A Convex Geodesic Selective Model for Image Segmentation
Abstract
Selective segmentation is an important application of image processing. In contrast to global segmentation in which all objects are segmented, selective segmentation is used to isolate specific objects in an image and is of particular interest in medical imaging—permitting segmentation and review of a single organ. An important consideration is to minimise the amount of user input to obtain the segmentation; this differs from interactive segmentation in which more user input is allowed than selective segmentation. To achieve selection, we propose a selective segmentation model which uses the edgeweighted geodesic distance from a marker set as a penalty term. It is demonstrated that this edgeweighted geodesic penalty term improves on previous selective penalty terms. A convex formulation of the model is also presented, allowing arbitrary initialisation. It is shown that the proposed model is less parameter dependent and requires less user input than previous models. Further modifications are made to the edgeweighted geodesic distance term to ensure segmentation robustness to noise and blur. We can show that the overall Euler–Lagrange equation admits a unique viscosity solution. Numerical results show that the result is robust to user input and permits selective segmentations that are not possible with other models.
Keywords
Variational model Partial differential equations Image segmentation Additive operator splitting Viscosity solution Geodesic1 Introduction
Segmentation of an image into its individual objects is one incredibly important application of image processing techniques. Segmentation can take two forms: firstly, global segmentation for isolation of all foreground objects in an image from the background and secondly, selective segmentation for isolation of a subset of the objects in an image from the background. A comprehensive review of selective segmentation can be found in [7, 19] and in [44] for medical image segmentation where selection refers to extraction of single organs.
Approaches to image segmentation broadly fall into two classes: region based and edge based. Some regionbased approaches are region growing [1], watershed algorithms [39], Mumford and Shah [29] and Chan and Vese [11]. The final two of these are partial differential equations (PDEs)based variational approaches to the problem of segmentation. There are also models which mix the two classes to use the benefits of the regionbased and edgebased approaches and will incorporate features of each. Edgebased methods aim to encourage an evolving contour towards the edges in an image and normally require an edge detector function [8]. The first edgebased variational approach was devised by Kass et al. [22] with the famous snakes model and further developed by Casselles et al. [8] who introduced the geodesic active contour (GAC) model. Regionbased global segmentation models include the wellknown works of Mumford and Shah [29] and Chan and Vese [11]. Importantly, they are nonconvex and hence a minimiser of these models may only be a local, not the global, minimum. Further work by Chan et al. [10] gave rise to a method to find the global minimiser for the Chan–Vese model under certain conditions.
This paper is mainly concerned with selective segmentation of objects in an image, given a set of points near the object or objects to be segmented. It builds in such user input to a model using a set \( {\mathcal {M}}=\{ (x_{i},y_{i})\in \varOmega , 1\le i\le k\} \) where \(\varOmega \subset {\mathbb {R}}^{2}\) is the image domain [4, 5, 17]. Nguyen et al. [30] considered marker sets \({\mathcal {M}}\) and \({\mathcal {A}}\) which consist of points inside and outside, respectively, the object or objects to be segmented. Gout et al. [17] combined the GAC approach with the geometrical constraint that the contour passes through the points of \({\mathcal {M}}\). This was enforced with a distance function which is zero at \({\mathcal {M}}\) and nonzero elsewhere. Badshah and Chen [4] then combined the Gout et al. model with [11] to incorporate a constraint on the intensity in the selected region, thereby encouraging the contour to segment homogeneous regions. Rada and Chen [35] introduced a selective segmentation method based on twolevel sets which was shown to be more robust than the Badshah–Chen model. We also refer to [5, 23] for selective segmentation models which include different fitting constraints, using coefficient of variation and the centroid of \({\mathcal {M}}\), respectively. None of these models have a restriction on the size of the object or objects to be detected, and depending on the initialisation, these methods have the potential to detect more or fewer objects than the user desired. To address this and to improve on [35], Rada and Chen [36] introduced a model combining the Badshah–Chen [4] model with a constraint on the area of the objects to be segmented. The reference area used to constrain the area within the contour is that of the polygon formed by the markers in \({\mathcal {M}}\). Spencer and Chen [38] introduced a model with the distance fitting penalty as a standalone term in the energy functional, unbounding it from the edge detector term of the Gout et al. model.
All of the above selective segmentation models discussed are nonconvex, and hence the final result depends on the initialisation. Spencer and Chen [38], in the same paper, reformulated the model they introduced to a convex form using convex relaxation and an exact penalty term as in [10]. Their model uses Euclidean distance from the marker set \({\mathcal {M}}\) as a distance penalty term; however, we propose replacing this with the edgeweighted geodesic distance from \({\mathcal {M}}\) (we call this simply the geodesic distance). This distance increases at edges in the image and is more intuitive for selective segmentation. The proposed model is given as a convex relaxed model with exact penalty term, and we give a general existence and uniqueness proof for the viscosity solution to the PDE given by its Euler–Lagrange equation, which is also applicable to a whole class of PDEs arising in image segmentation. We note that the use of geodesic distance for segmentation has been considered before [6, 33]; however, the models only use geodesic distance as the fitting term within the regulariser, so are liable to make segmentation errors for poor initialisation or complex images. Here, we take a different approach, by including geodesic distance as a standalone fitting term, separate from the regulariser, and using intensity fitting terms to ensure robustness.

