Fourier Reconstruction from Non-Uniform Spectral Data

Transcription

1 School of Electrical, Computer and Energy Engineering, Arizona State University With Profs. Anne Gelb, Doug Cochran and Rosemary Renaut Research supported in part by National Science Foundation grants DMS 583 and DMS (FRG). Computational and Applied Mathematics Proseminar Fall 9 Oct 6 9

2 Introduction Non-uniform Data Current Methods Alternate Approaches Motivating Eample Application Outline of the Talk Motivating Eample Template function.45 Fourier Transform f().5 ˆf(ω) ω (a) Template Function Figure: A motivating eample (b) Fourier Transform Fourier samples violate the quadrature rule for discrete Fourier epansion Computational issue no FFT available Mathematical issue given these coefficients, can we/how do we reconstruct the function? Resolution what accuracy can we achieve given a finite (usually small) number of coefficients?

3 Introduction Non-uniform Data Current Methods Alternate Approaches Motivating Eample Application Outline of the Talk Motivating Eample Template function.45 Fourier Transform.5.4 Fourier transform Acquired samples f().5 ˆf (ωk) ω k (a) Template Function Figure: A motivating eample (b) Fourier Coefficients, N = 3 Fourier samples violate the quadrature rule for discrete Fourier epansion Computational issue no FFT available Mathematical issue given these coefficients, can we/how do we reconstruct the function? Resolution what accuracy can we achieve given a finite (usually small) number of coefficients?

4 Introduction Non-uniform Data Current Methods Alternate Approaches Motivating Eample Application Outline of the Talk Motivating Eample Template function.45 Fourier Transform.5.4 Fourier transform Acquired samples f().5 ˆf (ωk) ω k (a) Template Function Figure: A motivating eample (b) Fourier Coefficients, N = 3 Fourier samples violate the quadrature rule for discrete Fourier epansion Computational issue no FFT available Mathematical issue given these coefficients, can we/how do we reconstruct the function? Resolution what accuracy can we achieve given a finite (usually small) number of coefficients?

5 Introduction Non-uniform Data Current Methods Alternate Approaches Motivating Eample Application Outline of the Talk Application Magnetic Resonance Imaging Fourier Coefficients Spiral scan trajectory ky fˆ(ωk, ωky ) ωky ωk (a) Acquired Fourier Samples k (b) Sampling Trajectory Reconstructed phantom Non-Cartesian sampling trajectories have some advantages greater resistance to motion artifacts instrumentation concerns ease in generating gradient waveforms a (c) Reconstructed Image Figure: MR Imaginga Sampling pattern courtesy Dr. Jim Pipe, Barrow Neurological Institute, Phoeni, Arizona.5

6 Introduction Non-uniform Data Current Methods Alternate Approaches Motivating Eample Application Outline of the Talk In this Talk We will discuss Issues with non-harmonic Fourier reconstruction Conventional reconstruction methods Accuracy vs Sampling Density Spectral Re-projection methods Incorporating edge information in the reconstruction

7 Introduction Non-uniform Data Current Methods Alternate Approaches Problem Formulation The Non-harmonic Kernel Eamples Outline Introduction Motivating Eample Application Outline of the Talk The Non-uniform Data Problem Problem Formulation The Non-harmonic Kernel Reconstruction Results using the Non-harmonic Kernel 3 Current Methods Reconstruction Methods Error Characteristics 4 Alternate Approaches Spectral Re-projection Incorporating Edge Information

8 Introduction Non-uniform Data Current Methods Alternate Approaches Problem Formulation The Non-harmonic Kernel Eamples Problem Formulation Let f be defined on R and supported in ( π, π) It has a Fourier transform representation, ˆf(ω), defined as ˆf(ω) = π Z π π f()e iω d, ω R Objective Recover f given a finite number of its non-harmonic Fourier coefficients, ˆf(ω k ), k = N,..., N ω k not necessarily Z We will refer to {ω k } N N as the sampling pattern/trajectory We will be particularly interested in sampling patterns with variable sampling density We will pay special attention to piecewise-smooth f

