1. A lane marker recognition apparatus for recognizing a lane marker on a road frontward or rearward of a vehicle in repeated cycles, the apparatus comprising: a characteristic object detection unit detecting a characteristic object on a road; a lane marker extraction unit extracting data on a lane marker from data on the detected characteristic object; a centerline calculation unit calculating a centerline position and a centerline shape on the road from the extracted data; a width calculation unit calculating a lane width of the road from the extracted data; and a parameter follow-up unit performing a follow-up process of parameters defining the centerline position, the centerline shape, and the lane width based on the calculated centerline positions, calculated centerline shape, and calculated lane width, wherein when the lane marker extraction unit extracts the data on the lane marker at a present cycle, the lane marker extraction unit uses a result from the follow-up process performed at a past cycle by the parameter follow-up unit; wherein the width calculation unit includes an opposing position calculation unit calculating opposing positions with respect to the centerline in an axle direction of the vehicle in the data extracted by the extraction unit by using a centerline position and a centerline shape estimated at the past cycle or projected for the present cycle in the follow-up process performed by the parameter follow-up unit; a frequency calculation unit dividing a coordinate axis indicating the opposing positions into segments, assigns each of the calculated opposing positions into a corresponding segment included in the segments, and calculating frequencies in the individual segments; and an auto-correlation function calculation unit performing a calculation of an auto-correlation function with the opposing positions regarded as a shift parameter, with respect to a distribution of the calculated frequencies, wherein the width calculation unit calculates a lane width of the road based on a result from the calculation of the auto-correlation function; and wherein each of the characteristic object detection unit, the lane marker extraction unit, the centerline calculation unit, the width calculation unit, the parameter follow-up unit, the opposing position calculation, the frequency calculation unit and the auto-correlation function calculation unit includes a hardware unit or a hardware unit and a software program recorded on a non-transitory computer readable memory.