3D Linearised Calderón Problem: Exact Reconstruction Method with Zernike Basis

Core Concepts

Reconstruction of unbounded perturbations in 3D using a Zernike basis.

Abstract

The article discusses the linearised Calderón problem in three dimensions, focusing on exact direct reconstruction of any L3 perturbation from linearised data. The method utilizes a 3D Zernike basis for efficient implementation and requires only a subset of boundary measurements for reconstruction compared to a full L2 basis. The content is structured as follows:

1. Introduction to the conductivity problem in a unit ball.
2. Neumann-to-Dirichlet map and its properties.
3. Linearised Calderón problem for reconstructing η from knowledge of Fη.
4. Detailed explanation of the 3D Zernike basis functions and their orthonormality.
5. Theorem 1.1 providing a formula for reconstructing η from linearised data Fη.
6. A numerical example demonstrating the ill-posedness of the problem.
7. Regularisation techniques and numerical simulations for accurate measurements.
8. Discussion on stability, numerical computations, and practical applications.

The appendix extends the linearised problem to bounded smooth domains in higher dimensions, emphasizing Fréchet differentiability with respect to complex-valued perturbations.

Stats

Recently an algorithm was given for exact direct reconstruction of any L2 perturbation from linearised data in two-dimensional Calderón problem.
The method uses a 3D Zernike basis to obtain exact direct reconstruction of any L3 perturbation from linearised data.

Quotes

Key Insights Distilled From

Linearised Calderón problem

by Henrik Garde... at arxiv.org 03-26-2024

https://arxiv.org/pdf/2403.16588.pdf

Deeper Inquiries

How does the use of a Zernike basis contribute to the efficiency of reconstruction methods

Zernike basis plays a crucial role in enhancing the efficiency of reconstruction methods due to its orthonormal properties and compact representation capabilities. By using Zernike basis functions, such as spherical harmonics and 3D radial Zernike polynomials, the perturbations can be expanded into a series of coefficients that can be directly reconstructed from linearized data. This expansion allows for a more straightforward forward substitution method for exact direct reconstruction, making it computationally efficient and numerically implementable. Additionally, the triangular structure of the solution formulas for coefficients in Zernike basis functions simplifies the reconstruction process by reducing redundancy in measurements and providing stability even with inaccurate data.

What are the implications of extending the linearised problem to bounded smooth domains in higher dimensions

Extending the linearised problem to bounded smooth domains in higher dimensions has significant implications on practical applications. In these scenarios, where the domain is not limited to simple shapes like balls but includes more complex geometries like general bounded regions in multiple dimensions, the ability to reconstruct unbounded perturbations becomes essential. The extension allows for studying conductivity problems beyond traditional settings and enables researchers to explore real-world situations involving irregular boundaries or varying conductivities within different regions of interest. This advancement opens up possibilities for applying these reconstruction methods in diverse fields such as medical imaging (e.g., electrical impedance tomography) or material science.

How can these reconstruction methods be applied practically in real-world scenarios beyond theoretical mathematics

Practically speaking, these reconstruction methods based on Zernike bases have various applications outside theoretical mathematics. For instance:

Medical Imaging: In electrical impedance tomography (EIT), these methods can be used for non-invasive monitoring of physiological processes within biological tissues.
Material Characterization: These techniques are valuable for analyzing material properties through non-destructive testing.
Geophysical Exploration: They can aid in mapping subsurface structures by analyzing electromagnetic responses.
Industrial Quality Control: Applications include detecting defects or inconsistencies in manufactured products using conductivity variations.
By leveraging advanced mathematical concepts like Zernike bases and extending them to higher-dimensional domains with real-world complexities, researchers and practitioners can enhance their understanding of physical phenomena while developing innovative solutions across various industries.