Abstract

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.

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References

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  1. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1999).
  2. K. Howell, “Fourier transforms,” in The Transforms and Applications Handbook, 2nd ed., A.D.Poularikas, ed. (CRC Press, 2000), pp. 2.1-2.159.
  3. G. Chirikjian and A. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups (CRC Press, 2001).
  4. Y. Xu, M. Xu, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography--II: Cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829-833 (2002).
    [CrossRef] [PubMed]
  5. A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmonic Anal. 21, 145-167 (2006).
    [CrossRef]
  6. R. Piessens, “The Hankel transform,” in The Transforms and Applications Handbook, 2nd ed., A.D.Poularikas, ed. (CRC Press, 2000), pp. 9.1-9.30.
  7. J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2004).
  8. G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier, 2005).
  9. A. Oppenheim and R. Schafer, Discrete-time Signal Processing (Prentice-Hall, 1989).
  10. A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446-460 (1972).
    [CrossRef]

2006

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

2002

Y. Xu, M. Xu, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography--II: Cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829-833 (2002).
[CrossRef] [PubMed]

1972

A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446-460 (1972).
[CrossRef]

Arfken, G.

G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier, 2005).

Averbuch, A.

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

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1999).

Chirikjian, G.

G. Chirikjian and A. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups (CRC Press, 2001).

Coifman, R. R.

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

Donoho, D. L.

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

Elad, M.

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

Goodman, J.

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

Howell, K.

K. Howell, “Fourier transforms,” in The Transforms and Applications Handbook, 2nd ed., A.D.Poularikas, ed. (CRC Press, 2000), pp. 2.1-2.159.

Israeli, M.

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

Jackson, A. D.

A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446-460 (1972).
[CrossRef]

Kyatkin, A.

G. Chirikjian and A. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups (CRC Press, 2001).

Maximon, L. C.

A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446-460 (1972).
[CrossRef]

Oppenheim, A.

A. Oppenheim and R. Schafer, Discrete-time Signal Processing (Prentice-Hall, 1989).

Piessens, R.

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

Schafer, R.

A. Oppenheim and R. Schafer, Discrete-time Signal Processing (Prentice-Hall, 1989).

Wang, L. V.

Y. Xu, M. Xu, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography--II: Cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829-833 (2002).
[CrossRef] [PubMed]

Weber, H.

G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier, 2005).

Xu, M.

Y. Xu, M. Xu, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography--II: Cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829-833 (2002).
[CrossRef] [PubMed]

Xu, Y.

Y. Xu, M. Xu, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography--II: Cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829-833 (2002).
[CrossRef] [PubMed]

Appl. Comput. Harmonic Anal.

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

IEEE Trans. Med. Imaging

Y. Xu, M. Xu, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography--II: Cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829-833 (2002).
[CrossRef] [PubMed]

SIAM J. Math. Anal.

A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446-460 (1972).
[CrossRef]

Other

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1999).

K. Howell, “Fourier transforms,” in The Transforms and Applications Handbook, 2nd ed., A.D.Poularikas, ed. (CRC Press, 2000), pp. 2.1-2.159.

G. Chirikjian and A. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups (CRC Press, 2001).

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

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

G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier, 2005).

A. Oppenheim and R. Schafer, Discrete-time Signal Processing (Prentice-Hall, 1989).

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Tables (1)

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Table 1 Summary of Fourier Transform Relationships in Polar Coordinates

Equations (102)

