An Object Splitting Model Using Higher-Order Active Contours for Single-Cell Segmentation

Abstract

Determining the number and morphology of individual cells on microscopy images is one of the most fundamental steps in quantitative biological image analysis. Cultured cells used in genetic perturbation and drug discovery experiments can pile up and nuclei can touch or even grow on top of each other. Similarly, in tissue sections cell nuclei can be very close and touch each other as well. This makes single cell nuclei detection extremely challenging using current segmentation methods, such as classical edge- and threshold-based methods that can only detect separate objects, and they fail to separate touching ones. The pipeline we present here can segment individual cell nuclei by splitting touching ones. The two-step approach merely based on energy minimization principles using an active contour framework. In a presegmentation phase we use a local region data term with strong edge tracking capability, while in the splitting phase we introduce a higher-order active contour model. This model prefers high curvature contour locations at the opposite side of joint objects grow “cutting arms” that evolve to one another until they split objects. Synthetic and real experiments show the strong segmentation and splitting ability of the proposed pipeline and that it outperforms currently used segmentation models.

Appendix

The derivation of the Euler-Lagrange equation associated to functional (2) is straightforward albeit long. Here we only provide the result. The notations used in the equations are the following: the point at contour parameter t is designated by position vector $\mathbf {r}\left( t\right)$. The bound variable of the integrals is denoted by $\tau$, hence the invariant arc length: $ds=\left| \dot{\mathbf {r}}\right| d\tau$ (the derivatives of the position vector w.r.t. the contour parameter are denoted by dots: $\dot{\mathbf {r}},\,\mathbf {\ddot{r}}\ldots$). The unit tangent and normal vectors of the contour are denoted by $\mathbf {e},\,\mathbf {n}$, where $\mathbf {e}=\frac{\dot{\mathbf {r}}}{\left| \dot{\mathbf {r}}\right| }$. Using right handed coordinate system, they are related with $\mathbf {n}=\mathbf {k}\times \mathbf {e}$, where $\mathbf {k}$ is the normal of the image plane, hence the contour normals point inward. The (signed) curvature of the contour is given by $\kappa =\frac{\mathbf {\ddot{r}}\cdot \mathbf {n}}{\left| \mathbf {\dot{r}}\right| ^{2}}$. The unit direction vector between points given by contour parameters t and $\tau$ is denoted by $\mathbf {e}_{t\tau }=\frac{\mathbf {r}_{\tau }-\mathbf {r}_{t}}{d_{t\tau }}$, where $d_{t\tau }=\left| \mathbf {r}_{\tau }-\mathbf {r}_{t}\right|$ is their Euclidean distance. We also use the normal to this direction vector with definition $\mathbf {n}_{t\tau }=\mathbf {k}\times \mathbf {e}_{t\tau }$. The alignment between contour points is defined by $a_{t\tau }\doteq \mathbf {n}\left( t\right) \cdot \mathbf {e}_{t\tau }+\mathbf {n}\left( \tau \right) \cdot \mathbf {e}_{\tau t}$ ($\mathbf {e}_{\tau t}=-\mathbf {e}_{t\tau }$). $f,\,g,\,l$ are the appropriately chosen functions of the curvature, alignment and distance respectively, for their derivatives we use prime mark $f^{\prime },\,f^{\prime \prime }\ldots$ The first and second derivatives of the curvature w.r.t. the arc length are denoted by $\frac{d\kappa }{ds},\,\frac{d^{2}\kappa }{ds^{2}}$. The curvature and its derivative are scalars. Note that the derivative w.r.t. the arc lenght of any scalar quantity q can be calculated on the planar grid using the formula: $\left( \nabla q\right) \cdot \mathbf {e}$, where $\nabla$ is the gradient operator with components being the partial derivatives $\left( \frac{\partial }{\partial x},\frac{\partial }{\partial y}\right)$ w.r.t. image coordinates $x,\,y$.

The formula: $A=\mathbf {e}_{t\tau }\cdot \left( \mathbf {e}\left( t\right) -\mathbf {e}\left( \tau \right) \right)$, $B=\mathbf {e}_{t\tau }\cdot \left( \mathbf {n}\left( t\right) -\mathbf {n}\left( \tau \right) \right)$,$C=\left[ \frac{l\left( d_{t\tau }\right) }{d_{t\tau }}-l^{\prime }\left( d_{t\tau }\right) \right]$, $D=\mathbf {e}_{t\tau }\cdot \mathbf {n}\left( t\right)$, $E=\mathbf {e}_{t\tau }\cdot \mathbf {e}\left( t\right)$, $F=\mathbf {n}\left( \tau \right) \cdot \mathbf {e}\left( t\right)$ are introduced to simplify the equation. The Euler-Lagrange equation associated to the half of (2) at point t has the form $\left| \dot{\mathbf {r}\left( t\right) }\right| Q\mathbf {n}\left( t\right) =\mathbf {0}$, where Q is given by the following sum:

