Tutorial. The two- dimensional Fourier transform. θ =. For example, if f( x, y) = cos[2 π( xcosθ + ysin θ)]

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1 Tutorial Many of the linear transforms in common use have a direct connection with either the Fourier or the Laplace transform [- 7]. The closest relationship is with the generalizations of the Fourier transform [8] to two or more dimensions, and with the Hankel transforms [9] of the zero and higher orders, into which the multidimensional Fourier transforms degenerate under circumstances of symmetry. The two- dimensional Fourier transform The variable x may stand for some physical quantity such as time or frequency, which is essentially one- dimensional, or it may be the coordinate in a one- dimensional physical system such as a stretched string or an electrical transmission line. However, in cases which are two dimensional- stretched membranes, antennas and arrays of antennas, lenses and diffraction gratings, pictures on television screens, and so on, more general formulas apply. A two- dimensional function f( x, y ) has a two- dimensional transform Fuv (, ), and between the two the following relations exist: i π ( ux+ vy) F( u, v) = f ( x, y) e dxdy i π ( ux+ vy) f ( x, y) = F( u, v) e dudv These equations describe an analysis of the two- dimensional function f( x, y ) into components of the form exp[ i π ( ux+ vy)]. Since any such component can be split into cosine and sine parts, we may begin by considering a cosine componentcos[ π ( ux + vy)]. The magnitude of Fu (, v ) tells us how much f( x, y ) looks like a cosine function heading in the direction arctan( v u ) θ =. For example, if f( x, y) = cos[ π( xcosθ + ysin θ)], then Fuv (, ) is a pair of impulses located at an angle θ with respect to the u - axis (see Figure ).

2 Figure - Dimensional Fourier transform pair As a second example of a two- dimensional function, consider the height of the ground at the geographical point ( xy, ), over the area occupied by the mountain which is conventionally represented in Figure by contours of constant height. The function cos[ π ( ux + vy )] represents a consinusoidally corrugated land surface whose contours of constant height coincide with lines whose equation is ux + vy = const The corrugations face in a direction that makes an angle arctan( v u) with the x axis and their wavelength is ( ) u + v. If a section is made through the corrugations, in the x direction, it will undulate with a frequency of u cycles per unit of x. Similarly, v may be interpreted as the number of cycles per unit of u, in the y direction. In Figure, a prominent Fourier component of the mountain is shown. In the transform domain the complex component is characterized in wavelength and orientation by the point ( uv, ) in the u v plane and its amplitude by Fuv (, ). The interpretation of u and v as spatial frequencies is emphasized by dimensioning u and v, the wavelengths of sections taken in the x and y direction, respectively (see Figure ). The second of the Fourier relations quoted above asserts that a summation of corrugations of appropriate wavelengths and orientations, taken with suitable amplitudes, can reproduce the original mountain. The sinusoidal components, which must also be included, allow for the possibility that the corrugations may have to be slid into appropriate spatial phases.

3 y y v - u - x x Figure A mountain(left)and a prominent Fourier component thereof (right) Schwarz s Inequality Another way to think about the meaning of Fu (, v ) as a large or small number compared to a sinusoid, is to consider Schwarz s Inequality. Let ψ ( x) and ( x) integrable function in [ ab,, ] then Schwarz s Inequality is given by When equality iff* ψ x x dx x dx x dx b ψ( ) ψ( ) b [ ψ( )] b [ ψ( )] a a a ψ ( x) αψ ( x) iπ ( ux+ vy) =e, then (, ) ψ be any two real = with α a constant. Thus, if f ( x y) F u v reaches a maximum when (, ) sinusoid with wavelengths and angle matching (, ) the Cauchy- Schwarz inequality or Buniakowsky inequality. ψ = and, f x y approaches a u v. Schwarz s Inequality is also called l iff* : If and only if (i.e., necessary and sufficient). The terms "just if" or "exactly when" are sometimes used instead.

4 Theorems for the two- dimensional Fourier transform [] Theorem f( x, y ) Fuv (, ) Similarity f ( ax, by ) u v F, ab a b Addition f( x, y) + g( x, y) Fuv (, ) + Guv (, ) Shift f( x a, y b) πiau ( + bv) e F(,) u v Modulation f( x, y)cosω x ω ω F u+, v + F u, v π π Convolution f( x, y) g( x, y) FuvGuv (, ) (, ) Autocorrelatio n f(, x y) f ( x, y) F( u, v ) Rayleigh Power Parseval Differentiation f( x, y) dxdy = F( u, v) dudv f( x, y) g ( x, y) dxdy= F( u, v) G ( u, v) dudv + + Where m x y x y f( x, y) f( x, y) = f( x, y) a mn, Fuv a u mv n (,) = mn[ δ(, )] n f( x, y) ( ) m n πiu ( π iv) F( u, v) π iuf( u, v) π ivf( u, v) f( x, y) x f( x, y) y xy f( x, y) 4 π ufuv (, ) 4 π vfuv (, ) 4 π uvf( u, v)

