huge amounts of 3D data Even with cheap hardware and software, on can easily generate 3D colored meshes Strong demand of algorithms for the editing of 3D colored meshes (for simpliﬁcation, sharpness enhancement, etc.) 4 / 23
smoothing ﬁlters within a hierarchical framework An image is decomposed into several layers from coarse to ﬁne details Very few approaches have addressed this problem for the editing 3D colored meshes Our proposal : a structure-preserving ﬁlter with adaptive p-Laplacian on directed graphs 6 / 23
domain represented by a graph (a triangulated mesh). A graph G = (V, E) consists in a set V = {v1, . . . , vnv } of vertices and a set E ⊂ V × V of edges connecting vertices. A graph signal is a function that associates real-valued vectors to vertices of the graph f : G → Rnc . The graph signal of nv elements is represented by a real matrix F = [fi,c ]i∈Nv ,c∈Nc , with Nv = {1, . . . , nv } and Nc = {1, . . . , nc } The i-th element can be represented by a vector fi = [fi,c ]c∈Nc The graph is represented by its weighted vertex-vertex adjacency matrix W = [wi,j ]i∈Nv ,j∈Nv , with wi,j ∈ (0, +∞) We consider directed graphs and wi,j ∈ E does not imply that wj,i ∈ E 8 / 23
for the p-Laplacian on directed graphsa. We consider here the unnormalized case aZ. Abu-Aisheh, S. Bougleux, O. L´ ezoray, p-Laplacian regularization of signals on directed graphs, International Symposium on Visual Computing, Vol. LNCS 11241, pp. 650-661, 2018 The Gradient Gradient of a graph signal F is given by ∇WF = (∂W j F)j∈Nv , with ∂W j F = [ √ wi,j (fj,c − fi,c )]i∈Nv ,c∈Nc the directional diﬀerences according to vertex j The gradient at a vertex i is thus given by ∇W i F = [ √ wi,j (fj,c − fi,c )]j∈Nv ,c∈Nc Its norm is given by |∇W i F| = nv j=1 wi,j nc c=1 (fj,c − fi,c )2 9 / 23
F is given by its p-total variation F pTV = |∇WF|p 1 = nv i=1 |∇W i F|p where p ∈ [1, +∞) controls the degree of regularity. The p-Laplacian The (directed) p-Laplacian is given by Lp,F = diag((Wp,F + WT p,F )1nv ) − (Wp,F + WT p,F ) with Wp,F = diag(|∇WF|p−2)W Can be rewritten as [Lp,F F]i,c = nv j=1 wi,j |∇W i F|2−p + wj,i |∇W j F|2−p (fi,c − fj,c ) One can show that ∇ F pTV = p Lp,F F For p = 2, L2,F = L and for p = 1, L1,F F deﬁnes the weighted (mean) curvature of F The p-Laplacian is non-linear, and can be viewed as a data-dependent Laplacian 10 / 23
an adaptive p-Laplacian and a guided preservation of gradient magnitudes. arg min F∈Rn×3 λd F − F0 2 ﬁdelity + λs Es (F, F0, α, S) structures + λr Er (F, F0, p, W) regularity with an initial graph signal F0 = (fi,c )i,c ∈ Rn×3 (colors) and two graphs S and W (same structure but diﬀerent weights). Structure Preservation: Es (F, F0, α, S) := 1 2N n i=1 αi |∇S i F|2 − |∇S i F0|2 2 guided by α = (αi )n i=1 ∈ [0, 1]n indicating ± the presence of structures on the non weighted directed graph S constructed from the mesh S0 Adaptive pTV regularization: Er (F, F0, p, W) := n i=1 1 pi ∇W i F pi p = (pi )n i=1 ∈ [1, 2]n is the regularity degree of each vertex p depends on F0, opposite role to α, indicating ± the abscence of structures on the weighted directed graph W constructed from S0 12 / 23
to the vertices {vi }nv i=1 « Set of colors Φτ i := (L∗a∗b∗(f0 j ))j∈N τ i (S0)∪{i} with Nτ i (S0) the set of vertices reached from i within a τ-hop in S0 « The Earth Mover Distance between the histograms of Φτ i is used to compare vertices si,j := 1 si j k-nn of i according to dEMD (H(Φτ i ), H(Φτ l )), l ∈ Nβ i (S0) 0 otherwise wi,j := 1 − dEMD (H(Φτ i ), H(Φτ j )) max l=1,...,n si,l =1 dEMD (H(Φτ i ), H(Φτ l )) if si,j = 1, wi,j := 0 otherwise « W and S have the same structure but diﬀerent weights. « Parameters: τ, β, k ∈ N>0 « The graphs are directed (k-nn) 14 / 23
indicator m := (mi )n i=1 mi := 1 |N ρ i (S0)| j∈N ρ i (S0) dEMD (H(Φρ−1 i ), H(Φρ−1 j )) pi :=1 + 1 1 + m2 i , αi := mi − minj mj δi (maxj mj − minj mj ) , δi out-degree of i in S « pi and αi are antagonists : one for smoothing the data, the other for preserving the main structures « Parameter: ρ ∈ N>0 15 / 23
signals relying on a data-ﬁtting term, a smoothness term and a structure-preserving term The energy is formalized on directed graphs with the p-Laplacian The ﬁltering process is adaptive with is use of a spatially-variant p-Total Variation norm The ﬁltering is guided with a structure indicator to preserve the main structures This enables to Obtain a coarse ﬁltering of 3D colored meshes : rough structures while preserving the edges Perform editing tasks such as sharpening and abstraction 22 / 23