3.1 Data and Preprocessing
The method described above is applied to inner cortical surfaces (GM/WM interface) that are reconstructed from T1-weighted SPGR brain images for ten subjects. For the T1-weighted volumetric MR image, we utilize the Oxford FSL tools for preprocessing and brain segmentation. First, non-cerebral tissues including bone, skin, fat, and etc. are removed by Oxford BET tool [17
]. Since only the cerebral cortex will be tested in our algorithm, we also remove the cerebellum and brain stem using the HAMMER package [16
]. We then use the automated segmentation tool FAST [17
] to segment the brain into GM, WM and CSF compartments, where the spatial bias is corrected simultaneously.
Reconstructing topologically correct and geometrically accurate inner surfaces is a challenging problem. An automated graph-based algorithm for topology correction in a binary volume image was proposed in [12
]. The graphs are constructed to represent the connectivity of both the foreground and background voxels respectively. Assuming that no cycle exists in the connectivity graphs if the expected surface is homeomorphic to a sphere, the handle removing problem can be considered as removing the cycles in the connectivity graphs. The graph-based method operates on the voxel membership set of the volume, and makes minimal changes to force it to have the topology of a sphere [12
]. After topology correction, a voxel-based method such as marching cubes algorithm [14
] can be used to reconstruct an isosurface of the WM volume.
In our experiment, the BrainSuite [13
] tools that implement the algorithm in [12
] were used to remove the topology errors and also to generate the inner cortical surface. Using these tools, the entire cortex can automatically be cut into two approximately symmetric parts. Then the whole inner cortical surfaces are parcellated into 76 anatomic regions with different colors marked for identification using the method in [7
3.2 Distortion Measurements
To measure the distortion between the original surface and the flat map, some measurements that are invariant to the similarity transformation are defined. We used a method similar to that used for measuring angular and metric distortions in [4
]. The angular distortion is defined as:
denote the “market share” [6
] angles vi1 vi2 vi3
The metric distortion is defined as:
) denotes the set of predefined neighbors of vi
. The symbol
denotes the normalized geodesic distance between vi
on the mesh and on its resulted map respectively, and s R+
is a scale parameter that is used to avoid the difference of similarity transformations. Only the directly connected neighbors (1-neighborhood) of vi
are used and illustrated in our experiments. The 2- and 3- neighborhoods are tested, but there is no significant difference (less than 1%), compared to the 1-neighborhood.
The area distortion is defined as:
) and AU
) are normalized geodesic areas of triangle ti
on original mesh and on its corresponding flat map respectively.
The three measurements defined above can well evaluate the local distortions over the surface. To measure the global distortion, we define the region-based area distortion as:
where Re is the set of 76 anatomic regions defined in section 3.1, ri
indicates each region in Re, and A
) and AU
) are normalized geodesic areas of region ri
on the original mesh and on its corresponding flat map respectively.
3.3 Spherical Mapping
We have applied the spherical conformal brain mapping method in section 2.4 to 10 cases of inner cortical surfaces, which are obtained using the method in section 3.1. As mentioned in section 3.1, each inner cortical surface is parcellated into 76 anatomic regions with different colors marked for identification using the method in [7
]. We map each of the cortical surface into spheres with different values of λ
. For visual evaluation, shows two examples of spherical mapping results. The results of this mapping for λ
= 0, λ
= 0.5, and λ
= 1.0 are shown in respectively. It is noted that the method of LSCM with spring energy is equivalent to the LSCM method when parameter λ
= 0. It is apparent that larger values of λ
produce less area distortions, as shown by the relative area ratios of labeled neuroanatomic regions.
Spherical mapping results of two subjects. (a) Cortical Surfaces. (b) Mapping results with λ= 0. (c) Mapping results with λ=0.5. (d) Mapping results with λ=1.0.
We have demonstrated a quantitative evaluation of the method of LSCM with spring energy in –. The values of angle distortion in the spherical mapping for 10 cases are presented in , where we have shown that the value of angle distortion increases as the value of parameter λ becomes larger. The results in – show that other three measurements of metric distortion, area distortion and region distortion decrease consistently as the value of parameter λ increases. Specifically, the average metric distortion over 10 cases decreases from 0.59 to 0.49, the average area distortion decreases from 0.82 to 0.76, and the average region distortion decreases from 0.79 to 0.76. This result indicates that the method of LSCM with spring energy reduces the metric and area distortions than LSCM method as expected. Based on the results in –, we can see that the method of LSCM with spring energy reduces metric and area distortion, while increasing the angle distortion. Note that in some applications, such as for visualization purposes, the problem of angle distortion is less important than the problem of metric and area distortion. Thus the method of LSCM with spring energy provides the opportunity of sacrificing angle distortion while prioritizing metric and area distortion.
Spherical mapping results of Dangle (U) (degree).
Spherical mapping results of Dregion (U).
Spherical mapping results of Dmetric (U).
It is noted that the cortical mapping process is very fast. For a typical cortical surface with about 50,000 vertices and 100,000 triangles, the average computation time on a P4 2.0GHz CPU is around 15 minutes. In terms of computation efficiency, this method is comparable to the LSCM method in [4
], as the method of LSCM with spring energy also employs a linear system similar to that of Eq. (16)
. Thus, comparing to other time-consuming methods [18
], this method is well-suited for fast visualization purposes.
3.4 Hemispheric Mapping
We have also applied the hemispheric conformal brain mapping method in section 2.5 to the same 10 cases of inner cortical surfaces presented in section 3.3. Each the hemispherical cortical surface is mapped to hemispheres with different values of λ. presents the hemispherical mapping results for two subjects. The mapping results with λ = 0, λ = 0.5, and λ = 1.0 are shown in respectively. Similarly, the method of LSCM with spring energy is equivalent to the LSCM method when the parameter λ is set to 0. As expected, larger λ produces less area distortions, as shown by the relative area ratios of labeled neuroanatomic regions.
Hemispherical mapping results of two subjects. (a) Cortical Surfaces. (b) Mapping results with λ=0. (c) Mapping results with λ=0.5. (d) Mapping results with λ=1.0.
– are the quantitative evaluations of the method of LSCM with spring energy. shows the angle distortion in the hemispherical mapping for 10 cases. The average angle distortion decreases slightly from 13.84 to 13.12, when the parameter λ changes from 0 to 0.1. But when λ is above 0.1, the angle distortion increases when the parameter λ becomes larger. The results in – demonstrate that the measurements of metric distortion, area distortion and region distortion decreases consistently with the increasing of parameter λ. Specifically, the average metric distortion over 10 cases decreases from 0.47 to 0.41, the average area distortion decreases from 0.71 to 0.56, and the average region distortion decreases from 0.70 to 0.63. This result further supports that the method of LSCM with spring energy reduces the metric and area distortions than LSCM method. Again, based on the results in –, it is apparent that the method of LSCM with spring energy reduces metric and area distortion, while at the same increases the angle distortion. This method of LSCM with spring energy provides the opportunity of being able to sacrifice angle distortion while prioritizing metric and area distortion. This is specially significant when angle distortion is not important in applications such as visualization of cortex.
Hemispherical mapping results of Dangle (U) (degree).
Hemispherical mapping results of Dregion (U).
Hemispherical mapping results of Dmetric (U).
In the hemispherical brain mapping, the procedure of automatically cutting the whole cortical surface into two hemispheres and their mappings to planar regions are fully automatic, and very fast. This step does not add much computational cost to the least-square conformal mapping with spring energy step. Thus the method of hemispherical brain mapping using LSCM with spring energy is desirable for visualization of human brain cortex.