tidimensional polynomail annihilation method [2] for given Fourier data by
projecting the Fourier co- efficients onto polynomial space. Further, limited
resolution ...
Jump Detection Using Gegenbauer Polynomials as a Basis for Polynomial Annihilation Aaron Jesse
[email protected]
Arizona State University
Anne Gelb
[email protected]
Abstract
Polynomial Annihilation Using Fourier Data
Subcell Resolution Enhancement
A common problem in image reconstruction is identifying jump locations from Fourier coefficients. While techniques exist which determine edges directly from Fourier data, they are inherently one-dimensional. Here, we adapt the multidimensional polynomail annihilation method [2] for given Fourier data by projecting the Fourier coefficients onto polynomial space. Further, limited resolution is enhanced without accumulating more Fourier data by using a new subcell resolution enhancement approach. Finally, the method is tested on Fourier data which may be corrupted by noise, or that may not be collected uniformly.
Now suppose you are given Fourier coefficients {fˆk }N k=−N so that
In addition to evenly spaced sampling of Fourier coefficients, data was sampled in two other patterns:
Polynomial Annihilation +
−
Let [f ](x) = f (x ) − f (x ) define a jump function for piecewise smooth f (x). The polynomial annihilation method is given by [2]: X 1 Lm f (x) = cj (x)f (xj ) → [f ](x) qm (x) xj ∈Sx
{f (xj )}N j=0
is a finite sampling of f (x), Sx is where a stencil defined as the nearest m + 1 points to x, and qm (x) normalizes the jump function approximation. The annihilation coefficients cj (x) satisfy X (m) cj (x)pi (xj ) = pi (x), i = 1, ..., m.
N X
SN f (x) =
1. A "Jittered" sampling, which starts with Fourier data given at integers and adds a random number between 0 and 1
ikπx ˆ fk e .
k=−N
In [4], it was shown that kPK (f − SN f )k → 0 exponentially for the Gegenbauer polynomials Ψk (x) = Ckλ (x), for large enough λ. Hence, define g˜k =
(1−x2 )λ−1/2
A sinc transform is used to effectively resample the data on evenly spaced Fourier coefficients, and the same polynomial annihilation method is applied.
and construct 1 Lm PK SN f (x) = qm (x)
K X
g˜k
k=m
X
2. Log spaced sampling, which effectively concentrates the data sampled near the low modes in a similar fashion to the known sampling pattern in MRI
λ cj (x)Ck (xj ).
xj ∈Sx
Notes:
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2 −1
1. To optimize the accuracy of the polynomial annihilation method, the reλ (x), thus ensuring a small construction grid is chosen to be the roots of CK Lebesgue constant. 2. This will cause the reconstruction grid to have points concentrated near the boundaries, which is not likely to be the natural place for edges to exist.
−0.5
0
0.5
−2 −1
1
a) Jittered
−0.5
0
0.5
1
b) Log Sampled
As can be seen, the edge detection given unevenly spaced Fourier coefficients matches that of the evenly spaced case. Additionally, noise was introduced which followed a Gaussian distribution across the original Fourier coefficients, in this case using a signal to noise ratio of 25. 2
1
f(x) fN
0.5
1
0
3. We can improve results by introducing new cells based on different values of λ (see Subcell Resolution Enhancement).
0
−0.5 −1
−1 −1.5
4. These methods are combined with nonlinear postprocessing techniques to preserve jump heights and add additional accuracy and resolution.
−2 −1
−0.5
0
0.5
1
a) Reconstruction w/ Noise
−2 −1
−0.5
0
0.5
1
b) Computed Jump Function
xj ∈Sx
Hence in smooth regions, where [f](x)=0 can be well approximated by a polynomial, we have m Lm f (x) = [f ](x) + O(h(x) ) where h(x) is the largest distance between neighboring gridpoints. Conversely, if x = ξ is a jump discontinuity, then Lm f (ξ) = [f ](ξ) + O(h(x)). Now, suppose instead N of {f (xj )}j=0 we are given K {ˆ gk }k=0
= {
w }k=0
{Ψk }K k=0
where is an orthogonal basis with respect to the weight function w, so that PK f (x) = PK ˆk Ψk (x). In [3], it was shown that k=0 g 1 Lm PK f (x) = qm (x) converges to [f ](x).
K X k=m
gˆk
X xj ∈Sx
cj (x)Ψk (x)
Subcell Resolution Enhancement
Open Research
References
The subcell resolution enhancement algorthm can now be defined as:
While the polynomial annihilation using Fourier to Gegenbauer coefficients method itself has been applied successfully to 2D, the Subcell Resolution Enhancement algorithm has not. The algorithm itself will remain unchanged, but cost effective implementation becomes more complicated due to the increased cost of projection from the upper grid to the lower grid with larger numbers of gridpoints. Additionally, other basis polynomials are being explored, as perhaps another basis will provide a nice set of points which can be added to the Subcell Resolution Enhancement algorithm.
[1] J. Hesthaven, S. Gottlieb, and D. Gottlieb, "Spectral Methods for Time-Dependent Problems" Cambridge University Press, 2007. [2] R. Archibald, A. Gelb, and J. Yoon, "Polynomial Fitting For Edge Detection In Irregularly Sampled Signals and Images," SIAM J. Appl. Numer. Anal., vol. 43(1), pp. 259-279, 2005. [3] R. Archibald, A. Gelb, R. Saxena, and D. Xiu, "Discontinuity Detection in Multivariate Space for Stochastic Simulations," J. Comput. Phys. vol. 228, pp. 2676-2689, 2009. [4] D. Gottlieb, C.-W. Shu, A. Solomonoff, and H. Vandeven, "On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function," J. Comp. Appl. Math., vol. 43, 1992.
λi XU
= (x1 , x2 , ..., xK ) : S 2. Define the lower grid XL = λi ∈Λ XUλi .
1. Define the upper grid
λi CK (xk )
= 0, λi ∈ Λ.
3. Use several annihilation orders m for each XUλi , postprocess results to reduce artificial oscillations. 4. Project each of these results onto XL , postprocess once again to achieve highly resolved jumps. 1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
−1
−1.5
−1.5
−1.5
−2 −1
−0.5
0
0.5
a) Upper Grid
1
−2 −1
−0.5
0
0.5
b) Lower Grid 2
1
−2 −1
SCRE Single λ −0.5
0
0.5
1
c) Resolution of MSRE vs. Standard
