Mathematical and Statistical Methods for Imaging: Nims by Habib Ammari, Josselin Garnier, Hyeonbae Kang, Knut Solna

By Habib Ammari, Josselin Garnier, Hyeonbae Kang, Knut Solna

This quantity includes the court cases of the NIMS Thematic Workshop on Mathematical and Statistical tools for Imaging, which was once held from August 10-13, 2010, at Inha college, Incheon, Korea. The target of this quantity is to offer the reader a deep and unified realizing of the sector of imaging and of the analytical and statistical instruments utilized in imaging. It deals a great evaluation of the present prestige of the sector and of instructions for extra learn. demanding difficulties are addressed from analytical, numerical, and statistical views. The articles are dedicated to 4 major parts: analytical research of robustness; speculation checking out and determination research, rather for anomaly detection; new effective imaging recommendations; and the consequences of anisotropy, dissipation, or attenuation in imaging.

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Additional resources for Mathematical and Statistical Methods for Imaging: Nims Thematic Workshop Mathematical and Statistical Methods for Imaging August 10-13, 2010 Inha University Incheon, Korea (Contemporary Mathematics)

Sample text

Suppose a small inclusion embedded in an homogenous medium scatters an incoming wave, the scattered field gives some information about the localization and details of the scatterer. The problem is to recover the location of the inclusion, along with its shape and scattering power. In general, this very general and well known problem is set with the scattered field given on the boundary of the medium. This context is in general what one encounters in engineering applications such as acoustic imaging, radar, seismic imaging, and connects very well with the mathematical theory of boundary value problems.

That is, up to a paraxial approximation, J(z) = 1 + R+h sin(kh) δ R ln( )−2 + η(R) + O( ), 2h R−h kh R where h = |z|, and limR→∞ η(R) = 0. 2. Resolution limits and connection with match filtering. The decomposition of the imaging functional into two different parts reflects well the two different information carried by the Green function for the Helmholtz problem: the amplitude and the phase. The phase functional sin(kh)/(kh) is very robust to changes of geometry of the domain, because of the Kirchoff-Helmholtz identity, and has an accuracy that is essentially related to the wavelength λ = 1/k.

To perform a resolution analysis, one can fix the ad hoc definitions for an estimator of the type zˆ = Argminz F (z). If ∂z F (ˆ z ) = 0 the estimator has finite resolution, and in this case, one can define the resolution δF using the following definition: δF = 1 2 F (h) < 0 or ∞} . min{h/ ∂|z| In other words, we consider the resolution to be the smallest distance two points, such that the convexity of F in the middle of those two points changes sign. In the case where the domain has dimensions larger than the wavelength, the phase functional has sharper information than the amplitude functional.

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