A Graph Downsampling Technique Based On Graph Fourier Transform
11 A GRAPH DOWNSAMPLING TECHNIQUEBASED On GRAPH FOURIER TRANSFORM
Nileshkumar Vaishnav and Aditya TatuDAIICT, Gandhinagar, India.
Abstract —In this paper, we provide a Graph Fourier Trans-form based approach to downsample signals on graphs. For ban-dlimited signals on a graph, a test is provided to identify whethersignal reconstruction is possible from the given downsampledsignal. Moreover, if the signal is not bandlimited, we provide aquality measure for comparing different downsampling schemes.Using this quality measure, we propose a greedy downsamplingalgorithm. Most of the prevailing approaches consider undirectedgraphs, and exploit the topological properties of the graphin order to downsample the grid, while the proposed methodexploits spectral properties of graph signals, and is applicableto directed graphs, undirected graphs, and graphs with negativeedge-weights. We provide several experiments demonstrating ourdownsampling scheme, and compare our quality measure withmeasures like normalized cuts.
Index Terms —Graph Signal Processing, Graph Downsampling,Graph coarsening, Graph Fourier Transform.
I. I
NTRODUCTION
There are many applications, where the domain of themeasured data can be modeled as a graph. Examples ofsuch data include weather data, seismic activity data, sensornetworks data, social network data, transportation data. Giventhe large scope of applications[1], analysis and processing ofsignals on graph is important. Signals on graph often comefrom a nonuniform grid, and there is no natural orderingof the vertices; rather the inter-relations between vertices isimportant. Defining concepts such as shift, Fourier transformand convolution is not trivial and diverges greatly from similarconcepts defined for uniform signals.Formally, a graph is a collection of vertices with a givenrelation structure between the vertices. The relation betweenvertices is given by a matrix called the graph adjacencymatrix. For an unweighted graph, the adjacency matrix hasbinary entries. For an undirected graph, the adjacency matrixis symmetric. Traditionally, spectral properties of graph signalsare derived using graph Laplacian . The study of eigenvaluesand eigenvectors of graph Laplacian is called
Spectral GraphTheory [2]. A recent approach[3] indicates that the spectralanalysis of graph signals can also be carried out effectivelyusing the graph adjacency matrix. This approach allows us towork with signals on directed graphs, which is not possiblewith Graph Laplacian based approach.Often we encounter signals on graph which are smooth in nature. Such signals exhibit low-pass behavior in spectraldomain. When a graph signal does not contain frequencycontent above a certain cut-off frequency, it is called a ban-dlimited signal (a formal definition is provided in section 2). If the graph signal is bandlimited, it can be reconstructed fromfewer samples in vertex-domain. The process of finding thecollection of vertices which can reconstruct the original signalis given various names: graph coarsening [4], [5], site perco-lation [6]. Graph downsampling is a special case of graph-coarsening, where we reduce the nodes by an integer factor(e.g. downsampling by a factor of implies removing halfvertices). Graph downsampling can be used for compressionand as a building block for multiresolution analysis for signalson graph[7]. In this paper, the term downsampling refers todownsampling by a factor of two, unless explicitly specified.Recently, research on graph downsampling has gained mo-mentum in the field of signal processing. Downsampling ofa graph with respect to bandlimited signals draws in analogyfrom the classical uniform sampling and downsampling pro-cess. In 1-D uniform case, there is an ordered set of vertices,and the downsampling process amounts to selecting every al-ternate vertex from the set of vertices. Spectrally, the selectionof every alternate vertex results in folding of spectrum exactlyby a factor of two. If the signal is bandlimited with upper-half of frequency content absent, then spectral folding doesnot introduce any aliasing. Thus, any signal which has nospectral content on the upper-half of the frequency spectrumcan be recovered from