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Structured Linear Algebra Problems: Analysis, Algorithms, and Applications Cortona, Italy - September 15-19, 2008 Structured matrices in nonlinear imaging Claudio Estatico Department of Mathematics and Informatics University of Cagliari, Italy


  1. Structured Linear Algebra Problems: Analysis, Algorithms, and Applications Cortona, Italy - September 15-19, 2008 Structured matrices in nonlinear imaging Claudio Estatico Department of Mathematics and Informatics University of Cagliari, Italy Joint work with: F. Di Benedetto (Dep. of Mathematics, University of Genova); J. G. Nagy (Dep. of Mathematics and Computer Science, Emory University); G. Bozza, M. Pastorino and A. Randazzo (Dep. of Biophysical and Electronic Engineering, University of Genova).

  2. Outline I - Nonlinear Inverse Problems in imaging (linear vs nonlinear case). II - The block matrix of the linearization iterative scheme. III - Exploiting the structure of the blocks: direct and iterative regularization blocks methods. IV - A three-level block splitting iterative regularization algorithm. V - SuperResolution post-processing enhancement. VI - Numerical results.

  3. Inverse Problem By the knowledge of some “observed” data y (i.e., the effect), find an approximation of some model parameters x (i.e., the cause). Usually, inverse problems are ill-posed, they need regularization technique. NonLinear Inverse Problem Given the data y ∈ Y , find (an approximation of) the unknown x ∈ X such that A ( x ) = y where A : X − → Y is a nonlinear operator (Fr´ echet differentiable), between the Hilbert spaces X and Y .

  4. A nonlinear inverse problem: The Microwave Inverse Scattering (nonlinear imaging) y ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� Source ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� Ω obj x ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� Ω inv Ω obs Input: scattered electromagnetic field on Ω obs (observation domain). Output: dielectric properties, i.e. the object, in Ω inv (investigation domain). The model leads to a nonlinear integral equation -particles’ interaction-. Features: very low degree of invasivity; provide information about the dielec- tric properties (instead of density); microwave cheap and easy to generate. Applications: medical imaging, nondestructive evaluations of materials, sub- surface prospecting,...

  5. Linear Imaging vs Nonlinear Imaging Inverse problem in Imaging: to reconstruct the true image x from the know- ledge of the acquired image A ( x ), where A is an integral operator. Linear imaging (Image Deblurring) � G ( r − r ′ ) x ( r ′ ) dr ′ ( A x )( r ) = Ω inv ∀ r ∈ Ω inv , i.e. the 2D investigation domain. Nonlinear imaging (Inverse Scattering) � G ( r − r ′ ) ( N ( x ))( r ′ ) dr ′ ( A ( x ))( r ) = Ω inv ∀ r ∈ Ω obs , i.e. the 2D observation domain, where N is a nonlinear functional (sometimes not completely known).

  6. Linear Imaging 50 50 100 100 150 150 200 200 250 250 50 100 150 200 250 50 100 150 200 250 True object to restore Input data for restoration Nonlinear imaging y 4 λ 0 λ 0 260 ���������� ���������� ���������� ���������� Reconstructed values ���������� ���������� ���������� ���������� ���������� ���������� 0.5 λ Actual values ���������� ���������� ���������� ���������� 0 ���������� ���������� 240 ���������� ���������� ���������� ���������� 4 λ 0 ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� 220 ���������� ���������� ���������� ���������� x 2 λ 0 ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� |E tot | [V/m] 200 ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� 180 ���������� ���������� ���������� ���������� ���������� ���������� 160 140 120 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 θ [rad] True object to restore Input data for restoration

  7. Microwave Inverse Scattering → E , where D = L 2 ( R 2 ) the Forward non-linear integral operator A : D − Dielectric permittivity, and E = L 2 ( R 2 ) the scattered Electric field, � G ( r − r ′ ) [( A ( χ ))( r ′ ) + u inc ( r ′ )] χ ( r ′ ) dr ′ ( A ( χ ))( r ) = Ω inv ∀ r ∈ Ω obs . χ ∈ D is the scatterer (i.e., the unknown to retrieve), A ( χ ) is the scattered field, unknown in Ω inv , u inc is the incident field, known everywhere, G is the known integral kernel. Inverse Scattering Problem INPUT: the scattered field u scat = A ( χ ) on the observation domain Ω obs OUTPUT: the scattering potential χ in the investigation domain Ω inv by solving of the nonlinear equation A ( χ ) = u scat

  8. Coupled formulation for both scatterer and scattered field Since u tot = u scat + u inc is not known inside the investigation domain Ω inv , we have to consider it as unknown too (together with the actual unknown χ to retrieve). We obtain two coupled integral equations. In the observation domain Ω obs (i.e., measured data): � G ( r − r ′ ) u tot ( r ′ ) χ ( r ′ ) dr ′ = u scat ( r ) ∀ r ∈ Ω obs . Ω inv In the investigation domain Ω inv : � G ( r − r ′ ) u tot ( r ′ ) χ ( r ′ ) dr ′ = u inc ( r ) u tot ( r ) − ∀ r ∈ Ω inv . Ω inv Remarks The apparatus is rotated (multiple views), in order to provide different acquisitions of u scat onto Ω obs . In the model we have to consider an index p = 1 , . . . , P related to the particular view.

  9. Full Formulation for both scatterer and scattered field By introducing the nonlinear operator A defined as   Ω inv G ( r − r ′ ) u 1 tot ( r ′ ) χ ( r ′ ) dr ′ � . . .     Ω inv G ( r − r ′ ) u P tot ( r ′ ) χ ( r ′ ) dr ′ �     A ( u 1 tot , . . . , u P   tot , χ )( r ) =   u 1 Ω inv G ( r − r ′ ) u 1 tot ( r ′ ) χ ( r ′ ) dr ′   � tot ( r ) −     . . .     u P Ω inv G ( r − r ′ ) u P tot ( r ′ ) χ ( r ′ ) dr ′ � tot ( r ) − � T � u 1 scat , . . . , u P scat , u 1 inc , . . . , u P and the data vector b = , inc the inverse scattering problem becomes: find χ ∈ L 2 (Ω inv ) and u s tot ∈ L 2 (Ω inv ), s = 1 , . . . , P , such that A ( u 1 tot , . . . , u P tot , χ ) = b

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