Two Dimensional Edge Detection From Fourier DataAaron Jesse aaron.jesse@asu.edu Arizona State University Anne Gelb anne.gelb@asu.edu

AbstractA common problem in image reconstruction isidentifying jump locations from Fourier coeffi-cients. While techniques exist which determineedges directly from Fourier data, they are inher-ently one-dimensional. Here, we adapt the mul-tidimensional polynomail annihilation method [2]for given Fourier data by projecting the Fourier co-efficients onto polynomial space. Further, limitedresolution is enhanced without accumulating moreFourier data by using a new subcell resolution en-hancement approach.

Polynomial AnnihilationLet [f ](x) = f(x+)− f(x−) define a jump functionfor piecewise smooth f(x). The polynomial anni-hilation method is given by [2]:

Lmf(x) =1

qm(x)

∑

xj∈Sx

cj(x)f(xj) → [f ](x)

where {f(xj)}Nj=0 is a finite sampling of f(x), Sx is

a stencil defined as the nearest m + 1 points to x,and qm(x) normalizes the jump function approxi-mation. The annihilation coefficients cj(x) satisfy

∑

xj∈Sx

cj(x)pi(xj) = p(m)i (x), i = 1, ...,m.

Hence in smooth regions, where [f](x)=0 can bewell approximated by a polynomial, we haveLmf(x) = [f ](x) + O(h(x)m) where h(x) is thelargest distance between neighboring gridpoints.Conversely, if x = ξ is a jump discontinuity, thenLmf(ξ) = [f ](ξ) + O(h(x)). Now, suppose insteadof {f(xj)}

Nj=0 we are given

{gk}Kk=0 = {< f,Ψk >w}

Kk=0

where {Ψk}Kk=0 is an orthogonal basis with re-

spect to the weight function w, so that PKf(x) =∑K

k=0 gkΨk(x). In [3], it was shown that

LmPKf(x) =1

qm(x)

K∑

k=m

gk∑

xj∈Sx

cj(x)Ψk(x) (1)

converges to [f ](x).

Polynomial Annihilation From Fourier DataNow suppose you are given Fourier coefficients {fk}

Nk=−N so that

SNf(x) =

N∑

k=−N

fkeikπx.

In [4], it was shown that ‖PK(f−SNf)‖ → 0 exponentially for the Gegenbauerpolynomials Ψk(x) = Cλ

k (x), for large enough λ. Hence, define

gk =< SNf(x), Cλk (x) >(1−x2)λ−1/2

and construct

LmPKSNf(x) =1

qm(x)

K∑

k=m

gk∑

xj∈Sx

cj(x)Cλk (xj).

Notes:

1. To optimize the accuracy of the polynomial annihilation method, the re-construction grid is chosen to be the roots of Cλ

K(x), thus ensuring a smallLebesgue constant.

2. This will cause the reconstruction grid to have points concentrated nearthe boundaries, which is not likely to be the natural place for edges toexist.

3. We can improve results by introducing new cells based on different val-ues of λ (see Subcell Resolution Enhancement).

4. These methods are combined with nonlinear postprocessing techniquesto preserve jump heights and add additional accuracy and resolution.

Two DimensionsIn two dimensions, the polynomial annihilation method is written as [2]:

Lmf(x) =1

qm(x)

∑

xj∈Sx

cj(x)f(xj).

Sx is now the set of the closest m2−3 points to x, where m2 =(

m+22

)

and cj(x),j = 1, ...,m2, are chosen so that

∑

xj∈Sx

cj(x)pi(xj) =∑

|α|1=m

p(α)i (x), i = 1, ...,m2.

Hereα is a two-dimensional vector of any positive integers, with |α|1 = α1+α2.

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a) Sample Cell f(x) MM(LM f(x)) MM(LMf(x)) Top View

Subcell Resolution EnhancementThe multi-polynomial subcell resolution enhancement algorithm for P sets ofbasis polynomials can now be defined as:

1. Define each basis polynomial as ϕpk(x), p = 1, ..., P .

Define xj as the cell value between [xj , xj+1].

2. For p = 1 : P , Define an upper grid XpU = (x1, x2, ..., xK) such that

ϕpk(xk) = 0. (Gaussian point distribution)

3. Define the lower grid XL =⋃

p∈P XpU .

4. For p = 1 : P , Compute LMPpk f(xj), xj ∈ X

pU using (1). Use initial post-

processing if desired. Compute LMPpk f(xj), xj ∈ XL for each X

pU by

determining the appropriate upper cell and projecting it onto the corre-sponding lower cell.

5. Use postprocessing (e.g. Minmod) to combine LMPpk f(xj), xj ∈ X

pL, p =

1, ..., P .

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f(x)

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a) Original Signal b) Upper Grid

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SCRESingle λ

b) Lower Grid c) Resolution of MSRE vs. Standard

Open ResearchWhile the polynomial annihilationusing Fourier to Gegenbauer coeffi-cients method itself has been appliedsuccessfully to 2D, the Subcell Res-olution Enhancement algorithm hasnot. The algorithm itself will re-main unchanged, but cost effectiveimplementation becomes more com-plicated due to the increased cost ofprojection from the upper grid to thelower grid with larger numbers ofgridpoints. Additionally, other basispolynomials are being explored, asperhaps another basis will provide anice set of points which can be addedto the Subcell Resolution Enhance-ment algorithm.

References[1] J. Hesthaven, S. Gottlieb, and D. Gottlieb,

"Spectral Methods for Time-DependentProblems" Cambridge University Press,2007.

[2] R. Archibald, A. Gelb, and J. Yoon, "Poly-nomial Fitting For Edge Detection In Ir-regularly Sampled Signals and Images,"SIAM J. Appl. Numer. Anal., vol. 43(1), pp.259-279, 2005.

[3] R. Archibald, A. Gelb, R. Saxena, andD. Xiu, "Discontinuity Detection in Multi-variate 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 phe-nomenon I: recovering exponential accu-racy from the Fourier partial sum of a non-periodic analytic function," J. Comp. Appl.Math., vol. 43, 1992.