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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Determining $S_2$ for the considered 2D example. Obtain test cubic factors $CT_j$, by dilating $C_i$ with $T_j$, where j = 1, 2, and 3 for 2D SE. As illustrated above, we can then obtain a set of candidate $RSE$, $RT_j$, and candidate sparse factors $ST_j$. Determine the subset of $ST_j$ that is indeed sparse and from this subset pick the ``best'' one and assign it to $S_i$. We apply a greedy criterion for ``best'', which is to pick the sparse $ST_j$ that has the largest number of pixels/voxels. The implementation tests each $ST_j$ for sparseness. Of course, once a particular $ST_j$ is selected, we can simply assign its corresponding $RT_j$ to $RSE_i$.}}{467}{figure.1.5}}
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\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces The origin coincides with $S_3$ and is dark gray, $P_3$ is the union of light gray and the origin. $C_3$ is cube-7, which coincides with $P_3$. The region of $P_1 \cup P_2$ that does not overlap with $P_3$ is white. The union of all foreground voxels is the union of all three partitions and is the radius-5.5 Euclidean isotropic sphere.}}{469}{figure.1.8}}
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\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Processing times for gray-level dilation (binary SE) using sphere SE for a $512 \times 512 \times 418$ chest CT image. Results obtained on and Intel Pentium IV Xeon Dual CPU 2.00 GHz platform with 2.00 GB of RAM.}}{469}{figure.1.9}}
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\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces [Left] 2-partition decomposition of a 2D SE using the proposed method. Cubic factors are square-3 and square-7. Dark gray - origin (center): not part of $S_1$, however $S_2$ is the pixel at the origin. Light and medium gray together - the partition, medium gray (only in iteration 1) - $S_1$, white - region of $P_1$ that does not overlap with $P_2$. [Right] 4-factor decomposition for the same SE from Example 1 in \cite {Vaz:Park:1995:SE}. The proposed method is more efficient.}}{470}{figure.1.10}}
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\bibcite{Vaz:Anelli:1998:DeArBiSE}{{2}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces Number of comparison ops required per output voxel for MM using sphere SE. Note, op count presented on a log scale. Data obtained analytically. Proposed implementation follows Equation \ref  {xxx:eqn:2d} and implements the sparse factors directly. A possible optimization would be to decompose the sparse factors, but this would require another interim copy of the image volume.}}{471}{figure.1.11}}
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\bibcite{Vaz:Hashimoto:2003:SE}{{8}{}}
\bibcite{Vaz:Jones:1994:SE}{{9}{}}
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\bibcite{Vaz:Soille:2003:MatMedImg}{{17}{}}
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\bibcite{Vaz:Vaz:2006:EfMatMor}{{20}{}}
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