5 Definite integral + f( x, y) x y f( x, y) dxdy= F(,) First moments xf ( x, y) dxdy F ' u (,) = πi ( ) ( xcosθ + ysin θ) f x, y dxdy = u + π i + 4 π ( u v ) F( u, v) [ cos θ F' (,) sin F' (,)] Center of F '(,) u F ' v(,) gravity x = y = πif(,) πif(, ) Second F '' uu (,) x f( x, y) dxdy moments = 4π F '' uv(,) xyf ( x, y) dxdy = 4π ( x y ) f( x, y) dxdy F'' (,) '' (,) uu F + = + vv 4π Equivalent width f ( x, y) dxdy F(,) = f (,) F( u, v) dudv Finite differences (, ) v [ ] Δ f x y isin π uf( u, v) x f xy ( x, y ) Δ 4sin πu sin πvf ( u, v ) (, ) Δ xx f x y 4(sin πu) F( u, v) Running means Separable product x y Π Π f a b ( x, y) f( x) g( y ) FuGv ( ) ( ) absin causin cbvf ( u, v ) The finite differences in the table are defined as follows: Δ f ( x, y ) = f ( x+ x, ) (, ) y f x y Δ f ( x, y ) = f ( x+ xy, ) (, ) (, ) (, ) y+ f x y+ f x+ y + f x y ( ) Δ = + + f x, y f ( x, y xx ) f ( x, y ) f ( x, y )

6 The following table illustrates a number of two- dimensional Fourier transforms. f ( x, y ) F( u, v ) cosπ x δ u, v + δ u, v cos πxcos π y πδ ( u π ) δ ( v π ) + πδ ( u+ π ) δ ( v π ) + πδ π δ + π + πδ + π δ + π ( u ) ( v ) ( u ) ( v ) 3 ( ) δ ( x) III( v ) III y

7 4 δ ( xy, ) 5 sinc sinc x y ( uv, ) 6 x + y a ( π + ) aj a u v u + v

8 7 sinc xsinc y Λ( u) Λ ( v) 8 π ( + ) exp π ( u + v ) exp x y 9 sinc xsinc y Λ( u) ( v) x y exp[ π + ] A a ( ) Aaexp[ π A u + a v ]

9 δ ( y) δ ( u) cosπyδ ( x) π π δ π ( v) + δ( π + v) The Hankel Transform of Zero Order [] Two- dimensional systems may often show circular symmetry; for example, optical systems are often constructed from components that, in themselves, are circularly symmetrical. Then again, waves spreading out in two dimensions from a source of energy exhibit symmetry for natural reasons. It may be expected that in these cases a simplification will result, for one radial variable

10 will suffice in place of the two independent variables x and y. The appropriate expression of such problems is in terms of the Hankel transform, a one- dimensional transform with Bessel function kernel. When circular symmetry exists, that is, when f( x, y) = f( r), Where r = x + y, Then Fuv (, ) proves also to be circularly symmetrical; that is. Fuv (, ) = Fq ( ), Where q = u + v, To show this, change the transform formula to polar coordinates and integrate over the angular variable. Then the relations between the two one- dimensional functions f() r and Fq ( ) are ( π ) F( q) = π f ( r) J qr rdr ( π ) f () r = π F() q J qr qdq We refer to Fq ( ) as the Hankel transform(of zero order) of f() r and note that the transformation is strictly reciprocal, as was the case when the kernels were cos and sin. The kernel J, together with cos, sin, and others, is referred to as a Fourier kernel in the broad sense of a kernel associated with a reciprocal transform. The factors π in the above formulas may be canceled by suitable redefinition of the variables, but their retention follows logically from the form adopted for Fourier transforms. In physical situations the π in parentheses will be found to result from the measurement of q in whole cycles per unit of r. The π before the integral sign comes from the element of area π rdr. Theorems for the Hankel transform Theorem f() r Fq ( ) Similarity f( ar ) q a F a Addition f() r + g() r Fq ( ) + Gq ( ) Shift Shift of origin destroys circular symmetry Convolution π FqGq ( ) ( ) f( r') g( R) r' dr' dθ ( R = r + r' rr'cos θ) Rayleigh f() r rdr = F() q qdq

11 Power Differentiation f() r g*() r rdr = F() q G*() q qdq r r m f() r f() r r f() r m ( π iq) F( q) π iqf( q) 4 π qfq ( ) Definite integral π f ( r) rdr = F() Second moment F ''() π r f( r) rdr = π Equivalent width π f ( r) rdr F() = f () π F( q) qdq References. Erdélyi, A. and H. Bateman, Tables of integral transforms. vol.. 954, McGraw- Hill: New York Bracewell, R.N., The Fourier transform and its applications. 3rd ed., Boston: WCB/McGraw- Hill. 3. Andrews, L.C., B.K. Shivamoggi, and Society of Photo- optical Instrumentation Engineers., Integral transforms for engineers. 999, Bellingham, Wash.: SPIE Optical Engineering Press. x, 353 p. 4. Andrews, L.C. and R.L. Phillips, Mathematical techniques for engineers and scientists. 3, Bellingham: SPIE- The International Society for Optical Engineering. XV, 797 p. 5. Poularikas, A.D., The handbook of formulas and tables for signal processing. The electrical engineering handbook series. 999, Boca Raton, Fla. u.a.: CRC Press u.a. Getr. Zählung [ca. 8 S.]. 6. Parker, K.J., Lecture Notes, Medical Imaging: Theory and Implementation. 3, University of Rochester. 7. Bracewell, R.N., Two- dimensional imaging. Prentice- Hall signal processing series. 995, Englewood Cliffs, N.J.: Prentice Hall. xiv, 689 p. 8. Wikipedia. Fourier transform. 9 August 3 : UTC Available from: 9. Wikipedia. Hankel transform 9 August 3 : UTC Available from: Waag, R.C., J.A. Campbell, J. Ridder, and P.R. Mesdag, Cross- sectional Measurements and Exploitations of ultrasonic fields. 985, IEEE Transactions on Sonics and Ultrasonics 3()6-35.

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