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Efficient Global Minimization Methods for Image Segmentation Models with Four Regions

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Abstract

We propose an exact global minimization framework for certain variational image segmentation models, such as the Chan–Vese model, involving four regions. A global solution is guaranteed if the data term satisfies a certain condition. We give a theoretical analysis of the condition for \(L^p\) type of data terms, such as in the Chan–Vese model and Mumford Shah model for \(p=2\). We show experimentally that the condition tends to hold in practice for \(p \ge 2\) and also holds in many cases for more advanced data terms. If the condition is violated, convex and submodular relaxations are proposed which are not guaranteed to produce exact solutions, but tend to do so in practice. We also build up simple convex relaxations for some other four region segmentation models, including Potts’ model. Algorithms are proposed which are very efficient compared to related work due to the simple and compact formulations.

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Acknowledgments

We thank the anonymous reviewers for valuable feedback. This research has been supported by the Norwegian Research Council eVita project 214889 and eVita project 166075.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, CA, USA

    Egil Bae

  2. Department of Mathematics, University of Bergen, Bergen, Norway

    Xue-Cheng Tai

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Correspondence to Egil Bae.

Appendices

Appendix 1: A Proofs of Prop 2–5

1.1 Appendix 1.1: Proof of Prop 4

Proof

Let \(\frac{c_2+c_1}{2} \le I \le \frac{c_4+c_3}{2}\). Then

$$\begin{aligned} |c_2-I|^\beta \le |c_1-I|^\beta \;\;\; \text {and} \;\;\; |c_3-I|^\beta \le |c_4-I|^\beta , \end{aligned}$$

for any \(\beta \ge 1\). Therefore, adding these two inequalities

$$\begin{aligned} |c_2 - I|^\beta + |c_3 - I|^\beta \le |c_1 - I|^\beta + |c_4 - I|^\beta . \end{aligned}$$

\(\square \)

1.2 Appendix 1.2: Proof of Prop 2

When \(\frac{c_1+c_2}{2} \le I \le \frac{c_4+c_3}{2}\), the result follows from Prop 4. Consider \(I < \frac{c_2+c_1}{2}\), then

$$\begin{aligned} |I-c_1|^{\beta _0} \le |I-c_2|^{\beta _0} \le |I-c_3|^{\beta _0} \le |I-c_4|^{\beta _0}. \end{aligned}$$

Together with (64), this implies

$$\begin{aligned} 0 < |I-c_2|^{\beta _0} - |I-c_1|^{\beta _0} \le |I-c_4|^{\beta _0} - |I-c_3|^{\beta _0}. \end{aligned}$$

Therefore, there exists \(\theta _1 \ge \theta _2 > 1\) such that

$$\begin{aligned} |I-c_4|^{\beta _0} = \theta _1 |I-c_3|^{\beta _0} \quad \text {and} \quad |I-c_2|^{\beta _0} = \theta _2 |I-c_1|^{\beta _0}. \end{aligned}$$

For \(\beta \ge {\beta _0}\)

$$\begin{aligned}&|I-c_4|^\beta = \theta _1^{\beta - {\beta _0}} |I-c_3|^\beta \quad \text {and}\\&\quad |I-c_2|^\beta = \theta _2^{\beta - {\beta _0}} |I-c_1|^\beta , \end{aligned}$$

where \(\theta _1^{\beta - {\beta _0}} \ge \theta _2^{\beta - {\beta _0}} > 1\), hence

$$\begin{aligned}&|I-c_2|^\beta + |I-c_3|^\beta = \theta _2^{\beta - {\beta _0}} |I-c_1|^\beta + \frac{1}{\theta _1^{\beta - {\beta _0}}} |I-c_4|^\beta \\&\quad \le \theta _1^{\beta - {\beta _0}} |I-c_1|^\beta + \frac{1}{\theta _1^{\beta - {\beta _0}}} |I-c_4|^\beta \le \theta _1^{\beta - {\beta _0}} |I-c_1|^\beta \\&\quad + \frac{1}{\theta _1^{\beta - {\beta _0}}} |I-c_4|^\beta , \end{aligned}$$

where the last inequality follows from \(|I-c_1|^\beta \le |I-c_4|^\beta \). Exactly the same argument can be used to show Prop 2 when \(I > \frac{c_4+c_3}{2}\).

