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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred S PARSITY E NFORCING E DGE D ETECTION M ETHOD FOR BLURRED AND NOISY F OURIER D ATA Rosemary Renaut This is

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

SPARSITY ENFORCING EDGE DETECTION METHOD

FOR BLURRED AND NOISY FOURIER DATA

Rosemary Renaut This is joint work with Wolfgang Stefan, Rice University, Aditya Viswanathan, Cal Tech, and Anne Gelb, Arizona State University

TEMPLE UNIVERSITY 2011

April 27, 2011

National Science Foundation: Division of Computational Mathematics 1 / 41

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Outline

1

Problem Statement and Test Examples

2

Background

3

Jump detection using the Concentration method

4

The Matching Waveform

5

l1 minimization to detect edges in blurred signals

6

Extension for Edge Detection from Non-harmonic Coefficients

7

Conclusions

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Imaging from Spectral Data Spectral methods enjoy great popularity in a vast array of applications. Their superior numerical properties, however, break down when the underlying function is piecewise-analytic. Processing of such functions from spectral data requires special attention. Goal:

leverage the interplay between local features, primarily jump discontinuities, and Fourier coefficients, to enable accurate and efficient processing of such data.

National Science Foundation: Division of Computational Mathematics 3 / 41

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Motivating Application – Magnetic Resonance Imaging I We acquire spectral data. Data may be acquired along non-Cartesian sampling trajectories.

resistance to motion artifacts ease in generating field gradients

Data can be degraded by blur. Data can be noisy

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Motivating Application – Magnetic Resonance Imaging II

(a) Acquired Fourier Sam- ples (b) Spiral Sampling Trajec- tory (c) Basic Recon- structed Shepp Logan

Figure: MR Imaging: Example without blur or noise, but non Cartesian Data1

1Sampling pattern courtesy Dr. Jim Pipe, Barrow Neurological Institute,

Phoenix, Arizona

National Science Foundation: Division of Computational Mathematics 5 / 41

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Brief Overview Objective Estimate the edges in a piecewise smooth function from blurred and noisy Fourier data on non-equispaced grids. Assume a finite number of Fourier Coefficients available for a piecewise function. These may be noisy The function may be blurred Data may be collected at non cartesian grid points Desire accurate and robust detection of jump discontinuities. Aim improve signal reconstructions, restorations and classification. Illustrate by counting true classifications of edges in data.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Background

The function f is represented by a finite number of spectral coefficients f is 2π-periodic and piecewise-smooth in [−π, π). It has Fourier series coefficients ˆ fk = 1 2π π

−π

f(x)e−ikxdx , k ∈ [−N, N] ˆ f is a global representation; i.e., ˆ fk are obtained using values of f over the entire domain [−π, π). Assume f is piecewise smooth Its jump function is defined by [f](x) := f(x+) − f(x−) A jump discontinuity is a local feature; i.e., the jump function at any point x

nly depends on the values of f at x+ and x−.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Concentration Factor Edge Detection Method (Gelb, Tadmor)

Concentrating the edges using convolution with Cσ

N(x)

Approximate [f](x) using generalized conjugate partial Fourier sum Sσ

N[f](x) = i N

k=−N

ˆ fk sgn(k) σ |k| N

eikx = (f ∗ Cσ

N)(x)

(1) (1) converges when concentration factors σk,N(η) = σ( |k|

N ) satisfy admissibility

properties: (i)

σk,N sin(kx) (ii) σ(η)

∈ C2(0, 1) (iii) 1

σ(η) η → −π, is odd ǫ = ǫ(N) > 0 is small Convergence Sσ

N[f](x) = [f](x) + O(ǫ), depends on σ and the distance between x

and a discontinuity of f.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Classical Concentration Factors

Factor Expression Trigonometric σT (η) = π sin(α η) Si(α) Si(α) = Z α sin(x) x dx Polynomial σP (η) = −p π ηp p is the order of the factor Exponential σexp(η) = C η exp „ 1 α η (η − 1) « C - normalizing constant; α > 0 - order C = π R 1− 1

