Before any work under this article petitioned to be done is commenced, either under contract or by the Town itself, the Mayor shall ascertain the respective frontages of lots abutting on such portion of such street or avenue, and upon the completion of the work, the cost thereof shall be assessed against the owners of the respective properties abutting on such portion of such street in accordance with the terms of their petition, and the Finance Director is hereby authorized and directed to collect such assessments in accordance with such petition.