After the completion of any improvements, the engineer shall file with the clerk of the district a complete statement of all the costs of such improvement, a plat of the property in the district specially benefited thereby, and a schedule of the amount proposed to be assessed against each separate piece of property as a special assessment. A copy of the plat and a schedule of the proposed special assessment shall be filed in the office of the county clerk of the county in which the greater portion of the area of the district is located for public inspection. The trustees of the district shall then order the clerk of the district to give notice that the plat and schedule are on file with the county clerk where the plat and schedule are kept for examination, and that all objections thereto or to prior proceedings on account of errors, irregularities, or inequalities not made in writing and filed with the clerk of the district within twenty days after first publication of the notice shall be deemed to have been waived. Such notice shall be given by publication, once each week during two consecutive weeks, in a newspaper of general circulation in the district and whenever possible by giving notice in writing by either registered or certified mail to the owner of each separate piece of property against which a special assessment is proposed. The notice shall state the time and place where objections are to be filed. The time of such hearing shall be determined in the manner stated in section 39-1624. Any objections so filed shall be considered by the trustees of the district.