$$\begin{aligned} -\left[ f^{\prime \prime \prime }\left( \frac{d\kappa }{ds}\right) ^{2}+f^{\prime \prime }\frac{d^{2}\kappa }{ds^{2}}\right] \oint g\left( a_{t\tau }\right) l\left( d_{t\tau }\right) f\left( \kappa \left( \tau \right) \right) ds\\ -2f^{\prime \prime }\frac{d\kappa }{ds}\oint g^{\prime }\left( a_{t\tau }\right) l\left( d_{t\tau }\right) \frac{AD}{d_{t\tau }}f\left( \kappa \left( \tau \right) \right) ds\\ +2f^{\prime \prime }\frac{d\kappa }{ds}\kappa \oint g^{\prime }\left( a_{t\tau }\right) l\left( d_{t\tau }\right) Ef\left( \kappa \left( \tau \right) \right) ds\\ +2f^{\prime \prime }\frac{d\kappa }{ds}\oint g\left( a_{t\tau }\right) l^{\prime }\left( d_{t\tau }\right) Ef\left( \kappa \left( \tau \right) \right) ds\\ -f^{\prime }\oint g^{\prime \prime }\left( a_{t\tau }\right) l\left( d_{t\tau }\right) \left[ \frac{\left( B-\kappa \left( t\right) d_{t\tau }\right) E+F}{d_{t\tau }}\right] ^{2}f\left( \kappa \left( \tau \right) \right) ds\\ +f^{\prime }\kappa \oint g^{\prime }\left( a_{t\tau }\right) l\left( d_{t\tau }\right) \frac{AE-2}{d_{t\tau }}f\left( \kappa \left( \tau \right) \right) ds\\ +f^{\prime }\kappa ^{2}\oint g^{\prime }\left( a_{t\tau }\right) l\left( d_{t\tau }\right) Df\left( \kappa \left( \tau \right) \right) ds\\ +f^{\prime }\oint g^{\prime }\left( a_{t\tau }\right) l\left( d_{t\tau }\right) \frac{BD^{2}}{d_{t\tau }^{2}}f\left( \kappa \left( \tau \right) \right) ds\\ -2f^{\prime }\oint g^{\prime }\left( a_{t\tau }\right) \frac{ACDE}{d_{t\tau }}f\left( \kappa \left( \tau \right) \right) ds\\ +2f^{\prime }\kappa \oint g^{\prime }\left( a_{t\tau }\right) CE^{2}f\left( \kappa \left( \tau \right) \right) ds\\ -f^{\prime }\kappa ^{2}\oint g\left( a_{t\tau }\right) l\left( d_{t\tau }\right) f\left( \kappa \left( \tau \right) \right) ds\\ +f^{\prime }\kappa \oint g\left( a_{t\tau }\right) l^{\prime }\left( d_{t\tau }\right) Df\left( \kappa \left( \tau \right) \right) ds\\ -f^{\prime }\oint g\left( a_{t\tau }\right) l^{\prime }\left( d_{t\tau }\right) \frac{D^{2}}{d_{t\tau }}f\left( \kappa \left( \tau \right) \right) ds\\ -f^{\prime }\oint g\left( a_{t\tau }\right) l^{\prime \prime }\left( d_{t\tau }\right) E^{2}f\left( \kappa \left( \tau \right) \right) ds\\ +f\kappa \oint g^{\prime \prime }\left( a_{t\tau }\right) l\left( d_{t\tau }\right) E^{2}f\left( \kappa \left( \tau \right) \right) ds\\ -f\oint g^{\prime \prime }\left( a_{t\tau }\right) l\left( d_{t\tau }\right) \frac{ADE}{d_{t\tau }}f\left( \kappa \left( \tau \right) \right) ds\\ -f\kappa \oint g^{\prime }\left( a_{t\tau }\right) l\left( d_{t\tau }\right) Df\left( \kappa \left( \tau \right) \right) ds\\ -f\oint g^{\prime }\left( a_{t\tau }\right) l\left( d_{t\tau }\right) \frac{D^{2}-AE}{d_{t\tau }}f\left( \kappa \left( \tau \right) \right) ds\\ +f\oint g^{\prime }\left( a_{t\tau }\right) l^{\prime }\left( d_{t\tau }\right) E^{2}f\left( \kappa \left( \tau \right) \right) ds\\ +f\oint g\left( a_{t\tau }\right) \left[ \kappa \left( t\right) l\left( d_{t\tau }\right) +l^{\prime }\left( d_{t\tau }\right) D\right] f\left( \kappa \left( \tau \right) \right) ds.\\ \end{aligned}$$

The factors outside the integrals are calculated at parameter t (i.e. independent of the bound variable). Note that any factor depending only on parameter t, could be brought before the integrals, but would lead more complicated expression.