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Table 1 Relationship between binary region number and characteristic function

From: The Vese-Chan model without redundant parameter estimation for multiphase image segmentation

Ωi\({b_{1}^{i - 1}}\)\({b_{2}^{i - 1}}\)\({b_{3}^{i - 1}}\)Characteristic function \({{\chi _{i}} = \prod \limits _{j = 1}^{m} \left ({b_{j}^{i - 1} + {{\left ({ - 1} \right)}^{b_{j}^{i - 1}}}{\phi _{j}}} \right)}\)
Ω1000χ1=ϕ1ϕ2ϕ3
Ω2001χ2=ϕ1ϕ2(1−ϕ3)
Ω3010χ3=ϕ1(1−ϕ2)ϕ3
Ω4011χ4=ϕ1(1−ϕ2)(1−ϕ3)
Ω5100χ5=(1−ϕ1)ϕ2ϕ3
Ω6101χ6=(1−ϕ1)ϕ2(1−ϕ3)
Ω7110χ7=(1−ϕ1)(1−ϕ2)ϕ3
Ω8111χ8=(1−ϕ1)(1−ϕ2)(1−ϕ3)