THE CURVELET TRANSFORM FOR IMAGE FUSION

More documents

Recommendations

Info

2II. RIDGELET AND CURVELET TRANSFORMSIn this section, we review the theory of ridgelet and curvelettransforms, presented in [5]–[10], which are used in thesubsequent sections.A. Continuous ridgelet transformLet ψ be in L 2 (R) with sufficient decay and satisfying theadmissibility condition,∫ |2 ˆψ(ξ)|K ψ :=|ξ| 2 dξ < ∞,In this paper, we adopt Meyer wavelet ψ, which has highsmoothness and a compact support in the frequency domain.We refer to [11], [12] for the definition and the basic propertiesof Meyer wavelet. We will suppose that ψ is normalized sothat K ψ = 1.For each a > 0, b ∈ R, and θ ∈ [0, 2π], ridgelet basisfunctions are defined byψ a,b,θ (x) := a −1/2 ψ((x 1 cos θ + x 2 sin θ − b)/a).The Radon transform for f ∈ L 2 (R 2 ) is given by∫Rf(θ, t) = f(x 1 , x 2 )δ(x 1 cos θ + x 2 sin θ − t)dx 1 dx 2 ,where δ is the Dirac distribution. For f ∈ L 1 (R 2 ), thecontinuous ridgelet coefficient is given by∫R f (a, b, θ) = f(x)ψ a,b,θ (x)dx. (1)Then any function f ∈ L 1 (R 2 ) ∩ L 2 (R 2 ) is represented as acontinuous superposition of ridgelet functionsf(x) =∫ 2π ∫ ∞ ∫ ∞0−∞0R f (a, b, θ)ψ a,b,θ (x) da dθdba3 4πNotice that (1) is also given by the wavelet transform of theRadon transform f:∫R f (a, b, θ) = Rf(θ, t)a −1/2 ψ((t − b)/a)dt.Deans found that the two-dimensional Fourier transform of fis equal to the one-dimensional Fourier transform of the Radontransform of f [13]:∫ˆf(ξ cos θ, ξ sin θ) = Rf(θ, t)e −iξt dt.Hence the Radon transform of f is obtained by the onedimensionalinverse Fourier transform of ˆf(ξ cos θ, ξ sin θ) asa function of ξ.B. Curvelet transformLet Q denote a dyadic square [k 1 /2 s , (k 1 + 1)/2 s ) ×[k 2 /2 s , (k 2 + 1)/2 s ) and let Q s be the collection of all dyadicsquares of scale s. We also let w Q be a window near Q,obtained by dilation and translation of a single w, satisfying∑Q∈Q sw 2 Q = 1.We define multiscale ridgelets {w Q T Q ψ a,b,θ : s ≥ s 0 , Q ∈Q s , a > 0, b ∈ R, θ ∈ [0, 2π)}, where T Q f(x 1 , x 2 ) =2 s f(2 s x 1 − k 1 , 2 s x 2 − k 2 ). Then we have the reconstructionformulaf = ∑Q∈Q sfw 2 Q= ∑Q∈Q s∫ 2π ∫ ∞ ∫ ∞0−∞0da dθ〈f, w Q T Q ψ a,b,θ 〉w Q T Q ψ a,b,θ dba3 4πThe curvelet transform is given by filtering and applying multiscaleridgelet transform on each bandpass filters as follows:1) Subband Decomposition. The image is filtered into subbands:f → (P 0 f, ∆ 1 f, ∆ 2 f, · · · ),where a filter P 0 deals with frequencies |ξ| ≤ 1 and thebandpass filter ∆ s is concentrated near the frequencies[2 s , 2 2s+2 ], for example,∆ s = Ψ 2s ∗ f, ˆΨ 2s (ξ) = ˆΨ(2 −2s ξ);2) Smooth Partitioning. Each subband is smoothly windowedinto “squares” of an appropriate scale∆ s f → (w Q ∆ s f) Q∈Qs ;3) Renormalization. Each resulting square is renormalizedto unit scaleg Q = (T Q ) −1 (w Q ∆ s f), Q ∈ Q s ;4) Ridgelet Analysis. Each square is analysed via the discreteridgelet transform.For improved visual and numerical results of the digitalcurvelet transform, Starck et al. presented the following discretecurvelet transform algorithm [6], [14]:1) apply the à trous algorithm with J scales:J∑I(x, y) = C J (x, y) + w j (x, y),j=1where c J is a coarse or smooth version of original imageI and w j represents “the details of I” at scale 2 −j ;2) set B 1 = B min ;3) for j = 1, · · · , J doa) partition the subband w j with a block size B j andapply the digital ridgelet transform to each block;b) else B j+1 = B j .III. THE IMAGE FUSION METHOD BASED ON THECURVELET TRANSFORMThe following is the specific operational procedure for theproposed curvelet-based image fusion approach. While theoperational procedure may be generally applied , LandsatETM+ images are utilized as an example in order to illustratethe method.1) The original Landsat ETM+ panchromatic and multispectralimages are geometrically registered to each other.
3Fig. 1.Curvelet-based image fusion method.2) Three new Landsat ETM+ panchromatic images I 1 , I 2and I 3 are produced. The histograms of these images arespecified according to the histograms of the multispectralimages R, G and B, respectively.3) Using the well-known wavelet-based image fusionmethod, we obtain fused images I 1 + R, I 2 + G andI 3 + B, respectively.4) I 1 , I 2 and I 3 , which are taken from 2), are decomposedinto J + 1 subbands, respectively, by applying “à trous”subband filtering algorithm. Each decomposed imageincludes C J , which is a coarse or smooth version of theoriginal image, and w j , which represents “the details ofI” at scale 2 −j .5) Each C J is replaced by a fused image obtained from 3).For example, (C J for I 1 ) is replaced by (I 1 + R).6) The ridgelet transform is then applied to each block inthe decomposed I 1 , I 2 and I 3 , respectively.7) Curvelet coefficients (or ridgelet coefficients) are modifiedusing hard-thresholding rule in order to enhanceedges in the fused image.8) Inverse curvelet transforms (ICT) are carried out for I 1 , I 2and I 3 , respectively. Three new images ( F 1 , F 2 and F 3 )are then obtained, which reflect the spectral informationof the original multispectral images R, G and B, and alsothe spatial information of the panchromatic image.9) F 1 , F 2 and F 3 are combined into a single fused imageF .In this approach, we can obtain an optimum fused image thathas richer information in the spatial and spectral domainssimultaneously. Therefore, we can easily find small objectsin the fused image and separate them. As such, the curveletsbasedimage fusion method is very efficient for image fusion.IV. EXPERIMENTAL STUDY AND ANALYSISA. Visual analysisSince the wavelet transform preserves the spectral informationof the original multispectral images, it has the highspectral resolution in contrast with the IHS-based fusion result,which has some colour distortion. But the wavelet-basedfusion result has much less spatial information than that ofthe IHS-based fusion result.To overcome this problem, we use the curvelet transform inimage fusion. Since the curvelet transform is well adapted torepresent panchromatic image containing edges, the curveletbasedfusion result has both high spatial and spectral resolution.From the curvelet-based fusion result in the Landsat ETM+image fusion presented in Figure 2, it should be noted that both
Page 1: 1THE CURVELET TRANSFORM FOR IMAGEFU
Page 5: 5TABLE IA COMPARISON OF IMAGE FUSIO

THE CURVELET TRANSFORM FOR IMAGE FUSION

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?