If the county court authorizes the improvement, it shall be done under the direction and supervision of the county surveyor, or the county engineer, who shall set all necessary grade stakes and bench marks. The owner of any of the lands through which or on which any portion of the improvement is to be constructed may, at the expense of the owner, perform such construction work under the supervision of the county surveyor, or the county engineer. If the owner does not elect to do such work, it shall be done by the county under the direction and supervision of the county surveyor, or the county engineer. [Amended by 1965 c.287 § 2]