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9. Hankel Transform, Fourier Bessel Transform, and Radon Transform

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9. Hankel Transform, Fourier Bessel Transform, and Radon Transform 9. Hankel Transform, Fourier Bessel Transform, and Radon Transform J. Scott Tyo OPTI 512R College of Optical Sciences University of Arizona Last Updated August 8, 2011 1. HANKEL TRANSFORM Many of the two-dimensional functions that we would be interested in in our optics class will have azimuthal symmetry. Examples are circular lenses and apertures. For the most part, we will consider only functions that do not vary with azimuth at all. However, the analysis tools that we develop here are generally applicable to other types of symmetry. If we consider a function that is azimuthally symmetric, we can write g(x, y)=g(r)=g x2 + y2 . (1) To compute the Fourier transform, we start with the typical method ∞ F (ξ,η)= g(x, y)e−j2π(ξx+ηy)dxdy (2) −∞ ∞ = g x2 + y2 e−j2π(ξx+ηy)dxdy. (3) −∞ We now make the change of variables x = r cos θ, y = r sin θ. We also define a cylindrical coordinate system for the frequency variables as ξ = ρ cos φ and η = ρ sin φ. Substituting these into Eq. 2 yields F (ρ, φ)= g(r)e−j2πρr(cosθ cos φ+sin θ sin φ)rdrdθ (4) ∞ 2π = rg(r) e−j2πρr cos(θ−φ)dθdr. (5) r=0 θ=0 One of the definitions of the Bessel function is given as 2π 1 −jx cos(θ−φ) J0(x)= e dθ. (6) 2π 0 We recognize this as the interior integral in Eq. 5, and we can now write ∞ G(ξ,η)=2π g(r)J0(2πρr)rdr = G(ρ). (7) 0 We see then that the Fourier transform of a circularly symmetric function is itself azimuthally symmetric. While this is simply a special case of the Fourier transform, it has a special name knows as the zero-order Hankel transform. Equation 7 is the forward Hankel transform, and the inverse transform is ∞ g(r)=2π G(ρ)J0(2πρr)ρdρ. (8) 0 It is important to note that the zero-order Hankel transform is simply a special way of evaluating the two-D Fourier transform for functions that are azimuthally symmetric. Therefore, all of the properties of the two-D Fourier transform also apply to the zero-order Hankel transform. 1 The three most important of azimuthally symmetric functions we will encounter in this course are the cyl function, the Gaus function, and the somb function. Actually performing these integrals is beyond the scope of this course. However, knowledge of the transforms is important. First, we can relate the cyl function to the somb function through π cyl(r) ↔ somb (ρ) . (9) f If we think about this, it should make sense. We know already that the rect function transforms to the sinc function. Since the cyl function is like a rect function in cylindrical coordinates, we should expect its Fourier transform to look like a sinc function in cylindrical coordinates, which describes the sombrero function. Likewise we can write the zero-order Hankel transform of the Gaus function as Gaus (r) ↔ Gaus (ρ) . (10) This follows from our knowledge of the Fourier transform of the Gaus function in rectangular coordinates. 2. FOURIER-BESSEL TRANSFORM The derivation in the previous section was specific to functions that were azimuthally symmetric in the (x, y)- plane. However, many classes of functions do not have this symmetry. Some of these functions can be written in terms of an azimuthal Fourier series as ∞ jmθ f(r, θ)= fm(r)e . (11) m=−∞ The orthogonality of the exponential functions allows us to compute fm(r) by multiplying both sides of Eq. 11 by e−inθ and integrating from 0 to 2π as 2π 2π ∞ 1 −jnθ 1 j(m−n)θ f(r, θ)e dθ = fm(r)e dθ (12) 2π θ=0 2π θ=0 m=−∞ ∞ 2π 1 i(m−n)θ = fm(r) e dθ (13) 2π m=−∞ θ=0 ∞ 1 = fm(r)2πδnm (14) 2π m=−∞ = fm(r). (15) With this representation we can once again compute the 2-D Fourier transform ∞ 2π ∞ jmθ −j2πρr(cos θ cos φ+sin θ sin φ) F (ρ, φ)= fm(r)e e rdrdθ (16) r=0 θ=0 m=−∞ ∞ ∞ 2π jmθ −j2πρrcos(θ−φ) = fm(r)e e rdrdθ (17) m=−∞ r=0 θ=0 ∞ ∞ 2π jmθ −j2πρrcos(θ−φ) = fm(r)r e e dθdr. (18) m=−∞ r=0 θ=0 To evaluate this we need to make the variable transformation α = θ − φ − π (19) dα = dθ, (20) 2 which allows us to write Eq. 18 as ∞ ∞ 2π jmφ jmπ imα j2πρr cos α F (ρ, φ)= e e fm(r)r e e dαdr. (21) m=−∞ r=0 0 We can now use the identity 2π 1 jx cos θ jnθ Jn(x)= n e e dθ (22) 2πj 0 to write ∞ ∞ jmφ m F (ρ, φ)=2π e (−j) fm(r)Jm(2πρr)rdr (23) m=−∞ r=0 ∞ jm(φ−π/2) = e Fm(ρ), (24) m=−∞ where ∞ Fm(ρ)=2π fm(r)Jm(2πρr)rdr. (25) r=0 The orthogonality of the Bessel functions gives us ∞ fm(r)=2π Fm(ρ)Jm(2πρrρdρ, (26) ρ=0 and we see that the Fourier-Bessel transform is in general self reciprocal. We see from Eq. 24 that the Hankel transform is a single term in the more general Fourier-Bessel transform. It should be noted that the Fourier-Bessel series is in general not complete, since the expansion is only in terms of Bessel functions of the first kind. However, the Bessel functions of the second kind diverge as r → 0, meaning that the functions that are left out of the set are undefined at the origin. 3. RADON TRANSFORM Let’s consider a very special case of a two-dimensional convolution. Consider the convolution of a two-D function with a one-D delta function f(x, y) ∗ δ(x)= f(α, β)δ(x − α)dαdβ (27) = f(x, β)dβ = p(x). (28) The notation p(x) is chosen to represent a one-dimensional projection of a two-dimensional function. The concept is illustrated in Fig. 1. Let’s consider now the Fourier transform of this projection F{p(x)} = P (ξ)δ(η). (29) However, we can use the convolution theorem to write that F{p(x)} = F{f(x, y) ∗ δ(x)} (30) = F (ξ,η)δ(η) (31) = F (ξ,0)δ(η). (32) 3 Figure 1. Convolution of the two-D function with a one-D delta function produces a projection function. Equation 32 is extremely important. It is known as the Radon transform, and tells us that the Fourier transform of a one-dimensional projection of a two-dimensional function gives us a one-dimensional slice of the two- dimensional Fourier transform. For this reason, the Radon transform is sometimes known as the projection-slice theorem. Now, if we rotate the object by some amount and take a second projection, we end up with a second slice of the Fourier transform at a different angle. In this way we can build up the two-D Fourier transform slice-by-slice by taking various projections of the object at different angles as shown in Fig. 2. This is the basic underlying principle of computed tomography (although the reconstruction these days is usually performed using a different inversion technique). 4 Figure 2. The basic principle of Computed Tomography relies on the Radon transform. Figure 4.9 from Barrett & Myers. 5.
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