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F ̂ n ( ρ ) = H n ( f ( r ) ) = 0 f ( r ) J n ( ρ r ) r d r ,
f ( r ) = 0 F n ( ρ ) J n ( ρ r ) ρ d ρ .
0 s d s 0 f ( r ) J n ( s r ) J n ( s u ) r d r = 1 2 [ f ( u + ) + f ( u ) ] .
F ( ω ) = F ( ω x , ω y ) = f ( x , y ) e j ( ω x x + ω y y ) d x d y .
f ( r ) = f ( x , y ) = 1 ( 2 π ) 2 F ( ω x , ω y ) e j ω r d ω x d ω y ,
F ( ρ , ψ ) = 0 π π f ( r , θ ) e i r ρ cos ( ψ θ ) r d r d θ .
e i ω r = n = i n J n ( ρ r ) e i n θ e i n ψ ,
e i ω r = n = i n J n ( ρ r ) e i n θ e i n ψ .
F ( ρ , ψ ) = 0 r f ( r ) d r π π e i r ρ cos ( ψ θ ) d θ .
J 0 ( x ) = 1 2 π π π e i x cos ( ψ θ ) d θ = 1 2 π π π e i x cos α d α ,
F ( ρ ) = F 2 D { f ( r ) } = 2 π 0 f ( r ) J 0 ( ρ r ) r d r ,
F ( ρ ) = F 2 D { f ( r ) } = 2 π H 0 { f ( r ) } .
f ( r ) = f ( r , θ ) = n = f n ( r ) e j n θ ,
f n ( r ) = 1 2 π 0 2 π f ( r , θ ) e j n θ d θ .
F ( ω ) = F ( ρ , ψ ) = n = F n ( ρ ) e j n ψ
F n ( ρ ) = 1 2 π 0 2 π F ( ρ , ψ ) e j n ψ d ψ .
F ( ω ) = f ( r ) e j ω r d r = 0 0 2 π m = f m ( r ) e j m θ n = i n J n ( ρ r ) e i n θ e i n ψ d θ r d r = n = 2 π i n e i n ψ 0 f n ( r ) J n ( ρ r ) r d r .
0 2 π e i n θ d θ = 2 π δ n 0 = { 2 π if n = 0 0 otherwise } ,
F n ( ρ ) = 2 π i n 0 f n ( r ) J n ( ρ r ) r d r = 2 π i n H n { f n ( r ) } .
f ( r ) = 1 ( 2 π ) 2 0 0 2 π F ( ω ) e j ω r d ψ ρ d ρ .
f ( r ) = n = 1 2 π i n e i n θ 0 F n ( ρ ) J n ( ρ r ) ρ d ρ
f n ( r ) = i n 2 π 0 F n ( ρ ) J n ( ρ r ) ρ d ρ = i n 2 π H n { F n ( ρ ) } .
f ( r ) = δ ( r r 0 ) = 1 r δ ( r r 0 ) δ ( θ θ 0 ) .
f n ( r ) = 1 2 π 0 2 π 1 r δ ( r r 0 ) δ ( θ θ 0 ) e i n θ d θ = 1 2 π r δ ( r r 0 ) e i n θ 0 ,
F ( ω ) = n = F n ( ρ ) e j n ψ = n = 2 π i n e i n ψ 0 f n ( r ) J n ( ρ r ) r d r = n = 2 π i n e i n ψ 0 δ ( r r 0 ) 2 π r e i n θ 0 J n ( ρ r ) r d r = n = i n J n ( ρ r 0 ) e i n θ 0 e i n ψ = e i ω r 0 ,
F n ( ρ ) = i n J n ( ρ r 0 ) e i n θ 0 .
δ ( r ) = 1 2 π r δ ( r ) ,
1 2 π r δ ( r r 0 ) .
F ( ρ ) = 2 π 0 { 1 2 π r δ ( r r 0 ) } J 0 ( ρ r ) r d r = J 0 ( ρ r 0 ) .
