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j j d d i j m 2 l t 0 d i 0 l t j t j 1 d i j l d l d d max d i j m 2 r j j 1 j 0 d error j 1 max Clearly it requires order n time to compute error jd from the d i j alone It is possible however to compute error jd 1 from error jd in constant time To accomplish this we express the calculation of error jd in terms of the following sums jd d i j t j l d 1 j 0 j d i j 2 j j 1 d jd t2 j l d d i j 2 3 j 0 j t2 j r d max d i j 2 4 j j 1 d From these error jd is calculated as follows t2 j t2 j r d l d 2 2 j j t t r d l d j j j 1 max d d error j d j 1 max 5 Since Eqns 1 4 can be computed recursively it takes only a constant number of operations to compute tl tr t2l t2r and error at jd 1 given their values at jd We can make the computation faster still by computing instead of error jd some monotonic 1 to 1 function of error jd such as the following j f j d t 0 r max d i j 8 j 1 t j t j 1 d i j r d r d d 2 2 t j t j r d l d f j d j j j 1 max d d 9 This additional speedup results in a roughly 30 increase in cycle rate compared to computing error j d directly from Eqn 5 References max t j r d j 7 2 error j j 1 d max max d i j 6 j 0 which can be computed recursively using the following three equations 1 Billingsley J and Schoenfisch M Vision Guidance of Agricultural Vehicles Autonomous Robots 2 pp 65 76 1995 2 Reid J F and Searcy S W An Algorithm for Separating Guidance Information from Row Crop Images Transactions of the ASAE Nov Dec 1988 v 31 6 pp 16241632 3 Hayashi M and Fujii Y Automatic Lawn Mower Guidance Using a Vision System Proceedings of the USA Japan Symposium on Flexible Automation New York NY July 1988 4 Jahns Gerhard Automatic Guidance in Agriculture A Review ASAE paper NCR 83 404 St Joseph MI 1983 5 Ross Bill A Practical Stereo Vision System Proceedings of IEEE Conference on Computer Vision and Pattern Recognition CVPR 93 New York NY June 1993 6 Tomasi Carlo Kanade Takeo Detection and Tracking of Point Features Technical Report CMU CS 91 132 Carnegie Mellon



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