Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates

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Natalie Baddour

Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates Natalie Baddour Department of Mechanical Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, Ontario, K1N 6N5, Canada (nbaddour@uottawa.ca) Received February 24, 2009; revised May 26, 2009; accepted June 11, 2009; posted June 12, 2009 (Doc. ID 108008); published July 10, 2009 For functions that are best described in terms of polar coordinates, the two-dimensional Fourier transform can be written in terms of polar coordinates as a combination of Hankel transforms and Fourier series—even if the function does not possess circular symmetry. However, to be as useful as its Cartesian counterpart, a polar version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, etc. This paper derives the requisite polar version of the standard Fourier operations. In particular, convolution—two dimensional, circular, and radial one dimensional—is discussed in detail. It is shown that standard multiplication/convolution rules do apply as long as the correct definition of convolution is applied. © 2009 Optical Society of America OCIS codes: 070.6020, 070.4790, 350.6980, 100.6950.

1. INTRODUCTION The Fourier transform needs no introduction, and it would be an understatement to say that it has proved to be invaluable in many diverse disciplines such as engineering, mathematics, physics, and chemistry. Its applications are numerous and include a wide range of topics such as communications, optics, astronomy, geology, image processing, and signal processing. It is known that the Fourier transform 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. Specifically, the standard Fourier toolset consists of results for scaling, translation (spatial shift), multiplication, and convolution, along with the basic transforms of the Dirac-delta function and complex exponential that are so essential to the derivation of the shift, multiplication, and convolution results. This basic toolset of operational rules is well known for the Fourier transform in single and multipledimensions [1,2].

As is also known, the Fourier transform in two dimensions can be developed in terms of polar coordinates [3] instead of the usual Cartesian coordinates, most usefully in the case where the function being transformed is naturally describable in polar coordinates. For example, this has seen application in the field of photoacoustics [4], and attempts have been made to translate ideas from the continuous domain to the discrete domain by developing numerical algorithms for such calculations [5]. However, to the best of the author’s knowledge, a complete interpretation of the standard Fourier operational toolset in terms of polar coordinates is missing from the literature. Some results are known, such as the Dirac-delta function in both polar and spherical polar coordinates, but the results 1084-7529/09/081768-11/$15.00

on shift, multiplication, and in particular convolution, are incomplete.

This paper thus aims to develop the Fourier operational toolset of Dirac-delta, exponential, spatial shift, multiplication, and convolution for the two-dimensional (2D) Fourier transform in polar coordinates. Of particular novelty is the treatment of the shift, multiplication, and convolution theorems, which can also be adapted for the special cases of circularly symmetric functions that have no angular dependence. It is well known from the literature that 2D Fourier transforms for radially symmetric functions can be interpreted in terms of a (zeroth-order) Hankel trasnform. It is also known that the Hankel transforms do not have a multiplication/convolution rule, a rule that 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 zeroth-order 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 it in terms of the curvilinear coordinates so that the standard multiplication/convolution rule will once again be applicable.

The outline of the paper is as follows. For completeness, the Hankel transform and the interpretation of the 2D Fourier transform in terms of a Hankel transform and a Fourier series are introduced in Sections 2 and 3. Sections 4 and 5 treat the special functions of the Dirac-delta and complex exponential. Sections 6–8 address the multiplication, spatial shift, and convolution operations. In particular, the nature of the spatial shift and its role in the ensuing convolution theorem are discussed. Sections 9 and 10 discuss the spatial shift and convolution operators © 2009 Optical Society of America

Natalie Baddour

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

when dealing with the special case of radially (circularly) symmetric functions. Sections 11 and 12 discuss the special cases of angular or radial convolution only—that is, not a full 2D but a special one-dimensional (1D) convolution as restricted to convolving over only one of the variables of the polar coordinates. In particular, it is shown that while the angular convolution yields a simple convolution relationship, the radial-only convolution does not.

Section 13 derives the Parseval relationships. The derived operational toolset is summarized in a table at the end of the paper.

2. HANKEL TRANSFORM The nth-order Hankel transform is defined by the integral [6] Fˆn共␳兲 = Hn共f共r兲兲 =

冕

⬁

f共r兲Jn共␳r兲rdr,

1769

Polar coordinates can be introduced as x = r cos ␪, y = r sin ␪ and similarly in the spatial frequency domain as ␻x = ␳ cos ␺ ␻y = ␳ sin ␺. It then follows that the 2D Fourier transform can be written as

冕冕 ⬁

F共␳, ␺兲 =

␲

f共r, ␪兲e−ir␳ cos共␺−␪兲rdrd␪ .

共6兲

−␲

0

Thus, in terms of polar coordinates, the Fourier transform operation transforms the spatial position radius and angle 共r , ␪兲 to the frequency radius and angle 共␳ , ␺兲. The usual polar coordinate relationships apply in each domain so that r2 = x2 + y2, ␪ = arctan共y / x兲 and ␳2 = ␻2x + ␻2y , ␺ = arctan共␻y / ␻x兲. Using rជ to represent 共r , ␪兲 in physical polar coordinates and ␻ ជ to denote the frequency vector 共␳ , ␺兲 in frequency polar coordinates, the following expansions are valid [3]: ⬁

共1兲

兺 i J 共␳r兲e

ei␻ជ ·rជ =

0

n

n

in␪

e−in␺ ,

共7兲

n=−⬁

where Jn共z兲 is the nth-order Bessel function and the overhat indicates a Hankel transform as shown in Eq. (1).

Here, n may be an arbitrary real or complex number.

However, an integral transform needs to be invertible in order to be useful, and this restricts the allowable values of n. If n is real and n 艌 1 / 2, the transform is selfreciprocating and the inversion formula is given by f共r兲 =

冕

⬁

Fn共␳兲Jn共␳r兲␳d ␳ .

共2兲

0

The inversion formula for the Hankel transform follows immediately from Hankel’s repeated integral, which states that under suitable boundary conditions and subject to the condition that 兰0⬁f共r兲冑rdr is absolutely convergent, then for n 艌 1 / 2

冕 冕 ⬁

⬁

sds

0

0

1

f共r兲Jn共sr兲Jn共su兲rdr = 关f共u + 兲 + f共u − 兲兴. 共3兲 2

The most important cases correspond to n = 0 or n = 1. The Hankel transform exists only if the Dirichlet condition is satisfied, that is, if the following integral exists: 兰0⬁兩r1/2f共r兲兩dr. The Hankel transform is particularly useful for problems involving cylindrical symmetry.