9 Introduction Non-uniform Data Current Methods Alternate Approaches Problem Formulation The Non-harmonic Kernel Eamples k y Problem Formulation Let f be defined on R and supported in ( π, π) It has a Fourier transform representation, ˆf(ω), defined as ˆf(ω) = π Z π π f()e iω d, ω R Objective Recover f given a finite number of its non-harmonic Fourier coefficients, ˆf(ω k ), k = N,..., N ω k not necessarily Z We will refer to {ω k } N N as the sampling pattern/trajectory We will be particularly interested in sampling patterns with variable sampling density We will pay special attention to piecewise-smooth f Sampling scheme ω Spiral scan trajectory k

10 Introduction Non-uniform Data Current Methods Alternate Approaches Problem Formulation The Non-harmonic Kernel Eamples Problem Formulation Let f be defined on R and supported in ( π, π) It has a Fourier transform representation, ˆf(ω), defined as ˆf(ω) = π Z π π f()e iω d, ω R Objective Recover f given a finite number of its non-harmonic Fourier coefficients, ˆf(ω k ), k = N,..., N ω k not necessarily Z We will refer to {ω k } N N as the sampling pattern/trajectory We will be particularly interested in sampling patterns with variable sampling density We will pay special attention to piecewise-smooth f f() f()

11 Introduction Non-uniform Data Current Methods Alternate Approaches Problem Formulation The Non-harmonic Kernel Eamples Sampling Patterns Jittered Sampling: ω k = k ± τ k, τ k U[, θ], k = N, (N ),..., N Log Sampling: ω k is logarithmically distributed between v and N, with v > and N + being the total number of samples ω (e) Jittered Sampling ω (f) Log Sampling Figure: Non-uniform Sampling Schemes (right half plane), N = 6

12 Introduction Non-uniform Data Current Methods Alternate Approaches Problem Formulation The Non-harmonic Kernel Eamples The Non-harmonic Reconstruction Kernel Consider standard (harmonic) Fourier reconstruction. The Fourier partial sum S N f() = ˆf(k)e ik can be k N written as S N f() = (f D N )() where D N () = e ik is the Dirichlet kernel. k N Now consider non-harmonic Fourier reconstruction using the sum S N f() = ˆf(ω k )e iωk. We may again k N write S N f() = (f AN )() where A N () = is the non-harmonic kernel. k N The non-harmonic kernels do not constitute a basis for span {e ik, k N} e iω k D N () Dirichlet Figure: The Dirichlet Kernel plotted on the right half plane, N = 64

13 Introduction Non-uniform Data Current Methods Alternate Approaches Problem Formulation The Non-harmonic Kernel Eamples The Non-harmonic Reconstruction Kernel Consider standard (harmonic) Fourier reconstruction. The Fourier partial sum S N f() = ˆf(k)e ik can be k N written as S N f() = (f D N )() where D N () = e ik is the Dirichlet kernel. k N Now consider non-harmonic Fourier reconstruction using the sum S N f() = ˆf(ω k )e iωk. We may again k N write S N f() = (f AN )() where A N () = is the non-harmonic kernel. k N The non-harmonic kernels do not constitute a basis for span {e ik, k N} e iω k A N () A N () Non harmonic (a) Jittered Sampling Non harmonic (b) Log Sampling Figure: Non-harmonic Kernel, N = 64

14 Introduction Non-uniform Data Current Methods Alternate Approaches Problem Formulation The Non-harmonic Kernel Eamples The Non-harmonic Reconstruction Kernel Consider standard (harmonic) Fourier reconstruction. The Fourier partial sum S N f() = ˆf(k)e ik can be k N written as S N f() = (f D N )() where D N () = e ik is the Dirichlet kernel. k N Now consider non-harmonic Fourier reconstruction using the sum S N f() = ˆf(ω k )e iωk. We may again k N write S N f() = (f AN )() where A N () = is the non-harmonic kernel. k N The non-harmonic kernels do not constitute a basis for span {e ik, k N} e iω k (a) Jittered Sampling (b) Log Sampling Figure: Autocorrelation plot of the kernels

