k l,l fl⬙ ⬘共u兲Sl⬙ ⬘共u,r,r0兲u2du.

共34兲

Hence, the coefficients for the shifted function are

The expression for a Fourier series shifted in space is found by finding the inverse Fourier transform of the exponential-weighted transform. In other words, f共rជ − rជ 0兲 is defined from f共rជ − rជ 0兲 = F 兵e −1

−i ជ ·rជ 0

F共 ជ 兲其.

兺 兺 l+l⬘

⫻

兺

8共i兲l−l⬘Yl⬘⬘共r0, r0兲 k

l⬘=0 k⬘=−l⬘

共31兲

The reason for defining the shifted function from (31) is that we have already found the expansion for the complex exponential as well as the rules for product of two expansions. It is not sufficient to find any expression for the spatial shift, but rather we seek the expression that is (i) in the form of a spherical harmonic series and (ii) in terms of the unshifted coefficients of the original function, as this builds the rule for what to do to the coefficients when a shift is desired. Thus, by building on the previously found results, the relevant spatial shift result can be found in the desired form. Using the inverse Fourier transform in Eq. (12), along with expansions (8) and (11), then the desired quantity is given by

l⬘

⬁

关f共rជ − rជ 0兲兴lk =

共− i兲l⬙cl⬙共l,k,l⬘,k⬘兲

l⬙=兩l−l⬘兩

⫻

冕

⬁

0

k−k l,l fl⬙ ⬘共u兲Sl⬙ ⬘共u,r,r0兲u2du.

共35兲

The preceding equation thus defines the rule for finding the coefficients of the shifted function 关f共rជ − rជ 0兲兴lk, once the coefficients of the original function f共r兲lk are known. This is the shift rule while remaining in the spatial domain. The “shift operator” in Eq. (33) is non-trivial and has been the topic of several papers, e.g., [17–19]. The most important observation regarding these results is that Sl ⬘共u , r , r0兲 is zero unless 共u , r , r0兲 can form a triangle or a ⬙ degenerate triangle [that is to say 共u , r , r0兲 are collinear]. l,l

l,l That is, Sl ⬘共u , r , r0兲 is zero unless 兩r − r0兩 ⱕ u ⱕ r + r0, which

⬙

Natalie Baddour

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

should be geometrically intuitive. If 共u , r , r0兲 cannot form a triangle, the proposed geometry is not physically realizable. Mathematics (and intuition) indicates that the shift operator evaluates to zero. Corresponding coefficients of the 3D transform of the shifted function can also be found. That is, if we define h共rជ 兲 = f共rជ − rជ 0兲, then H共 ជ 兲 = e−iជ ·rជ 0F共 ជ 兲, and Hlk共兲 are sought. Knowing that multiplication in the Fourier domain implies convolution of the coefficients, with the convolution operation defined in Eq. (29), and also knowing the coefficients for the complex exponential from (21), then Hlk共兲 = 关4共− i兲ljl共r0兲Ylk共r0, r0兲兴 ⴱ Flk共兲,

冕冕 冕兺 2

⬁

h共rជ 兲 =

0

共36兲

l

兺

兺 兺

兺兺

l+l⬘

兺

⫻

l⬙=兩l−l⬘兩

⫻

8共i兲lYlk共r, r兲

k−k cl⬙共l,k,l⬘,k⬘兲Fl⬙ ⬘共兲,

共37兲

⫻

⬁

l

l=0 k=−l

l⬘=0 k⬘=−l⬘

8共i兲lYlk共r, r兲

l⬘

兺 兺

l⬘=0 k⬘=−l⬘

共− i兲l⬘

l+l⬘

兺

共− i兲l⬙cl⬙共l,k,l⬘,k⬘兲

l⬙=兩l−l⬘兩

冕冕 ⬁

0

As indicated in Eq. (40), the integrals in the last term can be recognized as the definition of the spherical harmonic coefficients of the Fourier transform of G and H. Thus, the preceding equation can be written in the form of Eq. (15) as ⬁

l

共i兲l

兺 兺 4 l=0 k=−l ⬁

=

再冕 2

冎

⬁

Hlk共兲jl共r兲2d Ylk共r, r兲

0

l

兺 兺 h 共r兲Y 共 , 兲, k l

k l

r

共41兲

r

l=0 k=−l

where ⬁

Hlk共兲 =

l⬘

兺 兺

l⬘=0 k⬘=−l⬘

=

Glk共兲

ⴱ

Gl⬘⬘共兲 k

Flk共兲.

共− i兲l⬘Yl⬘⬘共r0, r0兲 k

共− i兲l⬙cl⬙共l,k,l⬘,k⬘兲

⬁

l+l⬘

兺

l⬙=兩l−l⬘兩

k−k l,l fl⬙ ⬘共u兲Sl⬙ ⬘共u,r,r0兲u2du

⫻sin r0dr0dr0r02dr0 .

