If the theory is correct that what we now know as the Shroud was previously known as the "Image of Edessa" and the "Holy Mandylion", then it s fairly well documented.
The image would have been brought to Edessa in the years after the Resurrection, then brought to Constantinople, then stolen by Crusaders during the sack of that city in 1204, after which it was hidden until the mid-1300s.
http://www.johnsanidopoulos.com/2013/08/synaxarion-for-holy-mandylion-of-edessa.html