2.4 Discussion
3.2.2 Non-rigid registration
Non-rigid spatial mapping between brain images is related by a set of Wendland’s RBFs with different levels of support extents. Fig. 3.5 shows the flowchart of the pro-posed non-rigid registration procedure. Non-brain structures are first removed because the inter-subject variation of these regions is relatively large compared to that of brain tissues and hence could interfere with registration efficacy [36]. Therefore, we use a brain mask to extract the brain area as well as the boundary of the brain in the TDOG image, referred
(a) (b)
(c) (d) (e)
O
O
Figure 3.4: Segmentation of CC on MSP. (a) The circular area centered at the gravity of brain MSP, O, is considered in the segmentation of CC. (b) Only one connected region, the CC on MSP, is found in the first thresholding step locates the CC. (c) Excess non-brain tissues can bias the estimation of the circular area considered in the CC segmentation. (d) The intensity differences between CC and the surrounding tissues are not significant and thus many connected regions, the candidates of CC, are found in the first thresholding step.
(e) Applying Otsu’s method [67], the calculated intensity threshold can distinguish the CC from other tissues.
Target Image
Figure 3.5: Flowchart of the proposed non-rigid registration method. Non-brain regions of the target brain are first removed to obtain a brain mask, which is then used to extract the brain region in the TDOG image of the target brain and construct the brain-only TDOG.
The deformation field is modeled by a set of Wendland’s RBFs hierarchically distributed at the anatomical structures revealed in the brain-only TDOG.
as the brain-only TDOG. Though the structures revealed in the brain-only TDOG are quite rough, they provide a guidance to deploy RBFs near the brain boundary, the boundary between GM and WM, and the boundary between CSF and GM/WM. Furthermore, the deformation field is progressively estimated by optimizing the coefficients of each RBF in a coarse-to-fine manner.
Non-rigid transformation model
We use a combination of 𝐾 RBFs to model the non-rigid deformation field, Tl(⋅),
Tl(p) =
∑𝐾 𝑖=1
𝛼𝑖𝜙(∥p − c𝑖∥) , (3.2)
0 0.2 0.4 0.6 0.8 1
Figure 3.6: Wendland’s RBF 𝜙(𝑟) adopted in this work.
where ∥ ⋅ ∥ is the Euclidean norm and the one-argument function 𝜙 : ℜ+ → ℜ is the RBF centered at c𝑖 with coefficients 𝛼𝑖 ∈ ℜ3, 𝑖 = 1, . . . , 𝐾. There are different types of RBFs.
In this work, we use one of the Wendland’s 𝜓-functions, 𝜓3,1, as the RBF 𝜙 due to its low computational complexity and the compact support property [68, 69]. This function 𝜙 is shown in Fig. 3.6 and is formulated as
𝜙(𝑟) = RBFs determine the 3-D displacement vector Tl(p) for each point p in the image volume.
With the compact support property, the influence of each RBF is restricted to a local region which is a unit sphere around the RBF center. It is a preferred characteristic in non-rigid registration because the deformation of one area should not affect remote regions.
This property also greatly alleviates the computational complexity of spatial transformation in contrast to the functions with global support, such as TPS and multi-quadrics, because only a few RBFs are involved in the calculation of displacement vector for an image point.
To accommodate various extents of compact support, the argument 𝑟 of the function 𝜙(⋅)
is scaled by the shape parameter, 𝑠 [129]. The Wendland’s RBF with support extent 𝑠 is formulated as
𝜙𝑠(𝑟) = 𝜙(𝑟
𝑠) . (3.4)
Hierarchical decomposition of deformation field
For the 𝐾 RBFs involved in our non-rigid deformation model, there are 𝐿 different support extents, 𝑠𝑗, 𝑗 = 1, . . . , 𝐿, each with 𝐾𝑗 RBFs and 𝐾1 + 𝐾2 + ⋅ ⋅ ⋅ + 𝐾𝐿 = 𝐾.