⬁

0

gl⬘⬘共r0兲jl⬘共r0兲r02dr0 k

=

h共rជ 兲 =

兺 兺

共39兲

Integrating over the angular variables and using the orthogonality of the spherical harmonics gives l = l⬘ and k = k⬘. This and the definition of the shift operator give

The 3D convolution of two functions is defined by

兺兺

冕

0

8. CONVOLUTION

⬁

l⬘

l⬙=兩l−l⬘兩

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

h共rជ 兲 =

兺

k

⬁

l

l+l⬘

4共− i兲l⬘jl⬘共r0兲Yl⬘⬘共r0, r0兲 k

l⬘=0 k⬘=−l⬘

共38兲

gl共r0兲Yl共r0, r0兲 k

0 l=0 k=−l

0

⬁

⬁

l=0 k=−l

Hlk共兲 =

g共rជ 0兲f共rជ − rជ 0兲drជ 0 .

The triple-star notation of ⴱ ⴱ ⴱ is used to emphasize that this is a 3D convolution and to distinguish it from onedimensional (1D) and 2D convolutions, and it follows the convention in [8]. Using the spherical harmonic expansion for g and the shifted version of f given by Eq. (34), Eq. (38) is

or more explicitly l⬘

⬁

−⬁

⫻

⬁

冕

h共rជ 兲 = f共rជ 兲 ⴱ ⴱ ⴱ g共rជ 兲 =

2149

k−k cl⬙共l,k,l⬘,k⬘兲Fl⬙ ⬘共兲

共42兲

Thus, the convolution of two functions in space is the convolution of the spherical harmonic transform coefficients

il⬘ k⬘ G 共兲 4 l⬘

冕

⬁

0

k−k fl⬙ ⬘共u兲jl⬙共u兲u2dujl共r兲2d .

=

il⬙ k−k⬘ F 共兲 4 l⬙

共40兲

of their 3D Fourier transform. From the section on the multiplicative property, it is known that convolving the coefficients is equivalent to multiplication of the functions themselves; or, in other words, the result that Hlk共兲 = Glk共兲 ⴱ Flk共兲 is equivalent to stating that H共 ជ兲 = G共 ជ 兲F共 ជ 兲, as would be expected. This is, of course, a well known Fourier result and serves to confirm the accuracy of the development. However, for the purposes of this development, the main result we seek is that of Eq. (41), which gives the values of hlk共r兲 in terms of glk共r兲 and flk共r兲 (or rather the spherical harmonic transform of those) and essentially defines the convolution operation for functions given as a spherical harmonic series. This equation defines the convolution operation so that to find the convolution of two functions given in spherical harmonic series form, one must first find the 共l , k兲th coefficient of the 3D Fourier transform of each function, namely, Glk共兲 and Flk共兲, and then subsequently convolve the resulting series as per Eq. (42) to get Hk共兲. The final step is then to inverse spherical Hankel transform the result to finally obtain hlk共r兲. It is pointed out that convolving the two func-

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Natalie Baddour

tions f共rជ 兲 and g共rជ 兲 is not equivalent to convolving their series—the convolution of the series was shown previously to be equivalent to the multiplication of the functions themselves.

f共rជ 兲 ⴱ g共rជ 兲 = f共r兲 ⴱ ⴱ ⴱ g共r兲 =

冕

⬁

g共r0兲⌽3D共r − r0兲r0dr0 ,

0

共48兲 with

9. CONVOLUTION OF SPHERICALLY SYMMETRIC FUNCTIONS Spherical harmonic series expansions of spherically symmetric functions include the (0,0) term only. The 3D convolution of the two spherically symmetric functions is defined as usual as h共rជ 兲 = f共rជ 兲 ⴱ ⴱ ⴱ g共rជ 兲 =

冕

⬁

g共rជ 0兲f共rជ − rជ 0兲drជ 0 ,

共43兲

−⬁

where it is noted that the integration is over all rជ 0, which includes all possible values of radial and angular variables. For spherically symmetric functions, Eq. (43) should be properly interpreted as a 3D convolution along with Eq. (34) for the shifted function. In other words, Eq. (43) is not to be interpreted as

冕

⬁

g共r0兲f共r − r0兲dr0 ,

共44兲

0

which would be a 1D convolution in the radial variable. Using the fact that since the functions are spherically symmetric only the l = k = 0 term is retained in the series and that Y00共 , 兲 = 1 / 冑4, then the convolution of two spherically symmetric functions becomes h共rជ 兲 = f共rជ 兲 ⴱ ⴱ ⴱ g共rជ 兲 =

冕

⬁

g共r0兲8

−⬁

l⬘

⬁

⫻

兺 兺

l⬘=0 k⬘=−l⬘

⫻

冕

⬁

0

冑4 l⬘

k

兺

共− i兲l⬙cl⬙共0,0,l⬘,k⬘兲

l⬙=兩−l⬘兩

−k 0,l fl⬙ ⬘共u兲Sl⬙ ⬘共u,r,r0兲u2dudrជ 0 .

