dilation and erosion. Dilation δ of a function f 0 : Ω ⊂ R2 → R consists in replacing the function value by the maximum value within a structuring element B such that: δB f 0(x, y) = max f 0(x + x , y + y )|(x , y ) ∈ B Erosion is computed by: B f 0(x, y) = min f 0(x + x , y + y )|(x , y ) ∈ B 4 / 27
is considered as a grid graph G = (V, E) Vertices V = {v1, . . . , vm} correspond to pixels Edges eij = (vi , vj ) connect vertices with 8-adjacency Images are represented as graph signals where real-valued vectors are associated to vertices: f : G → T ⊂ Rn The set T = {v1, · · · , vm} represents all the vectors associated to all vertices To each vertex vi ∈ G is associated a vector f (vi ) = vi = T [i] 5 / 27
vectors: a complete lattice (T , ≤) MM is problematic for vector images since there is not natural ordering for vectors The framework of h-orderings can be considered for that : construct a mapping h from T to L where L is a complete lattice equipped with the conditional total ordering h : T → L and v → h(v), ∀(vi , vj ) ∈ T × T vi ≤h vj ⇔ h(vi ) ≤ h(vj ) . ≤h denotes such an h-ordering 6 / 27
account spatial and spectral information to construct the complete lattice (T , ≤h ) Considers several complete lattices and combines them Can beneﬁt from several graph constructions and vector distances Naturally extends to nonlocal processing 7 / 27
ordering ≤h on T ) ⇔ (space ﬁlling curve in T ) ⇔ (One dimensional permutation of T ) We propose to construct a one dimensional permutation ordering on the grid graph G (spatial constraints) by taking into account the graph signal similarity (spectral constraints) The constructed one dimensional permutation ordering will correspond to an Hamiltonian path on the graph 10 / 27
T is deﬁned as P = PT with P a permutation matrix of size m × m σ denotes a permutation of the index set I = {1, · · · , m} Any permutation is not of interest and constraints have to be taken into account We search for the smoothest permutation expressed by the Total Variation of its elements: T TV = m−1 i=1 vi − vi+1 (1) The optimal permutation operator P can be obtained by minimizing the total variation of PT : P∗ = arg min P PT TV (2) 11 / 27
solving the traveling salesman problem, which is very computationally demanding We consider a greedy approximation using a stochastic version of nearest neighbors heuristics This algorithm starts from an arbitrary vertex and continues by ﬁnding the two nearest unexplored neighbor vertices and choosing one of them at random. A new representation of the graph signal is obtained in the form of the pair (I, P) with I(vi ) = σ(i) The original graph signal can be recovered: f (vi ) = P[I(vi )] f : G → T I P = P∗T 12 / 27
signal f at vertex vi ∈ G by a structuring element Bk ⊂ G are deﬁned as: Bk (f )(vi ) = {P[∧I(vj )], vj ∈ Bk (vi )} δB (f )(vi ) = {P[∨I(vj )], vj ∈ Bk (vi )} A structuring element Bk (vi ) of size k deﬁned at a vertex vi corresponds to the set of vertices that can be reached from vi in k walks: Bk (vi ) = {vj ∼ vi } ∪ {vi } if k = 1 Bk−1 (vi ) ∪ ∪∀vl ∈Bk−1(vi ) B1 (vl ) if k ≥ 2 13 / 27
the graph, its topology has strong inﬂuence The graph can consider only spatial or spectral information: 8-adjacency grid graph (denoted G0 ): connects 8 spatially closest nearest neighbors. B-adjacency graph (denoted GB ): connects each vertex vi to all the vertices contained in a square box of size (2B + 1) × (2B + 1) around vi . K-Nearest Neighbor graph (denoted Gs K ): connects each vertex vi to its K nearest neighbors (in terms of spectral distance) within the set of all vertices. The graph can also consider both information and can use patches pw i around each vertex for distance computation G0 G10 G0 ∪ Gs 20 with vi G0 ∪ Gs 20 with p3 i Figure: Illustration of the inﬂuence of graph construction on the obtained permutation ordering. 16 / 27
arbitrary vertex Diﬀerent results with diﬀerent starting vertices Idea: combine several orders hi Three diﬀerent aggregation strategies are considered: Instant-Runoﬀ: determines the ﬁnal order according to majority ranking votes Borda-Count: assigns each item a score Bi (vj ) = 1 − hi (vj )−1 m based on the positions and ranks the elements according to mean aggregation of the scores Weighted Borda Count: takes into account the smoothness of the order Bi s (vj ) = Bi (vj ) × ∇Pi (vj ) 17 / 27
P4 P5 Original image Instant-Runoﬀ P Borda count P Weighted BC P Weighted BC I Figure: Consensus combination of diﬀerent stochastic permutations. 18 / 27
5 × 5 square structuring element with 5 combined permutations and graph G10 . Original image Instant-Runoﬀ Borda count Weighted BC WBC - G10 ∪ Gs 20 with vi WBC - G10 ∪ Gs 20 with p5 i 21 / 27
of images Several stochastic orderings are constructed to obtain smooth paths on graphs The graph construction can beneﬁt from both spatial and spectral constraints The permutation orderings are combined using weighted borda count Enables nonlocal processing 26 / 27