Fig. 5

Relaxed problems in DCDP. Here, a segmentation problem (1) with (n,m,ς)=(8,12,1) is considered. We use an m-sided polygon to model the boundary of a glomerulus. The vertices are restricted to be on any of the n points lying on the m rays from the center of the glomerulus. The boundary likeliness is computed on each of the n points, and the configuration that maximizes the sum of the boundary likeliness is found, as described in (1). In Panel (a), the sizes of the green circles indicate the quantities of boundary likeliness. As in (1), the feasible configurations of the polygon are restricted to be in \({\mathcal {S}}({\mathcal {I}}_{0})\), where \({\mathcal {I}}_{0}={\mathbb {N}}_{n}\). Overlapping all feasible configurations yields the gray edges in Panel (a). The optimal polygon is drawn with red edges. DCDP relaxes the feasible region \({\mathcal {S}}({\mathcal {I}}_{0})\) to get \({\mathcal {S}}_{\text {L}}({\mathcal {I}}_{0})\). In Panel (b), the relaxed feasible region \({\mathcal {S}}_{\text {L}}({\mathcal {I}}_{0})\) is depicted. The blue polygon is the optimal configuration for the relaxed problem \({\boldsymbol {p}}_{\text {L},0}=\operatorname *{argmax}_{{\boldsymbol {p}}\in {\mathcal {S}}_{\text {L}}({\mathcal {I}}_{0})}J({\boldsymbol {p}})\). The proposed algorithm DCDP divides the problem into many sub-problems. The relaxed versions of the four sub-problems with \({\mathcal {S}}_{\text {L}}(\{7,8\})\), \({\mathcal {S}}_{\text {L}}(\{1,2,\dots,6\})\), \({\mathcal {S}}_{\text {L}}(\{5,6,7,8\})\), and \({\mathcal {S}}_{\text {L}}(\{1,2,3,4\})\) are illustrated in Panels (c), (d), (e), and (f). The blue polygons in (c), (d), (e), and (f) are the optimal solutions of the four relaxed problems, respectively. See the main text for details