We incorporate the geodesic distance as a distance penalty term within the variational framework.

We propose a convex selective segmentation model using this penalty term and demonstrate how it can achieve results which cannot be achieved by other models.

We improve the geodesic penalty term, focussing on improving robustness to noise and improving segmentation when object edges are blurred.

We give an existence and uniqueness proof for the viscosity solution for the PDEs associated with a whole class of segmentation models (both global and selective).
The paper is structured as follows: in Sect. 2, we review some global and selective segmentation models. In Sect. 3, we discuss the geodesic distance penalty term, propose a new convex model and also address weaknesses in the naïve implementation of the geodesic distance term. In Sect. 4, we discuss the nonstandard AOS scheme, introduced in [38], which we use to solve the model. In Sect. 5, we give an existence and uniqueness proof for a general class of PDEs arising in image segmentation, thereby showing that for a given initialisation, the solution to our model is unique. In Sect. 6, we compare the results of the proposed model to other selective segmentation models and show that the proposed model is less parameter dependent than other models and is more robust to user input. Finally, in Sect. 7, we provide some concluding remarks.
2 Review of Variational Segmentation Models
Although we focus on selective segmentation, it is illuminating to introduce some global segmentation models first. Throughout this paper, we denote the original image by z(x, y) with image domain \(\varOmega \subset {\mathbb {R}}^{2}\).
2.1 Global Segmentation
2.2 Selective Segmentation
Selective segmentation models make use of user input, i.e. a marker set \({\mathcal {M}}\) of points near the object or objects to be segmented. Let \( {\mathcal {M}}=\{ (x_{i},y_{i})\in \varOmega , 1\le i\le k\} \) be such a marker set. The aim of selective segmentation is to design an energy functional where the segmentation contour \(\varGamma \) is close to the points of \({\mathcal {M}}\).
3 Proposed Convex Geodesic Selective Model
3.1 Computing the Geodesic Distance Term \({\mathcal {D}}_{M}(x,y)\)
3.2 Comparing Euclidean and Geodesic Distance Terms
 1.
Parameter robustness. The Geodesic Model is more robust to the choice of the fitting parameter \(\theta \), as the penalty on the inside of the shape we want segmented is consistently small. It is only outside the shape where the penalty is large. However, with the Euclidean distance term, we always have a penalty inside the shape we actually want to segment. This is due to the nature of the Euclidean distance which does not discriminate on intensity—this penalty can also be quite high if our marker set is small and does not cover the whole object.
 2.
Robust to marker set selection. The geodesic distance term is far more robust to point selection; for example, we can choose just one point inside the object we want to segment and this will give a nearly identical geodesic distance compared to choosing many more points. This is not true of the Euclidean distance term which is very sensitive to point selection and requires markers to be spread in all areas of the object you want to segment (especially, at extrema of the object).
Remark 1
(Computational complexity) The main concern of using the geodesic penalty term, which we obtain by solving PDE (11), would be that it takes a significant amount of time to compute compared to the Euclidean distance. However, using the fast marching algorithm of Sethian [37], the complexity of computing \({\mathcal {D}}_{M}(x,y)\) is \({\mathcal {O}}(N\log (N))\) for an image with N pixels. This is only marginally more complex than computing the Euclidean distance which has \({\mathcal {O}}(N)\) complexity [28].
3.3 Improvements to Geodesic Distance Term
 1.
Not robust to noise. The computation of the geodesic distance depends on \(\nabla z(x,y)^{2}\) in f(x, y) [see (11)]. So, if an image contains a lot of noise, each noisy pixel appears as an edge and we get a misleading distance term.
 2.