15 Introduction Non-uniform Data Current Methods Alternate Approaches Problem Formulation The Non-harmonic Kernel Eamples Non-harmonic Kernels Dirichlet Non harmonic Non harmonic A N () A N () A N () (a) Dirichlet, N = 3 (b) Jittered, N = 3 (c) Log, N = Dirichlet Non harmonic Non harmonic A N () A N () A N () (d) Dirichlet, N = 8 (e) Jittered, N = 8 (f) Log, N = Dirichlet 5 Non harmonic 5 Non harmonic A N () A N () A N () (g) Dirichlet, N = 56 (h) Jittered, N = 56 (i) Log, N = 56

16 Introduction Non-uniform Data Current Methods Alternate Approaches Problem Formulation The Non-harmonic Kernel Eamples Reconstruction Eamples Function Reconstruction 5 Function Reconstruction True.5 4 Non equispaced sum.5 3 f() f().5.5 True Gridding (a) Jittered Sampling (b) Log Sampling (c) Spiral Sampling Figure: Non-harmonic Fourier sum Reconstruction,N = 8

17 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Outline Introduction Motivating Eample Application Outline of the Talk The Non-uniform Data Problem Problem Formulation The Non-harmonic Kernel Reconstruction Results using the Non-harmonic Kernel 3 Current Methods Reconstruction Methods Error Characteristics 4 Alternate Approaches Spectral Re-projection Incorporating Edge Information

18 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics f^ Conventional Reconstruction Methods Several approaches available to perform reconstruction Convolutional gridding most popular Uniform resampling Iterative Methods Fi the quadrature rule while evaluating the non-harmonic sum Trapezoidal Rule Unequal subintervals S N f() = N k= N α k ˆf(ωk )e iω k α k are density compensation factors e.g., α k = ω k+ ω k Evaluated using a non-uniform FFT 3 f^(ω p+ ) f^(ω p ) p ω p ω p ω f^(ω q ) q ω q ω q+ f^(ω q+ ) Figure: Evaluating the non-uniform Fourier sum Although there are distinct difference in methodology and computational cost, reconstruction accuracy is similar in most schemes. We will look at convolutional gridding to obtain an intuitive understanding of the problems in reconstruction

19 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Conventional Reconstruction Methods Several approaches available to perform reconstruction Convolutional gridding most popular Uniform resampling Iterative Methods.45.4 Fourier Transform Fourier transform Acquired samples.45.4 Fourier Coefficients True Interpolated.5 Function Reconstruction ˆf (ωk).5. F(k).5. f() True URS 3 3 ω k (a) Non-harmonic modes k (b) Obtain uniform modes Figure: Uniform Resampling 3 3 (c) (Filtered) Fourier sum Although there are distinct difference in methodology and computational cost, reconstruction accuracy is similar in most schemes. We will look at convolutional gridding to obtain an intuitive understanding of the problems in reconstruction

20 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Conventional Reconstruction Methods Several approaches available to perform reconstruction Convolutional gridding most popular Uniform resampling Iterative Methods Function Reconstruction ( iterations) Function Reconstruction (4 iterations) Function Reconstruction (5 iterations) Function Reconstruction (45 iterations) f() f() f() f() True CG.5 True CG.5 True CG.5 True CG (a) after iteration (b) after iteration 4 (c) after iteration 5 Figure: Iterative Reconstruction (d) after iteration 45 Although there are distinct difference in methodology and computational cost, reconstruction accuracy is similar in most schemes. We will look at convolutional gridding to obtain an intuitive understanding of the problems in reconstruction

21 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Conventional Reconstruction Methods Several approaches available to perform reconstruction Convolutional gridding most popular Uniform resampling Iterative Methods Function Reconstruction ( iterations) Function Reconstruction (4 iterations) Function Reconstruction (5 iterations) Function Reconstruction (45 iterations) f() f() f() f() True CG.5 True CG.5 True CG.5 True CG (a) after iteration (b) after iteration 4 (c) after iteration 5 Figure: Iterative Reconstruction (d) after iteration 45 Although there are distinct difference in methodology and computational cost, reconstruction accuracy is similar in most schemes. We will look at convolutional gridding to obtain an intuitive understanding of the problems in reconstruction

