Applications of the Curvelet Transform to ImagingFrancisco J. Blanco-Silva

Institute for Mathematics and its ApplicationsUniversity of Minnesotablancosi@ima.umn.edu

Department of MathematicsPurdue University

fbs@math.purdue.edu

Curvelet Transforms

Curvelets are band-limited complex-valued functions Φαβθ : R2 → Cparametrized in a scale(α > 0) / location(β ∈ R2) / rotation(θ ∈ S1) space:The graph of the modulus of a curvelet looks like a brush stroke of a giventhickness (as given by α > 0), location on the canvas (β ∈ R2), and direction(θ ∈ S1).

Graph of |Φαβθ|α = 210, β = (0, 0), θ = 120o

Of particular importance will be the curvelet coefficients: For a functionf ∈ L2(R2),

〈f,Φαβθ〉 =

∫R2

f (x)Φαβθ(x) dx

Curvelet Analysis

Square integrable functions f ∈ L2(R2) can be represented by curvelets intwo ways:

• Continuous Curvelet Transform:

f (x) =

∫ ∞

0

∫S1

∫R2〈f,Φαβθ〉Φαβθ(x) dβ dσ(θ) dα

‖f‖2L2(R2) =

∫ ∞

0

∫S1

∫R2|〈f,Φαβθ〉|2dβ dσ(θ) dα

•Discrete Curvelet Transform: A well-chosen discrete subset of curvelet func-tions, when weighted by appropriate constants, are used to construct atight frame in L2(R2) with frame bound 1:

f (x) =∑n∈Z

2π/ϕn∑k=1

∑z∈Z2

〈f, φnkz〉φnkz

‖f‖L2(R2) =∑n∈Z

2π/ϕn∑k=1

∑z∈Z2

|〈f, φnkz〉|2

• The order of approximation of any function by finite sums of curvelet coef-ficients gives global information on the smoothness of the function f .

This will be used to classify images by smoothness and/or by the occur-rence of fractal structures; in this way we attempt to improve upon currenttechniques for Noise Removal by means of curvelet coefficient shrinkage(second column), or Image Compression and Removal of Artifactsby either linear or nonlinear approximation of signals with curvelets (thirdcolumn).

• The behavior of the sequences of “atoms” {〈f,Φαβθ〉}α indicates smooth-ness (or lack of it!) of the function f in a neighborhood of β, along thedirections given by θ and its perpendicular.

This fact will be used to detect singularities and the curves alongwhich these arise. As an application, we will explore how curveletanalysis helps solve problems in Local Tomography (fourth column).

E. Candes and D. Donoho. New tight frames of curvelets and optimalrepresentation of objects with piecewise C2 singularities. Comm. PureAppl. Math. 57(2):219–266, 2004.——— Continuous Curvelet Transform: I & IIhttp://www-stat.stanford.edu/~donoho/Reports/2003/

Classification of Natural Images

We follow the Besov approach to differentiability, in which smoothness ismeasured in terms of the behavior of successive differences in any direction,rather than limits. Smoothness is then quantized by three parameters: Givens ≥ 0, f ∈ Lp is in Bs

q(Lp) if its successive r = dse differences are asymptot-ically similar to the function t 7→ ts in an Lq sense.We are particularly interested in Besov Spaces continuously embedded inL2(R2):

Theorem (DeVore, Popov)

The parameters s ≥ 0, 0 < q ≤ 2 of the Besov space Bsq(Lq(R2)) of

minimal smoothness continuously embedded in L2(R2) satisfy 1q = s+1

2 .

In this case, either parameter s or q indicate both smoothness of the imageand occurrence of fractal structures on its graph.

s ≈ 0.5255 s ≈ 0.7437q ≈ 1.311 q ≈ 1.147

This landscape presents severalfractal structures of different di-mension (bushes, trees, clouds,mountain ridges, terrain. . . ) Eachof those structures contributes tolowering the smoothness (≈ 1/2)and increasing the value of q.

In contrast, this cartoon-like imagepresents almost no visible fractalstructures; as a consequence, s iscloser to 3/4.Are you able to locate thearea(s) where the image isnot so smooth?

The current methods of computation of these parameters are based on non-linear approximation by wavelets.Given an image f with 256 grey scales, let fN be the image obtained bycoding the N wavelet coefficients with largest absolute value, and let ENbe the corresponding error (in least squares sense). The plot below presentslog EN (VERTICAL) vs.− log N (HORIZONTAL), and the slope of the regressionline of this data is approximately s/2.

− log N

log EN

−13 −12.5 −12 −11.5 −11 −10.5

14

20

26

The knowledge of the smoothness (s, q) of a given image allows custom-madealgorithms for near-best Noise Removal by means of shrinkage of waveletcoefficients.It is very desirable to have a similar procedure based on curvelet coef-ficients, since these offer a more appropriate tool to recognize and decodefractal structures; thus, it is natural to expect a more accurate computationof smoothness parameters and therefore better noise removal.

