Full-Vectorial 3D Microwave Imaging of Sparse Scatterers through a Multi-Task Bayesian Compressive Sensing Approach
<p>Geometry of the <span class="html-italic">3D-MI</span> microwave imaging problem.</p> "> Figure 2
<p><span class="html-italic">Sensitivity Analysis</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>50</mn> </mfenced> </mrow> </semantics></math> [dB])—Actual contrast function (<b>a</b>). Behavior of the total error, <math display="inline"><semantics> <msub> <mi>ξ</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </semantics></math>, versus the <span class="html-italic">MT-BCS</span> control parameters: (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <msup> <mn>10</mn> <mn>2</mn> </msup> </mfenced> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mi>β</mi> <mfenced separators="" open="(" close=")"> <mi>o</mi> <mi>p</mi> <mi>t</mi> </mfenced> </msup> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>) and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mfenced> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mi>α</mi> <mfenced separators="" open="(" close=")"> <mi>o</mi> <mi>p</mi> <mi>t</mi> </mfenced> </msup> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>).</p> "> Figure 3
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>)—<span class="html-italic">MT-BCS</span> reconstructions when processing the scattering data with (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo>=</mo> <mn>0.81</mn> </mrow> </semantics></math>); (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo>=</mo> <mn>0.79</mn> </mrow> </semantics></math>); (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo>=</mo> <mn>0.78</mn> </mrow> </semantics></math>), and (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo>=</mo> <mn>0.77</mn> </mrow> </semantics></math>).</p> "> Figure 4
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>1.0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>4.0</mn> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>50</mn> </mfenced> </mrow> </semantics></math> [dB])—Behavior of the (<b>a</b>) total (<math display="inline"><semantics> <msub> <mi>ξ</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </semantics></math>); (<b>b</b>) internal (<math display="inline"><semantics> <msub> <mi>ξ</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </semantics></math>); and (<b>c</b>) external (<math display="inline"><semantics> <msub> <mi>ξ</mi> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> </mrow> </msub> </semantics></math>) reconstruction errors when processing the scattering data with the <span class="html-italic">MT-BCS</span>.</p> "> Figure 5
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>4.0</mn> </mrow> </semantics></math>)—(<b>a</b>) Actual contrast function and (<b>b</b>,<b>c</b>) <span class="html-italic">MT-BCS</span> reconstructions when processing the scattering data with (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo>=</mo> <mn>3.40</mn> </mrow> </semantics></math>), (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> [dB] (unfiltered) (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo>=</mo> <mn>3.28</mn> </mrow> </semantics></math>), and (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> [dB] (filtered <math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>) (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo>=</mo> <mn>3.28</mn> </mrow> </semantics></math>).</p> "> Figure 6
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1.0</mn> <mo>−</mo> <mn>0.6</mn> <mi>j</mi> </mrow> </semantics></math>)—Real (<b>a</b>,<b>c</b>,<b>e</b>) and imaginary parts (<b>b</b>,<b>d</b>,<b>f</b>) of the (<b>a</b>,<b>b</b>) actual contrast function and of the (<b>b</b>–<b>f</b>) <span class="html-italic">MT-BCS</span> reconstructed profiles when processing the scattering data with (<b>c</b>,<b>d</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo>=</mo> <mn>0.84</mn> <mo>−</mo> <mn>0.49</mn> <mi>j</mi> </mrow> </semantics></math>), (<b>e</b>,<b>f</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo>=</mo> <mn>0.81</mn> <mo>−</mo> <mn>0.45</mn> <mi>j</mi> </mrow> </semantics></math>).</p> "> Figure 7
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>)—(<b>a</b>) Actual contrast function and <span class="html-italic">MT-BCS</span> reconstructions when processing the scattering data with (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.82</mn> </mrow> </semantics></math>), (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.84</mn> </mrow> </semantics></math>), (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.84</mn> </mrow> </semantics></math>), and (<b>e</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.83</mn> </mrow> </semantics></math>).</p> "> Figure 8
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> [dB])—(<b>a</b>,<b>c</b>,<b>e</b>) Actual contrast function and (<b>b</b>,<b>d</b>,<b>f</b>) <span class="html-italic">MT-BCS</span> reconstructions when the distance of the scatterers from the origin is equal to (<b>a</b>,<b>b</b>) <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <msub> <mi>d</mi> <mo movablelimits="true" form="prefix">min</mo> </msub> <mo>=</mo> <mn>0.11</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi>λ</mi> </semantics></math>] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.73</mn> </mrow> </semantics></math>), (<b>c</b>,<b>d</b>) <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.54</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi>λ</mi> </semantics></math>] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.83</mn> </mrow> </semantics></math>), and (<b>e</b>,<b>f</b>) <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <msub> <mi>d</mi> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.97</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi>λ</mi> </semantics></math>] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.84</mn> </mrow> </semantics></math>).</p> "> Figure 8 Cont.
