[Looking at the first and second stages of the process], there are three
ways of collecting data.
The bracketed element is a non-finite clause functioning as a supplementary adjunct. Supplements are not modifiers; rather, they have a semantic 'anchor' that they refer to. Here the anchor is the main clause "there are three ways of collecting data", though the supplement does not describe "there", which simply has an existential meaning here.
Like most non-finite clauses, it has no overt subject, though one is usually recoverable from the main clause. Here, however, no subject for the verb "looking" is recoverable from the main clause and hence it is similar to what prescriptivists call the 'dangling participle' construction.
Generally though, examples like yours are regarded as acceptable. The understood subject is roughly recoverable from the context as the speaker or the speaker and addressee together.
I don't know what you mean by the 'original version of the sentence', so I can't help you there.