Automatic Circle Detection on Images Based on an Evolutionary Algorithm That Reduces the Number of Function Evaluations

<table>Circle candidate (individual) built from the combination of points <svg height="12.3875" id="M418" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 11.5375 12.3875" width="11.5375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,8.1)"><path d="M403 126l24 -14q-44 -70 -88.5 -98t-109.5 -28q-90 0 -147 67t-57 171q0 112 57 180q58 69 152 69q83 0 128 -50q53 -59 59 -178h-252q6 -81 25 -128q24 -59 94 -59q32 0 58 15.5t57 52.5zM164 282h134v21q0 73 -9 92q-13 26 -24.5 36.5t-30.5 10.5q-37 0 -53.5 -37
t-16.5 -123z" id="x1D41E"></path></g>
<g transform="matrix(.012,-0,0,-.012,7.613,12.188)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" id="x1D456"></path></g>
</svg>, <svg height="15.35" id="M419" style="vertical-align:-5.76988pt" version="1.1" viewbox="0 0 13.325 15.35" width="13.325" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,8.1)"><path d="M403 126l24 -14q-44 -70 -88.5 -98t-109.5 -28q-90 0 -147 67t-57 171q0 112 57 180q58 69 152 69q83 0 128 -50q53 -59 59 -178h-252q6 -81 25 -128q24 -59 94 -59q32 0 58 15.5t57 52.5zM164 282h134v21q0 73 -9 92q-13 26 -24.5 36.5t-30.5 10.5q-37 0 -53.5 -37
t-16.5 -123z" id="x1D41E"></path></g>
<g transform="matrix(.012,-0,0,-.012,7.613,12.187)"><path d="M394 607q0 -25 -15.5 -43t-39.5 -18q-19 0 -32 13t-13 35t16 41.5t40 19.5q20 0 32 -13.5t12 -34.5zM360 349l-57 -283q-38 -188 -122 -263q-72 -64 -114 -64q-16 0 -30 10.5t-14 22.5q0 24 21 45q15 13 26 6q30 -20 54 -20q41 0 65 95q37 147 86 415q6 34 6.5 42.5
t-5.5 11.5q-18 0 -79 -54l-14 25q48 48 96.5 79t71.5 31q29 0 9 -99z" id="x1D457"></path></g>
</svg>, and <svg height="12.3875" id="M420" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 14.275 12.3875" width="14.275" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,8.1)"><path d="M403 126l24 -14q-44 -70 -88.5 -98t-109.5 -28q-90 0 -147 67t-57 171q0 112 57 180q58 69 152 69q83 0 128 -50q53 -59 59 -178h-252q6 -81 25 -128q24 -59 94 -59q32 0 58 15.5t57 52.5zM164 282h134v21q0 73 -9 92q-13 26 -24.5 36.5t-30.5 10.5q-37 0 -53.5 -37
t-16.5 -123z" id="x1D41E"></path></g>
<g transform="matrix(.012,-0,0,-.012,7.613,12.188)"><path d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1
l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" id="x1D458"></path></g>
</svg>.</table>

Mathematical Problems in Engineering

fig5

Figure 5

Figure 5: Automatic Circle Detection on Images Based on an Evolutionary Algorithm That Reduces the Number of Function Evaluations 