22 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Convolutional Gridding Gridding To evaluate S N f() = N k= N α k ˆf(ωk )e iω k efficiently Map the non-uniform modes to a uniform grid. A convolution operation is typically used. Compute a Fourier or filtered Fourier partial sum. (f φ)(ω) ω=..5 φ^(k ω) f^(ω) 3 If required, compensate for the mapping operation The new coefficients on the uniform grid are therefore given by ˆf ˆφ α m ˆf(ωm) ω=k ˆφ(k ω m) m st. k ω m q Figure: Gridding Choose the interpolating function φ to be essentially bandlimited, i.e., ˆφ(ω) ω > q, q R, small φ() > π φ() [ π, π] ω

23 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Convolutional Gridding 4 FFT Reconstruction To evaluate S N f() = N k= N α k ˆf(ωk )e iω k efficiently Map the non-uniform modes to a uniform grid. A convolution operation is typically used. Compute a Fourier or filtered Fourier partial sum. g() If required, compensate for the mapping operation. 3 3 The new coefficients on the uniform grid are therefore given by ˆf ˆφ α m ˆf(ωm) ω=k ˆφ(k ω m) m st. k ω m q Figure: Fourier Reconstruction Choose the interpolating function φ to be essentially bandlimited, i.e., ˆφ(ω) ω > q, q R, small φ() > π φ() [ π, π]

24 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Convolutional Gridding To evaluate S N f() = N k= N α k ˆf(ωk )e iω k efficiently Map the non-uniform modes to a uniform grid. A convolution operation is typically used. Compute a Fourier or filtered Fourier partial sum. f() Compensation g() φ() f() 4 3 If required, compensate for the mapping operation. 3 3 The new coefficients on the uniform grid are therefore given by ˆf ˆφ α m ˆf(ωm) ω=k ˆφ(k ω m) m st. k ω m q Figure: Compensation Choose the interpolating function φ to be essentially bandlimited, i.e., ˆφ(ω) ω > q, q R, small φ() > π φ() [ π, π]

25 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Convolutional Gridding To evaluate S N f() = N k= N α k ˆf(ωk )e iω k efficiently Map the non-uniform modes to a uniform grid. A convolution operation is typically used. Compute a Fourier or filtered Fourier partial sum. f() Compensation g() φ() f() 4 3 If required, compensate for the mapping operation. 3 3 The new coefficients on the uniform grid are therefore given by ˆf ˆφ α m ˆf(ωm) ω=k ˆφ(k ω m) m st. k ω m q Figure: Compensation Choose the interpolating function φ to be essentially bandlimited, i.e., ˆφ(ω) ω > q, q R, small φ() > π φ() [ π, π]

26 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Reconstruction Eamples True Gridding reconstruction f() f().8.5 True Gridding reconstruction 3 3 (a) Test Function reconstruction (b) Cross-section of a brain scan Figure: Gridding reconstruction, N = 8 (processed by a 4 th -order eponential filter)

27 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Gridding Error Theorem (Convolutional Gridding Error) Let ĝ = ˆf ˆφ denote the true gridding coefficients and ˆ g denote the approimate gridding coefficients. Let k be the maimum distance between sampling points and d k := k be the minimum sample density in the q-vicinity of k. Then, the gridding error at mode k is bounded by e(k) C, k = N,..., N for some positive constant C. Proof. d 3 k ĝ(k) ˆ g(k) = Z ˆf(ω) ˆφ(k ω)dω p st. k ω p q Error in approimating the integral in the interval (ω p, ω p+) is e p = Z ωp+ ω p ˆf(ω) ˆφ(k ω)dω ω p+ ω p α p ˆf(ωp) ˆφ(k ω p) ˆf(ωp) ˆφ(k ω p) + ˆf(ω p+) ˆφ(k ω p+)

