A&A 398, 785–800 (2003) Astronomy DOI: 10.1051/0004-6361:20021571 & c ESO 2003 Astrophysics Astronomical image representation by the curvelet transform J. L. Starck1, D. L. Donoho2, and E. J. Cand`es3 1 DAPNIA/SEDI-SAP, Service d’Astrophysique, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France 2 Department of Statistics, Stanford University, Sequoia Hall, Stanford, CA 94305, USA 3 Department of Applied Mathematics, Mail Code 217-50, California Institute of Technology, Pasadena, CA 91125, USA Received 12 April 2002 / Accepted 29 October 2002 Abstract. We outline digital implementations of two newly developed multiscale representation systems, namely, the ridgelet and curvelet transforms. We apply these digital transforms to the problem of restoring an image from noisy data and com- pare our results with those obtained via well established methods based on the thresholding of wavelet coefficients. We show that the curvelet transform allows us also to well enhance elongated features contained in the data. Finally, we describe the Morphological Component Analysis, which consists in separating features in an image which do not present the same morpho- logical characteristics. A range of examples illustrates the results. Key words. methods: data analysis – techniques: image processing 1. Introduction from wavelet-like systems. Curvelets and ridgelets take the the form of basis elements which exhibit very high directional The wavelet transform has been extensively used in astronom- sensitivity and are highly anisotropic. In two-dimensions, for ical data analysis during the last ten years. A quick search instance, curvelets are localized along curves, in three dimen- with ADS shows that around 600 papers contain the keyword sions along sheets, etc. Continuing at this informal level of “Wavelet” in their abstract, and all astrophysical domains were discussion we will rely on an example to illustrate the funda- concerned, from the sun study to the CMB analysis. mental difference between the wavelet and ridgelet approaches This large success of the wavelet transform (WT) is due to –postponing the mathematical description of these new sys- the fact that astronomical data presents generally complex hi- tems. erarchical structures, often described as fractals. Using multi- Consider an image which contains a vertical band embed- scale approaches such as the wavelet transform (WT), an image ded in white noise with relatively large amplitude. Figure 1 (top can be decomposed into components at different scales, and the left) represents such an image. The parameters are as follows: WT is therefore well-adapted to astronomical data study. the pixel width of the band is 20 and the SNRis set to be 0.1. A series of recent papers (Cand`es & Donoho 1999d; Note that it is not possible to distinguish the band by eye. The Cand`es & Donoho 1999c), however, argued that wavelets and wavelet transform (undecimated wavelet transform) is also in- related classical multiresolution ideas are playing with a lim- capable of detecting the presence of this object; roughly speak- ited dictionary made up of roughly isotropic elements occur- ing, wavelet coefficients correspond to averages over approxi- ring at all scales and locations. We view as a limitation the mately isotropic neighborhoods (at different scales) and those facts that those dictionaries do not exhibit highly anisotropic wavelets clearly do not correlate very well with the very elon- elements and that there is only a fixed number of directional el- gated structure (pattern) of the object to be detected. ements, independent of scale. Despite the success of the classi- cal wavelet viewpoint, there are objects, e.g. images that do not We now turn our attention towards procedures of a very exhibit isotropic scaling and thus call for other kinds of multi- different nature which are based on line measurements. To be scale representation. In short, the theme of this line of research more specific, consider an ideal procedure which consists in is to show that classical multiresolution ideas only address a integrating the image intensity over columns; that is, along the portion of the whole range of interesting multiscale phenom- orientation of our object. We use the adjective “ideal” to em- ena and that there is an opportunity to develop a whole new phasize the important fact that this method of integration re- range of multiscale transforms. quires a priori knowledge about the structure of our object. This method of analysis gives of course an improved signal to noise Following on this theme, Cand`es & Donoho introduced ratio for our linear functional better correlate the object in ques- new multiscale systems like curvelets (Cand`es & Donoho tion, see the top right panel of Fig. 1. 