The sentence can be simplified with the same basic structure:
A member of staff objects to something.
Somebody objects, and 'object' expects a prepositional phrase afterwards, the object of the objection. What follows is not an infinitive at all.
What is it that the staff member objects to? It is to a someone using the image, a gerund phrase. But this latter has been passivized. Instead of someone using the image, it is to the image being used.
So this separates into pieces:
(A member of staff) objects (to (their image being used)).
When I followed your link I found one more link possessive pronouns (also known as genitives)
Where I found one more example: They objected to the youngest girl's being given the command position.
– anton May 16 '18 at 10:19I will split it into parts:
"their image" - possessive pronoun 'their' with its subject;
"being used" - passive gerund
– anton May 16 '18 at 15:35