When the halves have different subjects, the opener keeps its own subject, as in weather permitting, time permitting or all things considered.
When the opener keeps its own subject
Mei: Are you running along the brook trail on Saturday?
Matilda: Weather permitting, yes. I leave at seven.
Mei: All things considered, that sounds like a good plan. I'll join you.
Not "Having been analyzed, the team presented the results." — say "The data having been analyzed, the team presented the results.". Keep the noun when the subjects differ, or the team becomes the thing analyzed.
Try it: If time allows, we will schedule a follow-up meeting. The two halves have different subjects, so the opener must keep its own. Build it. (Leave out the tile you don't need.) Answer: Time permitting, we will schedule a follow-up meeting. — The opener keeps its own subject, so the halves may differ: Time permitting, we will schedule a follow-up meeting.
Openers that start with the past participle
A passive opener can begin with just the past participle when the state held at the time and the subject matches.
Mei: Why was the puppy so quiet at the clinic?
Matilda: Frightened by the noise, it hid under the chair.
Mei: Delayed by a long phone call, I missed the start myself.
Try it: Because it was written for beginners, the guide avoids technical terms. Start the opener with the past participle. (Leave out the tile you don't need.) Answer: Written for beginners, the guide avoids technical terms. — The past participle starts the opener on its own: Written for beginners, the guide avoids technical terms.
Try it: Mei: I'm writing the formal entry for the new library in the city guide. Which opening suits this formal guide better? (Located beside the station, the new library is easy to reach by train. / Because it's right beside the station, the new library is easy to get to by train.) Answer: Located beside the station, the new library is easy to reach by train. — A reduced opener is a formal, written compression, so a city guide prefers "Located beside the station, the new library is easy to reach by train".