f ( r ) = e i ω 0 r = n = i n J n ( ρ 0 r ) e i n ψ 0 e i n θ ,
f n ( r ) = 1 2 π 0 2 π m = i m J m ( ρ 0 r ) e i m ψ 0 e i m θ e i n θ d θ = i n J n ( ρ 0 r ) e i n ψ 0 .
F ( ω ) = n = 2 π i n e i n ψ 0 f n ( r ) J n ( ρ r ) r d r = n = 2 π i n e i n ψ 0 { i n J n ( ρ 0 r ) e i n ψ 0 } J n ( ρ r ) r d r = n = 2 π e i n ψ 0 1 ρ δ ( ρ ρ 0 ) e i n ψ ,
F n ( ρ ) = 2 π e i n ψ 0 1 ρ δ ( ρ ρ 0 ) .
δ ( ψ ψ 0 ) = n = 1 2 π e i n ψ 0 e i n ψ ,
F ( ω ) = ( 2 π ) 2 1 ρ δ ( ρ ρ 0 ) δ ( ψ ψ 0 ) = ( 2 π ) 2 δ ( ω ω 0 ) .
F ( ω ) = ( 2 π ) 2 1 ρ δ ( ρ ) δ ( ψ ) = ( 2 π ) 2 δ ( ω ) ,
F ( ω ) = 2 π ρ δ ( ρ ) n = e i n ψ .
H ( ω ) = f ( r ) g ( r ) e i ω r d r = 0 0 2 π n = f n ( r ) e i n θ m = g m ( r ) e i m θ k = i k J k ( ρ r ) e i k θ e i k ψ d θ r d r .
H ( ω ) = k = 2 π i k e i k ψ 0 m = f k m ( r ) g m ( r ) J k ( ρ r ) r d r = k = H k ( ρ ) e i k ψ ,
H k ( ρ ) = 2 π i k 0 m = f k m ( r ) g m ( r ) J k ( ρ r ) r d r .
H k ( ρ ) = 2 π i k 0 h k ( r ) J k ( ρ r ) r d r ;
h k ( r ) = m = f k m ( r ) g m ( r ) ,
( f g ) k = f k g k ,
( f k g k ) ( r ) m = f k m ( r ) g m ( r ) .
f ( r r 0 ) = F 1 { e i ω r 0 F ( ω ) } .
f ( r r 0 ) = 1 ( 2 π ) 2 0 0 2 π m = i m J m ( ρ r 0 ) e i m θ 0 e i m ψ n = F n ( ρ ) e j n ψ × k = i k J k ( ρ r ) e i k θ e i k ψ d ψ ρ d ρ .
f ( r r 0 ) = 1 2 π n = k = i n e i ( k n ) θ 0 e i k θ 0 F n ( ρ ) J k n ( ρ r 0 ) J k ( ρ r ) ρ d ρ .
[ f ( r r 0 ) ] k ( r ) = 1 2 π n = i n e i ( k n ) θ 0 0 F n ( ρ ) J k n ( ρ r 0 ) J k ( ρ r ) ρ d ρ ,
F n ( ρ ) = 2 π i n 0 f n ( r ) J n ( ρ r ) r d r = 2 π i n H n { f n ( r ) } ,
f ( r r 0 ) = k = e i k θ n = e i ( k n ) θ 0 0 f n ( u ) S n k ( u , r , r 0 ) u d u ,
S n k ( u , r , r 0 ) = 0 J n ( ρ u ) J k n ( ρ r 0 ) J k ( ρ r ) ρ d ρ .
[ f ( r r 0 ) ] k = n = e i ( k n ) θ 0 0 f n ( u ) S n k ( u , r , r 0 ) u d u .
S n k ( u , r , 0 ) = 0 δ n k J n ( ρ u ) J n ( ρ r ) ρ d ρ = 1 u δ ( u r ) δ n k
H k ( ρ ) = [ i k J k ( ρ r 0 ) e i k θ 0 ] F k ( ρ )
H k ( ρ ) = m = i m J m ( ρ r 0 ) e i m θ 0 F k m ( ρ )
