Microlocal Analysis of the Geometric Separation Problem

Image data are often composed of two or more geometrically distinct constituents; in galaxy catalogs, for instance, one sees a mixture of pointlike structures (galaxy superclusters) and curvelike structures (filaments). It would be ideal to process a single image and extract two geometrically `pure' images, each one containing features from only one of the two geometric constituents. This seems to be a seriously underdetermined problem, but recent empirical work achieved highly persuasive separations. We present a theoretical analysis showing that accurate geometric separation of point and curve singularities can be achieved by minimizing the $\ell_1$ norm of the representing coefficients in two geometrically complementary frames: wavelets and curvelets. Driving our analysis is a specific property of the ideal (but unachievable) representation where each content type is expanded in the frame best adapted to it. This ideal representation has the property that important coefficients are clustered geometrically in phase space, and that at fine scales, there is very little coherence between a cluster of elements in one frame expansion and individual elements in the complementary frame. We formally introduce notions of cluster coherence and clustered sparsity and use this machinery to show that the underdetermined systems of linear equations can be stably solved by $\ell_1$ minimization; microlocal phase space helps organize the calculations that cluster coherence requires.

...but we assume fj is known and Pj and Cj are unknown to us. The following result shows that in a certain sense they are recoverable by solving (CSep). Theorem 1.1 ([10]): ASYMPTOTIC SEPARATION...
...Proposition 2.1 ([10]): Suppose that S can be decomposed as S = S0 1 +S0 2 so that each component S0 i is relatively sparse in �i, i = 1,2, i.e.,...
...In contrast, cluster coherence bou nds coherence between a single member of frame � and a cluster of members of frame �, clustered at S. Proposition 2.2 ([10]): We have...
...For the precise, technically quite involved definition we would lik e to refer to [10]...
...Corollary 3.1 ([10]): Suppose that the sequence of transform-space clusters (S1,j), and (S2,j) has both of the following two properties: (i) asymptotically negligible c luster coherences:...
...The paper [10] designs clusters of wavelet coefficients S1,j and curvelet coefficients S2,j inspired by these intuitions...
...For the tedious and involved proof we refer to [10]...
...Lemma 3.1 ([10]): The sequence of transform-space clusters (S1,j), and (S2,j) provides (i) asymptotically negligible cluster coherences:...

Keywords (7)

Microlocal Analysis of the Geometric Separation Problem,Computing Research Repository,David L. Donoho,Gitta Kutyniok

Citation Context (2)

David L. Donoho, et al. Analysis of 1 minimization in the Geometric Separation Problem

GITTA KUTYNIOK. WHAT IS APPLIED HARMONIC ANALYSIS?

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Citations (8)

Uncertainty Relations and Sparse Signal Recovery for Pairs of General Signal Sets (Citations: 1)

Data Separation by Sparse Representations

Compressed Sensing with Coherent and Redundant Dictionaries (Citations: 12)

Geometric Separation using a Wavelet-Shearlet Dictionary (Citations: 2)

Development of a digital shearlet transform based on Pseudo-Polar FFT (Citations: 2)


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