Wavelets on Graphs: Theory and Applications 1
Antonio Ortega
Signal and Image Processing InstituteDepartment of Electrical Engineering
University of Southern California
June 2011
1Supported in part by NASA (AIST-05-0081) and NSF (CCF-1018977).A. Ortega (USC) Wavelets on Graphs June 2011 1 / 71
Acknowledgements
Collaborators
- Dr. Alexandre Ciancio- Dr. Godwin Shen- Sunil Narang- Alfonso Sanchez Blanco- Javier Perez Trufero- Woo-Shik Kim- Prof. Bhaskar Krishnamachari- Dr. Sundeep Pattem
Funding
- NASA AIST-05-0081- NSF CCF-1018977
A. Ortega (USC) Wavelets on Graphs June 2011 2 / 71
Introduction
Next Section
1 Introduction
2 Distributed Transforms on Graphs
3 Image Compression with Graph-based Transforms
4 Wavelet Transforms on Arbitrary Graphs
5 Downsampling of Graph Signals
6 Conclusions
A. Ortega (USC) Wavelets on Graphs June 2011 3 / 71
Introduction
Motivation
Wavelets popular tool in many signal processing applications
- Signal analysis- Compression- Storage
Useful properties
- Multiresolution representation- Energy compaction/Sparsity- Computational efficiency
A. Ortega (USC) Wavelets on Graphs June 2011 4 / 71
Introduction
Wavelets in 2 slides – 1
(a) 2 Channel Filterbank (b) Tree-structured Filterbank
From Vetterli and Kovacevic, Wavelets and Subband Coding, ’95
A. Ortega (USC) Wavelets on Graphs June 2011 5 / 71
Introduction
Wavelets in 2 slides – 2
(a) Separable Transform (b) Example Image
From Vetterli and Kovacevic, [Ding’07]
A. Ortega (USC) Wavelets on Graphs June 2011 6 / 71
Introduction
From images to graphs
View an image as a graph with pixels as nodesCan wavelets be extended to irregular graphs?Can they be extended completely arbitrary graphs?
Extension
(a) (b)
(a) Regular lattice (b) Irregular Lattice with missing grid points
A. Ortega (USC) Wavelets on Graphs June 2011 7 / 71
Introduction
Extensions
- Distributed transform under transport cost constraints (e.g., sensornetworks)
Sensors located in arbitrary locations
- Arbitrary “directional” filtering of standard signals (e.g., images)?
Can we traverse images in an arbitrary way?
- Datasets defined on graphs (e.g., data in an online social network)
arbitrary connectivity between nodes in the graph.
A common theme: how to filter when sample locations have arbitrarylocation and connectivity?
A. Ortega (USC) Wavelets on Graphs June 2011 8 / 71
Introduction
This Talk
- Some theoretical results
- Lifting transforms- General transforms, invertibility conditions- New directions, downsampling
- Example applications
- Distributed data gathering in sensor network- Image compression- Graph data
A. Ortega (USC) Wavelets on Graphs June 2011 9 / 71
Distributed Transforms on Graphs
Next Section
1 Introduction
2 Distributed Transforms on Graphs
3 Image Compression with Graph-based Transforms
4 Wavelet Transforms on Arbitrary Graphs
5 Downsampling of Graph Signals
6 Conclusions
A. Ortega (USC) Wavelets on Graphs June 2011 10 / 71
Distributed Transforms on Graphs Data Gathering Problem
Data Gathering Problem
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t(3) = 2
t(9) = 1
t(13) = 3
t(12) = 4
t(10) = 5
t(8) = 6
t(6) = 7
t(4) = 8
t(2) = 9
t(11) = 10t(7) = 11
t(5) = 12
t(1) = 13
Multi-hop routing / broadcast comm.