Objects far from\({\mathcal {M}}\)with low penalty. As the geodesic distance only uses marker set \({\mathcal {M}}\) for its initial condition [see (11)], this can result in objects far from \({\mathcal {M}}\) having a low distance penalty, which is clearly not desired.
 3.
Blurred edges. If we have two objects separated by a blurry edge and we have marker points only in one object, the geodesic distance will be low to the other object, as the edge penalty is weakly enforced for a blurry edge. We would desire low penalty inside the object with markers and a reasonable penalty in the joined object.
Problem 1: Noise robustness
Problem 2: Objects far from\({\mathcal {M}}\)with low penalty
Problem 3: Blurred edges
3.4 The New Model and Its Euler–Lagrange Equation
Next, we discuss a numerical scheme for solving this PDE (20). However, it should be remarked that updating \(c_{1}(u), c_{2}(u)\) should be done as soon as u is updated; practically, \(c_1, c_2\) converge very quickly since the object intensity \(c_1\) does not change much.
4 An Additive Operator Splitting Algorithm
5 Existence and Uniqueness of the Viscosity Solution
In this section, we use the viscosity solution framework and the work of Ishii and Sato [20] to prove that, for a class of PDEs in image segmentation, the solution exists and is unique. In particular, we prove the existence and uniqueness of the viscosity solution for the PDE which is determined by the Euler–Lagrange equation for the Geodesic Model. Throughout, we will assume \(\varOmega \) is a bounded domain with \(C^{1}\) boundary.
From the work of [12, 20], we have the following Theorem for analysing the solution of a partial differential equation of the form \(F(\varvec{x},u,Du,D^{2}u)=0\) where \(F: {\mathbb {R}}^{n}\times {\mathbb {R}}\times {\mathbb {R}}^{n}\times {\mathscr {M}}^{n}\rightarrow {\mathbb {R}}\), \({\mathscr {M}}^{n}\) is the set of \(n\times n\) symmetrical matrices, Du is the gradient of u and \(D^{2}u\) is the Hessian of u. For simplicity, and in a slight abuse of notation, we use \(x := \varvec{x}\) for the vector of a general point in \({\mathbb {R}}^{n}\).
Theorem 2
(Theorem 3.1 [20]) Assume that the following conditions (C1)–(C2) and (I1)–(I7) hold. Then, for each \(u_{0}\in C({\overline{\varOmega }})\) there is a unique viscosity solution \(u\in C([0,T)\times {\overline{\varOmega }})\) of (23) and (24) satisfying (25).
 (C1)
\(F(t,x,u,p,X) \le F(t,x,v,p,X)\) for \(u\le v\).
 (C2)
\(F(t,x,u,p,X) \le F(t,x,u,p,Y)\) for \(X,Y\in {\mathscr {M}}^{n}\) and \(Y\le X\).
 (I1)
\(F \in C \left( [0,T] \times {\overline{\varOmega }} \times {\mathbb {R}} \times \left( {\mathbb {R}}^{n}\backslash \{ 0 \} \right) \times {\mathscr {M}}^{n} \right) \).
 (I2)
There exists a constant \(\gamma \in {\mathbb {R}}\) such that for each \((t,x,p,X)\in [0,T]\times {\overline{\varOmega }}\times \left( {\mathbb {R}}^{n}\backslash \{ 0\} \right) \times {\mathscr {M}}^{n}\) the function \(u\mapsto F(t,x,u,p,X)  \gamma u\) is nondecreasing on \({\mathbb {R}}\).
 (I3)F is continuous at (t, x, u, 0, 0) for any \((t,x,u)\in [0,T]\times {\overline{\varOmega }}\times {\mathbb {R}}\) in the sense thatholds. Here, \(F^{*}\) and \(F_{*}\) denote, respectively, the upper and lower semicontinuous envelopes of F, which are defined on \([0,T]\times {\overline{\varOmega }}\times {\mathbb {R}}\times {\mathbb {R}}^{n}\times {\mathscr {M}}^{n}\).$$\begin{aligned} \infty< F_{*}(t,x,u,0,0) = F^{*}(t,x,u,0,0)<\infty \end{aligned}$$
 (I4)
\(B\in C\left( {\mathbb {R}}^{n} \times {\mathbb {R}}^{n} \right) \cap C^{1,1}\left( {\mathbb {R}}^{n} \times \left( {\mathbb {R}}^{n}\backslash \{ 0 \}\right) \right) \), where \(C^{1,1}\) is the Hölder functional space.
 (I5)
For each \(x\in {\mathbb {R}}^{n}\), the function \(p\mapsto B(x,p)\) is positively homogeneous of degree one in p, i.e. \(B(x,\lambda p) = \lambda B(x,p)\) for all \(\lambda \ge 0\) and \(p\in {\mathbb {R}}^{n}\backslash \{ 0\}\).
 (I6)
There exists a positive constant \(\Theta \) such that \(\langle {\varvec{n}}(x), D_{p} B(x,p)\rangle \ge \Theta \) for all \(x\in \partial \varOmega \) and \(p\in {\mathbb {R}}^{n}\backslash \{ 0\}\). Here, \({\varvec{n}}(x)\) denotes the unit outward normal vector of \(\varOmega \) at \(x\in \partial \varOmega \).