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> [dB])—(<b>a</b>,<b>c</b>,<b>e</b>) Actual contrast function and (<b>b</b>,<b>d</b>,<b>f</b>) <span class="html-italic">MT-BCS</span> reconstructions when the distance of the scatterers from the origin is equal to (<b>a</b>,<b>b</b>) <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <msub> <mi>d</mi> <mo movablelimits="true" form="prefix">min</mo> </msub> <mo>=</mo> <mn>0.11</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi>λ</mi> </semantics></math>] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.73</mn> </mrow> </semantics></math>), (<b>c</b>,<b>d</b>) <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.54</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi>λ</mi> </semantics></math>] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.83</mn> </mrow> </semantics></math>), and (<b>e</b>,<b>f</b>) <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <msub> <mi>d</mi> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.97</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi>λ</mi> </semantics></math>] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.84</mn> </mrow> </semantics></math>).</p> "> Figure 9
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>50</mn> </mfenced> </mrow> </semantics></math> [dB])—Behavior of the total error, <math display="inline"><semantics> <msub> <mi>ξ</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </semantics></math>, versus the distance of the scatterers from the origin, <span class="html-italic">d</span>.</p> "> Figure 10
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>)—(<b>a</b>) Actual contrast function and <span class="html-italic">MT-BCS</span> reconstructions when processing the scattering data characterized by (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.84</mn> </mrow> </semantics></math>), (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.84</mn> </mrow> </semantics></math>), (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.80</mn> </mrow> </semantics></math>), and (<b>e</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.70</mn> </mrow> </semantics></math>).</p> "> Figure 11
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> [dB])—(<b>a</b>,<b>c</b>,<b>e</b>) Actual contrast function and (<b>b</b>,<b>d</b>,<b>f</b>) <span class="html-italic">MT-BCS</span> reconstructions when the objects distance from the origin is (<b>a</b>,<b>b</b>) <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <msub> <mi>d</mi> <mo movablelimits="true" form="prefix">min</mo> </msub> <mo>=</mo> <mn>0.11</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi>λ</mi> </semantics></math>] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.73</mn> </mrow> </semantics></math>), (<b>c</b>,<b>d</b>) <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.54</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi>λ</mi> </semantics></math>] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.70</mn> </mrow> </semantics></math>), and (<b>e</b>,<b>f</b>) <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <msub> <mi>d</mi> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.97</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi>λ</mi> </semantics></math>] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.71</mn> </mrow> </semantics></math>).</p> "> Figure 12
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>50</mn> </mfenced> </mrow> </semantics></math> [dB])—(<b>a</b>) Behavior of the total error, <math display="inline"><semantics> <msub> <mi>ξ</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </semantics></math>, as a function of the objects distance from the origin, <span class="html-italic">d</span> and (<b>b</b>) filtered (<math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>) <span class="html-italic">MT-BCS</span> reconstruction when the objects distance from the origin is <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <msub> <mi>d</mi> <mo movablelimits="true" form="prefix">min</mo> </msub> <mo>=</mo> <mn>0.11</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi>λ</mi> </semantics></math>] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>0.73</mn> </mrow> </semantics></math>).</p> "> Figure 13
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>)—(<b>a</b>) Actual contrast function and <span class="html-italic">MT-BCS</span> reconstructions when (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>1.08</mn> </mrow> </semantics></math>); (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>1.05</mn> </mrow> </semantics></math>); (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>1.09</mn> </mrow> </semantics></math>); and (<b>e</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>=</mo> <mn>1.37</mn> </mrow> </semantics></math>).</p> "> Figure 14
<p><span class="html-italic">Numerical Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>50</mn> </mfenced> </mrow> </semantics></math> [dB])—Behavior of the total reconstruction error (<math display="inline"><semantics> <msub> <mi>ξ</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </semantics></math>) when processing the scattering data with the <span class="html-italic">MT-BCS</span>.</p> "> Figure 15
<p><span class="html-italic">Comparative Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>2.0</mn> </mrow> </semantics></math>)—(<b>a</b>) Actual contrast function and retrieved solutions by the (<b>b</b>,<b>c</b>) <span class="html-italic">MT-BCS</span> and (<b>d</b>,<b>e</b>) <span class="html-italic">ST-BCS</span> when processing noisy data at (<b>b</b>,<b>d</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mrow> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mrow> <mi>M</mi> <mi>T</mi> <mo>−</mo> <mi>B</mi> <mi>C</mi> <mi>S</mi> </mrow> </msubsup> <mo>=</mo> <mn>1.96</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mrow> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mrow> <mi>S</mi> <mi>T</mi> <mo>−</mo> <mi>B</mi> <mi>C</mi> <mi>S</mi> </mrow> </msubsup> <mo>=</mo> <mn>0.33</mn> </mrow> </semantics></math>) and (<b>c</b>,<b>e</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mrow> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mrow> <mi>M</mi> <mi>T</mi> <mo>−</mo> <mi>B</mi> <mi>C</mi> <mi>S</mi> </mrow> </msubsup> <mo>=</mo> <mn>1.68</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mrow> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mrow> <mi>S</mi> <mi>T</mi> <mo>−</mo> <mi>B</mi> <mi>C</mi> <mi>S</mi> </mrow> </msubsup> <mo>=</mo> <mn>0.36</mn> </mrow> </semantics></math>).</p> "> Figure 15 Cont.
