Jan 22, 1988 - Functions, Princeton University Press, Princeton, 1971. 12. R. Bracewell, The Fourier Transform and Its Applications, Mc-. Graw-Hill, New York, ...
If Eq. (1) is generalized to a medium where the relative permittivity cr and relative permeability pr are functions of position, the wave equation (now written specifically for the magnetic field 77) can be converted into the conventional “weak” equation [8, 91
=
/p ( v ix
x 77)
where I? is the domain of interest and is the spherical surface bounding that region. This surface is chosen so that all inhomogeneities are inside ar. T ( x , y , z ) is a vector testing function, typically chosen in the context of the finite element method to be identical to the expansion functions used to represent the H field. The absorbing boundary condition may be incorporated through the surface integral in Eq. (26). Note that the radiation condition only applies to the “scattered” or outward radiating H field, and depending on the specific formulation the fields may require separation into “incident” and “scattered” parts. Using Eq. (22) and appropriate vector identities, the part of the surface integral involving the scattered field can be written as
5 . B. Enquist and A. Majda, “Absorbing Boundary Conditions for the Numerical Simulation of Waves,” Math. Comp., Vol. 31,1977, pp. 629-651. 6. A. Bayliss and E. Turkel, ‘‘Rachation Boundary Conditions for Wave-like Equations,” Comm. Pure Appl. Math., Vol. 33, 1980, pp. 707-725. 7. C. H. Wilcox, “An Expansion Theorem for ElectromagneticFields,” Comm. Pure Appl. Math., Vol. 9, 1956, pp. 115-134. 8. P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, Cambridge University Press, 1983. 9. J. N. Reddy, An Introduction to the Finite Element Method, MCGraw-Hill, New York, 1984. Received 1-22-88 Microwave and Optical Technology Letters, 1/2, 62-64 0 1988 John Wiley & Sons, Inc. CCC 0895-2477/88/$4.00
CONTINUOUS ORDER DERIVATIVE FOR OPTICAL SIGNALS AND IMAGES H. Rabal, L. Zerbino, and J. Ojeda-Castafieda* clop Casilla de Correos 124 1900 La Plata, Argentina
+ 7 ( r ) T . v tan ( P . F ) ]
(27)
Since the incorporation of Eq. (22) only involves first-order tangential derivatives of the H-field components, this condition appears especially appropriate for finite-element implementation using first-order linear interpolation functions, as discussed by Silvester and Ferrari [8]. Details of this type of implementation are available in the context of the scalar wave equation using the second-order Bayliss-Turkel boundary condition [2, 61. The second-order Bayliss-Turkel operator is somewhat analogous to Eq. (22) in that it only requires first-order tangential derivatives of the unknown function when implemented within a similar weak equation. It is worth noting that the second-order Bayliss-Turkel condition improves on the scalar Sommerfeld radiation condition by three powers of the variable r. Equation (22) is similar in that it improves on Eq. (4) by three powers of r. REFERENCES 1. A. F. Peterson, “A Comparison of the Volume MFIE and Unimoment Methods for TE Scattering from Lossy, Inhomogeneous Dielectric Cylinders,” Digest of the 1987 IEEE Antennas and Propagation Society International Symposium, Blacksburg, VA, June 1987, pp. 1118-1121. 2. A. F. Peterson, “A Comparison of Integral, Differential and Hybrid Methods for TE-Wave Scattering from inhomogeneous Dielectric Cylinders,” submitted to J . Electromagnetic Waves Appl. 3. G. Meltz, B. J. McCartin, and L. J. Bahrmasel, “Application of the Control Region Approximation to Electromagnetic Scattering,” Abstracts of the 1987 URSI Radio Science Meeting, Blacksburg, VA, June 1987, p. 185. 4. R. T. Ling, “Numerical Solution for the Scattering of Sound Waves by a Circular Cylinder,” AIAA Journal, Vol. 25, April 1987, pp. 560-566.
64
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS
KEY TERMS Signal processing, optical processing, pattern recognition, Fourier optics, systems and applications ABSTRACT For optically implementing new mathematical operations and for processing signals and images, we define a continuous order derivative operator in Fourier optics. We examine some examples and discuss some potential applications for this kind of operator. We describe an experimental setup for implementing opticafly the continuous order derivative of one-dimensional signals. 1. INTRODUCTION
In Fourier optics the concept of a derivative operation plays an important role for describing the propagation in free space [l-31 and the influence of wave aberrations [4], for visualizing phase structures [5, 61, and in general for edge enhancement [7], image deblurring [8], and pattern recognition [9]. In many of the preceding applications, the order of the derivative is an integer number, but it can be also a rational number [lo]. In principle, the order of the derivative operation can be a real number [ll]. That is, it is possible to define a continuous order derivative operation. The aim of this letter is to suggest the use of a continuous order derivative operation in Fourier optics. We discuss some examples. We also describe an optical method for implementing a one-dimensional continuous order derivative operation. In Section 2, we discuss the basic theory. In Section 3, we consider some examples. Computer generated pictures are employed for visualizing two examples. Finally, in Section 4, we propose an optical setup for performing continuous order derivatives of one-dimensional signals. *Permanent address: INAOE, Apdo. Postal 216, Puebla, 72000 Pue., Mkxico.
/ Vol. 1, No. 2, April 1988
2. BASIC THEORY
Let us represent a physically possible one-dimensional optical signal by the complex function f(x). Thus, f ( x ) can be expressed as f ( x) =
lm f( v)exp( i27rxv) dv -m
Now we define the following two-dimensional complex function F( x ; y )
=
/
m
( i 2 7 r ~ ) ~ fv)exp( l( i27rxv) dv
(2)
-m
as the continuous order, or y-fold order, derivative of f ( x ) . Note that for y = 0, we recover from Eq. (2) the initial optical signal; that is F ( x , Y = 0) =f(x)
For y = n with n rivative of f (x):
=
(3)
1 , 2 , 3 , . . . we obtain the nth order de-
F( x , y
= H) =
D ” f ( x)
(4)
We point out that for y < 0, the operation in Eq. (2) is the opposite operation of that for y > 0. Thus, in a lax manner,
Figure 2 lmage and x-axis derivatives of Halley’s comet: (a) Input, gray level picture; (b) the first-order derivative; (c) the ?i-order derivative
we speak of the y-fold order integral of f ( x ) when y < 0. Consequently, in an operator notation, the definition in Eq. (2) can be denoted equivalently as F( x , y )
=
d.’ D-”f( x ) where D-” = dxY
(5)
which is equally meaningful either for y > 0 or for y < 0. Of course. we have that Figure 1 Continuous order derivative of the signals (a) cosine function; (b) sine function
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS
/ Vol 1 , No
2, April 1988
65
'%
f I
f =2f,
7
Figure 3 Schematic diagram of the optical setup for implementing a one-dimensional continuous order derivative operation
2(b) shows its first order derivative along the x axis and Figure 2(c) its n - 3.1416-fold derivative, along the x axis. Note that in this later case, the boundaries between levels are smoother than in Figure 2(b). This can be applied for edge smoothness or phase visualization.
Let us discuss some examples. 3. EXAMPLES
If the Fourier spectrum of the optical signal is such that f(v) =o
for
Y
