The histogram matching can be applied at different phases of the synthesis process, either at the search phase or at the copy phase. In the index and texton histogram matching, the distance in the neighborhood search is manipulated based on the match between the specified histograms in the search phase of the approach. In the position histogram matching, it is used in the copy phase of the approach. As is shown in Equa- tion 6.2, weights weightu,i,w are needed to compute the color value of the synthesized voxel w at the copy phase. The modified weights are used as a means to match the histograms of the exemplar and synthesized textures.
In this section, the details of the computation of the modified weightsweight0u,i,w are explained.
1. Position histogram matching: To achieve position histogram matching, the histograms of the source position information for all the synthesized voxels are kept during texture synthesis. Then a re-weighting scheme is applied in the optimization (copy) phase of the approach:
weight0u,i,v = weightu,i,w 1 +max[0, Hpos(p(e
u,i,w))−θ)]
(6.3)
where p(eu,i,w) refers to the position of the pixel eu,i,w, Hpos is the position his-
togram,Hpos(p) is the value of the position pin the histogram Hpos, and θ is the histogram value when all the pixels from the exemplar completely evenly appear in the synthesized texture. This equation means that ifHpos(p)> θ, the weight is
reduced to make it less likely to selectpfor the voxelw. IfHpos(p)< θ, the weight
is increased to make it more likely to choose eu,i,w for the synthesized voxel w.
Since in this dissertation the focus is on synthesizing a region-specific solid texture inside a model as opposed to synthesizing an isotropic solid texture inside a cube, the value of θ is computed depending on the type of the synthesized texture and
the number of voxels inside it:
• Computation of θ for isotropic textures: If the synthesized texture is an isotropic solid texture, θ is computed as the number of voxels in the model divided by the number of pixels in the exemplar texture.
• Computation ofθ for region-specific textures: If the synthesized texture is a region-specific solid texture,θ is computed for each region separately. In that case,θfor a specific region is computed as the number of voxels in that region divided by the number of pixels in the exemplar texture for that region.
2. Index histogram matching: As opposed to the position histogram matching which is applied in the optimization phase of the approach, index histogram match- ing is used in the search phase to restrict the nearest neighborhood search. During the search phase, the distance between two neighborhoods is modified based on following equation:
d=weightd· ksw,i−ew,ik2 (6.4)
in which weightd is
weightd= 1 +max[0, Hindex(i(ew,i))−φ] (6.5)
where i(ew,i) refers to the nearest neighborhood index corresponding to the pixel
ew,i, Hindex is the index histogram, Hindex(i) is the histogram value of the index
i, and φ is the histogram value when all the indices completely equally distribute in the exemplar. This equation means that when Hindex(i(ew,i))> φ, the distance
d between two neighborhoods ew,i and sw,i will increase, making it less likely to
choose ew,i as the nearest neighborhood index for sw,i. Without this modification
in the distance function, the search phase can converge to the same position in the exemplar, making the preservation of the texture structures impossible.
3. Texton histogram adjusting: Using the texton histogram matching technique, the frequency of the textons can be increased or decreased in the synthesized texture. Texton histogram matching has an opposite effect from position histogram and index histogram matching. While in position and index histogram matching the aim is to make the pixels in the exemplar texture appear uniformly in the synthesized texture, in texton histogram matching the aim is to increase/decrease the frequency of some of the textons. The decision as to the frequency of textons can be determined depending on the specific organ or region inside the organ. As in the index histogram matching technique, in texton histogram matching during the search phase the distance between two neighborhoods is modified based on the following equation:
d=weightd· ksw,i−ew,ik2 (6.6)
in which weightd is
weightd= 1 +max[0, Hstexton(i(ew,i))−Hetexton(i(ew,i))] (6.7)
where i(ew,i) refers to the nearest neighborhood index corresponding to the pixel
ew,i, Hstexton is the texton histogram of the synthesized texture, Hstexton(i) is the
texton histogram value of the index i at the synthesized texture, Hetexton is the texton histogram of the exemplar texture, and Hetexton(i) is the texton histogram value of the index i at the exemplar texture. The equation for texton histogram matching is very similar to the equation for index histogram matching except for the usage of texton histogram of the exemplar texture (Htexton
e ). Hetexton is
precomputed using the binary image of the texton (Figure 6.2-(b)). For each pixel p outside the textons, φ, which is the histogram value when all the indices completely equally distribute in the exemplar, is assigned to that pixel inHetexton. For each pixel inside the textons, the value is computed based on whether the
application needs to increase or decrease the number of textons in the synthesized texture. If the goal is to increase the number of textons in the synthesized texture, then φ is multiplied by a large number. If the goal is to decrease the number of textons and increase the distance between the textons in the synthesized texture, then φ is multiplied by a small number.
Figure 6.5: In each column, the images in the left and in the middle columns show slices of the synthesized texture, and the images in the right column show the texton histogram for that synthesized texture. For each row, results for different Hetexton are illustrated. The top row shows the results when the values inside the texton regions are set to a high value in Htexton
e . The other rows show the results when these values are
set to successively lower values.
Figure 6.5 shows the results obtained using the texton matching method. As is shown, this method can be used for the illustration of hyperplasia (proliferation of cells
within an organ or tissue) in anatomical organs. Discussion of histogram matching
Color histogram matching, proposed in Kopf et al. (2007), is not always an appropri- ate technique since it only works for color but not for structural features available in the texture. In addition, when the texture channels are decorrelated it fails to preserve the color histograms. For the purpose of this dissertation, it is not appropriate for most of the anatomical textures, especially when the colors of the texture structures are similar to each other.
Using position and index histograms, both texture structures and color histograms can be preserved in the synthesized textures. Position histogram matching ensures that color histogram matching is also satisfied, since position histogram matching considers that all the pixels in the exemplar should have the same probability to appear in the result, which would preserve the color histograms and texture structures in the synthe- sized texture. The purpose of these matching techniques is to make sure that every pixel in the exemplar texture will appear same number of times in the synthesized tex- ture. These techniques are really useful if the aim is to have the same local and global statistics in both exemplar and synthesized textures. However, if we want to change the number of specific textons in the texture, then texton matching is a useful technique. It gives the user the ability to control the amount of specific textons in the synthesized texture. This can be used for visualizing hyperplasia (proliferation of cells within an organ or tissue) in an organ.