Abstract

This paper gives the definition and examples of an arbitrary order differentiation based on some properties of the Fourier transform. To physically implement the generalized differentiation a coherent processing system with binary synthetic filter was used. The experimental part of the paper shows models of the 1-D and 2-D binary filters and results of the filtering process.

© 1982 Optical Society of America

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References

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  1. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  2. N. A. Abel, “Solution de quelques problemes a l’aide d’integrales definies,” Mag. Naturvidenkaberne, 1823, Norway.
  3. J. Liouville, J. Ec. Polytech. 13, Sec. 21, p. 1 (1832).
  4. E. L. Post, Trans. Am. Math. Soc. 32, p. 723 (1930).
    [CrossRef]
  5. K. B. Oldham, J. Spanier, Fractional Calculus (Academic, New York, 1974).
  6. M. Ichise, Y. Nagayanagi, T. Kojima, J. Electroanal. Chem. Interfacial Electrochem. 33, p. 253 (1971).
    [CrossRef]
  7. K. B. Oldham, Anal. Chem. 45, p. 39 (1973).
    [CrossRef]
  8. H. Kasprzak, “Realizowalnosc optyczna operacji rozniczkowania rzedu rzeczywistego,” “An Optical Implementation of the Real Order Differentiation,” Ph.D. Dissertation, Technical University of Wroclaw, Report 105/1980 (1980), in Polish.
  9. H. Kasprzak, Opt. Appl. 10, p. 283 (1980).
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  11. J. Burch, Proc. IEEE 55, p. 599 (1967).
    [CrossRef]
  12. R. Sirohi, V. Ram Mohan, Opt. Acta 22, p. 207 (1975).
    [CrossRef]

1980

H. Kasprzak, Opt. Appl. 10, p. 283 (1980).

1975

R. Sirohi, V. Ram Mohan, Opt. Acta 22, p. 207 (1975).
[CrossRef]

1973

K. B. Oldham, Anal. Chem. 45, p. 39 (1973).
[CrossRef]

1971

M. Ichise, Y. Nagayanagi, T. Kojima, J. Electroanal. Chem. Interfacial Electrochem. 33, p. 253 (1971).
[CrossRef]

1967

J. Burch, Proc. IEEE 55, p. 599 (1967).
[CrossRef]

1930

E. L. Post, Trans. Am. Math. Soc. 32, p. 723 (1930).
[CrossRef]

1832

J. Liouville, J. Ec. Polytech. 13, Sec. 21, p. 1 (1832).

Abel, N. A.

N. A. Abel, “Solution de quelques problemes a l’aide d’integrales definies,” Mag. Naturvidenkaberne, 1823, Norway.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Burch, J.

J. Burch, Proc. IEEE 55, p. 599 (1967).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Ichise, M.

M. Ichise, Y. Nagayanagi, T. Kojima, J. Electroanal. Chem. Interfacial Electrochem. 33, p. 253 (1971).
[CrossRef]

Kasprzak, H.

H. Kasprzak, Opt. Appl. 10, p. 283 (1980).

H. Kasprzak, “Realizowalnosc optyczna operacji rozniczkowania rzedu rzeczywistego,” “An Optical Implementation of the Real Order Differentiation,” Ph.D. Dissertation, Technical University of Wroclaw, Report 105/1980 (1980), in Polish.

Kojima, T.

M. Ichise, Y. Nagayanagi, T. Kojima, J. Electroanal. Chem. Interfacial Electrochem. 33, p. 253 (1971).
[CrossRef]

Liouville, J.

J. Liouville, J. Ec. Polytech. 13, Sec. 21, p. 1 (1832).

Nagayanagi, Y.

M. Ichise, Y. Nagayanagi, T. Kojima, J. Electroanal. Chem. Interfacial Electrochem. 33, p. 253 (1971).
[CrossRef]

Oldham, K. B.

K. B. Oldham, Anal. Chem. 45, p. 39 (1973).
[CrossRef]

K. B. Oldham, J. Spanier, Fractional Calculus (Academic, New York, 1974).

Post, E. L.

E. L. Post, Trans. Am. Math. Soc. 32, p. 723 (1930).
[CrossRef]

Ram Mohan, V.

R. Sirohi, V. Ram Mohan, Opt. Acta 22, p. 207 (1975).
[CrossRef]

Sirohi, R.

R. Sirohi, V. Ram Mohan, Opt. Acta 22, p. 207 (1975).
[CrossRef]

Spanier, J.

K. B. Oldham, J. Spanier, Fractional Calculus (Academic, New York, 1974).

Anal. Chem.

K. B. Oldham, Anal. Chem. 45, p. 39 (1973).
[CrossRef]

J. Ec. Polytech.

J. Liouville, J. Ec. Polytech. 13, Sec. 21, p. 1 (1832).

J. Electroanal. Chem. Interfacial Electrochem.

M. Ichise, Y. Nagayanagi, T. Kojima, J. Electroanal. Chem. Interfacial Electrochem. 33, p. 253 (1971).
[CrossRef]

Opt. Acta

R. Sirohi, V. Ram Mohan, Opt. Acta 22, p. 207 (1975).
[CrossRef]

Opt. Appl.

H. Kasprzak, Opt. Appl. 10, p. 283 (1980).

Proc. IEEE

J. Burch, Proc. IEEE 55, p. 599 (1967).
[CrossRef]

Trans. Am. Math. Soc.