N 1 N

exp “

1 α τ (τ−1)

” dτ

Table: Examples of concentration factors

−80 −60 −40 −20 20 40 60 80 0.5 1 1.5 2 2.5 3 3.5 4 k Concentration Factors Trigonometric Polynomial Exponential

Figure: Envelopes of the Concentration Factors in Fourier Space

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Example for 2D: No Noise, No Blur, Cartesian Grid

−3 −2 −1 1 2 3 −2 −1.5 −1 −0.5 0.5 1 1.5 2 x f(x) f SN[f]

(a) Trigonometric Factor: Jump Response (b) Shepp Logan phantom Apply concentration to each dimension Sσ

N[f](x(¯

y)) = i

X

l=−N

sgn(l) σ „ |l| N « ·

X

k=−N

ˆ fk,l ei(kx+l¯

verbar represents constant dimension.

(c) Edge Map

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Illustration of Edge Detection N = 64. Black line is the jump function

Example Case: No Noise, no blur, Cartesian Grid f(x) = 8 > > < > > : 3/2 for − 3π

4 ≤ x < − π 2

7/4 − x/2 + sin(7x − 1/4) for − π

4 ≤ x < π 8

x11/4 − 5 for

3π 8 ≤ x < 3π 4

therwise.

(2) (d) Polynomial p=1 (σ1) (e) Polynomial p=2 (σ2) (f) Exponential (σexp)

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Observations Polynomial CFs only filter low frequencies Exponential also filters some high frequencies Fast convergence away from a jump leads to more oscillations around the jump Many false positive and false negatives with regard to classifying jumps.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Improving jump detection The minmod to improve the approximation (Gelb and Tadmor (2006)) Use the minmod function over different concentration functions minmod{a1, . . . , an} := s min(|a1|, |a2|, . . . , |an|) s := sgn(ai), ∀i

therwise

, (3) yielding the approximation obtained by finding the jump approximation with multiple σ SMM

N

[f](x) = minmod{Sσ1

N [f](x), Sσ2 N [f](x), . . . , Sσn N [f](x)}.

(4)

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Minmod CF edge detection for noisy and blurred functions: 2% threshold

(g) Under sampling (h) Blurring by a Gaussian (i) Noise contamination

Figure: False positives & negatives. (g) 10% missing Fourier Coefficients. (h) Gaussian blur of variance τ = 0.05, for point spread function coefficients ˆ hk = e− k2τ2

2 . (i) Noise of variance .015 applied to Fourier Coefficients.

For blurred functions the edges may be missed, for noisy functions or with missing data too many edges are determined.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

The Jump Response

Let r(x) denote the unit ramp function. r(x) =

x−π

2π

x < 0

π−x 2π

x > 0 , [r](x) = 1 x = 0 else Definition (Jump Response) The jump response, denoted by W σ

N(x), is defined as the jump

function approximation of the unit ramp as generated by the concentration sum, i.e., W σ

N(x) := Sσ N[r](x) = i

|k|≤N

ˆ r(k)sgn(k)σ |k| N

eikx

= 1 2π

0<|k|≤N

σ

eikx The jump response describes the unique oscillatory pattern of the jump function approximation in the immediate vicinity and away from jumps.

National Science Foundation: Division of Computational Mathematics 15 / 41

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Sample Jump Responses of the Concentration Factors

−3 −2 −1 1 2 3 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 SN[f](x) x f SN[f]

(a) Trigonometric Factor

−3 −2 −1 1 2 3 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 SN[f](x) x f SN[f]

(b) Polynomial Factor

−3 −2 −1 1 2 3 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 SN[f](x) x f SN[f]

(c) Exponential Factor

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Improvement: Matching Waveform (A. Gelb and D. Cates, 2008) Jump approximation at x = ξ depends on size [f] and location ξ, not f: Sσ

N[f](x) = [f](ξ)

π

N

σk,N cos k(x − ξ) k + O log N N

Use waveform (notice simplification) W σ

N(x) = N

σk,N cos kx

k

. Apply CF and correlate with CF applied to an indicator function Sσmw

N

[f](x) = 1 γmw (Sσ

N[f] ∗ W σ N)(x),

γmw = 1 π

N

σk,N k 2 (5) Gives admissible matching waveform concentration factor (MWCF) σmw |k| N

1 γmw σ |k| N π

−π

W σ

N(ρ) exp (−ikρ) dρ.

