Research Article
Direct Least Absolute Deviation Fitting of Ellipses
Table 3
Mean and standard deviation of the relative error and estimated ellipse parameters on simulated Laplace outlier data with tested methods.
| | Methods | Noise level | Relative error | | | | | |
| | DLS | 0.5 | 0.16%/0.12 | 72.00/0.02 | 64.00/0.02 | 70.00/0.03 | 50.00/0.03 | 30.01/0.06 | | 5 | 1.18%/0.47 | 71.99/0.29 | 63.98/0.24 | 69.99/0.34 | 50.23/0.29 | 29.85/0.79 | | 10 | 2.54%/0.99 | 71.91/0.54 | 63.96/0.47 | 69.99/0.67 | 50.67/0.57 | 29.92/2.14 | | 15 | 5.43%/2.35 | 72.06/1.02 | 64.01/1.19 | 70.00/1.35 | 51.80/1.38 | 30.07/4.90 | | 20 | 7.78%/3.16 | 72.00/1.50 | 64.04/1.64 | 70.35/2.14 | 52.80/1.97 | 30.94/6.14 | | 25 | 11.96%/4.21 | 71.71/2.31 | 64.25/2.51 | 70.87/2.58 | 54.54/2.15 | 32.08/10.98 |
| | HyperLS | 0.5 | 0.15%/0.12 | 72.00/0.02 | 64.00/0.02 | 70.00/0.03 | 50.00/0.03 | 30.01/0.06 | | 5 | 1.10%/0.49 | 71.99/0.29 | 63.98/0.24 | 70.03/0.35 | 50.02/0.28 | 29.85/0.80 | | 10 | 2.40%/0.99 | 71.91/0.54 | 63.96/0.48 | 70.12/0.74 | 49.87/0.55 | 29.92/2.27 | | 15 | 5.28%/3.03 | 72.05/1.06 | 63.99/1.30 | 70.25/1.97 | 49.86/1.36 | 30.10/5.65 | | 20 | 7.96%/4.48 | 71.99/1.69 | 64.02/1.83 | 70.98/4.08 | 49.42/2.32 | 31.33/7.77 | | 25 | 13.32%/9.16 | 71.80/2.74 | 64.54/4.51 | 72.98/8.74 | 48.96/3.04 | 32.77/14.60 |
| | SampsonLS | 0.5 | 0.17%/0.12 | 72.00/0.02 | 64.00/0.02 | 70.00/0.03 | 50.00/0.03 | 30.01/0.06 | | 5 | 1.05%/0.48 | 72.01/0.29 | 64.00/0.23 | 69.97/0.33 | 49.85/0.28 | 29.90/0.65 | | 10 | 3.08%/2.36 | 72.00/0.80 | 63.93/0.91 | 69.78/0.77 | 49.12/1.11 | 30.21/1.88 | | 15 | 6.69%/4.11 | 72.14/1.75 | 64.36/2.05 | 69.54/1.31 | 47.93/1.81 | 30.01/3.66 | | 20 | 10.34%/5.60 | 71.91/2.38 | 64.12/2.77 | 69.88/1.75 | 46.28/2.76 | 30.47/5.11 | | 25 | 15.09%/7.77 | 71.38/2.51 | 64.10/3.50 | 70.56/2.58 | 43.80/4.39 | 29.15/6.23 |
| | ElliFit | 0.5 | 0.13%/0.11 | 72.00/0.02 | 64.00/0.02 | 70.00/0.03 | 49.99/0.03 | 30.00/0.06 | | 5 | 1.37%/0.55 | 71.99/0.29 | 63.98/0.24 | 70.37/0.37 | 49.96/0.28 | 28.98/0.80 | | 10 | 3.72%/1.44 | 71.91/0.55 | 63.96/0.47 | 71.48/0.80 | 49.67/0.57 | 26.89/1.99 | | 15 | 8.35%/2.77 | 72.08/1.10 | 64.01/1.17 | 73.84/1.93 | 49.29/1.16 | 23.15/3.38 | | 20 | 13.41%/5.03 | 72.05/1.73 | 64.05/1.70 | 77.93/5.07 | 48.62/1.73 | 20.59/4.49 | | 25 | 20.12%/7.31 | 71.64/3.02 | 64.27/2.45 | 84.23/8.01 | 48.32/2.06 | 18.25/8.54 |
| | RANSAC | 0.5 | 0.66%/0.51 | 71.99/0.22 | 64.01/0.20 | 69.98/0.26 | 49.98/0.23 | 30.00/0.52 | | 5 | 1.39%/0.63 | 72.04/0.44 | 64.04/0.36 | 70.02/0.51 | 50.02/0.44 | 29.95/0.97 | | 10 | 1.59%/0.70 | 72.03/0.47 | 64.05/0.42 | 69.99/0.50 | 49.97/0.48 | 30.05/1.16 | | 15 | 1.57%/0.66 | 72.07/0.47 | 63.98/0.41 | 69.99/0.47 | 49.99/0.51 | 29.88/1.09 | | 20 | 1.61%/0.78 | 72.00/0.48 | 64.05/0.45 | 70.05/0.52 | 49.97/0.53 | 29.95/1.00 | | 25 | 1.56%/0.78 | 72.00/0.45 | 64.05/0.41 | 70.02/0.51 | 49.96/0.51 | 30.08/1.08 |
| | Proposed () | 0.5 | 0.14%/0.12 | 72.00/0.00 | 64.00/0.00 | 70.00/0.01 | 50.00/0.00 | 30.00/0.01 | | 5 | 0.12%/0.10 | 72.00/0.00 | 64.00/0.00 | 70.00/0.01 | 50.00/0.00 | 30.00/0.01 | | 10 | 0.18%/0.12 | 72.00/0.01 | 64.00/0.00 | 70.00/0.01 | 50.00/0.00 | 30.00/0.01 | | 15 | 0.17%/0.22 | 72.00/0.01 | 63.99/0.09 | 70.00/0.01 | 50.01/0.09 | 30.00/0.09 | | 20 | 0.20%/0.39 | 72.00/0.02 | 64.02/0.13 | 69.99/0.06 | 50.03/0.18 | 29.98/0.13 | | 25 | 1.62%/4.44 | 72.07/0.81 | 64.29/1.50 | 70.04/0.95 | 50.57/1.63 | 29.50/3.73 |
| | Proposed () | 0.5 | 0.42%/0.24 | 72.00/0.10 | 64.01/0.09 | 69.99/0.11 | 50.01/0.11 | 30.03/0.26 | | 5 | 0.30%/0.20 | 72.00/0.08 | 64.00/0.06 | 70.00/0.09 | 50.00/0.08 | 29.98/0.19 | | 10 | 0.29%/0.17 | 72.00/0.08 | 64.01/0.07 | 69.98/0.11 | 49.99/0.08 | 29.96/0.20 | | 15 | 0.38%/0.33 | 72.02/0.10 | 64.02/0.09 | 69.94/0.16 | 49.99/0.09 | 29.87/0.37 | | 20 | 0.70%/0.98 | 72.05/0.18 | 64.10/0.28 | 69.85/0.45 | 50.01/0.30 | 29.71/0.90 | | 25 | 1.44%/3.17 | 72.08/0.87 | 64.18/0.84 | 69.74/1.14 | 50.07/1.07 | 29.22/3.44 |
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