Therefore, the non-rigid deformation field, T𝑙(⋅), can be rewritten as
Tl(p) = For proper deployment of RBFs, brain volume is hierarchically divided into eight equal subregions, which are cubes in our implementation, and each subregion has an RBF placed at the center if this region contains any foreground voxels in the brain-only TDOG, as shown in Fig. 3.7. Although the maximum number of RBFs required at level 𝑗 is 8𝑗, the use of brain-only TDOG can help to place necessary RBFs near important anatomical fea-tures and avoid the dramatic increase of the number of RBFs. This deployment method is fairly beneficial for computational efficiency while maintaining high registration accuracy, particularly at the fine levels. The support extent 𝑠𝑗 at each level 𝑗 is a parameter which is set to be multiples of the width of subregions. Therefore, the RBFs from low to high levels are capable of modeling the deformation field from coarse to fine resolutions.
Since the deformation field model is hierarchically decomposed into RBFs with differ-ent levels of support extdiffer-ent, we estimate the coefficidiffer-ents of RBFs one-by-one while gradu-ally accumulating the deformation field in a coarse-to-fine manner. Consider the coefficient estimation for the 𝑚-th RBF at level 𝑙. The centered positions and coefficients of the RBFs at the coarser levels from 1 to 𝑙 − 1 and the RBFs from 1 to 𝑚 − 1 at level 𝑙 are all
deter-Level 1 Level 2 Level 3 Level 4
Figure 3.7: The whole brain volume is hierarchically divided into eight subregions. An RBF is deployed at the center of a subregion containing salient structure revealed by the brain-only TDOG. This figure illustrates the RBF distributions at four levels, in which a red square represents a subregion deployed with an RBF.
mined previously. We update the deformation field by adding a new RBF 𝜙𝑠𝑙(∥p − c𝑙,𝑚∥)
In this way, the displacement vector for the point p is progressively updated. The accumu-lation process terminates when 𝑙 = 𝐿 and 𝑚 = 𝐾𝐿, that is, Tl(p) = T𝐿,𝐾l 𝐿(p). Because only three coefficients of an RBF are estimated at a time, the proposed method avoids the searching in a huge parameter space and thus the whole optimization process is quite fast.
Objective function
Coefficient estimation for each RBF is an optimization process that minimizes an ob-jective function:
𝐶(𝐼𝑡, 𝐼𝑠, T) = −𝑆CR(𝐼𝑡, 𝐼𝑠, T) + 𝜆𝐸(T) , (3.7) where T is the spatial mapping between the target image 𝐼𝑡and the source image 𝐼𝑠, 𝑆CR
measures the image similarity by CR, the smoothness regularization function 𝐸 calculates
the deformation energy of T, and the parameter 𝜆 compromises the measurements 𝑆CR
and 𝐸. The Laplacian model and the thin-plate model are widely applied to regularize the structure deformation in image registration [91]. Due to the computational efficiency, we adopt the Laplacian model in this work:
𝐸(T) = 1
where 𝑉 is the volume involved in the estimation. The Nelder-Mead downhill simplex method is utilized to optimize the objective function. Fig. 3.2(e) demonstrates that the proposed non-rigid method can well register the corresponding anatomical structures.
Implementation issues
In the optimization process of non-rigid registration, iterative calculation of image transformation contributes to the major computation burden, particularly when registering high resolution image volumes. The hierarchical decomposition of the non-rigid transfor-mation model and the compact support property of Wendland’s RBFs are both beneficial to the alleviation of this heavy burden. However, the execution time of non-rigid registra-tion is still large. Some implementaregistra-tion techniques described below are helpful to further improve the efficiency. First, we construct a volume pyramid with 𝐿′ levels for each MR image volume, generally 𝐿′ ≤ 𝐿. When evaluating the objective function, the 𝐾𝑗 RBFs, 𝜙𝑠𝑗(⋅) with support extent 𝑠𝑗, 𝑗 = 1, . . . , 𝐿, are associated with the 𝑗′-th level of volume pyramid, where 𝑗′ = 𝑗 when 𝑗 ≤ 𝐿′ and 𝑗′ = 𝐿′ when 𝑗 > 𝐿′. This hierarchical architec-ture enables a coarse-to-fine optimization that helps to avoid the local traps and improves the computational efficiency. Second, one lookup table is constructed beforehand for each support extent level of Wendland’s RBFs to avoid the repeated function evaluations. There-fore, updating the deformation field with the RBF 𝜙𝑠𝑗(⋅) in Eq. (3.6) requires only three subtractions, one table lookup, three multiplications, and three additions.