(6) MWCF performs better in the presence of noise, does not remove oscillations. Performance deteriorates for nearby jumps.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Appealing to Sparsity : use MWCF for noisy, blurred, undersampled data Summary of The Approach

The concentration function applied in Fourier domain enhances scales at edges

We can find the Fourier expansion approximating jumps using concentration at the edges

Correlate the obtained jump approximation to a matching waveform. (use the MWCF)

Impose sparseness in space on the jump function and find sparse jump function which matches the jump function of the data.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Estimate Jump Function [f] given ˆ gk of Noisy Blurred f Appealing to sparsity Given ˆ gk for blur function h and noise n, ˆ gk = ˆ hk ˆ fk + ˆ nk Approximate [f] from [g], given ˆ gk : ( ˆ Sσ

N[g])k =

iσ|k|,Nsgn(k)
ˆ

gk Observe ˆ gk ≈ ˆ hk ˆ fk yields

iσ|k|,Nsgn(k)
ˆ

gk = ( ˆ Sσ

N[g])k ≈ ˆ

hk( ˆ Sσ

N[f])k

Seek sparse y which also approximates the jump function of f Convolve y with W σ

N(x) to approximate jump Sσ N[f](x)

( ˆ Sσ

N[f])k ≈ (

ˆ W σ

N ∗ y)k = ( ˆ

W σ

N)kˆ

yk, (7) Obtain for ( ˆ W σ

N)k = π |k|σ|k|,N, |k| ≤ N, k = 0

ˆ hk( ˆ W σ

N)kˆ

yk ≈ iσ|k|,Nsgn(k)ˆ gk

National Science Foundation: Division of Computational Mathematics 19 / 41

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

A Discrete Variational Formulation: for blur, noise and Cartesian

l1 minimization Introduce matrices describing components of the approximate equation Σ = diag „ σ „| − N| N « , · · · , 0, · · · , σ „|N − 1| N «« H = diag( π | − N| ˆ h−N, · · · , 0, · · · , π |N − 1| ˆ hN−1) and Fkj = 1 2N (−1)k exp(−iπjk N ) where ˆ y = Fy((x)). Find discrete approximation y to y(x) from second order cone problem. y = arg min

u u1

subject to Σ(HFu − b)2

2 ≤ δ,

(8) b = (−iˆ g−N, · · · , 0, · · · , iˆ gN−1). Σ weights the data fit term. Introduce λ and solve y = arg minu{λu1 + 1

2Σ(HFu − b)2 2} National Science Foundation: Division of Computational Mathematics 20 / 41

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Overview of Test Problems

One dimensional examples considered No noise, no blur, no under sampling: Best case scenario but still non trivial. No noise, no blur, but under sampling: Only partial Fourier data available. Fourier coefficients are deleted from the middle of the spectrum (symmetrically), i.e. both low as well as high frequencies still present. In context of minimization problems, missing band of Fourier data corresponds to under sampling. No noise, Gaussian blur, no under sampling: Fourier coefficients blurred by Gaussian filter ˆ hk = e− k2τ2

. Smooths edges in signal, edge detection using classical methods is difficult. Additive i.i.d. Gaussian noise, no blur, all samples: How does additive noise in Fourier coefficients impact edge detection? Non-harmonic Fourier data: Examine edge detection for efficient data collection, eg in MRI.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Experiments with N = 64 and under sampling but no noise and no blur.