0 J n 1 ( k 1 ρ ) J n 2 ( k 2 ρ ) J n 3 ( k 3 ρ ) ρ d ρ
S n k ( u , r , r 0 ) = S n 0 ( u , r , r 0 ) = 0 J n ( ρ u ) J n ( ρ r 0 ) J 0 ( ρ r ) ρ d ρ , r 0 0 .
f ( r r 0 ) = n = e i n θ 0 0 f n ( u ) S n 0 ( u , r , r 0 ) u d u .
[ f ( r r 0 ) ] n = 0 f n ( u ) S n 0 ( u , r , r 0 ) u d u .
[ f ( r r 0 ) ] n = 0 0 f n ( u ) J n ( ρ u ) u d u J n ( ρ r 0 ) J 0 ( ρ r ) ρ d ρ = ( i ) n 2 π 0 F n ( ρ ) J n ( ρ r 0 ) J 0 ( ρ r ) ρ d ρ ,
H n ( ρ ) = 2 π i n 0 h n ( r ) J n ( ρ r ) r d r = 2 π i n 0 [ ( i ) n 2 π 0 F n ( u ) J n ( u r 0 ) J 0 ( u r ) u d u ] J n ( ρ r ) r d r .
K n ( u , ρ ) = 0 J 0 ( u r ) J n ( ρ r ) r d r .
H n ( ρ ) = ( 1 ) n 0 F n ( u ) J n ( u r 0 ) K n ( u , ρ ) u d u .
h ( r ) = f ( r ) g ( r ) = g ( r 0 ) f ( r r 0 ) d r 0 .
h ( r ) = 0 0 2 π m = g m ( r 0 ) e i m θ 0 k = e i k θ n = e i ( k n ) θ 0 0 f n ( u ) S n k ( u , r , r 0 ) u d u d θ 0 r 0 d r 0 ,
h ( r ) = 1 2 π k = i k e i k θ 0 { n = G k n ( ρ ) F n ( ρ ) } J k ( ρ r ) ρ d ρ .
h ( r ) = 1 2 π k = i k e i k θ 0 H k ( ρ ) J k ( ρ r ) ρ d ρ = k = h k ( r ) e i k θ ,
H k ( ρ ) = n = G k n ( ρ ) F n ( ρ ) .
H ( ω ) = G ( ω ) F ( ω ) ,
f ( r r 0 ) = 1 2 π k = e i k θ 0 e i k θ 0 F 0 ( ρ ) J k ( ρ r 0 ) J k ( ρ r ) ρ d ρ ,
F 0 ( ρ ) = 2 π 0 f ( u ) J 0 ( ρ u ) u d u .
f ( r r 0 ) = k = e i k θ 0 f ( u ) 0 J 0 ( ρ u ) J k ( ρ r 0 ) J k ( ρ r ) ρ d ρ u d u .
f ( r r 0 ) = k = e i k ( θ θ 0 ) 0 f ( u ) S 0 k ( u , r , r 0 ) u d u .
F [ f ( r r 0 ) ] = k = i k J k ( ρ r 0 ) e i k θ 0 F 0 ( ρ ) e i k ψ ,
F [ f ( r r 0 ) ] = F 0 ( ρ ) k = i k J k ( ρ r 0 ) e i k θ 0 e i k ψ = F 0 ( ρ ) e i ω r 0 = F 0 ( ρ ) e i r 0 ρ cos ( ψ θ 0 ) ,
h ( r ) = f ( r ) g ( r ) = g ( r 0 ) f ( r r 0 ) d r 0 ,
h ( r ) = f ( r ) g ( r ) = 0 0 2 π g ( r 0 ) k = e i k θ 0 e i k θ 0 f ( u ) S 0 k ( u , r , r 0 ) u d u d θ 0 r 0 d r 0
0 g ( r 0 ) f ( r r 0 ) d r 0 ,
f ( r ) g ( r ) = f ( r ) g ( r ) = 0 g ( r 0 ) 0 f ( u ) S 0 0 ( u , r , r 0 ) u d u r 0 d r 0 ,
S 0 0 ( u , r , r 0 ) = 0 J 0 ( ρ u ) J 0 ( ρ r 0 ) J 0 ( ρ r ) ρ d ρ ,
