Quaternion Wavelet Analysis and Application in Image Denoising

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="493976.fig.006a" src="https://static.hindawi.com/articles/mpe/volume-2012/493976/figures/493976.fig.006a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b>Noisy image </td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="493976.fig.006b" src="https://static.hindawi.com/articles/mpe/volume-2012/493976/figures/493976.fig.006b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b> Hard</td></tr></table></td></tr><tr class="fig-image" id="c"><td><img alt="493976.fig.006c" src="https://static.hindawi.com/articles/mpe/volume-2012/493976/figures/493976.fig.006c.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(c) </b> LAWML</td></tr></table></td></tr><tr class="fig-image" id="d"><td><img alt="493976.fig.006d" src="https://static.hindawi.com/articles/mpe/volume-2012/493976/figures/493976.fig.006d.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(d) </b>SURE-LET </td></tr></table></td></tr><tr class="fig-image" id="e"><td><img alt="493976.fig.006e" src="https://static.hindawi.com/articles/mpe/volume-2012/493976/figures/493976.fig.006e.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(e) </b> P-Bivariate</td></tr></table></td></tr><tr class="fig-image" id="f"><td><img alt="493976.fig.006f" src="https://static.hindawi.com/articles/mpe/volume-2012/493976/figures/493976.fig.006f.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(f) </b>Our method</td></tr></table></td></tr></table>

<table>Denoised image using different denoising methods (<svg height="15.1375" id="M202" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 52.0625 15.1375" width="52.0625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,10.863)"><path d="M548 455q-32 -77 -74 -77q-37 0 -103 10l-3 -2q36 -30 48.5 -62t12.5 -80q0 -103 -75 -179.5t-175 -76.5q-70 0 -113 45t-43 128q0 115 83.5 200.5t209.5 85.5q31 0 77.5 -4t67.5 -4q36 0 62 30zM350 274q0 54 -17 86q-19 35 -57 35q-50 0 -90 -37.5t-59 -91.5t-19 -109
q0 -61 24.5 -96.5t65.5 -35.5q51 0 87.5 46.5t50.5 100.5t14 102z" id="x1D70E"></path></g>
<g transform="matrix(.012,0,0,-.012,9.663,14.938)"><path d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2
q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" id="x1D45B"></path></g>
<g transform="matrix(.017,0,0,-.017,20.988,10.863)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,0,0,-.017,35.674,10.863)"><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><g transform="matrix(.017,0,0,-.017,43.834,10.863)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g>
</svg>, Lena). </table>

Mathematical Problems in Engineering

Quaternion Wavelet Analysis and Application in Image Denoising

Figure 6