共45兲

Since l⬙ = l⬘ and then l⬘ = k⬘ = 0, because the spherically symmetric function only has the (0,0) entry, given by Eq. (10) as f00共r兲 = f共r兲冑4, Eq. (45) simplifies to

冕冕 冕 冕 2

⬁

h共rជ 兲 =

0

0

g共r0兲8

0

1

1

冑4 冑4

c0共0,0,0,0兲

⬁

f00共u兲S00,0共u,r,r0兲u2du sin r0dr0dr0r02dr0 .

⫻

f共u兲8S00,0共u,r,r0兲u2du

0

2

=

0

0

共46兲 From Eq. (26), c0共0 , 0 , 0 , 0兲 = 1 / 冑4 so that integration over the angular variables yields

=

冕

0

g共r0兲8

冕

f共rជ − rជ 0兲sin r0dr0dr0 .

共49兲

0

Thus the correct process of a full 3D convolution is that the function f is shifted from rជ to rជ 0 (destroying spherical symmetry), and then the resulting shifted function is integrated over all angular variables. The unshifted function g is still spherically symmetric and unaffected by the integration over angular variables so that the final result is given by Eq. (48). While it is tempting to write Eq. (44) for a 3D convolution, it is in fact incorrect. The correct version is given by Eq. (48). The 3D convolution of two radially symmetric functions yields another radially symmetric function; and moreover, by using the proper definition of a 3D convolution instead of the tempting definition of a 1D convolution, the wellknown relationship between convolutions and multiplications in space and frequency domains is preserved, namely, h共r兲 = f共r兲 ⴱ ⴱ ⴱ g共r兲 ⇒ H共兲 = F共兲G共兲.

共50兲

10. ANGULAR CONVOLUTIONS For any two functions f共rជ 兲 = f共r , , 兲 and g共rជ 兲 = g共r , , 兲 angular convolutions can also be defined as was done in two dimensions in [8]. These shall be discussed next. A. Rotation Formulas for Spherical Harmonics To consider angular convolutions, the rotation of a spherical harmonic needs to be considered. These formulas can be found in the literature, for example, [20,21], and the most relevant are presented here without proofs. We write a single-rotation operator as Rz共␥兲, with the subscript denoting the axis about which to take a rotation and the argument denoting the angle of rotation. Hence Rz共␥兲 defines a rotation about the z axis by an angle ␥. A general 3D rotation operator can written as R␣,,␥ = Rz共兲Ry共␣兲Rz共␥兲 and is essentially the standard Eulerangle rotations about axes z, y, and z by angles ␥, ␣, and . The reader is reminded that the order in which rotations are performed does matter. The rotation formula for spherical harmonics is given by [20]

h共rជ 兲 = f共r兲 ⴱ ⴱ ⴱ g共r兲 ⬁

The key point to this relationship is the proper definition of the convolution as a 3D convolution.

1

共− i兲l⬘Yl⬘⬘共r0, r0兲

冕 冕冕 ⬁

⌽3D共r − r0兲 =

l

⬁

f共u兲S00,0共u,r,r0兲u2dur02dr0 ,

共47兲

0

where the triple star emphasizes that this is a 3D convolution. Equation (47) can be thought of as

Ylm共R␣,,␥共, 兲兲 =

兺

m⬘=−l

Dmm⬘共␣, , ␥兲Ylm⬘共, 兲. l

共51兲

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

Natalie Baddour

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

m indices. However, the l indices are not mixed; rotations of spherical harmonics of order l are composed entirely of other spherical harmonics with order l. For a given order l, Dl is a matrix that tells how a spherical harmonic transforms under rotation, i.e., how to rewrite a rotated spherical harmonic as a linear combination of spherical harmonics of the same order. Analytical forms for Dl can be found in [21]. In particular, since R␣,,␥ = Rz共兲Ry共␣兲Rz共␥兲 dependences on  and ␥ are simple since they are rotations about the z axis, so Dmm⬘共␣, , ␥兲 = dmm⬘共␣兲eimeim⬘␥ , l

l

共52兲

where dl is a matrix that defines how a spherical harmonic transforms under rotation about the y axis. It can be shown [20] that dl satisfies

冕冕 2

l

dmm⬘共␣兲 =

0

Ylm共Ry共␣兲共, 兲兲Ylm⬘共, 兲sin共兲dd .

