We are investigating a new global framework for smart acquisition methods in biological imaging whereby it is possible with mathematical tools to recover a high quality image from very fewer samples than the full acquisition. It is based on theoretical tools from Compressed Sensing (CS), which is a recent mathematical theory for sampling and reconstructing signals in an efficient manner. The expected advances should lead to massively increased image acquisition rates with guaranteed resolution and improved image quality and features for a given acquisition rate. The use of this mathematical microscopy has already enabled the design of improved imaging protocols in digital holography microscopy dedicated to specific biological paradigms and experimental conditions. More recently, we have investigated how the framework of CS can be extended to perform an efficient joint reconstruction of a sequence of time correlated 2D images, using 3D total variation regularization.