Most of it is based on the Byzantine account of the origin of the Mandylion, which was supposedly discovered bricked-up in a wall of the city of Edessa, along with a "tile" bearing an exact duplicate of the image on the cloth. (A second history gives us both the tile and cloth, but depicts a much more fanciful account of their discovery.) The belief has always been that in the first century, the survivors of Abgar V sealed the image in the city's walls to protect it from his pagan successors, and that it stood there, undisturbed, for 500 years or so, until its discovery. So far as the tile was concerned, the belief was that the cloth image had miraculously transferred itself to the tile.
I'm not prepared to think that a soft image on cloth could produce an image on a piece of stone, but it's easier to think that the reverse might have been the case, particularly if the stone prototype contained significant amounts of radioactive elements, such as uranium and thorium. Given 400 or 500 years of undisturbed proximity, the relief image on the stone might well have "cooked" itself onto the cloth.
The most likely scenario is that Abgar V or someone else commissioned the statuary in remembrance of Jesus. The legend claims that Abgar had the cloth image placed above the city gates to be venerated by those entering the city. But a cloth image, exposed to the elements would quickly have deteriorated. It's more likely that the actual image was in stone, and this was the image that Abgar's successors bricked up into the walls to keep it from the sight of Abgar's pagan successors. This wouldn't have been necessary with a cloth image, which could simply have been folded up and removed.
The cloth itself, which ended up bearing a copy of the prototype, was probably intended as a drapery or covering for the statuary.
At least that's how I perceive it, and I'm not at all trying to push the notion that this is exactly how it would have happened. But I think there needs to be more of an attempt to explain the Shroud of Turin image as something other than either a miracle or a forgery.
The formation of the image, except for the vertical collimation, is explicable through natural processes of chemistry. Radiation, unless it is constrained by complicated technology, expands in a globular wave front from its sources. It has the same problem as a vertical collimated gas. What could have collimated the radiation so that the image is sufficiently resolved and not blurred out by radiation coming from other vectors???