We present a method for the analysis of heart motion from medical images. The algorithm exploits monogenic signal theory, recently introduced as an N-dimensional generalization of the analytic signal. The displacement is computed locally by assuming the conservation of the monogenic phase over time. A local affine displacement model is considered to account for typical heart motions as contraction/expansion and shear. A coarse-to-fine B-spline scheme allows a robust and effective computation of the model's parameters, and a pyramidal refinement scheme helps to handle large motions. Robustness against noise is increased by replacing the standard point-wise computation of the monogenic orientation with a robust least-squares orientation estimate. Given its general formulation, the algorithm is well suited for images from different modalities, in particular for those cases where time variant changes of local intensity invalidate the standard brightness constancy assumption. This paper evaluates the method's feasibility on two emblematic cases: cardiac tagged magnetic resonance and cardiac ultrasound. In order to quantify the performance of the proposed method, we made use of realistic synthetic sequences from both modalities for which the benchmark motion is known. A comparison is presented with state-of-the-art methods for cardiac motion analysis. On the data considered, these conventional approaches are outperformed by the proposed algorithm. A recent global optical-flow estimation algorithm based on the monogenic curvature tensor is also considered in the comparison. With respect to the latter, the proposed framework provides, along with higher accuracy, superior robustness to noise and a considerably shorter computation time.