of England Henri VI, when the region was English (third English university, after Oxford and Cambridge). Completely destroyed in and rebuilt in (the phoenix rising from its ashes). UMR CNRS, largest lab in Normandy ( persons), 6 research teams on digital sciences. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
contours on graphs . Morphological decomposition of graphs signals 6. Morphological segmentation of graph signals . Hand gesture recognition Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
contours on graphs . Morphological decomposition of graphs signals 6. Morphological segmentation of graph signals . Hand gesture recognition Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
graphs are everywhere: we are witnessing the rise of graphs in Big Data. Graphs occur as a very natural way of representing arbitrary data by modeling the neighborhood properties between these data. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 6 / 6
graphs are everywhere: we are witnessing the rise of graphs in Big Data. Graphs occur as a very natural way of representing arbitrary data by modeling the neighborhood properties between these data. Images (grid graphs), Image partitions (superpixels graphs) Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 6 / 6
graphs are everywhere: we are witnessing the rise of graphs in Big Data. Graphs occur as a very natural way of representing arbitrary data by modeling the neighborhood properties between these data. Meshes, D point clouds Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 6 / 6
graphs are everywhere: we are witnessing the rise of graphs in Big Data. Graphs occur as a very natural way of representing arbitrary data by modeling the neighborhood properties between these data. Social Networks: Facebook, LinkedIn Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 6 / 6
graphs are everywhere: we are witnessing the rise of graphs in Big Data. Graphs occur as a very natural way of representing arbitrary data by modeling the neighborhood properties between these data. Biological Networks, Brain Graphs Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 6 / 6
graphs are everywhere: we are witnessing the rise of graphs in Big Data. Graphs occur as a very natural way of representing arbitrary data by modeling the neighborhood properties between these data. Mobility networks : NYC Taxi, Velo’V Lyon Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 6 / 6
D: meshes, nD: manifolds) represented by graphs G = (V, E) carrying multivariate signals f : G → Rn Graphs can be oriented or undirected, and carry weights on edges. Their topology is arbitrary. f1 : G1 → Rn=3 f2 : G2 → Rn=3 f3 : G3 → Rn=21×21 f4 : G4 → Rn=∗ Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
theory · spectral analysis (for data processing : proximity graphs built from data) Variational and morphological methods (for signal and image processing: Euclidean graphs imposed by the domain) Emergence of a new research field called Graph Signal Processing (GSP) Objective: development of algorithms to process data that reside on the vertices (or edges) of a graph: signals on graphs Problem: how to process general (non-Euclidean) graphs with signal processing techniques? Many recent works aim at extending signal and image processing tools to graph-based signal processing: the same algorithm for any kind of graph signal ! Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 8 / 6
David I. Shuman, Sunil K. Narang, Pascal Frossard, Antonio Ortega, Pierre Vandergheynst, The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains. IEEE Signal Process. Mag. ( ): 8 - 8, . A. Ortega, P. Frossard, J. Kovaˇ cevi´ c, J. M. F. Moura and P. Vandergheynst, Graph Signal Processing: Overview, Challenges, and Applications, Proceedings of the IEEE, 6( ): 8 8-8 8, 8. W. Hu, J. Pang, X. Liu, D. Tian, C. -W. Lin and A. Vetro, Graph Signal Processing for Geometric Data and Beyond: Theory and Applications, in IEEE Transactions on Multimedia, . Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
levels Compression: · Wavelets for signals on graphs Completion: · Inpainting of signals on graphs Denoising : · Filtering of signals on graphs Manipulation: · Enhancement of signals on graphs Segmentation: ·Partitioning of signals on graphs Classification: · recognize graph signals types Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