⬁

兺i

e−i␻ជ ·rជ =

−n

Jn共␳r兲e−in␪ ein␺ .

共8兲

n=−⬁

These expansions can be used to convert the 2D Fourier transform into polar coordinates.

A. Radially Symmetric Functions If it is assumed that f is radially symmetric, then it can be written as function of r only and can thus be taken out of the integration over the angular coordinate so that Eq. (6) becomes F共␳, ␺兲 =

冕

⬁

rf共r兲dr

0

冕

␲

e−ir ␳ cos共␺−␪兲d␪ .

共9兲

−␲

Using the integral definition of the zero-order Bessel function, J0共x兲 =

1 2␲

冕

␲

e−ix cos共␺−␪兲d␪ =

−␲

1 2␲

冕

␲

e−ix cos ␣d␣ , 共10兲

−␲

Eq. (9) can be written as F共␳兲 = F2D兵f共r兲其 = 2␲

冕

⬁

f共r兲J0共␳r兲rdr,

共11兲

0

3. CONNECTION BETWEEN THE 2D FOURIER TRANSFORM AND THE HANKEL TRANSFORM The 2D Fourier transform of a function f共x , y兲 is defined as

冕冕 ⬁

F共␻ ជ 兲 = F共␻x, ␻y兲 =

−⬁

F共␳兲 = F2D兵f共r兲其 = 2␲H0兵f共r兲其.

⬁

f共x,y兲e−j共␻xx+␻yy兲dxdy.

共4兲

−⬁

The inverse Fourier transform is given by f共rជ 兲 = f共x,y兲 =

1 共2␲兲2

冕冕 ⬁

−⬁

which can be recognized as 2␲ times the Hankel transform of order zero. Thus the special case of the 2D Fourier transform of a radially symmetric function is the same as the Hankel transform of that function:

⬁

F共␻x, ␻y兲ej␻ជ ·rជ d␻xd␻y ,

共5兲

−⬁

where the shorthand notation of ␻ ជ = 共␻x , ␻y兲, rជ = 共x , y兲 has been used.

共12兲

With reference to Eq. (12), f共r兲 and F共␳兲 are functions with radial syummetry in a 2D regime, F2D兵·其 is an operator in the 2D regime, while H0兵·其 is an operator in 1D regime.

B. Nonradially Symmetric Functions When the function f共r , ␪兲 is not radially symmetric and is a function of both r and ␪, the preceding result can be generalized. Since f共r , ␪兲 depends on ␪, it can be expanded into a Fourier series,

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Natalie Baddour

⬁

兺

f共rជ 兲 = f共r, ␪兲 =

⬁

fn共r兲ejn␪ ,

共13兲

n=−⬁

n=−⬁

where fn共r兲 =

冕

1 2␲

f共r, ␪兲e−jn␪ d␪ .

共14兲

0

⬁

兺 F 共␳兲e

jn␺

共15兲

F共␳, ␺兲e−jn␺d␺ .

共16兲

F共␻ ជ 兲 = F共␳, ␺兲 =

n

n=−⬁

and 1 2␲

冕

2␲

0

Note that Fn共␳兲 is not the Fourier transform of fn共r兲. In fact, it is the relationship between fn共r兲 and Fn共␳兲 that we seek to define.

1. Forward Transform Expansions (8) and (13) are substituted into the definition of the forward Fourier transform to give

冕 冕冕 ⬁

F共␻ ជ兲 =

f共rជ 兲e−j␻ជ ·rជ drជ

−⬁ ⬁

=

0

⬁

2␲

⬁

兺

fm共r兲ejm␪

m=−⬁

0

=

兺 2␲i

−n in␺

e

n=−⬁

−n

Jn共␳r兲e−in␪ ein␺d␪ rdr

冕

⬁

fn共r兲Jn共␳r兲rdr.

共17兲

0

This follows since

冕

兺i

n=−⬁

⬁

2␲

ein␪d␪ = 2␲␦n0 =

0

再

2␲ if n = 0 0

冎

otherwise

,

共18兲

where ␦nm denotes the kronecker delta function.

By comparing Eqs. (15) and (17), we obtain the expression for Fn共␳兲, the Fourier coefficients of the Fourier domain expansion. Using Eq. (1), this can also be interpreted in terms of a Hankel transform as Fn共␳兲 = 2␲i−n

冕

⬁

fn共r兲Jn共␳r兲rdr = 2␲i−nHn兵fn共r兲其. 共19兲

0

2. Inverse Transform The corresponding 2D inverse Fourier transform is written as f共rជ 兲 =

1 共2␲兲

冕冕 ⬁

2

0

1 2␲

in ein␪

冕

⬁

Fn共␳兲Jn共␳r兲␳d ␳

共21兲

0

so that 2␲

This transform is well suited to functions that are separable in r and ␪. This case is extensively treated in Goodman’s widely used text on Fourier optics [7].

Similarly, the 2D Fourier transform F共␳ , ␺兲 can also be expanded into its own Fourier series so that

F n共 ␳ 兲 =

兺

f共rជ 兲 =

fn共r兲 =

in 2␲

0

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

共20兲

Fn共␳兲Jn共␳r兲␳d ␳ =

0

in 2␲

Hn兵Fn共␳兲其.

共22兲

Thus, it can be observed that the nth term in the Fourier series for the original function will Hankel transform into the nth term of the Fourier series of the Fourier transform function. However, it is an nth-order Hankel transform for the nth term, so all the terms are not equivalently transformed. Furthermore, we recall that the general 2D Fourier transform for radially symmetric functions was shown to be equivalent to the zeroth-order Hankel transform. Therefore the mapping from fn共r兲 to Fn共␳兲, which is an nth-order Hankel transform, is not a 2D Fourier transform.

Most importantly, it has been shown that the operation of taking the 2D Fourier transform of a function is equivalent to first (1) finding its Fourier series expansion in the angular variable and then (2) finding the nth-order Hankel transform (of the spatial radial variable to the spatial frequency radial variable) of the nth coefficient in the Fourier series and appropriately scaling the result.

Clearly, for functions with cylindrical-type symmetries that are naturally described by cylindrical coordinates, the operation of taking a three-dimensional (3D) Fourier transform will be equivalent to (a) a regular 1D Fourier transform in the z coordinate, then (2) a Fourier series expansion in the angular variable, and then (3) nth-order Hankel transform (of the radial variable to the spatial radial variable) of the nth coefficient in the Fourier series. It should further be noted that since each of these operations involves integration over one variable only with the others being considered parameters vis-à-vis the integration, the order in which these operations are performed is interchangeable.

