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Mathematical Methods in Elasticity Imaging$
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Habib Ammari, Elie Bretin, Josselin Garnier, Hyeonbae Kang, Hyundae Lee, and Abdul Wahab

Print publication date: 2015

Print ISBN-13: 9780691165318

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691165318.001.0001

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Imaging from Internal Data

Imaging from Internal Data

(p.160) Chapter Ten Imaging from Internal Data
Mathematical Methods in Elasticity Imaging

Habib Ammari

Elie Bretin

Josselin Garnier

Hyeonbae Kang

Hyundae Lee

Abdul Wahab

Princeton University Press

This chapter introduces efficient methods for reconstructing both the shape and the elasticity parameters of an inclusion using internal displacement measurements. It first considers the inverse problem of recovering the shape and the (constant) shear modulus of an inclusion from internal measurements. The small-volume asymptotic framework is used to separate the information in the measurements into a near-field and a far-field part. A discrepancy function is then presented and a regularization is discussed. The chapter proceeds by describing the more general case of shear distributions, taking into account the discrepancy between the measured and computed displacement fields. In order to write down a descent gradient scheme in the case of a heterogeneous shear distribution, the derivative of the discrepancy function with respect to the shear modulus is computed.

Keywords:   shape, elasticity, inclusion, internal displacement measurement, shear modulus, shear distribution, gradient scheme, heterogeneous shear distribution, discrepancy function

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