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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 University Pittsburgh PA April 1991 7 Krumm John Shafer Steven Segmenting textured 3D surfaces using the space frequency representation Spatial Vision Vol 8 No 2 pp 281 308 1994 8 Young Tzay Calvert Thomas Classification Estimation and Pattern Recognition American Elsevier Publishing Co 1974 pp 138 142 4 Results Using the algorithm described above we have successfully harvested approximately one acre of alfalfa autonomously This occurred during one week of testing at a site in Hickory Pennsylvania in sparse crop with curved crop lines Not all of our testing involved actual harvesting to avoid cutting all of our available crop many tests were conducted by simply driving next to the crop line using the same perception system to track the line Our peak speed while harvesting was approximately 4 5 miles an hour the average speed was approximately 3 miles an hour estimate that the crop line was tracked successfully to within a tolerance of roughly one foot The images in Figure 5 were processed using a discriminant of red green The images shown are 640 x 480 pixels typically when running in real time we use only a 400 x 300 window in the center of the image The first is from El Centro California the second is from our harvester testbed during a tracking run in Hickory Pennsylvania The black dots indicate the location of the computed crop cut boundary 5 Future work We plan to improve Demeter s perception system in the future on several fronts First we are examining improvements to this algorithm such as detecting and removing shadow noise and using a custom built filtered camera instead of an RGB camera Second we plan to integrate the crop line tracker with GPS and inertial sensing in order to provide additional information about the location of the crop cut line and also to help with tasks such as end ofrow detection Finally we plan to continue our investigation into alternative ways of sensing the crop line such as by texture segmentation Acknowledgments The authors would like to thank Nick Collela and Red Whittaker for their technical input Regis Hoffman and Tom Pilarski for their coding assistance and Kerien Fitzpatrick Henning Pangels and Simon Peffers for assistance with the Demeter harvester experiments This work was supported by NASA under contract number NAGW3903 Appendix Algorithm derivation Let jd be the rightmost column to the left of the discontinuity in the step function and suppose that the column numbers vary from 0 to jmax Then the ml mr and error terms can be calculated as functions of jd as follows these are defined for jd from 0 to jmax 1 j Figure 5 Sample image segmentations The RGB camera was mounted at the level of the top of the cab about 4 meters high and about 2 meters to the side directly over the crop line This allowed us to control the harvester without the need for camera calibration our control algorithm was merely based on steering to keep the cut line in the center of the image Steering commands were temporally smoothed over a one second time interval to prevent jerky motion We currently have no quantitative means for evaluating the precision of the cut however we d d i j 0 m j j l d j 1 d j max d i j j j 1 d m j r d j j max d between these regions is a single valued function of the row coordinate and that this boundary does not intersect either the left or right edge of the image This boundary function is represented explicitly by the set of pixels which lie on it so that nothing further is assumed about the shape of the boundary definition of best fit allows us to compute this step function rapidly d i j mr Figure 2 shows a representable segmentation and Figure 3 ml cut uncut jd j Figure 4 A model plot of d i j as a function of j for a single scan line i Figure 2 A sample segmentation shows some non representable segmentations This representation was chosen not because it accurately characterizes all the images which might need to be segmented but because it can be computed rapidly Although images such as the ones in Figure 3 do occur our limited representation …

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