the downsampled vertices without anyerror. Thus the spectral view of the signal coincides with thetopological view in case of classical signal processing.Downsampling on graph, however, differs from the traditionalview in both the domains (the vertex domain and the spectraldomain). This is because of the fact that a graph does notprovide any topology in which the vertices are ordered (exceptin special cases), hence selecting every alternate vertex isnot a meaningful operation. Moreover, the spectrum of agraph (i.e. eigenvalues of Laplacian/adjacency matrix) doesnot necessarily show symmetry, indicating that the spectral-folding phenomena is not the same as that in classical signalprocessing. Another challenge in downsampling on graphs ishow to determine the inter-relations among the reduced setof vertices. The determination of new adjacency relation inthe reduced graph is essential to obtain a multi-resolution ongraph [8], [9], [10].In this paper, we obtain a sampling scheme which takesinto account the spectral properties of the graph in order todownsample signals. The approach presented can be appliedto both directed as well as undirected graphs. The approach a r X i v : . [ s t a t . O T ] D ec is also applicable to the graphs with negative edge-weights .In case the signal is not bandlimited, we provide a measurethat allows to choose a scheme with minimum reconstructionerror.The paper is organized as follows. Section 2 providesthe background and related work in the field of graph-downsampling. In section 3 and 4, we provide our proposeddownsampling method for band-limited and non band-limited(low pass) signals. A greedy algorithm to implement theproposed method is presented in Section 5, followed byseveral experiments to validate our claims in Section 6. Weconclude the paper in Section 7, in which some future researchdirections are also listed.II. R ELATED W ORK
We begin by introducing notations and terms that are usedfrequently in the paper.
A. Definitions and Notations
A graph G is denoted as ( V , A ) , where V is the set ofvertices { v , ..., v N } with a specified order and A is the graphadjacency matrix which provides the relation structure betweenthe set of vertices. For matrix A , each element a i,j is theweight connecting vertex v j to vertex v i .A graph signal is defined as the vector ¯ x = [ x , x , · · · , x N ] T ,where x i ’s are scalar values sampled on vertices v i ’s respec-tively. Thus a signal ¯ x can be thought of as an element in C N . For undirected graphs, Graph Laplacian is defined as L = D − A , where D is a diagonal matrix with d i,i beingthe sum of edge-weights connecting vertex v i . NormalizedGraph Laplacian is defined as L n = D − / LD − / . Givengraph-Laplacian L = V Σ V T , where Σ is a diagonal matrixand V is an orthogonal matrix, V T is the designated GraphFourier Transform based on Laplacian , denoted by
GF T L .Given normalized graph-Laplacian L n = V Σ V T , where Σ is a diagonal matrix and V is an orthogonal matrix, V T isthe designated Graph Fourier Transform based on normalizedgraph Laplacian , denoted as
GF T N . Following [3], the matrix V − in A = V JV − , which puts the given adjacency matrix A into its Jordan Normal Form (JNF) J is designated as the Adjacency matrix based Graph Fourier Transform , denotedas
GF T A . In the case of GF T L and GF T N , the ascendingfrequency order correspond to usual ascending order on therespective eigenvalues, while for GF T A the ascending fre-quency order corresponds to an ascending order on | − λλ max | ,where λ is the respective eigenvalue and λ max is the maximumeigenvalue, for details refer [11]. In this paper, unless specified,we assume Adjacency Matrix Based GFT. The terms nodes and vetrices are used interchangeably.Having defined a GFT, one can now define bandwidthof a signal. Consider an undirected graph, with GF T N asthe designated GFT. For normalized graph Laplacian, alleigenvalues are real and non-negative and lie in interval [0 , .The bandwidth of a signal, in this context, can be interpreted Negative edge-weights usually indicate a negative correlation betweensignals on two vertices, e.g. in a Social Network, two individuals can beconnected by an inverse relation resulting in a negative edge-weight. as a real number in the interval [0 , [12]. Another way tointerpret bandwidth of a graph signal, is to count the numberof eigenvalues which lie below a certain cut-off threshold. Thisnumber itself can be interpreted as bandwidth. We define thebandlimited of a graph signal as follows. Definition