contours on graphs . Morphological decomposition of graphs signals 6. Morphological segmentation of graph signals . Hand gesture recognition Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
w) consists in a finite set V = {v1 , . . . , vN } of N vertices Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
w) consists in a finite set V = {v1 , . . . , vN } of N vertices and a finite set E = {e1 , . . . , eN } ⊂ V × V of N weighted edges. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
w) consists in a finite set V = {v1 , . . . , vN } of N vertices and a finite set E = {e1 , . . . , eN } ⊂ V × V of N weighted edges. eij = (vi , vj ) is the edge of E that connects vertices vi and vj of V. Its weight, denoted by wij = w(vi , vj ), represents the similarity between its vertices. Similarities are usually computed by using a positive symmetric function w : V × V → R+ satisfying w(vi , vj ) = 0 if (vi , vj ) / ∈ E. w w w w w w w w w w w Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
w) consists in a finite set V = {v1 , . . . , vN } of N vertices and a finite set E = {e1 , . . . , eN } ⊂ V × V of N weighted edges. eij = (vi , vj ) is the edge of E that connects vertices vi and vj of V. Its weight, denoted by wij = w(vi , vj ), represents the similarity between its vertices. Similarities are usually computed by using a positive symmetric function w : V × V → R+ satisfying w(vi , vj ) = 0 if (vi , vj ) / ∈ E. The notation vi ∼ vj is used to denote two adjacent vertices. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
Hilbert spaces of graph signals: real-valued functions defined on the vertices or the edges of a graph G. A function f : V → R of H(V) assigns a real value xi = f(vi) to vi ∈ V. By analogy with functional analysis on continuous spaces, the integral of a function f ∈ H(V), over the set of vertices V, is defined as V f = V f Both spaces H(V) and H(E) are endowed with the usual inner products: f, h H(V) = vi∈V f(vi)h(vi), where f, h : V → R F, H H(E) = vi∈V vj∼vi F(vi, vj)H(vi, vj) where F, H : E → R Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
of classical continuous differential geometry. The difference operator of f, dw : H(V) → H(E), is defined on an edge eij = (vi, vj) ∈ E by: (dwf)(eij) = (dwf)(vi, vj) = w(vi, vj)1/2(f(vj) − f(vi)) . ( ) The adjoint of the difference operator, d∗ w : H(E) → H(V), is a linear operator defined by dwf, H H(E) = f, d∗ w H H(V) and expressed by (d∗ w H)(vi) = −divw(H)(vi) = vj∼vi w(vi, vj)1/2(H(vj, vi) − H(vi, vj)) . ( ) Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
derivative) of f, at a vertex vi ∈ V, along an edge eij = (vi, vj), is defined as ∂f ∂eij vi = ∂vj f(vi) = (dwf)(vi, vj) = w(vi, vj)1/2(f(vj) − f(vi)) Weighted finite difference correspond to forward differences on grid-graph images with w(vi, vj) = 1 h2 with h2 the discretization step: ∂f ∂x (vi) = f(vi + h2) − f(vi) h2 Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
f ∈ H(V), at a vertex vi ∈ V, is the vector operator defined by (∇wf)(vi) = [(dwf)(vi, vj) : vj ∈ V]T . ( ) · The gradient considers all vertices vj ∈ V and not only vj ∼ vi . The Lp norm of this vector represents the local variation of the function f at a vertex of the graph (It is a semi-norm for p ≥ 1): (∇wf)(vi) p = vj∼vi wp/2 ij f(vj)−f(vi) p 1/p . ( ) Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 6 / 6
H(V), noted ∆i w,p : H(V) → H(V), is defined by: (∆i w,p f)(vi) = 1 2 d∗ w ( (∇wf)(vi) p−2 2 (dwf)(vi, vj)) . ( ) The p-Laplace operator of f ∈ H(V), at a vertex vi ∈ V, can be computed by: (∆i w,p f)(vi) = 1 2 vj∼vi (γi w,p f)(vi, vj)(f(vi) − f(vj)) , (6) with (γi w,p f)(vi, vj) = wij (∇wf)(vj) p−2 2 + (∇wf)(vi) p−2 2 . ( ) The p-Laplace operator is nonlinear, except for p = 2 (corresponds to the combinatorial Laplacian). For p = 1, it corresponds to the weighted curvature of the function f on the graph. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
contours on graphs . Morphological decomposition of graphs signals 6. Morphological segmentation of graph signals . Hand gesture recognition Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 8 / 6