 (I7)For each \(R>0\), there exists a nondecreasing continuous function \(\omega _{R}:[0,\infty )\rightarrow [0,\infty )\) satisfying \(\omega _{R}(0)=0\) such that if \(X,Y\in {\mathscr {M}}^{n}\) and \(\mu _{1},\mu _{2}\in [0,\infty )\) satisfythen$$\begin{aligned} \begin{bmatrix} X&0 \\ 0&Y \\ \end{bmatrix} \le \mu _{1} \begin{bmatrix} I&I \\ I&I \\ \end{bmatrix} +\mu _{2} \begin{bmatrix} I&0\\ 0&I \\ \end{bmatrix}\end{aligned}$$(26)for all \(t\in [0,T], x,y\in {\overline{\varOmega }}, u\in {\mathbb {R}}\), with \(u\le R\), \(p,q\in {\mathbb {R}}^{n}\backslash \{ 0\}\) and \(\rho (p,q) = \min \left( \frac{pq}{\min (p,q)},1 \right) \).$$\begin{aligned} \begin{aligned}&F(t,x,u,p,X)  F(t,y,u,q,Y) \ge \\&\quad \omega _{R}\Big ( \mu _{1}\left( xy^{2}+\rho (p,q)^{2} \right) + \mu _{2} + pq \\&\quad + xy\left( \max (p , q) +1 \right) \Big )\\ \end{aligned} \end{aligned}$$
5.1 Existence and Uniqueness for the Geodesic Model
We now prove that there exists a unique solution for the PDE (20) resulting from the minimisation of the functional for the Geodesic Model (18).
Remark 3
It is important to note that although the values of \(c_{1}\) and \(c_{2}\) depend on u, they are fixed when we solve the PDE for u and therefore the problem is a local one and Theorem 2 can be applied. Once we update \(c_{1}\) and \(c_{2}\), using the updated u, then we fix them again and apply Theorem 2. In practice, as we near convergence, we find \(c_{1}\) and \(c_{2}\) stabilise so we typically stop updating \(c_{1}\) and \(c_{2}\) once the change in both values is below a tolerance.
Theorem 4
(Theory for the Geodesic Model) The parabolic PDE \(\frac{\partial u}{\partial t} + F(t,x,u,Du,D^{2}u) = 0 \) with \(u_{0} = u(0,x)\in C({\overline{\varOmega }})\), F as defined in (28) and Neumann boundary conditions has a unique solution \(u=u(t,x)\) in \(C([0,T)\times {\overline{\varOmega }})\).
Proof
By Theorem 2, it remains to verify that F satisfies (C1)–(C2) and (I1)–(I7). We will show that each of the conditions is satisfied. Most are simple to show, the exception being (I7) which is nontrivial.
(C1): Equation (28) only has dependence on u in the term k(u); we therefore have a restriction on the choice of k, requiring \(k(v)\ge k(u)\) for \(v\ge u\). This is satisfied for \(k(u)=\alpha \nu '_{\varepsilon }(u)\) with \(\nu '_{\varepsilon }(u)\) defined as in (7).
(C2): We find for arbitrary \(s=(s_{1},s_{2})\in {\mathbb {R}}^{2}\) that \(s^{T} A(x,p) s \ge 0\) and so \( A(x,p)\ge 0\). It follows that \(\text {trace}(A(x,p) X) \le \text {trace}(A(x,p) Y)\); therefore, this condition is satisfied.
(I1): A(x, p) is only singular at \(p=0\); however, it is continuous elsewhere and satisfies this condition.
(I2): In F, the only term which depends on u is \(k(u)=\alpha \nu '_{\varepsilon }(u)\). With \(\nu '_{\varepsilon }(u)\) defined as in (7), in the limit \(\varepsilon \rightarrow 0\) this function is a step function from \(2\) on \((\infty ,0)\), 0 on [0, 1] and 2 on \((0,\infty )\). So we can choose any constant \(\varepsilon < 2\). With \(\varepsilon \ne 0\), there is smoothing at the end of the intervals; however, there is still a lower bound on L for \(\nu '_{\varepsilon }(u)\) and we can choose any constant \(\gamma < L\).
(I3): F is continuous at (x, 0, 0) for any \(x \in \varOmega \) because \(F^{*}(x, 0, 0) = F_{*} (x, 0, 0) = 0.\) Hence, this condition is satisfied.
(I5): By the definition above, \(B(x,\lambda \nabla u) = \langle {\varvec{n}} , \lambda \nabla u \rangle =\lambda \langle {\varvec{n}} , \nabla u \rangle = \lambda B(x,\nabla u) \). So this condition is satisfied.
(I6): As before, we can use the definition, \(\langle \varvec{n}(x), D_{p}B(x,p)\rangle = \langle \varvec{n}(x), \varvec{n}(x) \rangle = \varvec{n}(x)^{2}\). So we can choose \(\Theta = 1\) and the condition is satisfied.
(I7): This is the most involved condition to prove and uses many other results. For clarity of the overall paper, we postpone the proof to ‘Appendix A’.
5.2 Generalisation to Other Related Models
Theorems 2 and 4 can be generalised to a few other models. This amounts to writing each model as a PDE of the form (28) where k(u) is monotone and f(x), k(u) are bounded. This is summarised in the following corollary:
Corollary 5