0

共53兲 l dmm 共␣兲 ⬘

Closed form formulas for can be found in [21]. In order to work on a sphere, to be able to “sweep out” an entire sphere using only rotations of the north pole at (0,0,1), it suffices to use rotations about the y and z axes. Hence, for rotation on the sphere, the rotation operator becomes R␣, = R␣,,0 = Rz共兲Ry共␣兲,

共54兲

of scenes for computer graphics [16,20,22]. In computer graphics, the interaction between the incident illumination and the BRDF is a basic building block in most rendering algorithms. In computer vision, it is often desired to undo the effects of the reflection operator: to invert the interaction between the BRDF and lighting. In other words, it is desired to perform inverse rendering—the estimation of material and lighting properties from real photographs. Ramamoorthi and Hanrahan presented a mathematical theory of reflection for general complex lighting environments and arbitrary BRDFs [20]. They formalized the notion of convolution as it applied to reflection and showed the operation of reflection to be a (spherical) convolution [20] over angular coordinates. Thus, the mathematical operations involved are those of convolution and de-convolution of spherical harmonic expansions. Rephrasing these last results in terms of the terminology used here, the “convolution” which these computer vision researchers refer to is the equivalent of a convolution over only the angular variables in 3D spherical polar coordinates. This angular convolution over the latitudinal and longitudinal angles but not over the radial variable implies that this is a convolution on the sphere, and not a full 3D convolution. Based on these previous works, we define the angular convolution on the sphere as h共rជ 兲 = f共rជ 兲ⴱ共,兲g共rជ 兲 =

and the representation matrices are Dmm⬘共␣, 兲 = Dmm⬘共␣, ,0兲 = dmm⬘共␣兲eim . l

l

l

冕 冕 2

=0

=

␣=0

l 共Dmn 共␣, 兲兲共Dm⬘ ⬘n共␣, 兲兲sin ␣d␣d

4 2l + 1

l

␦ll⬘␦mm⬘ .

共56兲

Note that there is no orthogonality over the index n since the integration is over two rotations 共␣ , 兲 instead of the full Eulerian angles 共␣ ,  , ␥兲. Using the orthogonality of eim over the interval 关0 , 2兴, Eq. (56) in terms of the l dmm 共␣兲 is ⬘

冕

␣=0

l l⬘ 共dmn 共␣兲兲共dmn 共␣兲兲sin

␣d␣ =

2 2l + 1

␦ .

共57兲

共2兲2

0

f共r, 0, 0兲

0

共58兲

The subscript on ⴱ serves to denote the type of convolution so that there is no confusion with single-variable or full multidimensional convolutions. The notation for g共r , − 0 , − 0兲 is that of a shift in terms of the angular variables, which is in effect a rotation of the function g共r , , 兲 by 共−0 , −0兲. To evaluate the integral in Eq. (58), the integration is to be performed over the variables with the zero subscript 共0 , 0兲, and it is desired to have the final result in terms of the unsubscripted variables 共r , , 兲. Therefore, it is desirable to expand the shifted function g共r , − 0 , − 0兲 in terms of unshifted spherical harmonics. In other words, although the natural expansion for the rotated g共r , − 0 , − 0兲 is given in terms of the rotated spherical harmonics as ⬁

ll⬘

冕冕 2

1

⫻g共r, − 0, − 0兲sin 0d0d0 .

共55兲

These representation matrices satisfy orthogonality conditions

2151

g共r, − 0, − 0兲 =

l⬘

兺 兺

l⬘=0 k⬘=−l⬘

gl⬘⬘共r兲Yl⬘⬘共 − 0, − 0兲, k

k

共59兲 B. Convolution on the Sphere Spherical harmonics and the accompanying spherical harmonic expansions of a function can be considered to be a spherical Fourier transform as opposed to a full-fledged 3D Fourier transform, which would require an additional spherical Hankel transform of order n. These spherical Fourier transforms have recently found many applications in computer graphics and computer vision. Specifically, they have been used to efficiently represent the bidirectional reflection distribution function (BRDF) of materials [22] and to improve the modeling of the lighting

where gl ⬘共r兲 is found from Eq. (10), the expansion that we ⬘ really need is that of the rotated (“shifted”) function g共r , − 0 , − 0兲 in un-rotated spherical harmonics: k

⬁

g共r, − 0, − 0兲 =

l

兺 兺 ␥ 共r兲Y 共, 兲. k l

k l

共60兲

l=0 k=−l

The coefficients ␥lk共r兲 can be found in the normal way by the usual definition (10) along with the function whose coefficients are sought [Eq. (59)], that is,

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

冕 冕 再兺 兺 2

␥lk共r兲

=

0

⫻Ylk共, 兲sin

l⬘=0 k⬘=−l⬘

⫻

− 0, − 0兲

冎

冕冕

=

As previously mentioned, Yl ⬘共 − 0 , − 0兲 is a spherical ⬘ harmonic that has been rotated by 共−0 , −0兲 so that the rotation formula for spherical harmonics [Eq. (51)] can be used. Using the rotation formula for the spherical harmonics, the integral in Eq. (61) can be written as k

冕冕 冕冕

0

兺

=

0 k⬙=−l⬘

0

共62兲

However, the orthogonality of the spherical harmonics gives that the integral on the right hand side of Eq. (62) is nonzero only for l = l⬘ and k = k⬙ so that the result becomes

冕冕 0

0

glk⬘共r兲Dk⬘k共− 0,− 0兲Ylk共, 兲 l

册 共66兲

Yl⬙⬙共0, 0兲Dk⬘k共− 0,− 0兲sin 0d0d0 l

Yl⬙⬙共0, 0兲 k

0

0兲Ylk共, 兲sin

冕冕 2

0

Ylk⬘共 − 0,

0

dd sin 0d0d0 .