4. THE DIRAC-DELTA FUNCTION AND ITS TRANSFORM The unit-mass Dirac-delta function in 2D polar coordinates is defined as 1 f共rជ 兲 = ␦共rជ − rជ 0兲 = ␦共r − r0兲␦共␪ − ␪0兲. r

共23兲

To find the Fourier transform, the Fourier series expansion is required, followed by a Hankel transform, as previously discussed. Thus for the Dirac-delta function, the Fourier series expansions terms are fn共r兲 =

2␲

F共␻ ជ 兲ej␻ជ ·rជ d␺ ␳d␳ .

冕

⬁

1 2␲

冕

1 =

2␲r

2␲

0

1 r

␦共r − r0兲␦共␪ − ␪0兲e−in␪ d␪

␦共r − r0兲e−in␪0 ,

and then the full transform is given by

共24兲

Natalie Baddour

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

⬁

F共␻ ជ兲 =

兺

⬁

n=−⬁

兺

2␲i−n ein␺

n=−⬁

⬁

=

兺

Fn共␳兲ejn␺ =

2␲i−nein␺

n=−⬁

冕

⬁

fn共r兲Jn共␳r兲rdr

fn共r兲 =

0

␦共r − r0兲 2␲r

0

冕

⬁

兺

冕

兺

imJm共␳0r兲e−im␺0 eim␪ e−in␪d␪

m=−⬁

0

⬁

共25兲

n=−⬁

F共␻ ជ兲 =

兺 2␲i

−n

ein␺

n=−⬁

where Eq. (25) is the 2D linear exponential phase function. It can be seen that the Fourier transform of the Dirac delta function is the exponential function, as would be expected from the results in Cartesian coordinates.

More importantly, and the main result that we seek, is that the coefficients of the Fourier expansions of the transform are given by Fn共␳兲 = i−nJn共␳r0兲e−in␪0 .

共26兲

A. Dirac-Delta Function at the Origin If rជ 0 is at the origin, then it is multiply covered by the angular variable, and the Dirac-delta function in two dimensions is given by

␦共rជ 兲 =

1 2␲r

␦共r兲,

B. Ring-Delta Function An often-used function is the Ring-delta function given by 1

␦共r − r0兲.

共28兲

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

冕再 ⬁

0

1 2␲r

冎

␦共r − r0兲 J0共␳r兲rdr = J0共␳r0兲. 共29兲

From Eq. (7), the 2D complex exponential function can be written in polar coordinates as ⬁

兺 i J 共␳ r兲e n

n

0

−n

ein␺

n=−⬁

兺 2␲ e

⬁

−in␺0

ein␪ ,

兵inJn共␳0r兲e−in␺0其Jn共␳r兲rdr

0

⬁

=

fn共r兲Jn共␳r兲rdr

−in␺0

n=−⬁

1

␳

␦共␳ − ␳0兲ein␺ ,

共32兲

where the last line follows from the orthogonality of the Bessel functions [8]. This gives 1 Fn共␳兲 = 2␲ e−in␺0 ␦共␳ − ␳0兲. ␳

共33兲

It is also noted that the closure of the complex exponentials (or, equivalently, finding the Fourier series of the 1D Dirac-delta function) gives ⬁

␦ 共 ␺ − ␺ 0兲 =

兺

n=−⬁

1 2␲

e−in␺0 ein␺ ,

共34兲

so Eq. (32) actually gives the traditional Fourier transform of the complex exponential as it should: 1 F共␻ ជ 兲 = 共2␲兲2 ␦共␳ − ␳0兲␦共␺ − ␺0兲 = 共2␲兲2␦共␻ ជ −␻ ជ 0兲. 共35兲 ␳ A. Special Case As a special case of this, the Fourier transform of f共rជ 兲 = 1 can be computed by subsituting ␻0 = 0 into the above formulas. This gives 1 F共␻ ជ 兲 = 共2␲兲2 ␦共␳兲␦共␺兲 = 共2␲兲2␦共␻ ជ 兲, ␳

共36兲

or, alternatively, in series form the Fourier transform of f共rជ 兲 = 1 is given by F共␻ ជ兲 =

2␲

␳

⬁

␦共␳兲

兺e

in␺

.

共37兲

n=−⬁

6. MULTIPLICATION

5. COMPLEX EXPONENTIAL AND ITS TRANSFORM

f共rជ 兲 = ei␻ជ 0·rជ =

兺 2␲i

⬁

0

⬁

=

冕 冕

共27兲

where the difference in notation between ␦共rជ 兲 and ␦共r兲 is emphasized. The notation ␦共rជ 兲 is used to represent the Dirac-delta function in the appropriate multidimensional coordinate system, whose actual form may vary. The notation ␦共r兲 denotes the standard 1D scalar version of the Dirac-delta function that is most familiar. Either Eq. (11) or Eq. (26) can be used to calculate the Fourier transform, as they both yield the correct transform for the Diracdelta function at the origin, which is 1.

2␲r

共31兲

The Fourier transform is given by

i−nJn共␳r0兲e−in␪0 ein␺ = e−i␻ជ ·rជ 0 ,

F共␳兲 = 2␲

2␲

⬁

2␲

= inJn共␳0r兲e−in␺0 .

e−in␪0 Jn共␳r兲rdr

⬁

=

1

1771

共30兲

n=−⬁

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

We consider the product of two functions h共rជ 兲 = f共rជ 兲g共rជ 兲, ⬁ ⬁ fn共r兲ein␪ and g共rជ 兲 = 兺n=−⬁ gn共r兲ein␪. The where f共rជ 兲 = 兺n=−⬁ Fourier series coefficients fn共r兲 and gn共r兲 are given by Eq. (14), and we seek to find the equivalent coefficients hn共r兲 ⬁ hn共r兲ein␪. This is accomplished by of h共rជ 兲 = f共rជ 兲g共rជ 兲 = 兺n=−⬁ finding the Fourier transform of h共rជ 兲 and using the expan⬁ ⬁ fn共r兲ein␪ and g共rជ 兲 = 兺n=−⬁ gn共r兲ein␪ along sions f共rជ 兲 = 兺n=−⬁ with Eq. (8) for the polar form of the 2D complex exponential:

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

冕 冕冕

Natalie Baddour

⬁

H共␻ ជ兲 =

7. SPATIAL SHIFT

f共rជ 兲g共rជ 兲e−i␻ជ ·rជ drជ

The correct expression for a Fourier series shifted in space is obtained by finding the inverse Fourier transform of the complex exponential-weighted transform. In other words, f共rជ − rជ 0兲 is defined from

−⬁ ⬁

=

0

⬁

2␲

兺

⬁

fn共r兲ein␪

n=−⬁

0

兺

gm共r兲eim␪

m=−⬁

f共rជ − rជ 0兲 = F−1兵e−i␻ជ ·rជ 0 F共␻ ជ 兲其.

