We address a certain inverse problem in ultrasound-modulated optical tomography: the recovery of the amplitude of vibration of scatterers  in the ultrasound focal volume in a diffusive object from boundary measurement of the modulation depth (M) of the amplitude autocorrelation of light  traversing through it. Since M is dependent on the stiffness of the material, this is the precursor to elasticity imaging. The propagation of is described by a diffusion equation from which we have derived a nonlinear perturbation equation connecting and refractive index modulation  in the region of interest to M measured on the boundary. The nonlinear perturbation equation and its approximate linear counterpart are solved for the recovery of . The numerical results reveal regions of different stiffness, proving that the present method recovers with reasonable quantitative accuracy and spatial resolution.
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