共67兲

Changing the order of integration immediately yields that the integral is nonzero only for l⬙ = l, as previously discussed. Furthermore, with the form of the spherical harmonics from Eq. (4) and their dependence on the complex exponential, the integration over and 0 can be performed. Specifically, the dependence of the integrand in Eq. (67) is given by

Yl⬘⬘共 − 0, − 0兲Ylk共, 兲sin dd l

= ␦ll⬘Dk⬘k共− 0,− 0兲.

exp共ik⬙0兲exp共ik⬘共 − 0兲兲exp共− ik兲

⬘

l⬘

⬁

␥lk共r兲 =

兺 兺

l

兺

k⬘=−l

glk⬘共r兲Dk⬘k共− 0,− 0兲, l

共64兲

l

再

l

兺兺 兺 l=0 k=−l

k⬘=−l

l glk⬘共r兲Dk⬘k共−

0,− 0兲

冎

2

2

exp共i共k⬙ − k⬘兲0兲exp共i共k⬘ − k兲兲d0d

0

共69兲

Given these simplifications, Eq. (66) becomes

f共rជ 兲ⴱ共,兲g共rជ 兲 =

and the sought expansion is finally given as g共r, − 0, − 0兲

冕冕

= 共2兲2␦k⬙k⬘␦k⬘k .

l

=

共68兲

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

0

gl⬘⬘共r兲␦ll⬘Dk⬘k共− 0,− 0兲 k

l⬘=0 k⬘=−l⬘

= exp共i共k⬙ − k⬘兲0兲exp共i共k⬘ − k兲兲.

共63兲

In fact, Eq. (63) could be taken as the definition of l Dk k共−0 , −0兲. Given this result, Eq. (61) becomes

=

l

k

0

⬁

冕冕 冕冕 2

−

k

k

k

To evaluate the integration in Eq. (66), the definition of l Dk k共−0 , −0兲 is used, and it is observed that Ylk共 , 兲 can ⬘ come outside the integral. Hence, the integration over 共−0 , −0兲 reduces to evaluating

0

l

fl⬙⬙共r兲Yl⬙⬙共0, 0兲

⫻sin 0d0d0 .

2

Dk⬘⬘k⬙共− 0,− 0兲

⫻Yl⬘⬙共, 兲Ylk共, 兲sin dd .

2

冋兺 兺 兺

=

l⬘

l⬙

0 l⬙=0 k⬙=−l⬙

0

l

l=0 k=−l k⬘=−l

0

Yl⬘⬘共 − 0, − 0兲Ylk共, 兲sin dd

2

f共r, 0, 0兲

0

冕 冕兺兺

k

k

0

共2兲2

⫻

Yl⬘⬘共 − 0, − 0兲Ylk共, 兲sin dd . 共61兲

2

0

⬁

2

⬁

k

0

共2兲2

⫻g共r, − 0, − 0兲sin 0d0d0

dd

k gl⬘⬘共r兲

2 0

冕冕 2

1

f共rជ 兲ⴱ共,兲g共rជ 兲 =

1

l⬘

兺 兺

=

k k gl⬘⬘共r兲Yl⬘⬘共

l⬘=0 k⬘=−l⬘

0

⬁

l⬘

⬁

Natalie Baddour

l⬙

⬁

1

兺 兺

共2兲2 l

⬙=0 k⬙=−l⬙

fl⬙⬙共r兲 k

⬁

l

l

兺兺 兺

glk⬘共r兲

l=0 k=−l k⬘=−l

⫻共2兲 ␦ll⬙␦k⬙k⬘␦k⬘kYlk共, 兲 2

⬁

=

Ylk共, 兲.

l

兺 兺 f 共r兲g 共r兲Y 共, 兲. k l

k l

k l

共70兲

l=0 k=−l

共65兲

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

This is in fact a spectacularly useful result and is the spherical harmonic convolution theorem. What it says is that if we define the spherical convolution of two functions as Eq. (58), then

Natalie Baddour

Vol. 27, No. 10 / October 2010 / J. Opt. Soc. Am. A ⬁

h共rជ 兲 = f共rជ 兲ⴱ共,兲g共rជ 兲 =

l

兺 兺 h 共r兲Y 共, 兲 k l

k l

l=0 k=−l

⬁

=

l

兺 兺 f 共r兲g 共r兲Y 共, 兲, k l

k l

k l

共71兲

l=0 k=−l

or, in other words, the standard convolution result applies: hlk共r兲 = flk共r兲glk共r兲.

