This article focuses on the graph-based mathematical morphology operators presented in [J. Cousty... more This article focuses on the graph-based mathematical morphology operators presented in [J. Cousty et al, Morphological ltering on graphs, CVIU 2013]. These operators depend on a size parameter that species the number of iterations of elementary dilations/erosions. Thus, the associated running times increase with the size parameter. In this article, we present distance maps that allow us to recover (by thresholding) all considered dilations and erosions. The algorithms based on distance maps allow the operators to be computed with a single lineartime iteration, without any dependence to the size parameter. Then, we investigate a parallelization strategy to compute these distance maps. The idea is to build iteratively the successive level-sets of the distance maps, each level set being traversed in parallel. Under some reasonable assumptions about the graph and sets to be dilated, our parallel algorithm runs in O(n/p + K log 2 p) where n, p, and K are the size of the graph, the number of available processors, and the number of distinct level-sets of the distance map, respectively. two cases can be distinguish depending whether a nal composition with an operator from G • to G × (resp. from G × to G •) is considered or not. Denition 1 (Iterated dilations/erosions) Let λ be a nonnegative integer. Case 1 (even values of λ). If λ is even, the operators δ λ/2 and λ/2 are dened on G • by δ λ/2 = (δ • • δ ×) λ/2 and λ/2 = (• • ×) λ/2 ; the operators ∆ λ/2 and ε λ/2 are dened on G × by ∆ λ/2 = (δ × • δ •) λ/2 and ε λ/2 = (× • •) λ/2. Case 2 (odd values of λ). If λ is odd, the operators δ λ/2 and λ/2 are dened from G • to G × by δ λ/2 = δ × • (δ • • δ ×) (λ−1)/2 and λ/2 = × • (• • ×) (λ−1)/2 ; the operators ∆ λ/2 and ε λ/2 are dened from G × to G • by ∆ λ/2 = δ • •(δ × •δ •) (λ−1)/2 and ε λ/2 = • • (× • •) (λ−1)/2 .
This article focuses on the graph-based mathematical morphology operators presented in [J. Cousty... more This article focuses on the graph-based mathematical morphology operators presented in [J. Cousty et al, Morphological ltering on graphs, CVIU 2013]. These operators depend on a size parameter that species the number of iterations of elementary dilations/erosions. Thus, the associated running times increase with the size parameter. In this article, we present distance maps that allow us to recover (by thresholding) all considered dilations and erosions. The algorithms based on distance maps allow the operators to be computed with a single lineartime iteration, without any dependence to the size parameter. Then, we investigate a parallelization strategy to compute these distance maps. The idea is to build iteratively the successive level-sets of the distance maps, each level set being traversed in parallel. Under some reasonable assumptions about the graph and sets to be dilated, our parallel algorithm runs in O(n/p + K log 2 p) where n, p, and K are the size of the graph, the number of available processors, and the number of distinct level-sets of the distance map, respectively. two cases can be distinguish depending whether a nal composition with an operator from G • to G × (resp. from G × to G •) is considered or not. Denition 1 (Iterated dilations/erosions) Let λ be a nonnegative integer. Case 1 (even values of λ). If λ is even, the operators δ λ/2 and λ/2 are dened on G • by δ λ/2 = (δ • • δ ×) λ/2 and λ/2 = (• • ×) λ/2 ; the operators ∆ λ/2 and ε λ/2 are dened on G × by ∆ λ/2 = (δ × • δ •) λ/2 and ε λ/2 = (× • •) λ/2. Case 2 (odd values of λ). If λ is odd, the operators δ λ/2 and λ/2 are dened from G • to G × by δ λ/2 = δ × • (δ • • δ ×) (λ−1)/2 and λ/2 = × • (• • ×) (λ−1)/2 ; the operators ∆ λ/2 and ε λ/2 are dened from G × to G • by ∆ λ/2 = δ • •(δ × •δ •) (λ−1)/2 and ε λ/2 = • • (× • •) (λ−1)/2 .
Uploads
Papers by Raquel Lins