−2 2 −2 −1 1 2 3

x f; y 15 FP 0 FN

(d) (FC , λ) = 70%, 10−9

−2 2 −2 −1 1 2 3

x f; y 0 FP 6 FN

(e) (FC , λ) = 70%, 10−3

−2 2 −2 −1 1 2 3

x f; y 15 FP 4 FN

(f) (FC, λ) = 35%, 10−8

Figure: Edge detection using σexp. FC is percentage of Fourier Coefficients used, y is the thin line, unseen in (b). FP and FN are count of misidentified edges, either false positive or false negative, using 2% threshold on y.

One sees the effect of the regularization parameter comparing (a) and (b), and of reducing the number of Fourier Coefficients comparing (a) and (c).

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Under sampling, no noise, no blur. False Positives and False Negatives with Waveform

log10(lambda) % Fourier coefficients −9 −8 −7 −6 −5 −4 −3 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 False positives False negatives a b c d

Illustrating impact of choice of regularization parameter in relation to the number of Fourier Coefficients sampled, and impact on the number of False Positives and False Negatives Region (d) shows that there is a range of regularization parameters for which the method is robust with respect to correct identification of edges provided up to about 70% of coefficients are retained.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Polynomial Concentration N = 64, no noise, no blur, missing Fourier data

log10(lambda) % Fourier coefficients −9 −8 −7 −6 −5 −4 −3 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 False positives False negatives

(a) N=64, σ1

log10(lambda) % Fourier coefficients −9 −8 −7 −6 −5 −4 −3 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 False positives False negatives

(b) N=64, σ2 Higher order concentration factors perform better at capturing the edges correctly for a wider range of regularization parameters.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Edge detection in the presence of blur in the coefficients. N = 64.

log10(lambda) log10(σ) −9 −8 −7 −6 −5 −4 −3 −2 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 1 2 3 4 5 6 False positives False negatives

(c) σ1

log10(lambda) log10(σ) −9 −8 −7 −6 −5 −4 −3 −2 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 1 2 3 4 5 6 False positives False negatives

(d) σ2

log10(lambda) log10(σ) −9 −8 −7 −6 −5 −4 −3 −2 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 1 2 3 4 5 6 False positives False negatives

(e) σexp

Figure: Edge detection in blurred signals using σp, for p = 1, 2, and σexp. All plots show that the method can handle blurring where the traditional CF method fails.

Gaussian blur of variance τ = 0.05, for point spread function coefficients ˆ hk = e− k2τ2

. The higher order concentration factors again perform better at capturing the edges correctly for a wider range of regularization parameters.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Edge detection in the presence of additive noise in the coefficients. N = 64.

log10(lambda) noise level ν −5 −4.5 −4 −3.5 −3 2 4 6 8 10 12 14 16 18 x 10

−3

1 2 3 4 5 6 False positives False negatives

(a) σ1

log10(lambda) noise level ν −5 −4.5 −4 −3.5 −3 2 4 6 8 10 12 14 16 18 x 10

−3

1 2 3 4 5 6 False positives False negatives

(b) σ2

log10(lambda) noise level ν −5 −4.5 −4 −3.5 −3 2 4 6 8 10 12 14 16 18 x 10

−3

1 2 3 4 5 6 False positives False negatives

(c) σexp

Figure: Edge detection in signals with noise of variance .015 applied to Fourier

Coefficients. All plots show that the method can handle noise where the traditional

CF method fails.

In this case the higher order exponential concentration factor performs better than the quadratic, perhaps due to its inherent filtering of coefficients contaminated with noise.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Is waveform correlation required? Examples without waveform N = 64

log10(lambda) % Fourier coefficients −9 −8 −7 −6 −5 −4 −3 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 False positives False negatives

(a) σ1

log10(lambda) % Fourier coefficients −9 −8 −7 −6 −5 −4 −3 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 False positives False negatives

(b) σ3

log10(lambda) % Fourier coefficients −9 −8 −7 −6 −5 −4 −3 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 False positives False negatives

(c) σexp

Figure: No blur, no noise missing Fourier data: classification capability robustness with respect to choice of λ.