⬁

⫻

兺

i Jk共␳r兲e −k

−ik␪

e

ik␺

d␪ rdr.

共38兲

k=−⬁

Performing the integration over the angular variable yields ⬁

H共␻ ជ兲 =

兺

2␲i−keik␺

k=−⬁

冕

⬁

⬁

兺

fk−m共r兲gm共r兲Jk共␳r兲rdr

0 m=−⬁

⬁

=

兺 H 共␳兲e k

ik␺

共39兲

,

k=−⬁

where

Hk共␳兲 = 2␲i−k

冕

⬁

The shifted function is defined according to Eq. (45) since we have already found the expansion for the complex exponential, and the rules for finding the product of two expansions was found in the previous section. It is not sufficient to find any expression for the spatial shift, but rather the expression that is sought is one that is (i) in the form of a Fourier 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 if 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 definition of the inverse Fourier transform given in Eq. (20), along with the expansions in Eqs. (7), (8), and (15), the desired quantity is then given by f共rជ − rជ 0兲 =

⬁

兺

fk−m共r兲gm共r兲Jk共␳r兲rdr.

共40兲

0 m=−⬁

共45兲

1 共2␲兲2

冕冕 ⬁

0

⬁

2␲

兺

i−mJm共␳r0兲e−im␪0 eim␺

m=−⬁

0

⬁

⫻

兺 F 共␳兲e n

⬁

jn␺

⫻

n=−⬁

兺 i J 共␳r兲e k

k

ik␪

e−ik␺ d␺ ␳d ␳ .

k=−⬁

共46兲

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

Hk共␳兲 = 2␲i−k

冕

⬁

hk共r兲Jk共␳r兲rdr;

共41兲

0

Performing the integration over ␺ yields a nonzero value only if k − n − m = 0, so the preceding equation simplifies to f共rជ − rជ 0兲 =

⫻

兺 兺i

冕

n

e−i共k−n兲␪0 eik␪

⬁

Fn共␳兲Jk−n共␳r0兲Jk共␳r兲␳d ␳ .

共47兲

0

⬁

兺

fk−m共r兲gm共r兲,

共42兲

m=−⬁

which is in fact the convolution of the Fourier series of f共rជ 兲 and g共rជ 兲. In other words, we have that 共fg兲k = fk ⴱ gk ,

⬁

兺

fk−m共r兲gm共r兲.

It can be observed from Eq. (47) that we obtain the Fourier coefficients of the shifted function as

共44兲

m=−⬁

The definition of the discrete convolution of two series given in Eq. (44) is the same as the standard definition given in the literature [9].

1

关f共rជ − rជ 0兲兴k共r兲 =

⬁

兺i

2␲ n=−⬁ ⫻

共43兲

with the convolution of two series defined as

共fk ⴱ gk兲共r兲 ⬅

⬁

2␲ n=−⬁ k=−⬁

hence it follows that

hk共r兲 =

⬁

1

冕

n

e−i共k−n兲␪0

⬁

Fn共␳兲Jk−n共␳r0兲Jk共␳r兲␳d ␳ ,

共48兲

0

which gives the Fourier coefficients of the shifted function in r space in terms of the unshifted frequency-space coefficients Fn共␳兲. If the value of the shifted coefficeints are desired in terms of the original unshifted coefficients fn共r兲, ⬁ fn共r兲ejn␪, then the definition of the where f共rជ 兲 = 兺n=−⬁ Fourier-space coefficients from Eq. (19) can be invoked; Fn共␳兲 = 2␲i−n

冕

⬁

fn共r兲Jn共␳r兲rdr = 2␲i−nHn兵fn共r兲其, 共49兲

0

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

Natalie Baddour

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

⬁

f共rជ − rជ 0兲 =

⬁

兺

eik␪

k=−⬁

兺

e−i共k−n兲␪0

n=−⬁

冕

⬁

fn共u兲Snk共u,r,r0兲udu,

Snk共u,r,r0兲 = Sn0 共u,r,r0兲

0

=

共50兲 where Skn共u , r , r0兲 is defined as a shift-type operator given by the integral of the triple-Bessel product: Snk共u,r,r0兲 =

冕

⬁

Jn共␳u兲Jk−n共␳r0兲Jk共␳r兲␳d ␳ .

共51兲

0

Thus, the Fourier coefficients of the shifted function in terms of the unshifted function coefficients are given by ⬁

兺e

关f共rជ − rជ 0兲兴k =

−i共k−n兲␪0

n=−⬁

冕

冕

⬁

Jn共␳u兲J−n共␳r0兲J0共␳r兲␳d ␳,

fn共u兲Snk共u,r,r0兲udu. 共52兲

1

共57兲 emphasized that the result Skn共u , r , r0兲 = S0n共u , r , r0兲 is only for r0 ⫽ 0; otherwise, Skn共u , r , 0兲 = u1 ␦共u − r兲␦nk, as

It is true previously discussed. This result for the shift operator implies that Eq. (50) becomes ⬁

f共rជ − rជ 0兲 =

兺

ein␪0

关f共rជ − rជ 0兲兴n =

共54兲

or, explicitly, ⬁

H k共 ␳ 兲 =

兺

i−mJm共␳r0兲e−im␪0 Fk−m共␳兲

共55兲

A. Alternative Interpretation of the Expressions The preceding expressions permit a great deal of simplification by using the result derived in [10] that the tripleproduct integral given by

冕

⬁

Jn1共k1␳兲Jn2共k2␳兲Jn3共k3␳兲␳d ␳

共56兲

0

is nonzero only if n1 + n2 + n3 = 0. This is a restrictive constraint and thus permits the simplification of the shift operator so that Skn共u , r , r0兲 is nonzero only for k = 0, which gives

共58兲

冕

⬁

fn共u兲Sn0 共u,r,r0兲udu.

共59兲

0

Equation (59) becomes

冕冕 冕 ⬁

关f共rជ − rជ 0兲兴n =

⬁

fn共u兲Jn共␳u兲uduJ−n共␳r0兲J0共␳r兲␳d ␳

0

=

0

共− i兲n 2␲

⬁

Fn共␳兲Jn共␳r0兲J0共␳r兲␳d ␳ ,

共60兲

0

where the result J−n共x兲 = 共−1兲nJn共x兲 has been used, along with the definition of Fn共␳兲. Equation (60) is identical to Eq. (47) with the simplification k = 0 that became apparent only from the simplification of the shift operator.

