*3.3.3 Function-based method (FBM)*

The functional-based method (FBM) takes a set of input data and uses a function like polynomial to reconstruct 3D ultrasound volume [28]. The radial basis function (RBF) is one of the FBMs that used an estimate function to compute a spline that passes through the pixels that form a shape in the 2D ultrasound frames [9, 27]. The created splines need to be as identical and smooth as the original shapes in the 2D frames. The approximation requirement is required because of the existence of measurement errors, as well as to reduce the overshoots in order to have the gray-level range of interpolated voxels to be same as that of the original 2D ultrasound frames [27]. The mentioned measurement errors are such as the tissue motion, position sensor error, and calibration error during the data acquisition process. In addition, the overshoot is a situation in signal processing where the signal or function exceeds its supposed target.

Besides RBF, Bayesian framework can be used to infer the voxel values in a volume grid by assuming a 3D parametric function that has basic function centered at every voxel, and the volume grid is modeled using piecewise smooth Markov random field (PS-MRF) with typical 6-connected neighborhood system [7, 29]. The work of [7] showed that the PS-MRF can work with irregular spaced B-scan images and to reduce the speckle noise and preserve boundary. However, it requires extreme computation time and needs to use GPU and parallel programming to overcome this limitation. The FBMs able to create a high-quality 3D volume from the 2D ultrasound frames; however, they require intensive computational power as well as speed, which imply that these methods are not widely studied in the field of 3D ultrasound reconstruction.
