Figure 9 - uploaded by Diego Marcondes
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Selected elements in the lattice pF, ďq of the ASF architecture A 3 in Figure 7 when E " Z 2 and W " t´1, 0, 1u 2 . The number on top of the elements is the mean empirical loss of the set operator realized by them, and the colored edges represent the path of the LGDA starting from the empty sets after three epochs. The edges pointing to sets obtained by removing a point from a set were omitted for a better visualization. The returned elementˆCelementˆ elementˆC is outlined in green.
Source publication
A classical approach to designing binary image operators is Mathematical Morphology (MM). We propose the Discrete Morphological Neural Networks (DMNN) for binary image analysis to represent W-operators and estimate them via machine learning. A DMNN architecture, which is represented by a Morphological Computational Graph, is designed as in the clas...
Contexts in source publication
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... this example, F is isomorphic to the Boolean lattice PpW q 2 and an element pA, Bq P PpW q 2 realizes the ASF with two pairs of openings and closings in which the first has structuring element A and the second B. Denote by C A,B P F the element of F associated to pA, Bq. We have that N pC A,B q " tC C,B P F : dpC, Aq " 1u Y tC A,C P F : dpC, Bq " 1u , in which dpC, Aq and dpC, Bq are the respective distance in PpW q, so a neighbor of C A,B is obtained by adding or removing one point from A or B. This is an example in which the poset pF, ďq is actually a Boolean lattice, which is partially depicted in Figure 9. ■ Example 6.5. ...
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... 6.4). In Figure 9 we present selected elements of this Boolean lattice when E " Z 2 and W " t´1, 0, 1u 2 is the three by three square centered at the origin of E. The number on top of the elements in Figure 9 is the mean empirical loss in a sample D N of the set operator generated by the respective realization of the architecture. The colored edges represent the path of the LGDA starting from the empty sets after three epochs. ...
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... 6.4). In Figure 9 we present selected elements of this Boolean lattice when E " Z 2 and W " t´1, 0, 1u 2 is the three by three square centered at the origin of E. The number on top of the elements in Figure 9 is the mean empirical loss in a sample D N of the set operator generated by the respective realization of the architecture. The colored edges represent the path of the LGDA starting from the empty sets after three epochs. ...
Context 4
... this example, F is isomorphic to the Boolean lattice PpW q 2 and an element pA, Bq P PpW q 2 realizes the ASF with two pairs of openings and closings in which the first has structuring element A and the second B. Denote by C A,B P F the element of F associated to pA, Bq. We have that N pC A,B q " tC C,B P F : dpC, Aq " 1u Y tC A,C P F : dpC, Bq " 1u , in which dpC, Aq and dpC, Bq are the respective distance in PpW q, so a neighbor of C A,B is obtained by adding or removing one point from A or B. This is an example in which the poset pF, ďq is actually a Boolean lattice, which is partially depicted in Figure 9. ■ Example 6.5. ...
Context 5
... 6.4). In Figure 9 we present selected elements of this Boolean lattice when E " Z 2 and W " t´1, 0, 1u 2 is the three by three square centered at the origin of E. The number on top of the elements in Figure 9 is the mean empirical loss in a sample D N of the set operator generated by the respective realization of the architecture. The colored edges represent the path of the LGDA starting from the empty sets after three epochs. ...
Context 6
... 6.4). In Figure 9 we present selected elements of this Boolean lattice when E " Z 2 and W " t´1, 0, 1u 2 is the three by three square centered at the origin of E. The number on top of the elements in Figure 9 is the mean empirical loss in a sample D N of the set operator generated by the respective realization of the architecture. The colored edges represent the path of the LGDA starting from the empty sets after three epochs. ...