Bandlimited Signal On Graph
For a signal ¯ x on agiven graph with GFT ¯ b , if ¯ b ( i ) = 0 , ∀ i ≥ n , then the signalis called bandlimited with bandwidth n .Due to different behaviors in vertex-domain and spectral-domain, the downsampling on graph can be looked at fromboth topological and spectral viewpoints.If the signal is bandlimited in spectral domain, then it canbe downsampled without loss of data. One key problem ingraph downsampling is determining a sample-set, i.e., the setof vertices from which a bandlimited signal can be recoveredwithout any error. A method to determine the sample-set of anundirected graph is provided by Anis et al. [13], in which agreedy approach is used to add a vertex in every iteration to thesample-set, which provides the highest increase in bandwidth,until the cut-off threshold is reached. Due to the greedynature of the algorithm, the sample-set so obtained is notnecessarily optimal . As an example, every alternate sample isnot necessarily selected during downsampling of a standard 1-D uniform grid. It should be emphasized here that the sample-set (of a given cardinality) for a given bandwidth is not unique,and the algorithm indeed converges to one of those sample-sets. However, different sample-sets have different sensitivityto aliasing in case of signals which are not bandlimited, whichindicates that even among sample-sets, the quality of signal-reconstruction differs. The objective in [13] is to find theleast number of samples (and corresponding sample-set) for asignal with given bandwidth. On the other hand, the purposeof our proposed approach is to provide a way to select the best possible N/ vertices for a graph with N vertices.The eigenvector corresponding to highest frequency is usedto obtain a downsampling scheme in [14]. Based on polarityof eigenvector values, two equivalent sets of downsampledvertices are obtained. Other approaches to downsample includethose approaches which exploit the topological properties ofthe graph. One major class of graphs is called bipartite graphs ,which provide a natural way to downsample. An analysis ofdownsampling k -regular bipartite graphs is provided in [15].However, not all graphs exhibit bipartite structure, so to applythe downsampling to arbitrary graphs, a method proposed in[16] locally approximates the bipartite structure using a graph-colouring technique. On the other hand, Nguyen and Do[10]rely on Maximum Spanning Tree(MST) of a graph in order todownsample . A major limitation with topological approachesis that although the signal is assumed to be bandlimited inspectral domain, the actual process of finding the downsam-pling scheme does not take into account spectral properties ofthe graph in a direct way. Moreover, these approaches cannotbe applied to downsample a directed graph, or a graph withnegative edge-weights. The meaning of optimality will be provided in later sections It should be noted here that every tree is a bipartite graph.
Fig. 1. An example graph to be downsampled by a factor of 2Set of selected nodes
SDQM
Cut-Index { , , } , { , , } , { , , } , { , , } , { , , } .
19 0 . { , , } , { , , } , { , , } , { , , } , { , , } .
25 0 . { , , } , { , , } , { , , } , { , , } , { , , } .
40 0 . { , , } , { , , } , { , , } , { , , } , { , , } .
52 0 . TABLE IP
ROPOSED Q UALITY MEASURE ( SDQM ) FOR ALL POSSIBLE D OWNSAMPLING SCHEMES FOR THE GRAPH SHOWN IN F IGURE Another issue in graph downsampling is measuring the qualityof the affected partition on graph. Cut-index[10] is a popularobjective measure used to determine quality of the graphdownsampling scheme, and is defined as the ratio of sumof edgeweights of edges to be deleted in order to disconnecttwo selected partitions, and the total edgeweights in the graph.A downsampling scheme with higher cut-index is consideredto have better quality (and hence better signal reconstructionproperties). One major issue with cut-index is that a singlecut provides us two downsampling