changed into linear problems. In Wang's research, a
kernel-based method was introduced to positron
emission computed tomography (PET) to help the
image reconstruction.
21
Features obtained from
prior information are modeled and incorporated
into the forward function. Inspired by the well
performance of kernel-based method in PET re-
construction, similar approaches have been pro-
posed to improve reconstructions in optical
tomography in the last couple years.
Reconstructions in both DOT and FMT su®er
from the ill-posed and nonlinear inverse problems.
Anatomical guidance, as prior information, is of
great importance in improving the reconstruction
quality.
30,31
However, introducing anatomical in-
formation from other imaging modalities with high
spatial resolution calls for image segmentation and
registration ¯rst. Utilizing a kernel-based method,
optical absorption coe±cient at each ¯nite element
node is represented as a function of a set of features
obtained from anatomical images without any
preprocessing.
2.1. Kernel-based methods for DOT
As mentioned in Baikejiang et al.'s work, a Gauss-
ian kernel-based method was presented to involve
anatomical information in the forward model for the
DOT image reconstruction.
28,32
According to relative literatures, the theory of
Gaussian kernel-based reconstruction method for
DOT can be described as follows.
As DOT reconstruction is usually implemented
on ¯nite element method (FEM), the kernel func-
tion can be formulized as
k
m;n
¼
exp
jjf
m
f
n
jj
2
2
; f
n
2 knn of f
m
;
0; otherwise;
8
<
:
ð1Þ
where f
m
and f
n
are feature vectors corresponding
to ¯nite element nodes m and n extracted from
anatomical image, respectively. Feature vector f
m
consists of all the values of the neighbor voxels
surrounding the ¯nite element node m. knn denotes
the k closest neighbors of f
m
and can be acquired by
K nearest neighbors (KNN) search algorithm.
26
Ignoring the optical scattering coe±cient, the
absorption coe±cient vector can be written as
a
¼ K; ð2Þ
where K is the kernel matrix de¯ned by the kernel
function k
m;n
. Vector is the nth value of the vector
to be reconstructed.
In general, the objective function of inverse
problem for DOT can be written as
arg min F ð
a
Þ¼
1
2
jjy W ð
a
Þjj
2
; ð3Þ
where W is the system matrix representing the
forward model. By combining Eqs. (2) and (3), the
inverse problem for DOT can be described as
arg min F ðÞ¼
1
2
jjy W ðKÞjj
2
: ð4Þ
Then, the inverse problem of DOT can be trans-
ferred into a minimized problem of vector and the
prior information has been involved. Vector is
then obtained by conventional iterative method
such as Tikhonov regularization, and the absorption
coe±cient can be obtained using Eq. (2).
33
In the numerical simulation of Baikejiang's work,
it was shown that the voxel number of each corre-
sponding node and the number of nearest neighbors
had impact on the performance of kernel-based
method signi¯cantly. Di®erent values of k (16, 32,
or 64) and di®erent voxel numbers (3 3 3,
5 5 5, 7 7 7, or 9 9 9) were taken into
account to optimize the kernel-based method. Vol-
ume ratio (VR), Dice similarity coe±cient (Dice),
contrast-to-noise ratio (CNR), and mean square
error (MSE) are introduced as evaluation metrics.
34
Reconstruction results indicated that the kernel-
based method outperformed other cases whether
the value of k or the voxel number increased.
Then other three numerical experiments were
carried out to validate the performance of kernel-
based method. CT contrast e®ect in kernel-based
method for CT-guided DOT reconstruction, e®ect
of the false positive target in the kernel-based
method, and clinical breast CT image as anatomical
guidance were studied and compared to the
Laplacian-type soft prior method, respectively.
35
Numerical reconstruction results indicated that the
kernel-based method was robust to CT contrast and
the false positive targets in the guided CT image.
Although in some cases the reconstructions of ker-
nel-based method were not as good as that of soft
prior method, it still outperformed the method
without prior information and did not need seg-
mentation for the CT guidance.
Brief review on learning-based methods for optical tomography
1930011-3
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