R be the noisy version of a clean graph signal g : V → R defined on the vertices of a weighted graph G = (V, E, w). To recover g, seek for a function f : V → R regular enough on G, and close enough to f0, with the following variational problem: g ≈ min f:V→R Ew,p(f, f0, λ) = Rw,p(f) + λ 2 f − f0 2 2 , (8) where the regularization functional Rw,p is defined by the L2 norm of the gradient and is the discrete p-Dirichlet form of the function f ∈ H(V): Rw,p(f) = 1 p vi∈V (∇wf)(vi) p 2 = 1 p f, ∆i w,p f H(V) = 1 p vi∈V vj∼vi wij(f(vj) − f(vi))2 p 2 . ( ) Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
consider the following system of equations: ∂Ew,p(f, f0, λ) ∂f(vi) = 0, ∀vi ∈ V ( ) which is rewritten as: ∂Rw,p(f) ∂f(vi) + λ(f(vi) − f0(vi)) = 0, ∀vi ∈ V. ( ) Moreover, we can prove that ∂Rw,p(f) ∂f(vi) = 2(∆i w,p f)(vi) . ( ) The system of equations is then rewritten as : λ + vj∼vi (γi w,p f)(vi, vj) f(vi) − vj∼vi (γi w,p f)(vi, vj)f(vj) = λf0(vi). ( ) Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
to solve the previous systems. Let n be an iteration step, and let f(n) be the solution at the step n. Then, the method is given by the following algorithm: f(0) = f0 f(n+1)(vi) = λf0(vi) + vj∼vi (γ∗ w,p f(n))(vi, vj)f(n)(vj) λ + vj∼vi (γ∗ w,p f(n))(vi, vj) , ∀vi ∈ V. ( ) with (γi w,p f)(vi, vj) = wij (∇wf)(vj) p−2 2 + (∇wf)(vi) p−2 2 , ( ) It describes a family of discrete diffusion processes, which is parameterized by the structure of the graph (topology and weight function), the parameter p, and the parameter λ. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
p-Laplacian and guided gradient amplitude preservation arg min f λd f − f0 2 Fidelity +λr n i=1 1 pi (∇w f)(vi ) pi 2 Regularization +λs 1 2N n i=1 αi (∇w f)(vi ) 2 2 − (∇w f0)(vi ) 2 2 2 Structure preservation With an original signal f0 and graph weights s. Guided by αi ∈ {0, 1} that indicates ± the presence of structures ( : no structure, : structures) pi ∈ [1, 2] gives the regularity of each node ( : regular, : not regular) and indicates ± the absence of structures (opposite role to αi ). Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
from a structure indicator mi using the Earth Mover Distance between the histograms of ρ-hop vectors Ff0 ρ (vi) in a τ-hop: mi := 1 |Nτ (vi)| vj∈Nτ (vi) dEMD(H(Ff0 ρ (vi)), H(Ff0 ρ (vj))) ( 6) pi := 1 + 1 1 + m2 i ( ) αi := mi − minj mj δi(maxj mj − minj mj) , δi out-degree of i in S ( 8) « pi and αi are antagonists : one for smoothing the data, the other for preserving the main structures Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 8 / 6
contours on graphs . Morphological decomposition of graphs signals 6. Morphological segmentation of graph signals . Hand gesture recognition Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
with level sets. Geodesic Active Contours An energy is associated to a curve C(p): EGAC(C) = 1 0 g(I(C(p)))|C (p)|dp Active Contours without edges Considers two regions separated by a curve, and minimizes: EACW E(C, c1, c2) = µ·Length(C)+ν·Area(inside(C))+λ1 inside(C) |I(x)−c1|2dx+λ2 outside(C) |I(x)−c2|2dx Considered active contours: combines both E(C, c1, c2) = µ 1 0 g(C(p))|C (p)|dp + ν · inside(C) g(C(p))dA + λ1 d inside(C) I(x) − c1 2 2 dx + λ2 d outside(C) I(x) − c2 2 2 dx This can be solved using a gradient descent from the associated Euler-Lagrange equations with the level-set framework. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
« Demo Given f a level set function ft : V → {−1, +1} for inside/outside of the evolving propagating front, the solution is obtained by ft+1(vi ) = ft(vi ) + ∆tδf(vi,t) δt We express front propagation on graphs as δf(vi,t) δt = F(vi , t) (∇w f)(vi , t) p p with F(vi , t) a speed function We propose a front propagation function that solves the considered active contours with discrete calculus: F(vi , t) = νg(vi ) + µg(vi )(κw f)(vi , t) − λ1 d vi d2(Ff0 ρ (vi ), Fc1 ρ ) + λ2 d vi d2(Ff0 ρ (vi ), Fc2 ρ ) We consider local patches Ff0 ρ (vj ) on a ρ-hop subgraph to represent the regions (instead of vertex-based signal average) The potential function g(vi ) differentiates the most salient structures of a graph using patches comparison: the previous mi Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
(g) (h) Figure: From top to bottom, left to right : (a) Original mesh, (b) g(vi) (inverted) (c) Checkerboard initialization, (d) GSAC; g(vi), ρ = 0, (e) GSAC; g(vi), ρ = 2, (f) manual initialization (g) extracted region with GSAC; g(vi), ρ = 2, (h) re-colorisation of the extracted region. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