Chan–Vese [11]: \(f(x) = f_{\mathrm{CV}}(x):=\lambda _{1} (z(x)c_{1})^{2}  \lambda _{2}(z(x)c_{2})^{2}\), \(k(u) = 0\).

Chan–Vese (Convex) [10]: \(f(x) = f_{\mathrm{CV}}(x)\), \(k(u) = \alpha \nu '_{\varepsilon }(u)\).

Geodesic active contours [8] and Gout et al. [25]: \(f(x) = 0\), \(k(u) = 0\).

Nguyen et al. [30]: \(f(x) = \alpha \left( P_{\text {B}}(x,y)  P_{\text {F}}(x,y)\right) + \left( 1\alpha \right) \left( 12P(x,y)\right) \), \(k(u) = 0\).

Spencer–Chen (Convex) [38]: \(f(x) = f_{\mathrm{CV}}(x) + \theta {\mathcal {D}}_{E}(x)\), \(k(u) = \alpha \nu '_{\varepsilon }(u)\).
 (i)
Neumann boundary conditions \(\frac{\partial u}{\partial {\varvec{n}}} = 0\) (\({\varvec{n}}\) the outward normal unit vector)
 (ii)
k(u) satisfies \(k(u)\ge k(v)\) if \(u\ge v\)
 (iii)
k(u) and f(x) are bounded; and
 (iv)
\(G(x) = Id\) or \(G(x) = f(\nabla z(x)) = \frac{1}{1+\nabla z(x)^{2}}\),
Proof
The conditions (i)–(iv) are hold for all of these models. All of these models require Neumann boundary conditions and use the permitted G(x). The monotonicity of \(\nu '_{\varepsilon }(u)\) is discussed in the proof of (C1) for Theorem 4, and the boundedness of f(x) and k(u) is clear in all cases.
Remark 6
Theorem 4 and Corollary 5 also generalise to cases where \(G(x) = \frac{1}{1+\beta \nabla z^{2}}\) and to \(G(x) = {\mathcal {D}}(x) g(\nabla z)\) where \({\mathcal {D}}(x)\) is a distance function such as in [15, 16, 17, 38]. The proof is very similar to that shown in Sect. 5.1, relying on Lipschitz continuity of the function G(x).
Remark 7
We cannot apply the classical viscosity solution framework to the Rada–Chen model [36] as this is a nonlocal problem with \(k(u) = 2\nu \left( \int _{\varOmega }H_{\varepsilon }(u)\,\hbox {d}\varOmega A_{1} \right) \).
6 Numerical Results

