Perfusion imaging is of great clinical importance and is used to assess a wide range of diseases including strokes and brain tumors. Commonly used approaches for the quantitative analysis of perfusion images are based on measuring the effect of a contrast agent moving through blood vessels and into tissue. Contrast-agent free approaches, for example, based on intravoxel incoherent motion and arterial spin labeling, also exist, but are so far not routinely used clinically. Existing contrast-agent-dependent methods typically rely on the estimation of the arterial input function (AIF) to approximately model tissue perfusion. These approaches neglect spatial dependencies. Further, as reliably estimating the AIF is non-trivial, different AIF estimates may lead to different perfusion measures. In this work we therefore propose PIANO, an approach that provides additional insights into the perfusion process. PIANO estimates the velocity and diffusion fields of an advection-diffusion model best explaining the contrast dynamics without using an AIF. PIANO accounts for spatial dependencies and neither requires estimating the AIF nor relies on a particular contrast agent bolus shape. Specifically, we propose a convenient parameterization of the estimation problem, a numerical estimation approach, and extensively evaluate PIANO. Simulation experiments show the robustness and effectiveness of PIANO, along with its ability to distinguish between advection and diffusion. We further apply PIANO on a public brain magnetic resonance (MR) perfusion dataset of acute stroke patients, and demonstrate that PIANO can successfully resolve velocity and diffusion field ambiguities and results in sensitive measures for the assessment of stroke, comparing favorably to conventional measures of perfusion.