Image Processing by the Curvelet Transform Jean Luc Starck Dapnia/SEDI SAP, CEA Saclay, France. jstarck@cea.fr http://jstarck.free.fr
Collaborators: D.L. Donoho, Department of Statistics, Stanford E. Candès, department of Applied Mathematics, California Institute of Technology The Curvelet Transform for Image Denoising, IEEE Transaction on Image Processing, 11, 6, 2002. Gray and Color Image Contrast Enhancement by the Curvelet Transform, IEEE Transaction on Image Processing, in press. The Astronomical Image Representation by the Curvelet Transform, Astronomy and Astrophysics, in press. Experiments: http://jstarck.free.fr http://www stat.stanford.edu/~jstarck
Problems related to the WT 1) Edges representation: if the WT performs better than the FFT to represent edges in an image, it is still not optimal. 2) There is only a fixed number of directional elements independent of scales. 3) Limitation of existing scale concepts: there is no highly anisotropic elements.
Non Linear Approximation Wavelet Curvelet Width = Lengh²
Rate of Approximation Suppose we have a function f which has a discontinuity across a curve, and which is otherwise smooth, and consider approximating f from the best m terms in the Fourier expansion. The squarred error of such an m term expansion obeys: f f m F In a wavelet expansion, we have 2 m 1 2, m f f m W 2 m 1, m In a curvelet expansion (Donoho and Candes, 2000), we have f f m C 2 log m 3 m 2, m
The Curvelet Transform The curvelet transform can be seen as a combination of reversible transformations: à trous 2D isotropic wavelet transform partitionning ridgelet transform. Radon Transform. 1D Wavelet transform
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A trous algorithm: I x,y =c J J x,y j=1 w j,x,y
PARTITIONING
The ridgelet coefficients of an object f are given by analysis of the Radon transform via: R f a,b,ϑ = R f ϑ,t ψ t b a dt FFT IMAGE FFT2D FFF1D 1 WT1D Radon Transform Ridgelet Transform Angle Frequency
CONTRAST ENHANCEMENT y c x,σ =1 if x<c σ y c x,σ = x c σ c σ m c σ p 2c σ x c σ if x<2c σ Ĩ =C R y c C T I y c x,σ = m x p if 2c σ x<m y c x,σ = m x s if x>m Modified curvelet coefficient Curvelet coefficient
Contrast Enhancement
F
Color Images R,G,B ===> L, U, V We apply the curvelet transform to the three components At each scale and at each position, we calculate: Coefficients correction Inverse curvelet transform L,UV to RGB e= c L 2 c u 2 c v 2 c c c = y e c y e c y e c L, u, v c L, c u, c v
NOISE MODELING FILTERING For a positive coefficient: P=Prob W >w For a negative coefficient P = Prob W < w Given a threshold t: if P > t, the coefficient could be due to the noise. if P < t, the coefficient cannot be due to the noise, and a significant coefficient is detected. ỹ=c R δ C T y Hard Thresholding: δ c =c if c t =0 if c <t Soft Thresholding: δ c =sgn c c t +
FILTERING
PSNR Lenna Curvelet Decimated wavelet Undecimated wavelet Noise Standard Deviation
Barbara Curvelet Decimated wavelet Undecimated wavelet
Curvelet Wavelet Curvelet
RESTORATION: HOW TO COMBINE SEVERAL MULTISCALE TRANSFORMS? The problem we need to solve for image restoration is to make sure that our reconstruction will incorporate information judged as significant by any of our representations. Notations: Consider K linear transforms and α K the coefficients of x after applying :. T K T 1,...,T K α k =T k s, s=t 1 k α k
We propose solving the following optimization problem: min Complexity_penalty, subject to s C s Where C is the set of vectors which obey the linear constraints: s>0, positivity constraint T k s T k s l e, if T k s l is significant The second constraint guarantees that the reconstruction will take into account any pattern which is detected by any of the K transforms.
DECONVOLUTION: s=p s N We propose solving the following optimization problem: min Complexity_penalty, subject to s C s Where C is the set of vectors which obey the linear constraints: s>0, positivity constraint T k s T k P s l e, if T k s l is significant The second constraint guarantees that the reconstruction will take into account any pattern which is detected by any of the K transforms.
Morphological Component Analysis Given a signal s, we assume that it is the result of a sparse linear combination of atoms from a known dictionary D. φ γ γ Γ A dictionary D is defined as a collection of waveforms, and the goal is to obtain a representation of a signal s with a linear combination of a small number of basis such that: s= γ α γ φ γ Or an approximate decomposition: s= γ α γ φ γ R
Example Composed Signal 0.1 0.05 0 0.05 0.1 φ 1 1.0 0.1 0.05 0 0.05 0.1 1 0.5 0 1 0.5 0 0 64 128 φ 2 φ 3 φ 4 0.3 0.5 0.05 + 0.110 0 0 0.1 0.2 0.3 0.4 0.5 0.6 10 2 10 4 10 6 10 8 10 10 T{φ 1 +0.3φ 2 +0.5φ 3 +0.05φ 4 } T{φ1 +0.3φ 2} 0 20 40 60 80 100 120 DCT Coefficients 38
Example Desired Decomposition 10 0 10 2 10 4 10 6 10 8 10 10 0 40 80 120 160 200 240 DCT Coefficients Spike (Identity) Coefficients 39
Formally, the sparsest coefficients are obtained by solving the optimization problem: (P0) Minimize α subject to 0 s=φα It has been proposed to replace the l norm by the l¹ norm (Chen, 1995): (P1) Minimize α subject to 1 s=φα It can be seen as a kind of convexification of (P0). It has been shown (Donoho and Huo, 1999) that for certain dictionary, it there exists a highly sparse solution to (P0), then it is identical to the solution of (P1).
We consider now that the dictionary is built of a set of L dictionaries related to multiscale transforms, such wavelets, ridgelet, or curvelets. Considering L transforms, and α k the coefficents relative to the kth transform φ= φ 1,...,φ L, α= α 1,...,α L, L s=φα= k=1 φ k α k a solution α is obtained by minimizing a functional of the form: J α = s k=1 L φ k α k 2 2 λ α p
An efficient algorithm is the Block Coordinate Relaxation Algorithm (Sardy, Bruce and Tseng, 1998): Initialize all to zero Iterate j=1,...,m Iterate k=1,..,l Update the kth part of the current solution by fixing all other parts and minimizing: J α k α k L = s i=1, i k φ i α i φ k α k 2 2 λ α k 1 Which is obtained by a simple soft thresholding of : s r = s i=1, i k L φ i α i
a) Simulated image (gaussians+lines) b) Simulated image + noise c) A trous algorithm d) Curvelet transform e) coaddition c+d f) residual = e b
a) A370 b) a trous c) Ridgelet + Curvelet Coaddition b+c
Galaxy SBS 0335 052 Ridgelet Curvelet A trous WT
Galaxy SBS 0335 052 10 micron GEMINI OSCIR CTM = Ridgelet Curvelet A trous WT Residual + + + Rid+Curvelet = + Clean Data
Galaxy SBS 0335 052 20 micron GEMINI OSCIR Rid+Curvelet A trous WT Residual Clean Data +