Cleft lip pathology diagnosis and foetal landmark extraction via 3D geometrical analysis

Abstract

This work proposes a methodology to automatically diagnose and formalize prenatal cleft lip with representative key points and identify the type of defect (unilateral, bilateral, right, or left) in three-dimensional ultrasonography (3D US). Differential Geometry has been used as a framework for describing facial shapes and curvatures. Then, descriptors coming from this field are employed for identifying the typical key points of the defect and its dimensions. The descriptive accurateness of these descriptors has allowed us to automatically extract reference points, quantitative distances, labial profiles, and to provide information about facial asymmetry. Seventeen foetal faces, nine of healthy foetuses and eight with different types of cleft lips, have been obtained through a Voluson system and used for testing the algorithm. In case no defect is present, the algorithm detects thirteen standard facial soft-tissue landmarks. This would help ultrasonographists and future mothers in identifying the most salient points of the forthcoming baby. This algorithm has been designed to support practitioners in identifying and classifying cleft lips. The gained results have shown that differential geometry may be a valuable tool for describing faces and for diagnosis.

Appendix

The first and second fundamental forms are used to measure distance on surfaces and are defined by

$$\begin{aligned}&Edu^{2}+2Fdudv+Gdv^{2},\\&edu^{2}+2fdudv+gdv^{2}, \end{aligned}$$

respectively, where $E, F, G, e, f$ and $g$ are their coefficients. Curvatures are used to measure how a regular surface $x$ bends in $\mathrm{R}^{3}$. If $D$ is the differential and $N$ is the normal plane of a surface, then the determinant of DN is the product $\left( {-k_1 } \right) \left( {-k_2 } \right) =k_1 k_2$ of the principal curvatures, and the trace of DN is the negative $-\!\left( {k_1 +k_2 } \right)$ of the sum of principal curvatures. In point $P$, the determinant of $DN_P$ is the Gaussian curvature $K$ of $x$ at $P$. The negative of half of the trace of DN is called the mean curvature H of $x$ at $P.$ In terms of the principal curvatures can be written

$$\begin{aligned} K&= k_1 k_2 ,\\ H&= \frac{k_1 +k_2 }{2}. \end{aligned}$$

Some definitions of these descriptors are given. These are the forms implemented in the algorithm:

$$\begin{aligned} E&= 1+h_x^2 ,\\ F&= h_x h_y ,\\ G&= 1+h_y^2 ,\\ e&= \frac{h_{xx} }{\sqrt{1+h_x^2 +h_y^2 }},\\ f&= \frac{-h_{xy} }{\sqrt{1+h_x^2 +h_y^2 }},\\ g&= \frac{-h_{yy} }{\sqrt{1+h_x^2 +h_y^2 }},\\ K&= \frac{h_{xx} h_{yy} -h_{xy}^2 }{\left( {1+h_x^2 +h_y^2 } \right) ^{2}},\\ H&= \frac{\left( {1+h_x^2 } \right) h_{yy} -2h_x h_y h_{xy} +\left( {1+h_y^2 } \right) h_{xx} }{\left( {1+h_x^2 +h_y^2 } \right) ^{3/2}},\\ k_1&= H+\sqrt{H^{2}-K},\\ k_2&= H-\sqrt{H^{2}-K}, \end{aligned}$$

where $h$ is a differentiable function $z=h\left( {x,y} \right)$. It is, therefore, convenient to have at hand formulas for the relevant concepts in this case. To obtain such formulas let us parametrize the surface by

$$\begin{aligned} x\left( {u,v} \right) =\left( {u,v,h\left( {u,v} \right) } \right) ,\quad \left( {u,v} \right) \in U \end{aligned}$$

where $u=x, v=y$.

The most used descriptors are surely the shape and curvedness indexes $S$ and $C$, introduced by Koenderink and van Doorn [11]:

$$\begin{aligned} S&= -\frac{2}{\pi }arctan\frac{k_1 +k_2 }{k_1 -k_2 },\quad S\in \left[ {-1,1} \right] ,\quad k_1 \ge k_2 ,\\ C&= \sqrt{\frac{k_1^2 +k_2^2 }{2}}. \end{aligned}$$

For the role they play in the work, a little digression about their significance is needed. Their meaning is shown in Figs. 31, 32, 33 and in Table 5.

Fig. 31

Illustration of Shape Index scale divided into seven categories. Different subintervals of its range $[-1,1]$ correspond to seven geometric surfaces

Fig. 32

Curvedness index scale, whose range is $\left( {-\infty ,\infty } \right)$

Fig. 33

Indexes $(S,C)$ are viewed as polar coordinates in the $\left( {k_1 ,k_2 } \right)$-plane, with planar points mapped to the origin. The effects on surface structure from variations in the curvedness (radial coordinate) and Shape Index (angular coordinate) parameters of curvature, and the relation of these components to the principal curvatures ($k_1$ and $k_2$). The degree of curvature increases radially from the centre

Table 5 Topographic classes