options (i.e. both partitionsare considered equally good), selecting one of the two is anarbitrary choice.To understand the issues with topological downsampling,consider the graph as shown in Figure 1, with all edgeweightsset to . Table 1 provides all possible combinationsof downsampling the graph by a factor of , with thecorresponding proposed quality measure, which we refer toas SDQM (details provided in Section 4), and cut-index.Higher quality measure indicates lower reconstruction error.It should be noted from the table how the proposed qualitymeasure captures the symmetries which are present in thegraph. For example, selections {
1, 4, 6 } and {
1, 2, 5 } aretopologically symmetric, and hence they have equal qualitymeasure. The example under consideration also explains thelimitation of cut-index quality measure. It can be seen fromthe table that sets { } and { } have identical cut-indexmeasures. On the other hand, the proposed measure indicatesthat retaining { } would lead to a better reconstruction.Another major issue with topological downsamplingmethods (e.g. MST based method) is the fact that they relyon reducing the graph to a particular structure by deletingedges, which makes them sensitive to small changes inedge-weights. For example, changing the weights of edges (1 , , (1 , , (1 , , (2 , and (5 , from to . (i.e. a 1%change in weights) would lead MST to drop all edges withedge-weight . This in turn would result in downsampling partition { , , } , { , , } , none of which have the desiredcut-index. On the other hand, the proposed downsamplingmethod still yields { } (or equivalent) and thus is lesssensitive to small changes in edge-weights.In this paper, we emphasize on the fact that the graph signalis assumed to be bandlimited in spectral domain, hence theprocess of downsampling must take into account the spectral(Graph Fourier Transform) properties of the graph. With suchan approach, we propose a method for downsampling that canbe applied to undirected as well as directed graphs and alsoto the graphs with negative edge-weights. We also providean alternative way to determine quality of the downsamplingscheme, which allows us to determine which vertex-set to keepand which one to purge, i.e. both partitions may not be equiv-alent. The proposed algorithm optimizes this quality measureto obtain a downsampling scheme. We compare the proposedmethod with the spectral downsampling method (SVD based)and a topological downsampling method (MST based). Ananalysis of the proposed approach for bipartite graphs ifprovided, for which the topological and proposed approachcoincide; both approaches giving either of the disjoint set ofvertices as the downsampled graph.III. D OWNSAMPLING O F B ANDLIMITED S IGNALS A ND C ONDITION F OR P ERFECT R ECONSTRUCTION
For a graph G = ( V , A ) with N nodes, let the GFT matrixbe denoted by F , where F ∈ C N × N . N is assumed to be evenas we focus on downsampling the graph by two. If a signalon this graph is bandlimited with all the energy contained inthe lower half of the frequency spectrum, then the GFT ofthe signal is of form [ b , b , ..., b N/ , , ... T . The spectrumcan be expressed as [¯ b TL , ¯ b TH ] T , where ¯ b L = [ b , b , ..., b N/ ] and ¯ b H = [0 , ..., T . Let V p be the set of nodes to be purgedand V k be the set of nodes to be kept, both containing N/ nodes. For a given graph signal ¯ x , let ¯ x k and ¯ x p be the signalvalues taken from nodes in the sets V k and V p respectively.As both the sets are selections from V , we can write, P p ¯ x =¯ x p , P k ¯ x = ¯ x k where P p and P k are selection matrices. If wefix the order of nodes in V k and in V p , then P p and P k areunique. We can also write, P ¯ x = (cid:20) ¯ x k ¯ x p (cid:21) where P is an invertible permutation matrix, with inverse P T .Similarly, we can also define selection matrices P L and P H such that P L ¯ b = ¯ b L , P H ¯ b = ¯ b H .Since F ¯ x = ¯ b , F P T P ¯ x = ¯ b . ∴ F P (cid:20) ¯ x k ¯ x p (cid:21) = (cid:20) ¯ b L ¯ b H (cid:21) where F P = F P T . If we write F P as (cid:20) F F F F (cid:21) , then we get (cid:20) F F F F (cid:21) (cid:20) ¯ x k ¯ x p (cid:21) = (cid:20) ¯ b L ¯ b H (cid:21) (1) where F , F , F and F are N × N matrices. Given ¯ b H = 0 , ¯ x