f ( r ) g ( r ) = f ( r ) g ( r ) = 0 g ( r 0 ) Φ ( r r 0 ) r 0 d r 0 ,
Φ ( r r 0 ) = 0 f ( u ) S 0 0 ( u , r , r 0 ) u d u = 0 2 π f ( r r 0 ) d θ 0 .
h ( r ) = f ( r ) g ( r ) = 1 2 π 0 G 0 ( ρ ) F 0 ( ρ ) J 0 ( ρ r ) ρ d ρ ,
H k ( ρ ) = n = G k n ( ρ ) F n ( ρ ) = { G 0 ( ρ ) F 0 ( ρ ) k = 0 0 otherwise } .
h ( r ) = f ( r ) g ( r ) = f ( r ) g ( r ) H ( ρ ) = F ( ρ ) G ( ρ ) .
f ( r ) θ g ( r ) = 1 2 π 0 2 π f ( r , θ 0 ) g ( r , θ θ 0 ) d θ 0 .
h ( r , θ ) = f ( r ) θ g ( r ) = 1 2 π 0 2 π ( n = f n ( r ) e j n θ 0 ) ( m = g m ( r ) e j m ( θ θ 0 ) ) d θ 0 = n = m = f n ( r ) g m ( r ) e j m θ 1 2 π 0 2 π e j n θ 0 e j m θ 0 d θ 0 .
f ( r ) θ g ( r ) = m = f n ( r ) g n ( r ) e j n θ .
h ( r ) = f ( r ) r g ( r ) = 0 g ( r 0 , θ ) f ( r r 0 , θ ) r 0 d r 0 .
h ( r , θ ) = 0 ( m = g m ( r 0 ) e i m θ ) ( k = e i k θ n = 0 f n ( u ) S n k ( u , r , r 0 ) u d u ) r 0 d r 0 .
h ( r , θ ) = k = e i k θ 0 m = g k m ( r 0 ) n = 0 f n ( u ) S n m ( u , r , r 0 ) u d u r 0 d r 0 = k = e i k θ 0 m = g k m ( r 0 ) n = i n 2 π 0 F n ( ρ ) J m n ( ρ r 0 ) J m ( ρ r ) ρ d ρ r 0 d r 0 .
G k m m n ( ρ ) = 0 g k m ( r 0 ) J m n ( ρ r 0 ) r 0 d r 0 ,
h ( r , θ ) = k = e i k θ m = n = i n 2 π 0 G k m m n ( ρ ) F n ( ρ ) J m ( ρ r ) ρ d ρ .
f ( r ) = f ( r , θ ) = n = f n ( r ) e j n θ .
f ( r ) ¯ = f ( r , θ + π ) ¯ = n = f n ( r ) e j n ( θ + π ) ¯ = n = f n ( r ) ¯ ( 1 ) n e j n θ = n = f n ( r ) ¯ ( 1 ) n e j n θ ,
g ( r 0 ) f ( r 0 ) d r 0 = 2 π 0 { n = 0 g n ( r 0 ) J n ( ρ r 0 ) r 0 d r 0 0 f n ( u ) J n ( ρ u ) u d u } ρ d ρ ,
f ( r ) 2 d r = 1 2 π n = 0 F n ( ρ ) 2 ρ d ρ .
g ( r ) ¯ f ( r ) d r = 1 2 π n = 0 G n ( ρ ) ¯ F n ( ρ ) ρ d ρ .
f ( r ) g ( r ) ¯ = k = m = f m ( r ) g ( k m ) ( r ) ¯ e i k θ .
0 f ( r ) g ( r ) ¯ d r = 2 π m = 0 f m ( r ) g m ( r ) ¯ r d r ,
n = 0 f n ( r ) g n ( r ) ¯ r d r = 1 ( 2 π ) 2 n = 0 F n ( ρ ) G n ( ρ ) ¯ ρ d ρ .
n = 0 f n ( r ) 2 r d r = 1 ( 2 π ) 2 n = 0 F n ( ρ ) 2 ρ d ρ .

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