共72兲

It cannot be emphasized enough that these “standard convolution results,” by which we imply the equivalence of convolution in one domain to multiplication in the other domain, rely on the correct definition of a convolution. In particular, the result in Eq. (72) depends on the definition of convolution in Eq. (58). C. Full Rotational Convolution Equation (72) states that convolution on the sphere is equivalent to multiplication in the spherical Fourier domain; however this is not the same result as that obtained in the seminal paper by Driscoll and Healey [9]. The reason for this is that the convolution defined in the preceding section, namely, Eq. (58), is not the same as the convolution defined in [9], which is in fact a full rotational convolution, over all possible rotations in three dimensions. For each 3D rotation, R = R␣,,␥, Driscoll and Healey defined a rotation operator ⌳共R兲 on functions on the sphere: ⌳共R兲f共⍀兲 = f共R−1⍀兲.

共73兲

In Eq. (73), f共⍀兲 is a function on the sphere so that it is a function of ⍀ = 共 , 兲, and R−1 is the inverse rotation operator. Equation (73) indicates that the operator acts on the function on the sphere f共⍀兲 by rotating the angles 共⍀兲 by the rotation R−1. Driscoll and Healey [9] stated that any function on the sphere g may be used to define a convolution operator. This is accomplished by employing it as a weighting factor for the rotation operators. They defined the operation of left convolution by the function g as 共f ⴱR g兲共r,⍀兲 =

=

冕 冕

tioned, it bears repeating that the integrations in Eq. (74) are over all possible 3D rotations 共␣ ,  , ␥兲, which is a larger set of rotations than those given in the angular convolution of Eq. (58). The rotations in Eq. (58) are over the longitudinal and latitudinal angles only, and not the full set of Euler angles as implied in Eq. (74). Equation (74) can be interpreted in terms of the mathematics developed in this paper. To evaluate the integral in Eq. (74), we first need to write an expression for f共r , R兲. In f共r , R兲, the north pole (0,0,1) is rotated by R␣,,␥, and since it is only a vector it is thus unaffected by the final rotation ␥ and is only a function of 共␣ , 兲. In other words, f共r , R兲 can be written as

f共r,R兲g共r,R ⍀兲dR,

共74兲

where ⍀ is any point on the sphere, is the north pole (0,0,1), and the integration is taken over all 3D rotations R, which implies integrating over all possible Euler angles 共␣ ,  , ␥兲. There are several important observations to be made regarding the definition of convolution in Eq. (74). First, convolution in Eq. (74) does not necessarily commute so that f ⴱR g ⫽ g ⴱR f. This follows upon closer examination of the way the convolution operation in Eq. (74) is defined. Second, in Eq. (74), the function f is evaluated at the rotation of the north pole by R, whereas the function g itself is rotated by R−1. Thus, in this definition of convolution, the way the two functions are treated is not the same. Further, although this was already men-

兺 兺

l⬙=0 k⬙=−l⬙

fl⬙⬙共r兲Yl⬙⬙共␣, 兲. k

k

共75兲

On the other hand, for the second term in the integral in Eq. (74), the whole function g共r , ⍀兲 is rotated to g共r , R−1⍀兲 so that the rotation operator R␣,,␥ affects the whole function, and more importantly in this case all three rotations 共␣ ,  , ␥兲 affect the final result. Using the rotation results given in Eq. (65) and the simplification given in Eq. (52), we write g共r,R−1⍀兲 = g共r,R␣−1,,␥共, 兲兲 ⬁

=

l

l

兺兺 兺

l=0 k=−l k⬘=−l ⬁

=

glk⬘共r兲Dk⬘k共− ␣,− ,− ␥兲Ylk共, 兲 l

l

l

兺兺 兺

l=0 k=−l k⬘=−l

glk⬘共r兲e−ik⬘␥Dk⬘k共− ␣,− 兲Ylk共, 兲. l

共76兲 The benefit of writing g共r , R−1⍀兲 in the form of the previous equation is that it is identical to the formulation of the convolution on the sphere, with the exception of the additional e−ik⬘␥ term due to the full 3D rotation. Thus, with expressions for the two functions, we may proceed to evaluating the integral defining convolution in Eq. (74) to give

冕 冕 冕冕

f共r,R兲g共r,R−1⍀兲dR

2

−1

l⬙

⬁

f共r,R兲 = f共r, ␣, 兲 =

共f ⴱR g兲共r,⍀兲 =

f共r,R兲⌳共R兲dRg共r,⍀兲

2153

=

0

0

⬁

⫻

l

2

⬁

l⬙

兺 兺

l⬙=0 k⬙=−l⬙

0

fl⬙⬙共r兲Yl⬙⬙共␣, 兲 k

k

l

兺兺 兺

l=0 k=−l k⬘=−l

glk⬘共r兲e−ik⬘␥Dk⬘k共− ␣, l

− 兲Ylk共, 兲d␥ sin ␣d␣d .

