Positron Emission Tomography (PET) is a nuclear imaging technique of increasing importance e.g. in cardiovascular investigations. However, cardiac and respiratory motion of the patient degrade the image quality due to acquisition times in the order of minutes. Reconstructions without motion compensation are prone to spatial blurring and affected attenuation correction. These effects can be reduced by gating, motion correction and finally summation of the transformed images. This paper describes a new and systematic approach for the correction of both cardiac and respiratory motion. Key contribution is the splitting of the motion into respiratory and cardiac components, which are then corrected individually. For the considered gating scheme the number of registration problems is reduced by a factor of 3, which considerably simplifies the motion correction pipeline compared to previous approaches. The subproblems are stabilized by averaging cardiac gates for respiratory motion estimation and vice versa. The potential of the novel pipeline is evaluated in a group study on data of 21 human patients.