Automatic construction of statistical shape models using deformable simplex meshes with vector field convolution energy

Abstract

Background: In the active shape model framework, principal component analysis (PCA) based statistical shape models (SSMs) are widely employed to incorporate high-level a priori shape knowledge of the structure to be segmented to achieve robustness. A crucial component of building SSMs is to establish shape correspondence between all training shapes, which is a very challenging task, especially in three dimensions.

Methods: We propose a novel mesh-to-volume registration based shape correspondence establishment method to improve the accuracy and reduce the computational cost. Specifically, we present a greedy algorithm based deformable simplex mesh that uses vector field convolution as the external energy. Furthermore, we develop an automatic shape initialization method by using a Gaussian mixture model based registration algorithm, to derive an initial shape that has high overlap with the object of interest, such that the deformable models can then evolve more locally. We apply the proposed deformable surface model to the application of femur statistical shape model construction to illustrate its accuracy and efficiency.

Results: Extensive experiments on ten femur CT scans show that the quality of the constructed femur shape models via the proposed method is much better than that of the classical spherical harmonics (SPHARM) method. Moreover, the proposed method achieves much higher computational efficiency than the SPHARM method.

Conclusions: The experimental results suggest that our method can be employed for effective statistical shape model construction.

Keywords: Deformable models; Greedy algorithm; Shape correspondence establishment; Shape model construction; Simplex meshes; VFC energy.

Fig. 1

Diagram showing the duality between 2-simplex meshes (black lines) and triangular meshes (red lines)

Fig. 2

Flowchart of the proposed method for statistical shape model construction

Fig. 3

The greedy algorithm: a the energy function is calculated at vertex $P_i$ and voxels in the $w\times w\times w$ cubic window around $V_i$, and the point with the smallest energy is selected as the target position of $P_i$. b The vertex $P_i$ is moved only along its normal direction ${\mathbf{n}}_i$. For illustration purpose, only a 2-D window is showed

Fig. 4

3D visualization of the shape variation on the surface for the constructed femur shape model. The variation spans from small (blue) to large (red)

Fig. 5

The first two principal modes of variation for the constructed femur shape model. Each row shows the variation of a specific mode between $-3\mathit{\sigma }$ and $+3\mathit{\sigma }$

Fig. 6

Compactness for both SPHARM and our proposed method

Fig. 7

Generalization ability for both SPHARM and our proposed method

Fig. 8

Specificity for both SPHARM and our proposed method