We discuss and solve an inverse problem in nonlinear elasticity imaging in which we recover spatial distributions of hyperelastic material properties from measured displacement fields. This problem has applications to elasticity imaging of soft tissue because the strain dependence of the apparent stiffness may potentially be used to differentiate between malignant and normal tissues. We account for the geometric and material nonlinearity of the tissues by assuming a known hyperelastic model for

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... the soft tissue. We formulate the problem as the minimization of a cost function representing the difference between the measured and predicted displacement fields. We minimize the cost function with respect to the spatial distribution of material properties using a gradient-based (quasi-Newton) optimization approach. We calculate the gradient efficiently using the adjoint method and a continuation strategy in the material properties. We present numerical examples that demonstrate the feasibility of the approach.