<p><span class="html-italic">Comparative Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>2.0</mn> </mrow> </semantics></math>)—(<b>a</b>) Actual contrast function and retrieved solutions by the (<b>b</b>,<b>c</b>) <span class="html-italic">MT-BCS</span> and (<b>d</b>,<b>e</b>) <span class="html-italic">ST-BCS</span> when processing noisy data at (<b>b</b>,<b>d</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mrow> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mrow> <mi>M</mi> <mi>T</mi> <mo>−</mo> <mi>B</mi> <mi>C</mi> <mi>S</mi> </mrow> </msubsup> <mo>=</mo> <mn>1.96</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mrow> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mrow> <mi>S</mi> <mi>T</mi> <mo>−</mo> <mi>B</mi> <mi>C</mi> <mi>S</mi> </mrow> </msubsup> <mo>=</mo> <mn>0.33</mn> </mrow> </semantics></math>) and (<b>c</b>,<b>e</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mrow> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mrow> <mi>M</mi> <mi>T</mi> <mo>−</mo> <mi>B</mi> <mi>C</mi> <mi>S</mi> </mrow> </msubsup> <mo>=</mo> <mn>1.68</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mrow> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mrow> <mi>S</mi> <mi>T</mi> <mo>−</mo> <mi>B</mi> <mi>C</mi> <mi>S</mi> </mrow> </msubsup> <mo>=</mo> <mn>0.36</mn> </mrow> </semantics></math>).</p> "> Figure 16
<p><span class="html-italic">Comparative Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>1.0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>4.0</mn> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>10</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>20</mn> </mfenced> </mrow> </semantics></math> [dB])—Behavior of the total error, <math display="inline"><semantics> <msub> <mi>ξ</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </semantics></math>, as a function of the object contrast, <math display="inline"><semantics> <mi>τ</mi> </semantics></math>, when processing the scattering data with the <span class="html-italic">MT-BCS</span>, the <span class="html-italic">ST-BCS</span>, and the <span class="html-italic">CG</span> methods.</p> "> Figure 17
<p><span class="html-italic">Comparative Assessment</span> (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>2.0</mn> </mrow> </semantics></math>)—<span class="html-italic">CG</span> reconstructions when processing noisy data characterized by (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mrow> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mrow> <mi>C</mi> <mi>G</mi> </mrow> </msubsup> <mo>=</mo> <mn>0.33</mn> </mrow> </semantics></math>) and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> [dB] (<math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>τ</mi> <mo>^</mo> </mover> <mrow> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mrow> <mi>C</mi> <mi>G</mi> </mrow> </msubsup> <mo>=</mo> <mn>0.36</mn> </mrow> </semantics></math>).</p> ">
Abstract
:1. Introduction
2. Mathematical Formulation
3. Inversion Method
3D-CSI BCS-Based Problem Formulation—Starting from the measurement of (, ), and the knowledge of (), determine the sparsest guess of , as the maximum a-posteriori probability (MAP) estimateprovided that the support of , () is the same for all V different illuminations (i.e., ).
4. Numerical Assessment
5. Conclusions
- Reliable 3D reconstructions of the EM properties of the imaged domain are yielded processing scattering data also blurred with a non-negligible amount of additive noise;
- The inversion accuracy of the proposed CS-based approach depends on the degree of sparseness of the actual scenario with respect to the expansion basis at hand. However, it can be fruitfully and profitably applied when other/different (non-voxel) representations of the contrast source/contrast function are chosen [46];
- The MT implementation of the BCS-based inversion remarkably overcomes its single-task (ST-BCS) counterpart thanks to the profitable exploitation of the existing correlations between the V views and the scattered field components;
- The MT-BCS positively compares with other state-of-the-art approaches, also deterministic and non-CS, in terms of both reconstruction accuracy and computational efficiency.
Author Contributions
Funding
Conflicts of Interest
References
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Salucci, M.; Poli, L.; Oliveri, G. Full-Vectorial 3D Microwave Imaging of Sparse Scatterers through a Multi-Task Bayesian Compressive Sensing Approach. J. Imaging 2019, 5, 19. https://doi.org/10.3390/jimaging5010019
Salucci M, Poli L, Oliveri G. Full-Vectorial 3D Microwave Imaging of Sparse Scatterers through a Multi-Task Bayesian Compressive Sensing Approach. Journal of Imaging. 2019; 5(1):19. https://doi.org/10.3390/jimaging5010019
Chicago/Turabian StyleSalucci, Marco, Lorenzo Poli, and Giacomo Oliveri. 2019. "Full-Vectorial 3D Microwave Imaging of Sparse Scatterers through a Multi-Task Bayesian Compressive Sensing Approach" Journal of Imaging 5, no. 1: 19. https://doi.org/10.3390/jimaging5010019
APA StyleSalucci, M., Poli, L., & Oliveri, G. (2019). Full-Vectorial 3D Microwave Imaging of Sparse Scatterers through a Multi-Task Bayesian Compressive Sensing Approach. Journal of Imaging, 5(1), 19. https://doi.org/10.3390/jimaging5010019