Translation, Scale, and Deformation Weighted Polar Active Contours

Abstract

Polar active contours have proven to be a powerful segmentation method for many medical as well as other computer vision applications, such as interactive image segmentation or tracking. Inspired by recent work on Sobolev active contours we derive a Sobolev-type function space for polar curves, which is endowed with a metric that allows us to favor origin translations and scale changes over smooth deformations of the curve. The resulting translation, scale, and deformation weighted polar active contours inherit the coarse-to-fine behavior of Sobolev active contours as well as their robustness to local minima and are thus very useful for many medical applications, such as cross-sectional vessel segmentation, aneurysm analysis, or cell tracking.

Appendices

Appendix A: Solving the ODE

The computation of the gradient $\nabla _{\!E}\mathcal{H}$ requires the solution of the following ordinary differential equation, cf. also [25]:

$$g''(s) = \frac{1}{\lambda L^2}\bigl(f(s)-\bar{f}\bigr),\quad g(0)=g(L),\quad g'(0)=g'(L),$$

(34)

where we set g=k ^r and f=h ^r in order to keep the notation simple. Further, we denote the differentiation with respect to s is by ⋅′. At first, we integrate twice and obtain:

$$g(s) = g(0) + sg'(0) + \frac{1}{\lambda L^2}\int^s_0 \int^{\hat{s}}_0 f(\xi)-\bar{f}\,d\xi\,d\hat{s}.$$

(35)

Second, we perform integration by parts for the rightmost term:

(36)

(37)

(38)

(39)

Thus we end up with

$$g(s) = g(0) + sg'(0) + \frac{1}{\lambda L^2}\int^s_0(s-\hat{s})\bigl(f(\hat{s})-\bar{f}\bigr)\,d\hat{s}. $$

(40)

Now we set s=L and use the boundary condition g(0)=g(L) and obtain:

(41)

(42)

(43)

due to the definition of $\bar{f}$. Changing $\hat{s}$ to s eventually yields

$$g'(0) = -\frac{1}{\lambda L^3} \int^L_0s\bigl(f(s)-\bar{f}\bigr)\,ds.$$

(44)

Plugging this formula into (40), integrating both sides form 0 to L, and using again that $\bar{g}=\bar{f}$ we finally obtain

(45)

where K _γ (s) is the kernel defined in (26). Since the definition of the arc length allows us to choose the starting point arbitrarily we can conclude from (45) that g may also be obtained by g=K _γ f.

Appendix B: Deriving a First Order Correction for ϕ

Now we want to derive a first-order accurate correction ψ such that $\hat{\phi}(x)=\psi(x)\phi(x)$, where ϕ defined in (32), is close to a signed distance representation of c, which we denote by Φ(x). At first, we note that the level lines of a signed distance representation of c are solutions to c _t =n. Supposing that ϵ∈ℝ is sufficiently small one might say that the level line L _ϵ ={x∈Ω:Φ(x)=ϵ} can be approximated by evolving c with a forward Euler discretization of the flow c _t =n:

$$c_{0+\tau} = c_0+\tau \mathbf {n},$$

(46)

where c ₀=c and τ=ϵ—see also sketch below:

As we are only interested in the geometry of c we can add a tangential component to the flow c _t =n:

$$c_t = \mathbf {n}+ \frac{\mathbf {s}\cdot \mathbf {t}}{\mathbf {s}\cdot \mathbf {n}}\mathbf {t}= \frac{\mathbf {s}\cdot \mathbf {n}}{\mathbf {s}\cdot \mathbf {n}}\mathbf {n}+\frac{\mathbf {s}\cdot \mathbf {t}}{\mathbf {s}\cdot \mathbf {n}}\mathbf {t}=\frac{1}{\mathbf {s}\cdot \mathbf {n}}\mathbf {s},$$

(47)

since t and n are an orthonormal basis. From this we may conclude that the c _t =n and c _t =1/(s⋅n)s yield the same geometric curve. This means that the level line L _ϵ can also be approximated by performing a forward Euler step of the flow c _t =1/(s⋅n)s:

$$c_{0+\tau}=c_0+\frac{\tau}{ \mathbf {s}\cdot \mathbf {n}}\mathbf {s},$$

(48)

where again c ₀=c and τ=ϵ. As ϕ(x) gives us the signed distance ϵ/(s⋅n) we thus know that the correction ψ is given by

$$\psi(x)=\mathbf {s}\bigl(\theta(x)\bigr) \cdot \mathbf {n}\bigl(\theta(x)\bigr).$$

(49)

Of course, this correction is only first-order accurate in ϵ, because we derived it via a forward Euler discretization.