6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients
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Elena Beretta
Abstract
We consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients μ(x), D(x), from a single measurement of the absorbed energy E(x) = μ(x)u(x), where u satisfies the elliptic partial differential equation
−∇ ⋅ (D(x)∇u(x)) + μ(x)u(x) =0 in Ω ⊂ ℝN .
This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. Using similar ideas as in [31], we show that when the coefficients μ, D are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of μ, D, we suggest a variational method based on an Ambrosio-Tortorelli approximation of a Mumford-Shah-like functional, which we implement numerically and test on simulated two-dimensional data.
Abstract
We consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients μ(x), D(x), from a single measurement of the absorbed energy E(x) = μ(x)u(x), where u satisfies the elliptic partial differential equation
−∇ ⋅ (D(x)∇u(x)) + μ(x)u(x) =0 in Ω ⊂ ℝN .
This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. Using similar ideas as in [31], we show that when the coefficients μ, D are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of μ, D, we suggest a variational method based on an Ambrosio-Tortorelli approximation of a Mumford-Shah-like functional, which we implement numerically and test on simulated two-dimensional data.
Chapters in this book
- Frontmatter I
- Contents V
-
Part I
- 1. Second-order decomposition model for image processing: numerical experimentation 5
- 2. Optimizing spatial and tonal data for PDE-based inpainting 35
- 3. Image registration using phase–amplitude separation 84
- 4. Rotation invariance in exemplar-based image inpainting 108
- 5. Convective regularization for optical flow 184
- 6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients 202
- 7. On optical flow models for variational motion estimation 225
- 8. Bilevel approaches for learning of variational imaging models 252
-
Part II
- 9. Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems 295
- 10 The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls 314
- 11. Controllability of Keplerian motion with low-thrust control systems 344
- 12. Higher variational equation techniques for the integrability of homogeneous potentials 365
- 13. Introduction to KAM theory with a view to celestial mechanics 387
- 14. Invariants of contact sub-pseudo-Riemannian structures and Einstein–Weyl geometry 434
- 15. Time-optimal control for a perturbed Brockett integrator 454
- 16. Twist maps and Arnold diffusion for diffeomorphisms 473
- 17. A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I 496
- Index 517
Chapters in this book
- Frontmatter I
- Contents V
-
Part I
- 1. Second-order decomposition model for image processing: numerical experimentation 5
- 2. Optimizing spatial and tonal data for PDE-based inpainting 35
- 3. Image registration using phase–amplitude separation 84
- 4. Rotation invariance in exemplar-based image inpainting 108
- 5. Convective regularization for optical flow 184
- 6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients 202
- 7. On optical flow models for variational motion estimation 225
- 8. Bilevel approaches for learning of variational imaging models 252
-
Part II
- 9. Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems 295
- 10 The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls 314
- 11. Controllability of Keplerian motion with low-thrust control systems 344
- 12. Higher variational equation techniques for the integrability of homogeneous potentials 365
- 13. Introduction to KAM theory with a view to celestial mechanics 387
- 14. Invariants of contact sub-pseudo-Riemannian structures and Einstein–Weyl geometry 434
- 15. Time-optimal control for a perturbed Brockett integrator 454
- 16. Twist maps and Arnold diffusion for diffeomorphisms 473
- 17. A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I 496
- Index 517
