A Fast and Robust Ellipse-Detection Method Based on Sorted Merging

<table>Problems of regular ellipse fitting methods and the fitting results with different <svg height="11.9625" id="M238" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 28.5 11.9625" width="28.5" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg> should be merged with <svg height="11.6375" id="M240" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 40.174999 11.6375" width="40.174999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg> but is incorrectly merged with <svg height="15.7" id="M241" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 40.400002 15.7" width="40.400002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg> into a flat ellipse. (b) <svg height="11.6375" id="M242" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 35.674999 11.6375" width="35.674999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg> but is incorrectly merged with <svg height="12.4375" id="M244" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 39.587502 12.4375" width="39.587502" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M673 0h-245v28q47 6 58 16t1 39l-55 154h-213q-31 -87 -36 -129q-16 -47 -5 -61t68 -19v-28h-226v28q50 7 68.5 23.5t43.5 80.5l214 524l24 9l193 -536q23 -63 41.5 -79.5t68.5 -21.5v-28zM418 280l-81 255h-2q-74 -197 -101 -255h184z" id="x41"></path></g><g transform="matrix(.017,-0,0,-.017,11.809,12.162)"><path d="M181 342h2q63 107 121 107q24 0 41 -16t17 -34q0 -34 -32 -49q-17 -7 -26 2q-21 20 -43 20q-20 0 -42 -22t-38 -63v-183q0 -50 12.5 -62t69.5 -16v-26h-230v26q47 5 58 17t11 61v207q0 49 -9 60.5t-53 16.5v23q74 12 141 40v-109z" id="x72"></path></g><g transform="matrix(.017,-0,0,-.017,18.115,12.162)"><path d="M390 111l17 -21q-34 -45 -80 -73.5t-89 -28.5q-91 0 -146 62t-55 147q0 118 101 195q74 57 149 57h1q59 0 90 -27q16 -14 16 -30q0 -15 -12 -29t-21 -14q-8 0 -19 11q-44 41 -101 41q-52 0 -87.5 -42.5t-35.5 -117.5q0 -49 15 -87t39 -58t49 -30t48 -10q33 0 60.5 12
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</svg>. (c) Ellipse fitting results with <svg height="12.675" id="M245" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 75.25 12.675" width="75.25" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg>, <svg height="15.0875" id="M246" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 56.087502 15.0875" width="56.087502" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.8)"><path d="M393 379q-9 -16 -28 -29q-15 -10 -23 -2q-19 19 -36 19q-21 0 -52 -38q-57 -72 -82 -126l-40 -197q-23 -3 -75 -18l-7 7q49 196 74 335q7 43 -2 43q-7 0 -30 -14.5t-47 -37.5l-16 23q37 42 82 73t67 31q41 0 15 -113l-11 -50h4q41 71 85 117t77 46q29 0 45 -26
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</svg> by the proposed fitting method.</table>

The Scientific World Journal

A Fast and Robust Ellipse-Detection Method Based on Sorted Merging

Figure 8