M1 —the Nguyen et al. [30] model;

M2 —the Rada and Chen [36] model;

M3 —the convex Spencer–Chen [38] model;

M4 —the convex Liu et al. [26] model;

M5 —the reformulated Rada–Chen model with geodesic distance penalty (see Remark 8);

M6 —the reformulated Liu et al. model with geodesic distance penalty (see Remark 8);

M7 —the proposed convex Geodesic Model (Algorithm 1).
Remark 8

we extend M2–M5 simply by including the geodesic distance function \({\mathcal {D}}_{G}(x,u)\) in the functional.
 we extend M4–M6 with a minor reformulation to include data fitting terms. Specifically, the model M6 isfor \(\mu ,\lambda _{1},\lambda _{2}\) nonnegative fixed parameters, \(\alpha \) and \(\nu _{\varepsilon }(u)\) as defined in (7) and \(\omega \) as defined for the convex Liu et al. model. This is a convex model and is the same as the proposed Geodesic Model M7 but with weighted intensity fitting terms.$$\begin{aligned}&\min _{u,c_{1},c_{2}}\Big \{F_{CV\omega }(u,c_{1},c_{2})\nonumber \\&\quad = \int _{\varOmega }\omega ^{2}(x,y)\left[ \lambda _{1}(z(x,y)c_{1})^{2}\right. \nonumber \\&\qquad  \left. \lambda _{2}(z(x,y)c_{2})^{2} \right] u\,\hbox {d}\varOmega + \mu \int _{\varOmega }g(\nabla z))\nabla u\,\hbox {d}\varOmega \nonumber \\&\qquad + \theta \int _{\varOmega }{\mathcal {D}}_{G}(x,y)u \,\hbox {d}\varOmega + \alpha \int _{\varOmega }\nu _{\varepsilon }(u)\,\hbox {d}\varOmega \Big \} \end{aligned}$$(30)
Four sets of test results are shown below. In Test 1, we compare models M1–M6 to the proposed model M7 for two images which are hard to segment. The first is a CT scan from which we would like to segment the lower portion of the heart, the second is an MRI scan of a knee and we would like to segment the top of the Tibia. See Fig. 9 for the test images and the marker sets used in the experiments. In Test 2, we will review the sensitivity of the proposed model to the main parameters. In Test 3, we will give several results achieved by the model using marker and antimarker sets. In Test 4, we show the initialisation independence and marker independence of the Geodesic Model on real images.
For M7, we denote by \({\tilde{u}}\) the thresholded \(u > {\tilde{\gamma }}\) at some value \({\tilde{\gamma }}\in (0,1)\) to define the segmented region. Although the threshold can be chosen arbitrarily in (0, 1) from the work by [10, Theorem 1] and [38], we usually take \({\tilde{\gamma }}=0.5\).
Parameter choices and implementation. We set \(\mu = 1\), \(\tau = 10^{2}\) and vary \(\lambda = \lambda _{1} = \lambda _{2}\) and \(\theta \). Following [10], we let \(\alpha = \lambda _{1}(zc_{1})^{2}\lambda _{2}(zc_{2})^{2}+\theta {\mathcal {D}}_{G}(x,y)_{L^{\infty }}\). To implement the marker points in MATLAB, we use roipoly for choosing a small number of points by clicking and also freedraw which allows the user to draw a path of marker points. The stopping criteria used are the dynamic residual falling below a given threshold; i.e. once \(u^{k+1}u^{k}/u^{k} < \hbox {tol}\), the iterations stop (we use \(\hbox {tol} = 10^{6}\) in the tests shown).
Remark 9
Models M2–M7 are coded in MATLAB and use exactly the same marker/antimarker set. For model M1, the software of Nguyen et al. requires marker and antimarker sets to be input to an interface. These have been drawn as close as possible to match those used in the MATLAB code.
\(\varepsilon _{2}\)  Knee segmentation (Fig. 12)  Circle segmentation (Fig. 13) 