p can be uniquely determined from ¯ x k if and only if the sub-matrix F is invertible. Note that F = P H F P Tp . Moreover,using Schur Complement on the above equation, we obtain F kL such that F kL ¯ x k = ¯ b L ⇔ F kL = F − F F − F The matrix F kL can be understood as the GFT on the down-sampled graph. The signal on purged nodes, denoted as ¯ x p can be recovered from ¯ x k , using the following reconstructionrule obtained from Equation (1): ¯ x p = − F − F ¯ x k . (2)Thus, the procedure described above, allows us to find acondition for perfect reconstruction and at the same time,provides us with the GFT on the downsampled grid. The setof samples from which the given bandlimited signal can bereproduced without any error is called a sample-set . There canbe multiple sample-sets of same cardinality for a given graph.A similar analysis for condition for perfect reconstruction ofbandlimited signals is provided in [17], where the focus issolely on bandlimited signals. In this paper, we extend thesame principle for non-bandlimited signals which exhibit low-pass nature.IV. D OWNSAMPLING O F N ON -B ANDLIMITED L OWPASS S IGNALS
The discussion so far indicates that if the matrix F = P H F P Tp is invertible, then any bandlimited signal can bereconstructed without any error from nodes contained in V k .This raises a question: Are all possible node-selections withcorresponding invertible F , equivalent? As far as bandlimitedsignals are concerned, all sample-sets are equivalent. However,the property of bandlimitedness is highly restrictive. In theanalysis till now, we have assumed a perfectly bandlimitedsignal, i.e., (cid:107) ¯ b H (cid:107) = 0 . However, in real-world scenarios, weoften encounter situations where < (cid:107) ¯ b H (cid:107) = (cid:15) << (cid:107) ¯ b L (cid:107) .We refer to such signals as lowpass signals. In this section,we will analyze this scenario which will help in obtaining anoptimal downsampling scheme from the signal reconstructionpoint of view. From Equation (1) F ¯ x k + F ¯ x p = ¯ b H ⇒ ¯ x p = − F − F ¯ x k + F − ¯ b H . The reconstruction error e r is e r = F − ¯ b H ⇒ (cid:107) e r (cid:107) ≤ (cid:15)σ min ( F ) Here, σ min ( F ) denotes the minimum singular value of F and characterizes the sensitivity of the reconstruction error(from signal values on V k ) to high frequency content. Fora given partition V p , V k the value σ min ( F ) is referred to as SVD based Downsampling Quality Measure (abbreviated asSDQM). It should be observed here that if
SDQM = 0 , then F is not invertible and the signal cannot be reconstructed.Maximizing SDQM reduces the upper-bound on error. Asfar as bandlimited signals are concerned, all downsamplingschemes with SDQM (cid:54) = 0 are equivalent. However, whenthe signal is not bandlimited, they exhibit different amount of
Set of selected nodes
SDQM { , , } , { , , } . { , , } , { , , } , { , , } , ..., { , , } . Rest of the combinations ( in total) . TABLE IIQ
UALITY MEASURE FOR ALL POSSIBLE DOWNSAMPLING SCHEMES FORGRAPH IN F IGURE
2. T HE consecutive selection ( E . G . { } ) SHOWSLEAST
SDQM , WHILE every alternate node selection ( E . G . { } ) HASTHE LARGEST
SDQM . s s s s s s Fig. 2. (left) A six-node directed circulant graph, (right) Correspondingdownsampled graph. sensitivity towards the high frequency content of the signal.Thus, the goal of downsampling should be to find a sample-setthat maximizes
SDQM .With this analysis, the problem of downsampling can bestated as the following optimization problem, P opt = argmax P p ∈{ , } N/ × N { σ min ( P H F P Tp ) } In the above optimization, P H is known (selection of highfrequency components), F is the GFT of graph G and P p isto be found, which provides the selection of the nodes to bepurged.As we regard SDQM as a quality measure for a givendownsampling scheme, we explain the effect of this measureby an example on uniform 1-D grid (also called DFT grid[18]).Figure 2 shows the well-known downsampling on the grid andthe resultant smaller grid for N = 6 . The optimal solutionsbased on SDQM criteria are { , , } and { , , } . Table 2shows various selected nodes combinations and corresponding SDQM .We now show that the proposed method provides expecteddownsampling in case of bipartite graphs.
