Now looking at Mr. Manseau's sentence, we note it is passive ("had been borne") and that it is not followed by "by" ("of the techniques"). Therefore, the sentence cannot mean that "the techniques gave birth to the Shroud".
I think "born(e) of" has a similar meaning to "borne by", but with the emphasis on the idea of "originating from", rather than on the physical process of being conceived and born. "Born(e) of" can be used either literally (e.g. "Christ is born of Mary") or figuratively (e.g. "the theory of relativity was born of the genius of Einstein"; "the economic crisis was born of greed"). The Shroud example falls into the latter category.
"Born of" seems to be the correct spelling. On the Web it is about 15 times as common as "borne of".