\(10^{10}\)  0.97287  1.00000 
\(10^{8}\)  0.97287  1.00000 
\(10^{6}\)  0.97235  1.00000 
\(10^{4}\)  0.96562  1.00000 
\(10^{2}\)  0.94463  1.00000 
\(10^{0}\)  0.90660  1.00000 
\(10^{2}\)  0.89573  1.00000 
\(10^{4}\)  0.89159  1.00000 
Test 2—Test of M7’s sensitivity to changes in its main parameters. In this test, we demonstrate that the proposed Geodesic Model is robust to changes in the main parameters. The main parameters in (20) are \(\mu , \lambda _{1},\lambda _{2},\theta \) and \(\varepsilon _{2}\). In all tests, we set \(\mu = 1\), which is simply a rescaling of the other parameters, and we set \(\lambda = \lambda _{1} = \lambda _{2}\). In the first example, in Fig. 12, we compare the TC value for various \(\lambda \) and \(\theta \) values for segmentation of a bone in a knee scan. We see that the segmentation is very good for a larger range of \(\theta \) and \(\lambda \) values. For the second example, in Fig. 13, we show an image and marker set for which the Spencer–Chen model (M3) and modified Liu et al. model M6 cannot achieve the desired segmentation for any parameter range, but which can be attained for the Geodesic Model for a vast range of parameters. The final example, in Table 1, compares the TC values for various \(\varepsilon _{2}\) values with fixed parameters \(\lambda = 2\) and \(\theta = 2\). We use the images and ground truth as shown in Figs. 12 and 13 : on the synthetic circles image, we obtain a perfect segmentation for all values of \(\varepsilon _{2}\) tested, and in the case of the knee segmentation the results are almost identical for any \(\varepsilon _{2} < 10^{6}\), above which the quality slowly deteriorates.
Test 3—Further results from the Geodesic Model M7. In this test, we give some medical segmentation results obtained using the Geodesic Model M7. The results are shown in Fig. 14. In the final two columns, we use antimarkers to demonstrate how to overcome blurred edges and low contrast edges in an image. These are challenging, and it is pleasing to see the correctly segmented results.
Test 4—Initialisation and marker set independence. In the first example, in Fig. 15, we see how the convex Geodesic Model M7 gives the same segmentation result regardless of initialisation, as expected of a convex model. Hence, the model is flexible in implementation. In our experiments we find that the algorithm converges to the final solution faster when we initialise using the polygon formed from the marker points rather than an arbitrary initialisation. In the second example, in Fig. 16, we show intuitively how Model M7 is robust to the number of markers and the location of the markers within the object to be segmented. The Euclidean distance term, used in the Spencer–Chen model M3, is sensitive to the position and number of marker points; however, regardless of where the markers are chosen, and how many are chosen, the geodesic distance map will be almost identical.
7 Conclusions
In this paper, a new convex selective segmentation model has been proposed, using geodesic distance as a penalty term. This model gives results that are unachievable by alternative selective segmentation models and are also more robust to the parameter choices. Adaptations to the penalty term have been discussed which make it robust to noisy images and blurry edges whilst also penalising objects far from the marker set (in a Euclidean distance sense). A proof for the existence and uniqueness of the viscosity solution to the PDE given by the Euler–Lagrange equation for the model has been given (which applies to an entire class of image segmentation PDEs). Finally, we have confirmed the advantages of using the geodesic distance with some experimental results. Future works will look for further extension of selective segmentation to other frameworks such as using highorder regularizers [13, 45] where only incomplete theories exist.
Notes
Acknowledgements
The authors are grateful to Professor Joachim Weickert (Saarland, Germany) for fruitful discussions at the early stages of this work.
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