Circularly Symmetric Functions: Hankel Transforms of Zeroth Order, or, Fourier-Bessel Transforms

1. Circular symmetry: (no dependence on angle) Circularly Symmetric Functions: Hankel Transforms of Zeroth Order, or, Fourier-Bessel Transforms What if our function is expressed in polar coordinates? …and is separable in those coordinates? Why is this of interest? Useful for modeling circular sources, lesions, lenses, etc.

2. Transforming coordinates, u y  r  v  x rd rd d dr

3. Hankel Transform, continued transforming variables... where (Using Trig Identity) So, But, this can be simplified by using

4. Bessel functions Jn(x) is a Bessel function of the first kind of order n. Useful identities, and where J1(x) is a Bessel function of the first kind of order 1.

5. (from previous slide) Hankel Transform, continued (2) But, Subbing in Jo(2r) yields which is not a function of . Thus, the function is circularly symmetric in both domains.

6. The Inverse Hankel Transform Circularly symmetric in space Circularly symmetric in spatial frequency Notice no difference in sign between forward and inverse transforms

7. Example: Hankel transform of a circle Consider the Fourier Transform of a circle: assume (circular symmetry) y Consider g r(r) = circ(r) r = 1 x Let Then, and

8. Subbing in r’=2r yields Hankel Transform of Circle, continued Note: So

9. Jinc Function We define the jinc function as Similar to sinc function, only sinc has zeros are at equal intervals, jinc zeros vary

10. Hankel Transform of Circ circ(r) F.T { fft(circ(r))} = jinc(r) Log10(abs(fft(circ(r)))

11. Hankel Transform Scaling Property The only difference between the Fourier and Hankel transform scaling property is the scalar 1/a2 The scalar takes into account that the function is expanding or contracting in 2 dimensions.