1.3 Appendix 1.3: Proof of Prop 3

Proof

Assume first \(I > c_3\), then

$$\begin{aligned} |c_1 - I| > |c_2 - I| > |c_3 -I|. \end{aligned}$$

Therefore, there exists a \(\theta > 1\) s.t.

$$\begin{aligned} |I-c_1| = \theta \, |c_2-I|. \end{aligned}$$

Pick \(\mathcal {C}_I^1 \in \mathbb {N}\) s.t.

$$\begin{aligned} \theta ^\beta \ge 2, \;\;\;\;\; \forall \beta \ge \mathcal {C}_I^1. \end{aligned}$$

Then

$$\begin{aligned}&|c_1-I|^\beta + |c_4-I|^\beta \ge |c_1-I|^\beta \ge 2 |c_2-I|^\beta \\&\quad > |c_2-I|^\beta + |c_3-I|^\beta \;\; \forall \beta \ge \mathcal {C}_I^1 \end{aligned}$$

If \(I<c_2\), then

$$\begin{aligned} |c_4 - I| > |c_3 - I| > |c_2 -I| \end{aligned}$$

and thus the same argument can be used to show there exists \(\mathcal {C}_I^2 \in \mathbb {N}\) such that

$$\begin{aligned} |c_4-I|^\beta + |c_1-I|^\beta > |c_2-I|^\beta + |c_3-I|^\beta , \;\;\; \forall \beta \ge \mathcal {C}_I^2. \end{aligned}$$

In case \(c_2 \le I \le c_3\), the existence of such a \(\mathcal {C}\) was proved in Prop 4, e.g. \(\mathcal {C} = 1\).

Therefore the condition (64) is satisfied for any \(I \in [0,L]\) by choosing \(\beta \ge \mathcal {C} = \max _{I \in [0,L]} \max \{ \mathcal {C}^1_I,\mathcal {C}^2_I \}\). \(\square \)

1.4 Appendix 1.4 Proof of Prop 5

We will show the condition holds for \(\beta = 1\), which by Prop 2 implies it holds for all \(\beta \ge 1\). Observe that if \(c_1,c_2\) and \(c_3,c_4\) are equidistant it follows that \(c_1 + c_4 = c_2 + c_3\). For \(I < c_2\)

$$\begin{aligned}&|I-c_2| + |I-c_3| = (c_2 - I) + (c_3 - I) \\&\quad = -2I + (c_2 + c_3)\\&\quad = -2I + (c_1 + c_4) = (c_1 - I) + (c_4 - I)\\&\quad \le |I-c_1| + |I-c_4|. \end{aligned}$$

For \(I \ge c_3\)

$$\begin{aligned}&|I-c_2| + |I-c_3| = (I - c_2) + (I - c_3)\\&\quad = 2I - (c_2 + c_3)\\&\quad = 2I - (c_1 + c_4) = (I - c_1) + (I - c_4) \\&\quad \le |I-c_1| + |I-c_4|. \end{aligned}$$

Appendix 2: Relation Between Dual Problem (63) and Discrete Max-flow Problem

Recall that for each \(p \in \mathcal {P}\) the data term was represented by the six edges (31). The flow on each of these edges are constrained by the capacities defined in (35) and (34), i.e.

$$\begin{aligned} P_s^1(p) \le c(s,v_{p,1})&\quad \text {flow} \le \; \text {capacity on}\; (s,v_{p,1}) \nonumber \\ P_s^2(p) \le c(s,v_{p,2})&\quad \text {flow} \le \; \text {capacity on} \;(s,v_{p,2}) \nonumber \\ P_t^1(p) \le c(v_{p,1},t)&\quad \text {flow} \le \; \text {capacity on}\; (v_{p,1},t) \nonumber \\ P_t^2(p) \le c(v_{p,2},t)&\quad \text {flow} \le \; \text {capacity on}\; (v_{p,2},t) \nonumber \\ \tilde{P}^{12}(p) \le c(v_{p,1},v_{p,2})&\quad \text {flow} \le \; \text {capacity on}\; (v_{p,1},v_{p,2}) \nonumber \\ \tilde{P}^{21}(p) \le c(v_{p,2},v_{p,1})&\quad \text {flow} \le \; \text {capacity on}\; (v_{p,2},v_{p,1}). \nonumber \end{aligned}$$

