Feature Extraction from 3D Point Cloud Data Based on Discrete Curves

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="290740.fig.0020a" src="https://static-preview.hindawi.com/articles/mpe/volume-2013/290740/figures/290740.fig.0020a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b><svg height="16.125" id="M321" style="vertical-align:-0.23206pt" version="1.1" viewbox="0 0 18.4375 16.125" width="18.4375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5
t325.5 101.5q114 0 204 -38z" id="x1D436"></path></g>
<g transform="matrix(.012,-0,0,-.012,12.05,7.613)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg> discontinuity</td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="290740.fig.0020b" src="https://static-preview.hindawi.com/articles/mpe/volume-2013/290740/figures/290740.fig.0020b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b> Similarity indicator: <svg height="11.9625" id="M322" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 13.15 11.9625" width="13.15" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M352 391q0 -31 -27 -44q-14 -7 -24 6q-39 48 -84 48q-23 0 -39.5 -15t-16.5 -40q0 -43 73 -90q49 -32 70 -58t21 -57q0 -58 -62 -105.5t-129 -47.5q-40 0 -75.5 25t-35.5 52q0 28 32 46q7 4 15 3t11 -6q19 -31 48.5 -50.5t54.5 -19.5q34 0 54 19.5t20 42.5q0 43 -65 81
q-97 56 -97 123q0 50 51 96q19 17 58 32.5t62 15.5q37 0 61 -18t24 -39z" id="x1D460"></path></g>
<g transform="matrix(.012,-0,0,-.012,6.438,11.762)"><path d="M484 406q0 -21 -22 -38q-10 -7 -18 -1q-15 11 -30 11q-30 0 -77.5 -37t-74.5 -67l79 -168q11 -23 19.5 -34.5t13 -14t10.5 -2.5q9 0 30 17.5t40 42.5l17 -23q-31 -45 -57.5 -69.5t-40.5 -29.5t-31 -5q-18 0 -32 13.5t-35 55.5q-34 71 -69 160q-36 -32 -59 -56
q-12 -27 -36 -158q-44 -7 -77 -15l-5 7l82 349q6 25 -2 25q-16 0 -72 -41l-14 22q44 41 89 69.5t66 28.5q19 0 19 -32q0 -16 -6 -43l-35 -160l146 143q94 92 136 92q20 0 33 -12t13 -30z" id="x1D705"></path></g>
</svg></td></tr></table></td></tr></table>

<table>Similarity indicator <svg height="11.9625" id="M319" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 13.15 11.9625" width="13.15" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M352 391q0 -31 -27 -44q-14 -7 -24 6q-39 48 -84 48q-23 0 -39.5 -15t-16.5 -40q0 -43 73 -90q49 -32 70 -58t21 -57q0 -58 -62 -105.5t-129 -47.5q-40 0 -75.5 25t-35.5 52q0 28 32 46q7 4 15 3t11 -6q19 -31 48.5 -50.5t54.5 -19.5q34 0 54 19.5t20 42.5q0 43 -65 81
q-97 56 -97 123q0 50 51 96q19 17 58 32.5t62 15.5q37 0 61 -18t24 -39z" id="x1D460"></path></g>
<g transform="matrix(.012,-0,0,-.012,6.438,11.762)"><path d="M484 406q0 -21 -22 -38q-10 -7 -18 -1q-15 11 -30 11q-30 0 -77.5 -37t-74.5 -67l79 -168q11 -23 19.5 -34.5t13 -14t10.5 -2.5q9 0 30 17.5t40 42.5l17 -23q-31 -45 -57.5 -69.5t-40.5 -29.5t-31 -5q-18 0 -32 13.5t-35 55.5q-34 71 -69 160q-36 -32 -59 -56
q-12 -27 -36 -158q-44 -7 -77 -15l-5 7l82 349q6 25 -2 25q-16 0 -72 -41l-14 22q44 41 89 69.5t66 28.5q19 0 19 -32q0 -16 -6 -43l-35 -160l146 143q94 92 136 92q20 0 33 -12t13 -30z" id="x1D705"></path></g>
</svg> of the curvature at the <svg height="16.125" id="M320" style="vertical-align:-0.23206pt" version="1.1" viewbox="0 0 18.4375 16.125" width="18.4375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5
t325.5 101.5q114 0 204 -38z" id="x1D436"></path></g>
<g transform="matrix(.012,-0,0,-.012,12.05,7.613)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg> discontinuity with noise.</table>

Mathematical Problems in Engineering

fig20

Figure 20

Figure 20: Feature Extraction from 3D Point Cloud Data Based on Discrete Curves 