While this simplification benefits the expression for the shifted function, there are benefits to not simplifying too soon, as shall be seen in the following section on convolutions.

To find the simplified version of the forward transform of h共rជ 兲 = f共rជ − rជ 0兲, we invoke the definition to find

m=−⬁

as per the definition of the convolution operation for two Fourier series coefficients.

fn共u兲Sn0 共u,r,r0兲udu.

0

共53兲

Hk共␳兲 = 关i−kJk共␳r0兲e−ik␪0兴 ⴱ Fk共␳兲

冕

⬁

Equation (58) essentially points out that shifting the function to a new point involves a translation only and not a rotation. The translation does not materially affect the angular portion of the expansion, and it can thus be as easily expanded in terms of the angular coordinates with ein␪0 as in terms of ein␪. The translation portion of the shift is handled with the integration with the shift operator. Thus, from Eq. (58), we see that the coefficients in the expansion for the shifted function are

u

so that Eq. (52) returns the correct unshifted value of fn, as it should.

The corresponding coefficients of the 2D Fourier transform can be found. If we define h共rជ 兲 = f共rជ − rជ 0兲, then H共␻ ជ兲 = e−i␻ជ ·rជ 0F共␻ ជ 兲, and the Fourier coefficients Hn共␳兲 are sought.

Thus H共␻ ជ 兲 is defined as a product of the complex exponential and F共␻ ជ 兲, and it was previously shown that multiplication in the Fourier series domain implies convolution of the coefficients of the respective series, with the convolution operation defined in Eq. (42). Since the coefficients for the complex exponential are given in Eq. (8), it follows that

r0 ⫽ 0.

0

␦nkJn共␳u兲Jn共␳r兲␳d ␳ = ␦共u − r兲␦nk

0

⬁

⬁

Equation (52) describes the shift operation in terms of Fourier polar coordinates.

In the special case that rជ 0 = 0, then the definition of the shift operator along with the fact that Jn共0兲 = ␦n0 implies that in this case the shift operator becomes Snk共u,r,0兲 =

冕

n=−⬁

0

1773

Hn共␳兲 = 2␲i−n

⫻

冋

冕

⬁

hn共r兲Jn共␳r兲rdr = 2␲i−n

0

共− i兲 2␲

n

冕

冕

⬁

0

⬁

册

Fn共u兲Jn共ur0兲J0共ur兲udu Jn共␳r兲rdr.

0

共61兲 To evaluate this integral, the integration over r needs to be performed first, and in fact, this can be found in closed form [3] as the nth-order Hankel transform of J0共ur兲. Denote this integral with K so that we have Kn共u, ␳兲 =

冕

⬁

J0共ur兲Jn共␳r兲rdr.

共62兲

0

Given this definition, the expression for Hn共␳兲 becomes

1774

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

Hn共␳兲 = 共− 1兲n

冕

Natalie Baddour

⬁

Fn共u兲Jn共ur0兲Kn共u, ␳兲udu.

共63兲

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

0

冕冕 冕 ⬁

h共rជ 兲 =

8. CONVOLUTION

0

The 2D convolution of two functions is defined by h共rជ 兲 = f共rជ 兲 ⴱ ⴱ g共rជ 兲 =

冕

兺

⬁

gm共r0兲e

im␪0

m=−⬁

0

⬁

兺e 兺e ik␪

k=−⬁

−i共k−n兲␪0

n=−⬁

⬁

⬁

g共rជ 0兲f共rជ − rជ 0兲drជ 0 .

⬁

2␲

fn共u兲Snk共u,r,r0兲udud␪0r0dr0 ,

⫻

共64兲

共65兲

0

−⬁

The double-asterisk notation of ⴱⴱ is used to emphasize that this is a 2D convolution and to distinguish it from a 1D convolution.

which can be simplified by performing the integration over the angular variable and by using the definition of Skn共u , r , r0兲 to give

共66兲

With the simplifications in the brackets as shown in Eq. (66), this becomes h共rជ 兲 =

1

⬁

兺i

2␲ k=−⬁

冕 再兺 ⬁

⬁

k

e

ik␪

冎

Gk−n共␳兲Fn共␳兲 Jk共␳r兲␳d ␳ .

n=−⬁

0

共67兲

the resulting series as per Eq. (69) to get Hk共␳兲. The final step is then to inverse Hankel transform the result to finally obtain hk共r兲. It is important to note that convolving the two functions f共rជ 兲 and g共rជ 兲 is not equivalent to convolving their series: this in fact was shown previously in Section 6 to be equivalent to the multiplication of the functions themselves.

This can be written in the form h共rជ 兲 =

1

⬁

兺

2␲ k=−⬁

ik eik␪

冕

⬁

⬁

Hk共␳兲Jk共␳r兲␳d ␳ =

兺

hk共r兲eik␪ ,

k=−⬁

0

共68兲 where the coefficients Hk共␳兲 are clearly the convolution of the two series given by ⬁

H k共 ␳ 兲 =

兺G

k−n共␳兲Fn共␳兲.

共69兲

9. SPATIAL SHIFT OF RADIALLY SYMMETRIC FUNCTIONS From the previous discussion on radially symmetric functions embodied in Eq. (12) and the definition of the 2D Fourier transform as given in Eqs. (13) and (14), it is obvious that the Fourier series for a radially symmetric funcion and its transform include only the n = 0 term. Thus, for a radially symmetric function, the shifted function can be written from Eq. (47) as

n=−⬁

In the results of Section 7, it was shown that the convolution of two sets of Fourier coefficients is equivalent to multiplication of the functions so that Gk ⴱ Fk = 共GF兲k; hence it follows from Eq. (69) that H共␻ ជ 兲 = G共␻ ជ 兲F共␻ ជ 兲,

f共rជ − rជ 0兲 =

1

⬁

兺

2␲ k=−⬁

e−ik␪0 eik␪

F0共␳兲Jk共␳r0兲Jk共␳r兲␳d ␳ ,

0

共71兲 where

共70兲

as would be expected from the standard results of Fourier theory, and it serves to confirm the accuracy of the development. However, the main result we seek is Eq. (67), which gives the values of hk共r兲 in terms of gk共r兲 and fk共r兲 (or rather the Hankel transform of those) and essentially defines the convolution operation for functions given in Fourier series forms. This equation defines the convolution operation so that to find the convolution of two functions given in Fourier series form, one must first find the kth coefficient of the Fourier transform of each function, namely, Gk共␳兲 and Fk共␳兲, and then subsequently convolve

冕

⬁

F 0共 ␳ 兲 = 2 ␲

冕

⬁

f共u兲J0共␳u兲udu.