28 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Gridding Error Theorem (Convolutional Gridding Error) Let ĝ = ˆf ˆφ denote the true gridding coefficients and ˆ g denote the approimate gridding coefficients. Let k be the maimum distance between sampling points and d k := k be the minimum sample density in the q-vicinity of k. Then, the gridding error at mode k is bounded by e(k) C, k = N,..., N for some positive constant C. Proof. d 3 k ĝ(k) ˆ g(k) = Z ˆf(ω) ˆφ(k ω)dω p st. k ω p q Error in approimating the integral in the interval (ω p, ω p+) is e p = Z ωp+ ω p ˆf(ω) ˆφ(k ω)dω ω p+ ω p α p ˆf(ωp) ˆφ(k ω p) ˆf(ωp) ˆφ(k ω p) + ˆf(ω p+) ˆφ(k ω p+)

29 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Gridding Error Theorem (Convolutional Gridding Error) Let ĝ = ˆf ˆφ denote the true gridding coefficients and ˆ g denote the approimate gridding coefficients. Let k be the maimum distance between sampling points and d k := k be the minimum sample density in the q-vicinity of k. Then, the gridding error at mode k is bounded by e(k) C, k = N,..., N for some positive constant C. Proof. d 3 k ĝ(k) ˆ g(k) = Z ˆf(ω) ˆφ(k ω)dω p st. k ω p q Error in approimating the integral in the interval (ω p, ω p+) is e p = Z ωp+ ω p ˆf(ω) ˆφ(k ω)dω ω p+ ω p α p ˆf(ωp) ˆφ(k ω p) ˆf(ωp) ˆφ(k ω p) + ˆf(ω p+) ˆφ(k ω p+)

30 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Gridding Error Proof. From trapezoidal quadrature rule error analysis e p ωp+ ωp 3 v p The total error is the error is then given by ĝ(k) ˆ g(k) ĝ(k) ˆ g(k) p κ p st. k ω p q p st. k ω p q C 3 k, = C d 3 k e p p st. k ω p q ω p+ ω p 3 v p e p ω p+ ω p 3, κ = ma p for some positive constant C v p

31 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Gridding Error Proof. From trapezoidal quadrature rule error analysis e p ωp+ ωp 3 v p The total error is the error is then given by ĝ(k) ˆ g(k) ĝ(k) ˆ g(k) p κ p st. k ω p q p st. k ω p q C 3 k, = C d 3 k e p p st. k ω p q ω p+ ω p 3 v p e p ω p+ ω p 3, κ = ma p for some positive constant C v p

32 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Gridding Error Proof. From trapezoidal quadrature rule error analysis e p ωp+ ωp 3 v p The total error is the error is then given by ĝ(k) ˆ g(k) ĝ(k) ˆ g(k) p κ p st. k ω p q p st. k ω p q C 3 k, = C d 3 k e p p st. k ω p q ω p+ ω p 3 v p e p ω p+ ω p 3, κ = ma p for some positive constant C v p

33 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Gridding Error Proof. From trapezoidal quadrature rule error analysis e p ωp+ ωp 3 v p The total error is the error is then given by ĝ(k) ˆ g(k) ĝ(k) ˆ g(k) p κ p st. k ω p q p st. k ω p q C 3 k, = C d 3 k e p p st. k ω p q ω p+ ω p 3 v p e p ω p+ ω p 3, κ = ma p for some positive constant C v p

34 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Gridding Error Physical space reconstruction error e() g() S N g() = g() S N g() + S N g() S N g() = ĝ(k)e ik + ĝ(k) ˆ g(k) k >N {z } standard Fourier truncation k N e ik {z } gridding S N g suffers from Gibbs; the maimum error occurs in the vicinity of a jump (.9 of the jump value). There is also a reduced order of convergence with g S N g = O N /. Gridding Error S N g() S N g() = ĝ(k) ˆ g(k) e ik k N ĝ(k) ˆ g(k) k N C d 3 k N k.