共77兲

The first integral over the rotation ␥ is zero unless k⬘ = 0. This integration over ␥ was not present in the convolution on the sphere. Since the two functions in this full rotational convolution are not treated equally, the angle ␥ only appears in the expansion for the function g共r , R−1⍀兲, and after integrating over all ␥ this forces k⬘ = 0. We are now left with an integral to evaluate that is very similar to Eq. (66), with the exception that since k⬘ = 0, we need to

2154

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

Natalie Baddour

l evaluate D0k 共−␣ , −兲. An expression for this is derived and proved in the appendix of [20] as

l 共− D0k

␣,− 兲 =

This in fact simplifies to l D0k 共− ␣,− 兲 =

冑

冑

4 2l + 1

Ylk共␣, 兲e−ik .

cl⬙共0,0,l⬘,k⬘兲 =

1

2l + 1

= 共− 1兲k

冑

Ylk共␣, 兲e−ik 4

2l + 1

冕 冕兺兺 2

l⬙

⬁

0 l⬙=0 k⬙=−l⬙

0

⫻共− 1兲k

冑

4 2l + 1

Ylk共␣, 兲.

共79兲

fl⬙⬙共r兲Yl⬙⬙共␣, 兲 k

k

f ⴱ g兩rជ =0ជ =

⬁

兺 兺 g 共r兲 0 l

兺 兺 f 共r兲g 共r兲共− 1兲 k l

⫻

l=0 k=−l

Ylk共␣, 兲sin ␣d␣dYlk共, 兲.

0 l

l=0 k=−l

k

冑

4 2l + 1

Ylk共, 兲. 共81兲

冑

2l + 1

=

flk共r兲glk共r兲.

共83兲

11. PARSEVAL RELATIONSHIPS A Parseval relationship is important as it deals with the “power” of a signal or function in the spatial and frequency domains. In spherical polar coordinates if rជ = 共r , , 兲, then −rជ = 共r , − , + 兲 and Ylk共r , r兲 = 共−1兲kYl−k共r , r兲. Hence, it can be shown that, for any f, ⬁

f共− rជ 兲 =

⬁

l

兺兺

flk共r兲共− 1兲kYl−k共, 兲 =

l=0 k=−l

⫻共− 1兲−kYlk共, 兲.

冕冕 ⬁

冑4

冕

⬁

0

⬁

共85兲

⬁

8

l⬘

兺 冑4 l兺 ⬘=0 k⬘=−l⬘

共− 1兲l⬘共− 1兲−k⬘

gl⬘⬘共r0兲jl⬘共r0兲r02dr0 k

0

fl⬘ ⬘共u兲jl⬘共u兲u2du2d . −k

冕

⬁

g共rជ 兲f共rជ 兲drជ =

−⬁

共86兲

⬁

1

冕

l

兺兺

共2兲3 l=0

k=−l

⬁

Glk共兲Flk共兲2d . 共87兲

0

The integral on the left hand side of the preceding equation can be expressed in terms of the spherical harmonic expansion. To do this, use the multiplication property in Eq. (28) to write ⬁

f共rជ 兲g共rជ 兲 =

l

再

l⬘

⬁

兺兺 兺 兺 l=0 k=−l l+l⬘

兺

l⬙=兩l−l⬘兩

共82兲

,

whereas the spherical convolution of Eq. (58) yields 共fⴱ共,兲g兲lk

共− 1兲−k⬘␦l⬘l⬙ .

g共rជ 0兲f共− rជ 0兲drជ 0 =

⫻

4

−k

In particular, if the preceding convolution is performed for any function g共rជ 兲 and another function f共−rជ 兲, using the result of Eq. (84) the preceding equation gives one form of a generalized Parseval relationship as

This is precisely the result derived in [9] and, as previously noted, differs from the result given for convolution over the sphere as defined above in Eq. (58). In other words, Eq. (81) states that the full 3D rotational convolution gives 共f ⴱR g兲lk = flk共r兲gl0共r兲共− 1兲k

冑4

⬁

1

⫻

l

The orthonormality of the spherical harmonics guarantees that this integration over the angles ␣ and  gives ␦k⬙k␦l⬙l so that 共f ⴱR g兲共r,⍀兲 =

冕

0

l

k

0

−⬁

共80兲

⬁

0

Yl⬘⬘共r, r兲Yl⬙ ⬘共r, r兲

Using these simplifications, Eq. (40) for any two functions is

With this considerable simplification, the integral that remains to be evaluated is given by 共f ⴱR g兲共r,⍀兲 =

冑4

⫻sin rdrdr

共78兲 =

4

冕冕 2

1

fl⬘⬘共r兲 k

l⬘=0 k⬘=−l⬘

冎

k−k cl⬙共l,k,l⬘,k⬘兲gl⬙ ⬘共r兲 Ylk共r, r兲.

