Among the most fascinating and hotly-debated areas in contemporary astrophysics are the means by which planetary systems are assembled from the large rotating disks of gas and dust which attend a stellar birth. Although important work has already been, and is still being done both in theory and observation, a full understanding of the physics of planet formation can only be achieved by opening observational windows able to directly witness the process in action. The key requirement is then to probe planet-forming systems at the natural spatial scales over which material is being assembled. By definition, this is the so-called Hill Sphere which delineates the region of influence of a gravitating body within its surrounding environment. The Planet Formation Imager project (PFI; http://www.planetformationimager.org) has crystallized around this challenging goal: to deliver resolved images of Hill-Sphere-sized structures within candidate planethosting disks in the nearest star-forming regions. In this contribution we outline the primary science case of PFI. For this purpose, we briefly review our knowledge about the planet-formation process and discuss recent observational results that have been obtained on the class of transition disks. Spectro-photometric and multi-wavelength interferometric studies of these systems revealed the presence of extended gaps and complex density inhomogeneities that might be triggered by orbiting planets. We present detailed 3-D radiation-hydrodynamic simulations of disks with single and multiple embedded planets, from which we compute synthetic images at near-infrared, mid-infrared, far-infrared, and sub-millimeter wavelengths, enabling a direct comparison of the signatures that are detectable with PFI and complementary facilities such as ALMA. From these simulations, we derive some preliminary specifications that will guide the array design and technology roadmap of the facility.