We propose an image-based elastography method to measure the heterogeneous stiffness inside a cell and its nucleus. It uses a widely accessible setup consisting of plate compression imaged with fluorescence microscopy. Our framework recovers a spatial map of Young's modulus from images of the intracellular displacements. These displacements are measured with a novel optical-flow technique characterised by a Hessian-Schatten norm regularizer. The aim is to favor piecewise-linear displacements because they reproduce solutions to linear elasticity problems well when the underlying modulus is piecewise-constant, as is often the case in cells. Our computational approach is fast enough for long time-lapse acquisitions and 3D imaging. It is able to cope with two common pitfalls of biological elastography: high compressibility and small compressions to avoid damage. We show our method is faster and more accurate than the state-of-the-art.