For notational convenience, we use capital letters \(P,Q\) to denote flow on discrete edges. The dual variables \(p_s^1, p_s^2, p_t^1, p_t^2, \tilde{p}^{12}, \tilde{p}^{21}\) over the continuous domain \({\varOmega }\) are inspired by these discrete flow functions. They are constrained by the capacities \(C_s^1(x), C_s^2(x), C_t^1(x), C_t^2(x), C^{12}(x)\) and \(C^{21}(x)\) respectively. Note that \(\tilde{p}^{12}\) and \(\tilde{p}^{21}\) should satisfy

$$\begin{aligned} 0 \le \tilde{p}^{12}(x) \le C^{12}(x), \;\;\; 0 \le \tilde{p}^{21}(x) \le C^{21}(x) \end{aligned}$$

for all \(x \in {\varOmega }\), but can be merged in \(p^{12} = \tilde{p}^{12} - \tilde{p}^{21}\). The above two constraints then transform into (51).

For each pair of neighboring pixels \((p,q) \in \mathcal {N}\), two edges were constructed in the discrete graph \(\mathcal {G}\): \((v_{p,1},v_{q,1})\) and \((v_{p_2},v_{q,2})\). Let \(Q^1\) denote the flow function on \((v_{p,1},v_{q,1})\) and \(Q^2\) the flow function on \((v_{p,2},v_{q,2})\). These flows are constrained by

$$\begin{aligned} 0 \le Q^1(v_{p,1},v_{q,1}) \le w_{pq}, \quad 0 \le Q^2(v_{p_2},v_{q,2}) \le w_{pq}, \; \end{aligned}$$

for all \((p,q) \in \mathcal {N}\). In the same vein, the two spatial flow fields \(q^i \in (C^\infty ({\varOmega }))^N\) \(i=1,2\) are defined in the continuous setting, and should satisfy (52). Observe that the continuous counterpart allows to measure the magnitude of \(q^1,q^2\) with 2-norm, which in turn produces rotationally invariant results.

Flow conservation should hold at each \(v_{p,1}\) and \(v_{p,2}\), i.e.

$$\begin{aligned}&\sum _{q \in \mathcal {N}_p^k} Q^1(v_{p,1},v_{q,1}) - P_s^1(p) + P_t^1(p) - \tilde{P}^{12}(p) + \tilde{P}^{21}(p) \\&\quad = 0, \;\;\; \\&\sum _{q \in \mathcal {N}_p^k} Q^2(v_{p,2},v_{q,2}) - P_s^2(p) + P_t^2(p) + \tilde{P}^{12}(p) - \tilde{P}^{21}(p) \\&\quad = 0, \, \end{aligned}$$

\(\forall p \in \mathcal {P}\). Here \(N^+(v)\) is defined as all \(w \in \mathcal {V}\) such that \((v,w) \in \mathcal {E}\) and \(N^-(v)\) is defined as all \(w \in \mathcal {V}\) such that \((w,v) \in \mathcal {E}\). The final two constraints (61) and (62) are generalizations of discrete flow conservation to the continuous setting. The total amount of discrete flow in the graph is in this case

$$\begin{aligned} \sum _{p \in \mathcal {P}} P_s^1(p) + P_s^2(p) \, . \end{aligned}$$

The objective functional (63) is a generalization of the above, and measures the total amount of flow that originates from the source.

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Bae, E., Tai, XC. Efficient Global Minimization Methods for Image Segmentation Models with Four Regions. J Math Imaging Vis 51, 71–97 (2015). https://doi.org/10.1007/s10851-014-0507-2

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