Compensation. June 29, Arizona State University. Edge Detection from Nonuniform Data via. Convolutional Gridding with Density.

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1 Arizona State University June 29, 2012

2 Overview gridding with density compensation is a common method for function reconstruction nonuniform Fourier data. It is computationally inexpensive compared with methods that use an interpolation to get ˆf at integer points. Our goal is to use a convolutional gridding method for edge detection.

3 Summary of Gridding Method for Function Reconstruction Let ˆf denote the Fourier transform of f, f compactly supported and defined on [ π, π]. We are given ˆf (ω k ), at nonuniformly distributed ω k. φ(x) = e cxm is the window function for our convolution. Let g(x) = f (x)φ(x). Then ĝ(l) = (ˆf ˆφ)(l) = ˆf (τ) ˆφ(l τ) dτ

4 Gridding Summary Continued ĝ(l) α kˆf (ω k ) ˆφ(l ω k ) k K α k are quadrature weights based on the density of ω k. The sum is now truncated to reduce computational cost: ĝ(l) α kˆf (ωk ) ˆφ(l ω k ) k s.t. l ω k <q Now g(x) can be recovered using an FFT and this approximation φ is divided out to yield an approximation for f (x).

5 f (x) again compactly supported, defined on [ π, π], with a jump discontinuity at ξ ( π, π) Denote by [f ](x) the jump function of f (x). [f ](x) [f ](ξ)i ξ (x) [f ](x) = lim f (x) lim f (x) x ξ + x ξ I ξ (x) is an approximation of the indicator function. We want a way to express [f ](ω k ). [f ](ω k ) [f ](ξ)i ξ (ω k ) = [f ](ξ)î ξ (ω k )

6 [f ](ξ) without the ξ ˆf (n) = = π π ξ π = f (x)e inx in f (x)e inx dx π f (x)e inx dx + f (ξ )e inξ in [f ](ξ)e inξ in ξ π + f (x)e inx in f (ξ+ )e inξ in So we have [f ](ξ) iω kˆf (ω k )e iω kξ. f (x)e inx dx ξ + π π f (x)e inx in ξ + + O( 1 n 2 ) π dx

7 Back to [f ](ω k ) Recall [f ](ω k ) [f ](ξ)î ξ (ω k ). Substitute what we got for [f ](ξ). [f ](ω k ) iω kˆf (ωk )e iω kξ Î ξ (ω k ) We use a Gaussian indicator function approximation: I ξ (x) = e x ξ 2ε 2 Î ξ (ω k ) εe iω kξ e 1 2 ε2 ω 2 k π 2 Plugging this in yields [f ](ω k ) iω kˆf (ω k )εe 1 2 ε2 ω 2 k π 2.

8 Window Function φ(x) φ(x) should have several properties. 1 ˆφ(ω) should minimize computational cost of the convolution step. 2 φ(x) should be nonzero in the reconstruction interval. 3 φ(x) should result in minimal aliasing by being approximately zero outside of the reconstruction interval. 4 In order to sufficiently resolve jumps in all of the domain, φ(x) should be approximately 1 in ( π, π).

9 Window Function φ(x) φ(x) x 2λ cx Figure: Several suitable window functions. φ(x) = e

10 ˆφ(ω) ˆφ(ω) ω Figure: Fourier transform of φ(x) = e 1x10 12 x 26

11 Three Jumps of Equal Magnitude, No Noise y x Figure: logarithmically-spaced vs. jittered ω k, N = 256

12 Convergence with Logarithmically-spaced ω k and Noise y Figure: Jump reconstructions with complex Gaussian noise with variance.004 x

13 [Non-]Convergence with Jittered ω k and Noise y Figure: Random jitter of up to.2 each integer point in the spectral data. x

14 Tradeoff Between Jump Resolution and Convergence in Flat Regions y x Figure: Logarithmic sampling; N = 256; indicator I with variance ε 2

15 Apply optimization techniques to our method. Use different weights α k. Experiment with additional indicator function approximations I ξ.

16 Acknowledgments Dr. Anne Gelb Dr. Guohui Song Dr. Eric Kostelich CSUMS National Science Foundation