35 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Error Plots log e H(ω k,n) Error Normalized bound k N (a) Error bound, log sampling (b) S N g() S N g() vs N Figure: Error Plots.

36 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Error Plots.6.5 Fourier Coefficients True Interpolated F(k) k (a) Fourier coefficients High modes H(ω k,n) N (b) S N g() S N g() vs N Figure: Error Plots.

37 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Error vs Sampling Density The reconstruction error is e() ĝ(k)e ik + k >N st term decreases as N increases nd term increases as N increases k N ĝ(k) ˆ g(k) e ik.45 Error in Fourier Coefficients ˆf (k).5. Log E k k (a) Fourier coefficients (b) Coefficient error Figure: Error in uniform re-sampling For a given sampling trajectory and function, there is a critical value N crit beyond which adding coefficients does not improve the accuracy. While filtering decreases the error, the underlying problem is not solved.

38 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Error vs Sampling Density The reconstruction error is e() ĝ(k)e ik + k >N st term decreases as N increases nd term increases as N increases k N ĝ(k) ˆ g(k) e ik.45 Error in Fourier Coefficients ˆf (k).5. Log E k k (a) Fourier coefficients (b) Coefficient error Figure: Error in uniform re-sampling For a given sampling trajectory and function, there is a critical value N crit beyond which adding coefficients does not improve the accuracy. While filtering decreases the error, the underlying problem is not solved.

39 Introduction Non-uniform Data Current Methods Alternate Approaches Reconstruction Methods Error Characteristics Error vs Sampling Density The reconstruction error is e() ĝ(k)e ik + k >N st term decreases as N increases nd term increases as N increases k N ĝ(k) ˆ g(k) e ik.45 Normalized Error in Fourier Coefficients ˆf (k).5. Log E norm k k (a) Fourier coefficients (b) Coefficient error Figure: Error in uniform re-sampling For a given sampling trajectory and function, there is a critical value N crit beyond which adding coefficients does not improve the accuracy. While filtering decreases the error, the underlying problem is not solved.

40 Introduction Non-uniform Data Current Methods Alternate Approaches Spectral Re-projection Incorporating Edge Information Outline Introduction Motivating Eample Application Outline of the Talk The Non-uniform Data Problem Problem Formulation The Non-harmonic Kernel Reconstruction Results using the Non-harmonic Kernel 3 Current Methods Reconstruction Methods Error Characteristics 4 Alternate Approaches Spectral Re-projection Incorporating Edge Information

41 Introduction Non-uniform Data Current Methods Alternate Approaches Spectral Re-projection Incorporating Edge Information Piecewise-Smooth Functions 5 f(,35) (a) Brain scan (b) Cross-section of a scan Figure: Piecewise-smooth nature of medical images Due to the Gibbs phenomenon, we have non-physical oscillations at discontinuities, and, more importantly, reduced order of convergence (first order). Hence, we require a large number of coefficients to get acceptable reconstructions. However, by formulation of the sampling scheme and recovery procedure, the coefficients recovered at large ω are inaccurate. = we need more coefficients, but the coefficients we get are inaccurate!

42 Introduction Non-uniform Data Current Methods Alternate Approaches Spectral Re-projection Incorporating Edge Information Spectral Re-projection Spectral reprojection schemes were formulated to resolve the Gibbs phenomenon. They involve reconstructing the function using an alternate basis, Ψ (known as a Gibbs complementary basis). Reconstruction is performed using the rapidly converging series f() m l= c l ψ l (), where c l = fn, ψ l w ψ l, f N is the Fourier epansion of f w Reconstruction is performed in each smooth interval. Hence, we require jump discontinuity locations High frequency modes of f have eponentially small contributions on the low modes in the new basis