共72兲

0

It is clear from the previous two equations that the factors of 2␲ in the numerator and denominator will cancel out; however, they have been retained in order to maintain the formulas that have been developed in the previous sections. Although f共r兲 is radially symmetric and has only the n = 0 term in its Fourier series, f共rជ − rជ 0兲 is not radially symmetric and so needs the full complement of entries in its Fourier series. It is equally important to note that this is the case even if the new “center” rជ 0 to which

Natalie Baddour

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

the function f has been shifted is located on the radial axis so that ␪0 = 0. If this is the case, then Eq. (71) becomes ⬁

f共rជ − rជ 0兲 =

兺e

冕 冕 ⬁

ik␪

k=−⬁

⬁

J0共␳u兲Jk共␳r0兲Jk共␳r兲␳d ␳udu.

f共u兲

0

0

10. CONVOLUTION OF TWO RADIALLY SYMMETRIC FUNCTIONS Two radially symmetric functions are functions for which both Fourier series expansions include only the n = 0 term. The convolution of the two radially symmetric functions is defined as

共73兲 h共rជ 兲 = f共rជ 兲 ⴱ ⴱ g共rជ 兲 =

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

f共rជ − rជ 0兲 =

兺e

ik共␪−␪0兲

k=−⬁

冕

⬁

f共u兲S0k共u,r,r0兲udu.

共74兲

0

−k

g共rជ 0兲f共rជ − rជ 0兲drជ 0 ,

Jk共␳r0兲e−ik␪0 F0共␳兲eik␺ ,

冕冕 ⬁

h共rជ 兲 = f共rជ 兲 ⴱ ⴱ g共rជ 兲 =

⫻

冕

0

⬁

2␲

g共r0兲

兺e

−ik␪0

eik␪

k=−⬁

0

⬁

f共u兲S0k共u,r,r0兲udu d␪0r0dr0

and not interpreted as

冕

⬁

g共r0兲f共r − r0兲dr0 ,

共79兲

0

which would be a 1D convolution. Equation (78) can be simplified by performing the integration over the angular variable, which is nonzero only if k = 0, so that f共rជ 兲 ⴱ ⴱ g共rជ 兲 = f共r兲 ⴱ ⴱ g共r兲

共75兲

冕 冕 ⬁

=

which, using the expansion for the complex exponential given in Eq. (8), becomes

⬁

g共r0兲

0

f共u兲S00共u,r,r0兲udu r0dr0 ,

0

共80兲

⬁

兺i

共78兲

0

k=−⬁

F关f共rជ − rជ 0兲兴 = F0共␳兲

共77兲

where it is emphasized that the integration is over all rជ 0, which includes all possible values of radial and angular variables. In other words, Eq. (77) must be properly interpreted as a 2D convolution along with Eq. (71) as

⬁

兺i

冕

⬁

−⬁

Thus, we have the result that a function of r becomes a function of r and ␪ only once it has been shifted away from the origin or, in other words, that a radially symmetric function is no longer radially symmetric once shifted away from the origin. This may generate some confusion in that f共rជ − rជ 0兲 is the shift of the radially symmetric f共r兲 so that it is now centered at rជ 0. This is not the same as f共r − r0兲, which is a radially symmetric function. To illustrate these points, consider the Gaussian “dome” of f共r兲 2 2 2 = e−共r /4兲, f共r − 4兲 = e−共共r − 4兲 /4兲, and f共rជ − rជ 0兲 as f共r兲 = e−共r /4兲 shifted so that is it centered at rជ 0 = 共r = 4 , ␪ = 0兲. Clearly, the first two of these will be radially symmetric, while the third is not.

From Eq. (55), the Fourier transform of the shifted radially symmetric function is also not radially symmetric and is given by F关f共rជ − rជ 0兲兴 =

1775

−k

Jk共␳r0兲e

−ik␪0

e

ik␺

= F0共␳兲e

−i␻ ជ ·rជ 0

with

k=−⬁

= F0共␳兲e−ir0␳ cos共␺−␪0兲 ,

共76兲

as would be expected from the standard Fourier theory result for a shifted function. As previously mentioned, it is intuitively obvious from a 2D polar-coordinate view of the world that a radially symmetric function is no longer radially symmetric once shifted away from the origin. Thus, since the function is no longer radially symmetric, the 2D Fourier transform is no longer equivalent to a zerothorder Hankel transform as was the case for the radially symmetric function. In fact, Eq. (76) clearly shows that it is the e−i␻ជ ·rជ 0 term that destroys the symmetry of the 2D Fourier transform. In particular, e−i␻ជ ·rជ 0 = e−ir0␳ cos共␺−␪0兲 is a function of magnitude 1 but with a nonradially symmetric phase. Continuing with this argument, if Hankel transforms are considered to be merely transforming a function of a variable r to another function of a variable ␳, without consideration of its role in the 2D perspective, it would be reasonable to look for a Hankel analog of the shift and convolution theorems of standard Fourier theory. Clearly, a 1D shift/convolution rule for Hankel transforms does not exist, and it is only in considering the matter from the 2D perspective that the reason one cannot exist becomes obvious.

S00共u,r,r0兲 =

冕

⬁

J0共␳u兲J0共␳r0兲J0共␳r兲␳d ␳ ,

共81兲

0

so that Eq. (80) is hardly recognizable as a convolution operation. The double ⴱ notation has been used to highlight the fact that a 2D convolution is being taken. In fact, the proper, correct definition of the convolution, Eq. (78), can be interpreted from Eq. (80) as f共rជ 兲 ⴱ ⴱ g共rជ 兲 = f共r兲 ⴱ ⴱ g共r兲 =

冕

⬁

g共r0兲⌽共r − r0兲r0dr0 ,

0

共82兲 where ⌽共r − r0兲 =

冕

⬁

0

f共u兲S00共u,r,r0兲udu =

冕

2␲

f共rជ − rជ 0兲d␪0 .

0

共83兲 Thus, we observe that the definition of convolution that we are tempted to write, as given by Eq. (79), is almost correct. What needed to be done was to shift the function f from rជ to rជ 0 (thus destroying radial symmetry) and

1776

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

Natalie Baddour

then to integrate the resulting shifted function over all angular variables. The unshifted function g is still radially symmetric and thus is not affected by the integration over the angular variable, and the final result is the form given in Eq. (82).