and ) of the MNIST dataset. The colors around each image show the class it is affected to. The top row shows the initialization and bottom second row the final classification. 6 8 8.8 .8 . . . 6. . 6. Table: Classification scores for the 0 digit versus each other digit of the MNIST database. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 6 / 6
contours on graphs . Morphological decomposition of graphs signals 6. Morphological segmentation of graph signals . Hand gesture recognition Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
are dilation and erosion. Dilation δ of a function f0 : Ω ⊂ R2 → R consists in replacing the function value by the maximum value within a structuring element B such that: δB f0(x, y) = max f0(x + x , y + y )|(x , y ) ∈ B Erosion is computed by: B f0(x, y) = min f0(x + x , y + y )|(x , y ) ∈ B Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 8 / 6
vectors: a complete lattice (T , ≤) MM is problematic for multivariate data since there is no 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, it is a dimensionality reduction operation h : Rn → Rp with p < n. Advantage : the learned lattice depends of the signal content and is more adaptive. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
be linear since a distortion of the space is inevitable ! Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
be linear since a distortion of the space is inevitable ! Solution : Consider non-linear dimensionality reduction with Laplacian Eigenmaps that corresponds to learn the manifold where the vectors live. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
be linear since a distortion of the space is inevitable ! Solution : Consider non-linear dimensionality reduction with Laplacian Eigenmaps that corresponds to learn the manifold where the vectors live. × Problem : Non-linear dimensionality reduction directly on the set T of vectors is not tractable in reasonable time ! Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
be linear since a distortion of the space is inevitable ! Solution : Consider non-linear dimensionality reduction with Laplacian Eigenmaps that corresponds to learn the manifold where the vectors live. × Problem : Non-linear dimensionality reduction directly on the set T of vectors is not tractable in reasonable time ! Solution : Consider a more efficient strategy. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
be linear since a distortion of the space is inevitable ! Solution : Consider non-linear dimensionality reduction with Laplacian Eigenmaps that corresponds to learn the manifold where the vectors live. × Problem : Non-linear dimensionality reduction directly on the set T of vectors is not tractable in reasonable time ! Solution : Consider a more efficient strategy. Proposed Three-Step Strategy Dictionary Learning to produce a set D from the set of initial vectors T Laplacian Eigenmaps Manifold Learning on the dictionary D to obtain a projection operator hD Out of sample extension to extrapolate hD to T and define h Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
a sorted permutation P of T is constructed P = {v1 , · · · , vm } with vi ≤h vi+1 , ∀i ∈ [1, (m − 1)]. From the ordering, an index signal I : Ω ⊂ Z2 → [1, m] is defined as: I(pi ) = {k | vk = f(pi ) = vi} . Image of 256 colors Index Image (T , ≤h) The pair (I, P) provides a new graph signal representation (the index and the palette of ordered vectors). The original signal f can be directly recovered since f(pi ) = P[I(pi )] = vi . To process the graph signal: g(f(vi)) = P[g(I(vi))] with g an operation. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
into a base layer and several detail layers capturing each a given level of detail : f = l−2 i=0 fi + dl−1 Each layer is obtained by di = di−1 − FM(di−1) with d−1 = f the original graph signal Filtering is performed by a morphological filtering that creates flat zones (OCCO filter) The signal can be reconstructed after manipulating the layers: ˆ f(vk) = S0(f0(vk)) + m(vk) · l−1 i=1 Si(fi(vk)) avec fl−1 = dl−1 Each layer is manipulated with a nonlinear function Si(x) = 1 1+exp(−αix) and a guide m Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
contours on graphs . Morphological decomposition of graphs signals 6. Morphological segmentation of graph signals . Hand gesture recognition Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 6 / 6