共88兲

From Eq. (84), coefficients of the expansion for any function f共rជ 兲 are given by fl−k共r兲共−1兲k. Integrating both sides of Eq. (88) over all space requires the integration of Ylk共r , r兲 over all angles, which is nonzero only for l = k = 0, which yields 冑4. As previously calculated, the value of the required Slater coefficient is cl⬙共0 , 0 , l⬘ , k⬘兲 = 共1 / 冑4兲共−1兲−k⬘␦l⬘l⬙. All these simplifications together combine to yield

冕

⬁

⬁

−⬁

g共rជ 兲f共rជ 兲drជ =

l

兺兺

l=0 k=−l

冕

⬁

glk共r兲flk共r兲r2dr.

共89兲

0

l

兺兺f

Combining the results of Eqs. (87) and (89) gives another version of the generalized Parseval relationship as

−k l 共r兲

l=0 k=−l

共84兲

Convolution of any two functions is given by Eq. (40). The convolution of the two functions evaluated at r = 0 implies that l = k = 0 since jl共0兲 = ␦l0. Values of the Slater coefficients at l = k = 0 become

⬁

l

兺兺

l=0 k=−l

冕

⬁

0

glk共r兲flk共r兲r2dr =

1

⬁

l

兺兺

共2兲3 l=0

k=−l

冕

⬁

Glk共兲Flk共兲2d .

0

共90兲 Clearly, it then follows from the preceding equation that

Natalie Baddour ⬁

l

兺兺

l=0 k=−l

冕

⬁

兩flk共r兲兩2r2dr =

0

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

1

⬁

l

兺兺

共2兲3 l=0

k=−l

冕

⬁

兩Flk共兲兩22d .

0

共91兲

12. SUMMARY AND CONCLUSIONS In summary, this paper has considered the spherical polar-coordinate version of the standard 3D Fourier transform and derived the operational toolset required for standard Fourier operations. This version of the 3D Fourier transform is most useful for functions that are naturally described in terms of spherical polar coordinates. Additionally, Parseval relationships were also derived. The results of the paper are concisely collected in Table 1 at the end of the paper. Of particular interest are the results on convolution and spatial shift. Notably, standard convolution/multiplication rules do apply for 3D convolution, and in particular special results apply for the special cases of angular-only convolutions. The results for angular-only convolutions bear special mention as spherical convolutions and full rotational convolutions admit different results, and it is important to define the type of convolution required in order to apply the correct convolution/multiplication rule.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

17. 18.

REFERENCES 1. 2. 3. 4. 5.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1999). K. Howell, “Fourier transforms,” in The Transforms and Applications Handbook (CRC, 2000), pp. 2.1–2.159. G. S. Chirikjian and A. B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis (CRC Press, 2000). M. Xu and L. V. Wang, “Time-domain reconstruction for thermoacoustic tomography in a spherical geometry,” IEEE Trans. Med. Imaging 21, 814–822 (2002). A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M.

19. 20. 21. 22.

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Israeli, “Fast and accurate polar Fourier transform,” J. Appl. Comput. Harmonic Anal. 21, 145–167 (2006). M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, 1988). N. Baddour, “Fourier diffraction theorem for diffusionbased thermal tomography,” J. Phys. A 39, 14379–14395 (2006). N. Baddour, “Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates,” J. Opt. Soc. Am. A 26, 1767–1777 (2009). J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-sphere,” Adv. Appl. Math. 15, 202–250 (1994). R. Piessens, “The Hankel transform,” in The Transforms and Applications Handbook (CRC, 2000), pp. 9.1–9.30. “Spherical harmonics,” Wikipedia, the Free Encyclopedia. J. C. Slater, Quantum Theory of Atomic Structure, Vol. I of International Series in Pure and Applied Physics (McGrawHill, New York, 1960). “Slater integrals,” Wikipedia, the Free Encyclopedia. G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier Academic, 2005). E. W. Weisstein, “Spherical Harmonics,” Wolfram MathWorld. R. Basri and D. Jacobs, “Lambertian reflectance and linear subspaces,” in Proceedings of the Eighth IEEE International Conference on Computer Vision, 2001 (ICCV 2001) (2001), Vol. 2, pp. 383–390. R. Mehrem, J. T. Londergan, and M. H. Macfarlane, “Analytic expressions for integrals of products of spherical Bessel functions,” J. Phys. A 24, 1435–1453 (1991). V. Fabrikant, “Computation of infinite integrals involving three Bessel functions by introduction of new formalism,” Z. Angew. Math. Mech. 83, 363–374 (2003). A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446–460 (1972). R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for reflection,” ACM Trans. Graphics 23, 1004– 1042 (2004). T. Inui, Group Theory and Its Applications in Physics (Springer-Verlag, 1990). R. Ramamoorthi and P. Hanrahan, “An efficient representation for irradiance environment maps,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 2001), pp. 497–500.