# Topological Derivative Based Imaging of Small Inclusions in the Time-Harmonic Regime

# Topological Derivative Based Imaging of Small Inclusions in the Time-Harmonic Regime

This chapter introduces a topological derivative (TD) based imaging framework for detecting small inclusions in the time-harmonic regime. Based on a weighted Helmholtz decomposition of the TD based imaging functional, optimal resolution imaging is achieved. Its stability properties with respect to both medium and measurement noises are investigated. The chapter first considers the TD imaging functional resulting from the expansion of the filtered quadratic misfit with respect to the size of the inclusion. It shows that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. The chapter concludes by presenting the sensitivity analysis of a modified imaging functional based on the weighted Helmholtz decomposition of the TD.

*Keywords:*
topological derivative, small inclusion, time-harmonic regime, Helmholtz decomposition, stability, measurement noise, filtered quadratic misfit, imaging functional, pressure, shear wave

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