voxel and fills any remaining empty vox els. A variety of BFS and HFS
methods hav e been reported. In [17] and [1 8 ],amethodnamedpixel
nearest neighbor (PNN) is proposed. The algorit hm runs through each
pixel and assigns the pixel value into the nearest vo x el. Av eraging is
performed for multiple contributions to the same vo x el, but different
variants are possible, like the most recent value [1 9],thefirst value
[20] or the keeping the maximum value [17]. Instead of assigning the
pixel value to one voxel, the pixel value can spread among a local
neighborhood based on a kernel function. The kernel-based algorithm
uses a kernel around each pixel. A weighting function based on the
distance between current pixel and the target vo x el is used to weight
the contribution of the pixel val ue on the nearby vox e ls. Various
kernel functions have been reported for the BFS purpose, including
the ellipsoid truncated Gaussian kernel with Gaussian weighting [19],
the ellipsoid Gaussian kernel with exponential weighting [21],the
fix ed cubic kernel with linear weighting [22].Toperformthe3D
reconstruction with sparse ra w data, the three-order Bezier curves are
employed for approximating the voxels located in a 4 B-scans control
windows in the BFS in [23].
After BFS, there are usually gaps in the resulting volume,
especially if the scanned dataset is sparse. In the HFS, the
reconstructed volume is traversed and each empty voxel is
estimated with the information from the nearby filled voxels. A
variety of methods have been presented for this purpose, including
the averaging [24,25] or a median [26] of the filled voxels in a local
neighborhood, the interpolation between the two closest non-
empty voxels [27].In[28], an adaptive Gaussian kernel is intro-
duced into the HFS. Each bin-filled voxel is applied to the
neighboring voxels based on a spherical Gaussian kernel. The
variance of the kernel depends on the variance of the intensity of
the nearby bin-filled voxels. In [29], a pre-computed Gaussian
kernel is used to speed up the interpolation process. The algorit-
hm uses the graphics processing unit (GPU) to implement the
time-consuming reconstruction algorithm and the incremental
rendering computations. In [30], a reconstruction algorithm based
on fast marching method (FMM) is proposed. Instead of the linear
traversing in the conventional HFS, the algorithm advances the
interpolation boundary along its normal direction and fills the area
closest to known voxel points in first with the direction-weighted
interpolation scheme.
PBMs are one of the most popular reconstruction methods for
its high computation speed and low memory requirement. They
can provide physician a visible 3D dataset within a few seconds
after acquisition. However, obvious artifacts can be generated on
the boundaries between the highly detailed bin-filled region and
the smoothed hole-fi lled region [3]. Meanwhile, most hole-
filling
methods have a limit on how far from away from known voxels
the holes are filled. If the B-scan slices have not been scanned with
dense sampling or the hole-filling neighborhood is too small, there
will still be holes in the reconstructed volume [4].
2.3. Function-based reconstruction method
FBMs are another important means for voxel array creation.
They choose a particular function (e.g. a polynomial) and deter-
mine coefficients to make it pass through the input pixels. Then
the resulting volume can be created by evaluating the function at
the regular voxel grid. In [31], a method named radial basis
function (RBF) method is introduced. In the algorithm, an RBF is
used to create the spline approximation function for the under-
lying shape of the reconstructed volume data. Disadvantages for
the RBF interpolation method come from the existence of over-
fitting, especially for the ultrasound image data corrupted with
speckle noise. In [32], a statistical method named Rayleigh
reconstruction/interpolation with a Bayesian framework is pro-
posed. The algorithm uses the Rayleigh distribution to describe the
Fig. 1. The configuration for our freehand 3D ultrasound imaging system.
T. Wen et al. / Neurocomputing 168 (2015) 104–118106