A. Analysis Of The Proposed Downsampling Method ForBipartite Graphs
Consider a bipartite graph which has even number of nodes N , and equal nodes in bi-partition. An intuitive way todownsample the same is to select half of the nodes whichbelong to one of the partition of the bipartite structure. Weuse the adjacency matrix based GFT (i.e. GF T A ) in thegiven analysis. The same result can also be obtained usingnormalized Laplacian based GFT. The adjacency matrix of abipartite graph is given by, A = (cid:20) BB T (cid:21) . where B is a matrix of appropriate size. In our specific case(where we assume graph with equal number of nodes in bipar-tite structure), B is an N × N matrix. Let the SVD (SingularValue Decomposition) of matrix B be given by B = U Σ V ∗ ,where U and V are orthogonal matrices and Σ is a diagonal matrix containing singular values of matrix in ascending order.Following [2], the matrix A can be diagonalized as A = W Λ W ∗ where W = √ (cid:20) U UU − V (cid:21) and Λ = (cid:20)
Σ 00 − Σ (cid:21) . Withthe frequency ordering mentioned in [11], it can be seenthat the block corresponding to − Σ contains the upper-halfof frequencies. Hence, P H W ∗ = [ U ∗ − V ∗ ] . As U and V are orthogonal matrices, an optimal selection that maximizesSDQM is selecting the first set (or the second set) of disjointvertices.The 1-D directed uniform graph (DFT graph) is a specialcase of bipartite graph and selecting every alternative nodeis equivalent to selecting one set of disjoint vertices of thebipartite graph.V. A G REEDY A LGORITHM F OR D OWNSAMPLING B ASEDON
SDQMIn the formula F = P H F P Tp , the matrix P H is N/ × N rectangular matrix. Hence, P H F is also an N/ × N rectan-gular matrix. Let F H = P H F , then the desired optimizationturns into a column selection problem from F H such thatthe resultant matrix has maximum smallest singular value. Asimilar problem is discussed in [19], where the parameter tominimize is condition number of the selected columns from F H . The number of combinations to select the columns inmatrix F H are (cid:0) NN/ (cid:1) . An exhaustive search would requirecomputing minimum singular values (cid:0) NN/ (cid:1) times, which iscomputationally impractical for large values of N .To the best of our knowledge, there is no known optimalalgorithm to solve the given problem in polynomial timecomplexity. So, we propose a greedy strategy that may yielda suboptimal solution. The proposed greedy algorithm issummarized in Algorithm 1 .Let F i denote an i × N matrix obtained by selecting i columns from F H . Given F i , F i +14 is obtained by augmenting F i with a column from F H that maximizes the smallestsingular value F i +14 . Iterations continue till i = N/ . Theindices of the columns selected from F H forms the set V p ,the set of vertices to be purged.As discussed in section 4, the GFT for bipartite graphhas two sets of equal-norm orthogonal columns in the upper-half of frequency spectrum. This allows a single orthogonalcolumn being selected at every iteration of greedy algorithm,eventually converging to the optimal solution for any bipartitegraph with equal number of nodes in each partition.VI. E XPERIMENTAL V ALIDATION
In this section, we apply Algorithm 1 to solve the problemof downsampling for undirected and directed graphs. Themeasure of quality of downsampling scheme is given bythe reconstruction error, from the downsampled graph to theoriginal graph. In subsection 6.1, we observe the effect ofpresence of negative edges on various downsampling schemes.In subsections 6.2 and 6.3, we downsample undirected anddirected graphs respectively, and compare the reconstruction
Algorithm 1:
Downsampling On Graphs Using GFT
Input : F H , N Output: V k Procedure:
DownSample i ← V k ← { , ..., N }V p ← {} while i ≤ N/ do N d = getN odeT oDelete ( F H , V k , V p ) V p ← V p ∪ { N d }V k ← V k − { N d } i ← i + 1 return V k Input : F H , V k , V p Output: index
Procedure: getNodeToDeleteArray minSV D forall i ∈ V k do F iter ← columns f rom F H given by V p ∪ { i } minSV D ( i ) ← σ min ( F iter ) index = argmax i { minSV D } return index errors with existing downsampling schemes. Downsamplingof DCT-graphs is used in JPEG image compression standardfor the chrominance components of an image. In subsection6.4, we demonstrate that the downsampling scheme for DCT-graphs obtained using the SDQM measure outperforms theconventional downsampling.