From Eqs. (68) and (69), it follows that h共r兲 = f共r兲 ⴱ ⴱ g共r兲 =

1 2␲

冕

⬁

G0共␳兲F0共␳兲J0共␳r兲␳d ␳ , 共84兲

0

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

H k共 ␳ 兲 =

兺G

k−n共␳兲Fn共␳兲

=

n=−⬁

再

G0共␳兲F0共␳兲 k = 0 0

冎

otherwise

⬁

f共rជ 兲ⴱ␪g共rជ 兲 =

兺

fn共r兲gn共r兲ejn␪ .

In other words, the nth coefficient of the angular convolution of two functions is simply the product of the nth Fourier coefficient of each of the two functions. Mathematically, if h共r , ␪兲 = f共rជ 兲ⴱ␪g共rជ 兲, then hn共r兲 = fn共r兲gn共r兲. This is also the same result as obtained with Fourier series of (1D) periodic functions.

.

12. RADIAL CONVOLUTION

共85兲

Similarly to the definition of angular or circular convolution, we can define the notion of a radial convolution as

In the previous two equations, G0共␳兲 and F0共␳兲 are the zeroth-order coefficients in the expansions for F共␻ ជ 兲 and G共␻ ជ 兲 themselves. Equation (85) can also be obtained by observing that the integration over the angular variable in Eq. (78) forces k to be zero. Thus, the 2D convolution of two radially symmetric functions yields another radially symmetric function, as can be seen from Eq. (84). Moreover, by using the proper definition of a 2D convolution instead of using the tempting definition of a 1D convolution, the well-known relationship between convolutions in one domain leading to multiplication in the other domain is preserved, namely, that

h共rជ 兲 = f共rជ 兲ⴱrg共rជ 兲 =

冕

⬁

g共r0, ␪兲f共r − r0, ␪兲r0dr0 .

Such convolutions are less often seen than their 2D or angular counterparts; however, they are included herein for the sake of completeness. Using Eq. (50) for the radiallyonly shifted function yields

冕 冉兺 ⬁

⬁

h共r, ␪兲 =

gm共r0兲eim␪

m=−⬁

0

冉兺 ⬁

⫻

共86兲

⬁

e

兺

ik␪

k=−⬁

n=−⬁

冕

冊

⬁

For two functions f共rជ 兲 = f共r , ␪兲 and g共rជ 兲 = g共r , ␪兲, the notion of a circular or angular convolution can be defined. This is not the 2D convolution as previously discussed but rather a convolution over the angular variable only so that it may be defined as f共rជ 兲ⴱ␪g共rជ 兲 =

2␲

冕

冉兺 ⬁

⫻

⬁

2␲

兺 兺

h共r, ␪兲 =

兺e

k=−⬁

冕 冉兺 2␲

0

⬁

fn共r兲ejn␪0

n=−⬁

冊

冊

fn共r兲gm共r兲ejm␪

1 2␲

gk−m共r0兲

0 m=−⬁

⬁

⬁

fn共u兲Snm共u,r,r0兲udu r0dr0

⫻

n=−⬁

=

⬁

⬁

ik␪

0

⬁

⬁

n=−⬁ m=−⬁

冕兺 兺冕 兺 冕 兺 兺 冕 ⬁

0

⬁

⬁

e

ik␪

k=−⬁

⬁

⫻

n=−⬁

gk−m共r0兲

0 m=−⬁

in

2␲

⬁

Fn共␳兲Jm−n共␳r0兲Jm共␳r兲␳d ␳ r0dr0 .

0

共92兲

gm共r兲ejm共␪−␪0兲 d␪0

m=−⬁

=

1

Using the result for the product of two series so that the resulting Fourier coefficients are convolved as fk ⴱ gk ⬁ ⬅ 兺m=−⬁ fm共r兲gk−m共r兲 gives

共87兲

Note the notation of ⴱ␪ used to denote the angular convolution. A Fourier relationship can be defined for this operation of angular convolution: h共r, ␪兲 = f共rជ 兲ⴱ␪g共rជ 兲 =

共91兲

2␲

f共r, ␪0兲g共r, ␪ − ␪0兲d␪0 .

冊

fn共u兲Snk共u,r,r0兲udu r0dr0 .

0

The key point to this relationship is the proper definition of the convolution as a 2D convolution.

11. CIRCULAR (ANGULAR) CONVOLUTION

共90兲

0

h共rជ 兲 = f共rជ 兲 ⴱ ⴱ g共rជ 兲 = f共r兲 ⴱ ⴱ g共r兲 ⇒ H共␳兲 = F共␳兲G共␳兲.

1

共89兲

m=−⬁

Let us define

冕

2␲

ejn␪0 e−jm␪0 d␪0 .

m−n 共␳兲 = Gk−m

冕

⬁

gk−m共r0兲Jm−n共␳r0兲r0dr0 ,

共93兲

0

0

共88兲 The value of the integral in the preceding equation is 2␲ ␦mn, and the following simplification results:

which is the 共m – n兲th-order Hankel transform of the 共k – m兲th Fourier coefficient. With this definition, Eq. (92) becomes

Natalie Baddour

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

⬁

h共r, ␪兲 =

兺

⬁

eik␪

k=−⬁

⬁

in

兺 兺

m=−⬁ n=−⬁

2␲

冕

⬁

⬁ m−n Gk−m 共␳兲Fn共␳兲Jm共␳r兲␳d ␳ .

f共− rជ 兲 = f共r, ␪ + ␲兲 =

兺

This is the closest we can come to a convolution theorem for the radial convolution. Without the shift and integration over the angular portion of the function, the result is awkward because the order of the Hankel transform of one of the functions does not always correspond to the order of the Fourier coefficient as seen by the definition in Eq. (93). The angular shift and integration portion of the full 2D convolution effectively washes out all these cross terms to yield the nice result of the full 2D convolution, as previously discussed.