E. L. Post, Trans. Am. Math. Soc. 32, p. 723 (1930).
[CrossRef]

Other

K. B. Oldham, J. Spanier, Fractional Calculus (Academic, New York, 1974).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

N. A. Abel, “Solution de quelques problemes a l’aide d’integrales definies,” Mag. Naturvidenkaberne, 1823, Norway.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

H. Kasprzak, “Realizowalnosc optyczna operacji rozniczkowania rzedu rzeczywistego,” “An Optical Implementation of the Real Order Differentiation,” Ph.D. Dissertation, Technical University of Wroclaw, Report 105/1980 (1980), in Polish.

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Figures (8)

Fig. 1
Fig. 1

Arbitrary order derivative of the triangle function.

Fig. 2
Fig. 2

Cross section of the 1-D synthetic filter amplitude transmittance for the order of differentiation equal to ½.

Fig. 3
Fig. 3

Graph of the 2-D function (22).

Fig. 4
Fig. 4

Model of the 1-D ½-order differentiation binary filter.

Fig. 5
Fig. 5

Model of the 2-D binary filter to perform the operation (∂r1+r2)/(∂xr1yr2).

Fig. 6
Fig. 6

Result of the ½-order differentiation produced by the filter shown in Fig. 5.

Fig. 7
Fig. 7

Curves depicting the theoretical and experimental results.

Fig. 8
Fig. 8

Images differentiated by use of the filter shown in Fig. 6.

Equations (25)

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F ( u ) = - + f ( x ) · exp ( - i 2 π u x ) d x ,
f ( x ) = - + F ( u ) · exp ( i 2 π u x ) d u ,
f ( x ) = - + [ - + f ( t ) · exp ( - 2 π i u t ) d t ] exp ( i 2 π u x ) d u
f r ( x ) = - + ( 2 π u i ) r [ - + f ( x ) exp ( - 2 π i u x ) d x ] × exp ( i 2 π u x ) d u .
f ( x ) = e ( x ) + o ( x ) = 1 2 [ f ( x ) + f ( - x ) ] + 1 2 [ f ( x ) - f ( - x ) ] .
F ( u ) = E ( u ) - i O ( u ) .
f r ( x ) = 2 ( 2 π ) r { cos π r 2 0 u r [ E ( u ) cos 2 π u x - O ( u ) sin 2 π u x ] d u - sin π r 2 0 u r [ E ( u ) sin 2 π u x + O ( u ) cos 2 π u x ] d u } .
f r ( x ) = ( 2 π ) r 2 { cos π r 2 [ 0 u r - 2 sin π u ( 1 + 2 x ) · sin π u d u + 0 u r - 2 sin π u ( 1 - 2 x ) · sin π u d u ] + sin π r 2 [ 0 u r - 2 cos π u ( 1 + 2 x ) · sin π u d u - 0 u r - 2 cos π u ( 1 - 2 x ) · sin π u d u ] } .
1 = C ( r ) ( 1 + x 1 - r - x 1 - r ) , 2 = C ( r ) ( 1 - x 1 - r - x 1 - r ) , 3 = C ( r ) [ x 1 - r sgn ( x ) - 1 + x 1 - r ] , 4 = C ( r ) [ x 1 - r sgn ( - x ) - 1 - x 1 - r ] ,
C ( r ) = 2 1 - r π 2 - r 4 cos π r 2 Γ ( 2 - r ) .
f r ( x ) = { 0 for x - 1 , G ( r ) ( 1 + x ) 1 - r for - 1 < x 0 , G ( r ) [ ( 1 + x ) 1 - r - 2 x 1 - r ] for 0 < x 1 , G ( r ) [ ( 1 + x ) 1 - r - 2 x 1 - r + ( x - 1 ) 1 - r ] for x > 1 ,
G ( r ) = Γ ( r ) sin π r 1 - r .
lim r 1 G ( r ) = lim r 0 G ( r ) = 1.
x = 1 2 1 1 - r - 1 .
F ( cos ( p x ) ) = 1 2 [ δ ( u - P 2 π ) + δ ( u + p 2 π ) ] .
cos r ( p x ) = p r cos ( p x + r π 2 ) ;
sin r ( p x ) = p r sin ( p x + r π 2 ) .
I ( u ) = A · exp ( - i 2 π α u ) + ( 2 π u i ) r 2 ,
t ( u ) = C [ 1 + ( u u max ) r cos ( 2 π α u + r π 2 ) ] ,
η ( u ) = l ( u ) d [ sin π · l ( u ) d π · l ( u ) d ] 2 ,
l ( u ) d = 1 π arcsin η η max .
l ( u ) d = 1 π arcsin ( u u max ) r .
r 1 + r 2 x r 1 y r 2 f ( x , y ) < = > ( 2 π u i ) r 1 · ( 2 π u i ) r 2 · F ( u , v ) ,
l ( u , v ) d = 1 π arcsin u r 1 v r 2 u max r 1 v max r 2 .
rect 0.5 ( x ) 2 = 0 for x < - ½ , { [ 2 r Γ ( r ) sin π r π 1 ( 1 + 2 x ) r ] 2 { 2 r Γ ( r ) sin π r π [ 1 ( 1 + 2 x ) r - 1 ( 2 x - 1 ) r ] } 2 { for - ½ x < ½ , for x ½ ,

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