When using the low order polynomial concentration factor the method is quite robust, but for higher order concentration factors the method is more sensitive to the choice of λ and the ability to correctly detect edges is limited.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Non-harmonic Fourier data

Motivation: Modern MRI scanners optimize data collection Fourier data collected on non-cartesian representations of the k-space. Non-harmonic Fourier data, ˆ f(ωk), for piecewise-analytic f ∈ L2(R(−π, π)) are defined by ˆ f(ωk) := 1 2π Z π

−π

f(x)e−iωkxdx, ωk / ∈ Z. (9) Extension of convolution form of jump approximation (1) ˜ Sσ

N[f](x) = (f ∗ ˜

Cσ

N)(x) := i N

X

k=−N

αk ˆ f(ωk)sgn(ωk)σ „|ωk| N « eiωkx. Coefficients αk are weights for non-uniform trapezoidal rule approximation of inverse Fourier integral. (convolutional gridding).

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Examples: Non-harmonic sampling (right half plane), N = 16

2 4 6 8 10 12 14 16 0.5 0.5

equispaced

jittered

(a) Jittered sampling ωk = k ± ζk, ζk ∼ U [0, θ) , k = −N, −(N − 1), . . . , N.

2 4 6 8 10 12 14 16 0.5 0.5

equispaced

log

(b) Log sampling

Figure:

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Applying the Edge Detector with the non-harmonic concentration sum

3 2 1 1 2 3 1 0.5 0.5 1 1.5 x f, SN

[f]

Jittered sampling f SN

exp[f]

p1[f]

(a) Edges from jittered sampling using σ1 and σexp with α = 2.

3 2 1 1 2 3 1 0.5 0.5 1 1.5 x f, SN

[f]

Log sampling f SN

exp[f]

p1[f]

(b) Edges from log sampling using σ1 and σexp with α = 2.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Extending the Sparsity Approach

g = (ˆ g(ω−N), ..., ˆ g(ωN−1))T non-harmonic measurements. y, approximates [f] on equispaced grid xj = πj

N − π, j = 0, . . . , 2N − 1.

Introduce Σ diagonal matrix of concentration factors, H diagonal matrix of blur coefficients, F ∈ C2N×2N discrete non-harmonic Fourier matrix, and W a Toeplitz matrix whose rows contain shifted replicates of the jump waveform W σ

N(x)

Σ = idiag „ sgn(ωk) σ „|ωk| N «« , H = diag “ ˆ h(ωk) ” Fkj = exp » i „ −π + πj N « ωk – , k = −N, . . . , N − 1, j = 0, ..., 2N − 1. Compute the jump approximation by solving y = arg min

u {λu1 + 1

2HFWu − Σg2

2}.

(10)

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Example for exact data: Detects the location but not the height, N = 64

3 2 1 1 2 3 1 0.5 0.5 1 1.5 x f, SN

[f]

f SN

[f]

(c) Jittered sampling, σ1, λ = .0017

3 2 1 1 2 3 1 0.5 0.5 1 1.5 x f, SN

[f]

f SN

[f]

(d) Log sampling, σexp, α = 2, λ = .00091 Approximations: non-harmonic Fourier data using variational formulation

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Example for blurred and noisy non-harmonic Fourier data, N = 64

3 2 1 1 2 3 2 1 1 2 3 x SNf f SNf

(e) Fourier reconstruction of blurred noisy data

3 2 1 1 2 3 2 1 1 2 3 x f, SN [f] f SN [f]

(f) Jittered spectral data using σ1

3 2 1 1 2 3 2 1 1 2 3 x f, SN [f] f SN [f]

(g) Log spectral data using σexp, α = 2 Gaussian blur variance τ = .05. Additive white complex Gaussian noise, variance .015. Regularization parameters .002 in (e) and .0013 in (f). Solution is more sensitive to choice

f regularization parameter λ than

for the harmonic case. Determination of λ is harder for the log than jittered data.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