A. Downsampling Random Graphs
In this experiment, we randomly generate graphs with |V| =100 . We conduct the experiment for graphs with non-negativeedge-weights and for graphs which have negative as well aspositive edge-weights. For non-negative weights, each entry ofadjacency matrix is drawn from a uniform distribution U (0 , .For adjacency matrix with negative weights, each entry isdrawn from Gaussian distribution N (0 , . The adjacencymatrix thus obtained is made sparse by sparsity ratio in rangeof − . instances of such matrices are generated fornon-negative and negative-positive each. Table 3 summarizesaverage SDQM and cut-index measures for the trial using MSTbased approach, spectral approach and proposed approach (i.e.Algorithm 1).One can observe from the table that presence of negativeweights deteriorates performance of both MST based andspectral approach according to SDQM and cut-index measures.In case of spectral method, the difference is significant.This can be explained by the fact that the spectral methodattempts to affect a max-cut on the given graph, hence neg-ative edge-weights adversely affects the performance. On theother hand, the proposed approach, while optimizing SDQMalso maintains cut-index comparable to MST based approach.One more remarkable feature is that the performance ofproposed approach is unaffected by introduction of negative MST Spectral ProposedNonnegative Edge-weightsSDQM: 0.0196 0.0178
Cut Index: 0.5718 0.6158
Negative And Positive Edge-weightsSDQM: 0.0162 0.0119
Cut Index: 0.5710 0.5047
TABLE IIISDQM
AND CUT - INDEX FOR RANDOM UNDIRECTED GRAPHS edge-weights. This experiment establishes that the proposedapproach maximizes SDQM while maintaining a high cut-index and at the same time, it can also process graphs withnegative edge-weights.
B. Downsampling Undirected Graphs
The data used in the experiment is temperature data fromweather stations, publicly available on [20]. From the database,we consider nodes from which directed and undirectedgraphs are constructed. Data for year 2014 is considered withdata available on all nodes for days. Thus, we have graph signals with number of nodes being . To constructan undirected graph from the given temperature data, weuse similarity measure given by statistical correlation. Thediagonal entries of the correlation-matrix are all set to ,and the matrix is normalized with the largest eigenvalue. Thematrix is then designated as the adjacency matrix of the graph.This matrix is symmetric, and hence represents an undirectedgraph. We diagonalize the adjacency matrix in order to obtainthe GFT for the given graph .We obtain the downsampled grids using MST based approach,spectral method and proposed method. For each downsampledgrid, we reconstruct the graph signal on purged nodes using thevalues on kept nodes. The reconstruction accuracy is defined as
20 log (cid:16) (cid:107) ¯ x (cid:107)(cid:107) e r (cid:107) (cid:17) . Figures 3 and 4 provide the downsampled grids(both purged and kept nodes) and reconstruction accuracy. Thereconstruction accuracies indicate that the proposed algorithmoutperforms both the methods. The value of SDQM for spec-tral downsampling approach and MST based approach are . and . respectively, while the same for proposed approachis . . This fact reflects directly in the reconstruction errors. C. Downsampling Directed Graphs
For this experiment, we use the same dataset as used inSection 6.2. In order to create a directed graph for the tem-perature data, we first create an 8-neighborhood distance-basedadjacency matrix ˜ A , whose ( i, j ) entry is ˜ a i,j = e − dist ( i,j )2 d .Here, d is the mean distance over entire grid. Similarly, dist ( i, j ) is geometric (Euclidean) distance between latitudeand longitude of weather stations (nodes) numbered i and j .After this, each row of ˜ A is normalized to have unit norm Note that the choice between graph Laplacian based GFT and adjacencymatrix based GFT is arbitrary. The adjacency matrix based GFT is used here. Fig. 3. Result of downsampling undirected temperature data graph: (Top)MST Based approach (Bottom) SVD Based approach, + denotes purgednodes, ◦ denotes preserved nodes.Fig. 4. (Top) Result of downsampling undirected temperature data graphusing proposed method, + denotes purged nodes, ◦ denotes preserved nodes(Bottom) Reconstruction accuracy vs High frequency content in signal in order to obtain adjacency matrix A . This process makesthe adjacency matrix asymmetric, hence the adjacency matrixbased approach is used to obtain GFT for this graph. Usingthis GFT and Algorithm 1, we obtain a downsampling schemeon graph. For this downsampling, the reconstruction error forvarious levels of high-frequency content is shown in Figure 5.The value of SDQM for the obtained partition is . . Fig. 5. Reconstruction accuracy vs High Frequency content in signal(Directed Graph)