1999c) and ridgelets (Cand`es 1999) which are very different This example will make our point. Unlike wavelet Send offprint requests to: J. L. Starck, e-mail: [email protected] transforms, the ridgelet transform processes data by first Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20021571 786 J. L. Starck et al.: The curvelet transform Fig. 1. Top left, original image containing a vertical band embedded in white noise with relatively large amplitude. Top right, signal obtained by integrating the image intensity over columns. Bottom left, reconstructed image for the undecimated wavelet coefficient, bottom right, reconstructed image from the ridgelet coefficients. computing integrals over lines with all kinds of orientations and tion which is a solution of an optimization problem in this joint locations. We will explain in the next section how the ridgelet representation. transform further processes those line integrals. For now, we apply naive thresholding of the ridgelet coefficients and 2. The curvelet transform “invert” the ridgelet transform; the bottom right panel of Fig. 1 shows the reconstructed image. The qualitative difference with 2.1. The ridgelet transform the wavelet approach is striking. We observe that this method The two-dimensional continuous ridgelet transform in R2 can allows the detection of our object even in situations where the be defined as follows (Cand`es 1999). We pick a smooth uni- noise level (standard deviation of the white noise) is five times variate function ψ: R R with sufficient decay and satisfying superior to the object intensity. the admissibility condition→ The contrasting behavior between the ridgelet and the wavelet transforms will be one of the main themes of this pa- ψˆ(ξ) 2/ ξ 2 dξ< , (1) per which is organized as follows. We first briefly review some Z | | | | ∞ basic ideas about ridgelet and curvelet representations in the which holds if, say, ψ has a vanishing mean ψ(t)dt = continuum. In parallel to a previous article (Starck et al. 2002), 0. We will suppose a special normalizationR about ψ Sect. 2 rapidly outlines a possible implementation strategy. so that ∞ ψˆ(ξ) 2ξ 2dξ = 1. Sections 3 and 4 present respectively how to use the curvelet 0 | | − ForR each a > 0, each b R and each θ [0, 2π), we define transform for image denoising and and image enhancement. ∈ 2 ∈ the bivariate ridgelet ψa,b,θ: R R by We finally develop an approach which combines both the → 1/2 wavelet and curvelet transforms and search for a decomposi- ψa,b,θ(x) = a− ψ((x cos θ + x sin θ b)/a); (2) · 1 2 − J. L. Starck et al.: The curvelet transform 787 1 1 0.5 0.5 Z Z 0 0 -0.5 -0.5 6 4 4 2 4 2 6 0 4 2 0 Y 2 Y -2 0 X -2 0 -2 X -4 -4 -2 -4 -4 -6 -6 1 1 0.5 0.5 Z Z 0 0 -0.5 -0.5 6 6 4 4 2 6 2 6 4 0 0 2 24 Y Y -2 0 -2 0 X X -4 -2 -4 -2 -4 -4 -6 - -6 6 -6 Fig. 2. A few ridgelets. A ridgelet is constant along lines x1 cos θ + x2 sin θ = const. where δ is the Dirac distribution. Then the ridgelet transform Transverse to these ridges it is a wavelet. is precisely the application of a 1-dimensional wavelet trans- Figure 2 graphs a few ridgelets with different parameter val- form to the slices of the Radon transform where the angular ues. The top right, bottom left and right panels are obtained af- variable θ is constant and t is varying. ter simple geometric manipulations of the upper left ridgelet, This viewpoint strongly suggests developing approximate namely rotation, rescaling, and shifting. Radon transforms for digital data. This subject has received a Given an integrable bivariate function f (x), we define its considerable attention over the last decades as the Radon trans- ridgelet coefficients by form naturally appears as a fundamental tool in many fields of scientific investigation. Our implementation follows a widely f (a, b,θ) = ψ (x) f (x)dx. R Z a,b,θ used approach in the literature of medical imaging and is based We have the exact reconstruction formula on discrete fast Fourier transforms. The key component is to obtain approximate digital samples from the Fourier transform 2π da dθ f x ∞ ∞ a, b,θ ψ x b on a polar grid, i.e. along lines going through the origin in the ( ) = f ( ) a,b,θ( ) 3 d (3) Z0 Z Z0 R a 4π −∞ frequency plane. valid a.e. for functions which are both integrable and square integrable. Ridgelet analysis may be constructed as wavelet analysis in 2.2. An approximate digital ridgelet transform the Radon domain. Recall that the Radon transform of an ob- 2.2.1. Radon tranform ject f is the collection of line integrals indexed by (θ, t) ∈ [0, 2π) R given by A fast implementation of the RT can be performed in the × Fourier domain.