A Two Dimensional Example (Stefan and Yin)

(a) (b)

Figure: A modified Shepp logan phantom with gradients and a radial sampling pattern.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Some Two Dimensional Results (Stefan and Yin)

(a) (b) (c)

Figure: Edge detection using (a) Canny edge detector (matlab) after reconstruction from the radial samples using TV. (b) Wavelet edge detector on TV reconstruction. (c) A 5th order FD edge detector

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Conclusions

Use of the variational formulation which employs sparsity in the jump function approximation yields a robust approach for both noisy and blurred signals. The approach requires the matching waveform to improve robustness with respect to choice of the regularization parameter. Method is successful in the presence of missing Fourier data. (here sampled from the middle of the spectrum). The approach is a regularized deconvolution of the approximate jump function. Higher order exponential concentration function outperforms low order polynomial concentration functions. Method can be extended for non-harmonic data, edges are detected but the heights are not correct. Algorithm has been extended for two dimensional examples, Stefan and Yin (2010). Can be useful for accurate classification of edges in signals.

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

References

A. GELB AND E. TADMOR, Detection of Edges in Spectral Data, in Appl. Comp.

Harmonic Anal., 7 (1999), 101–135.

A. GELB AND E. TADMOR, Detection of Edges in Spectral Data II Nonlinear

Enhancement, in SIAM J. Numer. Anal., Vol. 38, 4 (2000), 1389–1408.

A. GELB AND E. TADMOR, Adaptive edge detectors for piecewise smooth data based
n the minmod limiter, in J. Sci. Comput., 28(2-3): (2006), 279–306.

A. GELB AND D. CATES, Detection of Edges in Spectral Data III -refinement of the

concentration method, in J. Sci. Comput., 36, 1 (2008), 1-43.

E. TADMOR AND J. ZOU, Novel edge detection methods for incomplete and noisy

spectral data, in J. Four. Analy. App. 14(5) (2008), 744-763.

W. STEFAN, A. VISWANATHAN, A. GELB, AND R. A. RENAUT, Sparsity enforcing

edge detection method for blurred and noisy Fourier data, (2010).

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

THANK YOU!

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Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Comparing the performance of the waveform correlation N = 64, for σ1

log10(lambda) % Fourier coefficients −9 −8 −7 −6 −5 −4 −3 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 False positives False negatives

(a) With W

log10(lambda) % Fourier coefficients −9 −8 −7 −6 −5 −4 −3 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 False positives False negatives

(b) Without W

Figure: No blur, no noise, missing data, first order with/without waveform weighting

Two approaches are comparable for low order polynomial concentration factor.

National Science Foundation: Division of Computational Mathematics 39 / 41

SLIDE 40

Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Comparing the performance of the waveform correlation N = 64, for σ3

log10(lambda) % Fourier coefficients −9 −8 −7 −6 −5 −4 −3 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 False positives False negatives

(a) With W

log10(lambda) % Fourier coefficients −9 −8 −7 −6 −5 −4 −3 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 False positives False negatives

(b) Without W No blur, no noise, missing data, third order with/without waveform. Clearly waveform is

required. Higher order polynomial CF introduces oscillations that need to be suppressed.

National Science Foundation: Division of Computational Mathematics 40 / 41

SLIDE 41

Problem Statement and Test Examples Background Concentration Method The Matching Waveform l1 minimization to detect edges in blurred

Comparing the performance of the waveform correlation N = 64, for σexp

log10(lambda) % Fourier coefficients −9 −8 −7 −6 −5 −4 −3 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 False positives False negatives

(c) With W

log10(lambda) % Fourier coefficients −9 −8 −7 −6 −5 −4 −3 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 False positives False negatives

(d) Without W No blur, no noise, missing data. Exponential concentration factor with/without the waveform. Waveform is required. Higher order CF introduces oscillations that need to be suppressed.

National Science Foundation: Division of Computational Mathematics 41 / 41