h-ordering based on an space filling curve on the graph · an Hamiltonian path Search a sorted permutation of the vectors of T : P = PT with P a permutation matrix of size m × m We search for the smoothest permutation expressed by the Total Variation of its elements: T TV = m−1 i=1 vi − vi+1 The optimal permutation operator P∗ can be obtained by minimizing the total variation of PT : P∗ = arg min P PT TV Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
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 its unexplored neighbor vertex f : G → T I P = P∗T Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 8 / 6
(Ii , Pi ) with i ∈ [1, M] Construct M watersheds from the minima of each ordering WSi (f) = WS(Ii , ∇f) Construct a pdf from the segmentations: pdf(f) = 1 M M i=0 G(WSi (f)) Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
gp Seeds Watershed with gc Watershed with gradient gp Figure: Segmentation examples with our stochastic permutation watershed. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
characters in the tapestry images A precise segmentation is required by using simple object/background seed labeling by point click to ease the end-users use The characters are visually easy to identify but the reduced number of colors, the fine embroidery as well as the texture differences in the linen fabric can make the segmentation hard. The permutation based stochastic watershed is performed with 7 × 7 patches Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
contours on graphs . Morphological decomposition of graphs signals 6. Morphological segmentation of graph signals . Hand gesture recognition Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
of a graph containing joints with associated coordinates (x, y, z)T . We represent this as a D grid-graph of size 4 × 5 from joints to ( joints for each finger). Figure: (left) graph of the hand joints, (right) D grid-graph representation. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
corresponds to a grid. They are processed using a deep network Figure: DeepSPDNet. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 6 / 6
D joints, ) Processing layers of 6 subsequences (original and split into and subsequences) for each finger with : a Gaussian mapping layer (characterizes the temporal variations by a covariance SPD matrix), a ReEig layer rectifying the eigenvalues in order to apply nonlinear transformations to the SPD matrices, a LogEig layer to project the SPD matrices into a Euclidean space, a VecMat layer to convert the SPD matrices into vectors, a Gaussian mapping layer that aggregates the vectors into SPD matrices, ) A linear mapping layer that aggregates the SPD matrices into one, ) LogEig, VecMat, FC and SoftMax layers for classification Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
. 6 Deep SDPNet (our approach) . 6 ST-TS-HGR-NET (our approach with spatio-temporal convolutions) . Table: Comparison of the performance of our approach with the state of the art on the First-Person Hand Action (FPHA) database which contains 1175 videos of 45 different hand gestures performed by 6 people. · Deep SDPNet is ten times faster than ST-TS-HGR-NET: . s to classify a gesture. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 8 / 6
changes in gestures without indication of the beginning and end of each gesture. The features of the BiMap layer of two SPDNet networks are exploited and their similarity measured by a multi-layer perceptron (Temporal Detection Network). Each Deep SDPNet receives as input clips (sequences of frames). Since the same gesture is found in several consecutive clips, the TDN network aims at estimating the similarity between the two clips. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing / 6
in the clips. The first Deep SPDNet has a fixed temporal position while the second moves temporally on the gesture stream. The similarity of the two clips is estimated by the TDN network and this allows to estimate if the two clips correspond to the same gesture. As soon as the TDN network detects a new gesture, a breakpoint is detected: the first SPDNet network is moved. Figure: Online gesture recognition: two Deep SDPNet networks move temporally along a gesture sequence and gesture breakpoints are located. CX stands for class X. Olivier L´ ezoray Variational, morphological and neural approaches to graph signal processing 6 / 6
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