43 Introduction Non-uniform Data Current Methods Alternate Approaches Spectral Re-projection Incorporating Edge Information Reducing the Impact of the High Mode Coefficients Filtered Fourier reconstructions N «k S N g() = σ ĝ(k)e ik N k= N Spectral re-projection Epansion coefficients are obtained using h λ l Z ( η ) λ / C λ l (η) k N ˆ g(k)e iπkη dη Damping of the high modes since Z ( η ) λ / Cl λ (η)e iπkη dη = h λ l Γ(λ) «λ i l (l + λ)j l+λ(πk) πk The gridding error can be shown to be «λ C ρ(m, λ) d 3 < k N k k {z } H(ω k,n,λ) σ(η) log ĝ(k) k (a) Filter function k (b) Gegenbauer Coefficients

44 Introduction Non-uniform Data Current Methods Alternate Approaches Spectral Re-projection Incorporating Edge Information Reducing the Impact of the High Mode Coefficients Filtered Fourier reconstructions N «k S N g() = σ ĝ(k)e ik N k= N Spectral re-projection Epansion coefficients are obtained using h λ l Z ( η ) λ / C λ l (η) k N ˆ g(k)e iπkη dη Damping of the high modes since Z ( η ) λ / Cl λ (η)e iπkη dη = h λ l Γ(λ) «λ i l (l + λ)j l+λ(πk) πk The gridding error can be shown to be «λ C ρ(m, λ) d 3 < k N k k {z } H(ω k,n,λ) H(ω k,n,λ) σ(η) k (c) Filter function N (d) H(ω k, N, λ)

45 Introduction Non-uniform Data Current Methods Alternate Approaches Spectral Re-projection Incorporating Edge Information Gegenbauer Reconstruction - Results.5 Function Reconstruction Log Error in Reconstruction.5 g().5.5 log f () g() True 6 Filtered Fourier(N=56) Gegenbauer(N=64) 7 Filtered Fourier(N=56) Gegenbauer(N=64) (e) Reconstruction (f) Reconstruction error Figure: Gegenbauer reconstruction Filtered Fourier reconstruction uses 56 coefficients Gegenbauer reconstruction uses 64 coefficients Parameters in Gegenbauer Reconstruction - m =, λ =

46 Introduction Non-uniform Data Current Methods Alternate Approaches Spectral Re-projection Incorporating Edge Information Error Plots 6 5 Gegenbauer nd order eponential 4 th order eponential 8 th order eponential f P m S N f 4 3 f PmSNf Gegenbauer nd order eponential 4 th order eponential 8 th order eponential N N (a) -norm error (b) Maimum-norm error Figure: Error Plots Filtered and Gegenbauer Reconstruction

47 Introduction Non-uniform Data Current Methods Alternate Approaches Spectral Re-projection Incorporating Edge Information Incorporating Edge Information Solve the following equation ˆf(k) = p P [f](ζ p) e ikζp πik f().5.5 Function Jump function Use the concentration method on the recovered coefficients N «k SN σ [f]() = i ˆf(k) sgn(k) σ e ik N k= N Solve for the jump function directly from the non-harmonic Fourier data min [f] s.t. F{[f]} ωk = i sgn(ω) σ ω N ˆf ωk f() Function Jump function 3 3 Figure: Edge Detection

48 Introduction Non-uniform Data Current Methods Alternate Approaches Spectral Re-projection Incorporating Edge Information Methods Incorporating Edge Information Compute the high frequency modes using the relation Compare ˆf(k) = p P [f](ζ p) e ikζp πik.5 Function Reconstruction.3 Fourier Coefficients True With edge information.5. f() F(k)..5 True.5 With edge information k (a) Reconstruction - Using edge informatiomation (b) The high modes - Using edge infor- Figure: Reconstruction of a test function using edge information

49 Introduction Non-uniform Data Current Methods Alternate Approaches Spectral Re-projection Incorporating Edge Information Summary We introduced the Fourier reconstruction problem for non-uniform spectral data We discussed the inherent problems associated with non-uniform Fourier data We briefly looked at conventional reconstruction methods We studied the error characteristics and relation to sampling density We looked at spectral re-projection and methods incorporating edge information to obtain better reconstructions Current Emphasis Incorporating edge detection into the reconstruction scheme