A. Chambolle, R. DeVore, N. Lee and B. Lucier. NonlinearWavelet Image Processing: Variational Problems, Compression, andNoise Removal Through Wavelet Shrinkage. IEEE Tran. Image Proc., 1998A. Deliu and B. Jawerth. Geometrical Dimension versus Smoothness.Constructive Approximation, 8, 1992

Detection and Classification of Singularities

The objects typically studied in Tomography are made up of regions ofnearly constant density, and so the images one treats in this field are essentiallylinear combinations of characteristic functions of sets. Insight into the natureof Local Tomography is obtained by studying the lambda operator onthese functions.

Phantom image f |Λf |To obtain images of projections of the brain and chest where, for instance, thearteries are brighter than the chambers (and thus stand out), Local Tomogra-phy is used in combination with the enhancement of the contrast of sets withsmall diameter.The reason why this method works better resides in the fact that the lambdaoperator preserves singularities and the curves in which these arise, but turnseverything else smooth:

Theorem (Faridani et al)

WF(f ) = WF(Λf ); in particular, sing supp f = sing supp Λf .If X is a measurable set in Rd (d ≥ 2), then Λ1X is an analytic function

on (∂X){ = Rd \ ∂X .

Curvelets also detect both the wavefront set and singular support of distribu-tions in R2:

Theorem (Candes, Donoho)

Let S(f ) be the set of points x ∈ R2 such that 〈f,Φαβθ〉 decays rapidlyfor β near x as α → ∞ or α → 0; then sing supp f is the complementof S(f ).LetM(f ) be the set of pairs (x, ω) ∈ R2×S1 such that 〈f,Φαβθ〉 decaysrapidly for (β, θ) near (x, ω) as α →∞ or α → 0. Then WF(f ) is thecomplement of M(f ).

The advantage of using curvelets comes from the fact that the order of decay ofthe coefficients {〈f,Φαβθ〉}α gives extra information about the type of singu-larity. The problem arises when we have actual data instead of a well definedfunction, since it is impossible (a priori) to compute asymptotic behavior ofthe curvelet coefficients.We seek to use the link between curvelet analysis and local tomography tofind solutions to the following two related research problems:

How can Curvelet Analysis help improve Local Tomogra-phy for limited-angle data?

How can analysis of the lambda operator help approximatethe asymptotic behavior of curvelet coefficients (and thus,the classification of singularities)?

A. Faridani, D. Finch, E. Ritman, K. Smith. Local Tomography.SIAM J. Appl. Math. 57 1095–1127, 1997.

Acknowledgements:

•Bradley J. Lucier (Purdue University)

•Adel Faridani (Oregon State University)

•Taufiquar Khan (Clemson University)

Giulio Ciraolo, Gloria Haro, Stacey Levine, Hstau Liao, AlisonMalcolm, Scott MacLachlan, Carl Toews, Alan Thomas.

Removal of ArtifactsConsider the following problem:

Given a synthetic image, detect groups of features orga-nized on linear structures (even if these features are notsegments or paths of almost-flat curves).

The image on the left is a phantom consisting of circles, some of which arearranged linearly. On the right, we have presented the reconstruction of theprevious image with only the curvelet coefficients of highest scaling level. Wewill call it, the Finest-Elements Component (FEC) of an image.Observe how the FEC shows the occurrence of segment-like structures fromthe original image.

Synthetic image: (FEC) Reconstruction withSets of cicles Curvelet coefficients

arranged linearly of highest scaling level

Curvelet Analysis is the right tool to locate artifacts with a linear struc-ture: The high scale curvelets code information of curvilinear features such aspath singularities, and also any other structure that can be well approximatedby ellipses with one axis considerably longer than the other.The information retrieved is then used to attack the following related problem:

Giving a synthetic image, remove groups of features organizedon linear structures.

Using the data from the FEC, we search for algorithms to remove artifactsorganized on linear structures. We briefly present some ideas:

•Combination of Reconstructions. By gathering curvelet coefficientsof the same scaling level, one obtains partial reconstructions of the origi-nal image where directional features of different thicknesses are enhanced.Linear combinations of these partial reconstructions are considered as ap-proximations to the solution of the problem of removal of artifacts. Weexpect to find good candidates by minimizing the L2–norm of the corre-sponding FEC.

Reconstruction of different scaling levels

•Modification of Curvelet Coefficients. FEC offers informationabout location of the conflicting structures; one may modify associatedcurvelet coefficients and reconstruct, again looking for reconstructions withFEC having the smallest L2–norm. In order to find good modificationformulas for the curvelet coefficients, we pose the previous question as aVariational Problem, and use techniques from the Calculus of Variations.

A copy of this poster can be retrieved at http://www.math.purdue.edu/~fbs/acting.pdf