13. PARSEVAL RELATIONSHIPS A Parseval relantionship is important, as it deals with the “power” of a signal or function in the spatial and frequency domains. As previously mentioned, it is known that we can write

兺f

=

n

jn␪

.

n

−n共r兲共−

1兲nejn␪ ,

共96兲

where the overbar indicates a complex conjugate. Thus if we define a function g共rជ 兲 = f共−rជ 兲, its coefficients are given by gn共r兲 = f−n共r兲共−1兲n. To derive the Parseval relation, we evaluate that convolution of two functions f共rជ 兲 and g共rជ 兲 with g共rជ 兲 = f共−rជ 兲 and where the convolution is evaluated at rជ = 0ជ implying r = 0, ␪ = 0. In other words, we evaluate 共f ⴱ g兲 as given by Eq. (64) at rជ = 0ជ . This is done by making use of Eq. (66), along with J−n共x兲 = 共−1兲nJn共x兲 and Jn共0兲 = ␦n0 so that

冕

⬁

冕 再兺 冕 冕 ⬁

⬁

⬁

g共− rជ 0兲f共rជ 0兲drជ 0 = 2␲

−⬁

n=−⬁

0

g−n共r0兲J−n共␳r0兲r0dr0

冎

0

⬁

共95兲

⫻

n=−⬁

fn共u兲Jn共␳u兲udu ␳d ␳ ,

0

It is noted that in polar coordinates if rជ = 共r , ␪兲, then −rជ = 共r , ␪ + ␲兲. Hence

n −jn␪

n=−⬁

⬁

兺 f 共r兲e

兺 f 共r兲共− 1兲 e

n=−⬁

⬁

共94兲

f共rជ 兲 = f共r, ␪兲 =

⬁

fn共r兲ejn共␪+␲兲 =

n=−⬁

0

1777

共97兲

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

共98兲

冕

Equation (98) furnishes the desired Parseval relationship as

冕

⬁

兩f共rជ 兲兩2drជ =

−⬁

⬁

1

兺

2␲ n=−⬁

冕

冕

g共rជ 兲f共rជ 兲drជ =

⬁

1

2␲ 兺

⬁

兩Fn共␳兲兩2␳d ␳ .

n=−⬁

−⬁

⬁

Gn共␳兲Fn共␳兲␳d ␳ .

共100兲

f共rជ 兲g共rជ 兲 =

m=−⬁

冕

⬁

fm共r兲gm共r兲rdr,

共102兲

0

which gives another version of the Parseval relationship in Eq. (100) as

⬁

兺

n=−⬁

冕

⬁

fn共r兲gn共r兲rdr =

0

1

⬁

兺

共2␲兲2 n=−⬁

冕

⬁

Fn共␳兲Gn共␳兲␳d ␳ .

0

0

共103兲

Furthermore, from the result on multiplication and observing from Eq. (96) that the coefficients of g共rជ 兲 are g−n共r兲, we have that ⬁

兺

共99兲

0

冕

f共rជ 兲g共rជ 兲drជ = 2␲

0

Following this same procedure and evaluating the convolution of any two well-behaved functions f共rជ 兲 and g共−rជ 兲 at 0ជ yields the generalized Parseval relationship: ⬁

⬁

⬁

Clearly, it then follows from the preceding equation that

⬁

兺 兺

fm共r兲g−共k−m兲共r兲eik␪ .

共101兲

k=−⬁ m=−⬁

Integrating both sides of the previous equation over all space gives

⬁

兺

n=−⬁

冕

⬁

0

兩fn共r兲兩 rdr = 2

1

⬁

兺

共2␲兲2 n=−⬁

冕

⬁

兩Fn共␳兲兩2␳d ␳ .

0

共104兲

1778

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

Natalie Baddour

Table 1. Summary of Fourier Transform Relationships in Polar Coordinates fn共r兲

F n共 ␳ 兲

F共␻ ជ兲

1 2␲ 兰 f共r , ␪兲e−jn␪ d␪ 2␲ 0

2␲i−n兰0⬁fn共r兲Jn共␳r兲rdr

⬁ F共␳ , ␺兲=兺n=−⬁ Fn共␳兲ejn␺

i−nJn共␳r0兲e−in␪0

e−i␻ជ ·rជ 0

ei␻ជ 0·rជ

␦共r − r0兲 −in␪ e 0 2␲r inJn共␳0r兲e−in␺0

1 2␲ e−in␺0 ␦共␳ − ␳0兲 ␳

共2␲兲2␦共␻ ជ −␻ ជ 0兲

f共rជ − rជ 0兲

⬁ n e−i共n−m兲␪0 ⫻ 兰0⬁fm共u兲Sm 共u , r , r0兲udu 兺m=−⬁

关i−nJn共␳r0兲e−in␪0兴 ⴱ Fn共␳兲 ⬁ i−mJm共␳r0兲e−im␪0 Fn−m共␳兲 =兺m=−⬁

e−i␻ជ ·rជ 0 F共␻ ជ兲

共−1兲n兰0⬁Fn共u兲Jn共ur0兲Kn共u , ␳兲udu

e−i␻ជ ·rជ 0 F共␻ ជ兲

2␲i−n兰0⬁fn共r兲Jn共␳r兲rdr

G共␻ ជ 兲 ⴱ H共␻ ជ兲

⬁ Fn共␳兲 = Hn共␳兲 ⴱ Gn共␳兲=兺m=−⬁ Gn−m共␳兲Hm共␳兲

G共␻ ជ 兲H共␻ ជ兲

f共rជ 兲 ⬁ fn共r兲ejn␪ f共r , ␪兲=兺n=−⬁

␦共rជ − rជ 0兲=

␦共r − r0兲 r

␦ 共 ␪ − ␪ 0兲

共−i兲n

f共rជ − rជ 0兲 (alternate) h共rជ 兲g共rជ 兲

2␲

兰0⬁Fn共␳兲Jn共␳r0兲J0共␳r兲␳d ␳

⬁ fn共r兲 = hn共r兲 ⴱ gn共r兲=兺m=−⬁ hn−m共r兲gm共r兲

h共rជ 兲 ⴱ ⴱ g共rជ 兲

n

i ⬁ 兰 F 共␳兲Jn共␳r兲␳d␳ 2␲ 0 n

14. SUMMARY AND CONCLUSIONS In summary, this paper has considered the polarcoordinate version of the standard 2D Fourier transform and derived the operational toolset required for standard Fourier operations. As previously noted, the polarcoordinate version of the 2D Fourier transform is most useful for functions that are naturally described in terms of polar coordinates. Additionally, Parseval relationships were also derived. The results of the paper are concisely collected in Table 1. Of particular interest are the results on convolution and spatial shift. With the application of a special result on the integral of a triple product of Bessel functions, the expression for spatial shift allowed for considerable simplification. Notably, standard convolution/ multiplication rules do apply for 2D convolution and 1D circular convolution but not for 1D radial convolution.

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ACKNOWLEDGMENTS

8.

This work was financially supported by the Natural Sciences and Engineering Research Council of Canada. The author also thanks one of the reviewers, whose thorough review and helpful comments contributed to improving this paper.

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