D. Downsampling DCT Graphs For Chromatic ComponentsOf Images
In this section, we have a look at the Discrete CosineTransform (DCT-type II) and its graph. A detailed study onthe graphs for which the DCT is GFT is provided in [21]. Thegraph for DCT-type II transform is given in Figure 6 (DCT-type II transform diagonalizes the adjacency matrix of thisgraph). The graph for DCT is undirected, and has self-loopsat the end-nodes. Looking at the structure of graph, intuitively,selecting every alternate node is a good strategy for obtainingthe downsampled grid. However, using the
SDQM measure,we find that there exists a better quality downsampling grid.For V = 16 , Algorithm converges to the sample-set: V k = { , , , , , , , } . Let us denote the selection ofevery alternate sample as set V r = { , , , , ..., } , whichserves as reference for comparing the results. SDQM for V k is . while for V r , it is . . According to our hypothesis, V k should outperform V r in signal reconstruction error.Downsampling of DCT-graphs is used in the JPEG compres-sion standard for color images. In the JPEG compression stan-dard, a color image is first converted into YUV components(Y is luminance, and U, V are chrominance components). Ashuman eye is less sensitive to chrominance, every × (non-overlapping) block of U and V components, are firstdownsampled to × block and then 2-D DCT is appliedon these blocks in order to compress the same . In thisexperiment, we change the downsampling set from V r to V k and show how the sample-set V k can reproduce original blockswith reduced error. For forward transform, 8-point DCT isused for V r and F kL (see section 4) is used for V k . Thereconstruction into × blocks is performed using 16-point2-D IDCT on the transformed blocks with appended zeros inboth the cases. We select three images namely Lena, Barbaraand Baboon images (all of size × ), which are shownin Figure 7. The blockwise average percentage errors in Uand V components are provided for all three images for boththe sample-sets in the table 4. The error for a single block is Note that a × pixel block forms a graph that is the cartesian productof the graph given in Figure 7 with itself. Hence GFT on the × nodegraph is the Kronecker product of GFT for graph in Figure 6, with itself. Fordetails refer [22] Fig. 6. Graph For DCT-type II ( |V| = 6 ). Notice the self-loops for end-nodes.Fig. 7. Images used: (left) Lena (center) Barbara (right) Baboon computed using 2-norm of the error block, and then the erroris averaged over all blocks to obtain blockwise average error.It can be seen that the sample-set derived using the proposedalgorithm reproduces the chromatic components with reducederror compared to standard DCT-IDCT method. The differencein SDQM explains the different performance of both schemes.It should be emphasized here that the purpose of this experi-ment is not to provide a new method of image-compression.Rather, the purpose is to show how underlying graph struc-ture provide non-intuitive downsampling schemes which arecaptured well by the proposed quality measure SDQM. Baboon Barbara Lena V r V k V r V k V r V k U-Component 4.2241
V-Component 3.6346
TABLE IVB
LOCKWISE A VERAGE P ERCENTAGE E RRORS F OR D OWNSAMPLED I MAGES
VII. C
ONCLUSION AND F UTURE W ORK
To summarize, the contributions of this paper are: (1) Weprovide a test for finding whether a signal can be perfectlyreconstructed from a given downsampled grid. (2) We proposea quality measure
SDQM , which can be used to determinequality of a downsampling grid. (3) Based on
SDQM , weobtain an optimization based formulation for downsampling anarbitrary graph. We also provide a greedy algorithm to solvethe optimization problem. The proposed method is applicableto undirected graphs, directed graphs, and graphs with negativeedge-weights.The proposed approach is computationally challenging forlarge graphs. To address this issue, we are presently workingon merging topological approaches with the proposed method.We are also working towards deriving the inter-relations be-tween the downsampled vertices based on spectral properties.VIII. C
OMMENT F ROM A UTHORS
This work was independently carried out by authors duringthe period of Nov 2015 to June 2016. Upon getting peerreviewed, it was pointed out that the work has significantoverlap with work presented in [23]. The purpose of this arXiv copy is to have a reference point for nomenclature andterminology. R
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