Collect samples from every node
Raw data gathering
- Simple but inefficient- Correlation not exploited- Broadcasts not used
Use de-correlating transforms
Use broadcasts
Reduce data nodes transmit
A. Ortega (USC) Wavelets on Graphs June 2011 11 / 71
Distributed Transforms on Graphs Distributed Transforms for WSN
Distributed Transforms for WSN
Design abstraction – A distributedtransform can be defined in terms of:
- Routing strategy- Processing strategy
Design transform first (a) →- Good de-correlation- Routing based on transform- Potentially inefficient routing
· Comm. away from sink
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(a) Transform design -> routing strategy
Design efficient routing first (b)
- Transform based on routing
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(b) Routing strategy -> transform design
A. Ortega (USC) Wavelets on Graphs June 2011 12 / 71
Distributed Transforms on Graphs Distributed Transforms for WSN
Distributed Transforms for WSN
Our strategy
- Design efficient routing first →- Transform along routing- Unidirectional transforms: no communication away from the sink
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y9 = T9 * [ x(9) ]y7 = T7 * [ x(7) ]
y5 = T5 * [ x(5) (y6)t (y8)
t ]
t
y8 = T8 * [ x(8) (y9)t ]
t
y6 = T6 * [ x(6) (y7)t (y9)
t ]
t
Less de-correlation (simpler filters), but more efficient overall (lowercommunication cost)
A. Ortega (USC) Wavelets on Graphs June 2011 13 / 71
Distributed Transforms on Graphs Problem Formulation
Problem Formulation
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t(3) = 2
t(9) = 1
t(13) = 3
t(12) = 4
t(10) = 5
t(8) = 6
t(6) = 7
t(4) = 8
t(2) = 9
t(11) = 10t(7) = 11
t(5) = 12
t(1) = 13
General class of transforms?
Key abstractions:
- Measurement x(n)- Routing tree T
· Parent ρ(n), children Cn
· Ancestors An, descendants Dn
· Depth h(n)
- Broadcast neighbors Bn
- Transmission schedule t(n)
Key assumptions:
- t(n) > t(m) for all m ∈ Dn ∪ Bn
- Dn = {n + 1, n + 2, . . . , n + |Dn|}· Preorder indexing [Valiente’02]
A. Ortega (USC) Wavelets on Graphs June 2011 14 / 71
Distributed Transforms on Graphs Problem Formulation
At each node n
nA n
D n
Bn
yDn
yBn
yn
yDn=[yt
c1yt
c2. . . yt
ck
]tyBn =
[yt
b1yt
b2. . . yt
bl
]t
Transform computations
nyn
c1
ck
b1
bl
Tn
yc1
yck
yb1
ybl
nx(n)
Algebraic representation
yn = Tn ·[x(n) yt
DnytBn
]tA. Ortega (USC) Wavelets on Graphs June 2011 15 / 71
Distributed Transforms on Graphs Problem Formulation
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t(3) = 2
t(9) = 1
t(13) = 3
t(12) = 4
t(10) = 5
t(8) = 6
t(6) = 7
t(4) = 8
t(2) = 9
t(11) = 10t(7) = 11
t(5) = 12
t(1) = 13
Unidirectional transmission
- Transmit forward on T
Causal computations
- n uses ym only if t(n) > t(m)
Critical-sampling- One coeff. transmission / sample- n transmits coeffs. for itself and Dn
General definition?
A. Ortega (USC) Wavelets on Graphs June 2011 16 / 71
Distributed Transforms on Graphs Definition of Unidirectional Transforms
Definition of Unidirectional Transforms
Unidirectional, causal, critically-sampled transforms
Assume t(n) is unique, i.e., t(n) 6= t(m) if n 6= m
yDn =[ytCn(1) . . . yt
Cn(|Cn|)
]t
yBn =[ytBn(1) . . . yt
Bn(|Bn|)
]t
yn = [An Bn] ·
x(n)yDn
yBn
(1)
yn : (1 + |Dn|)× 1 by critical sampling
An : (1 + |Dn|)× (1 + |Dn|)Bn : (1 + |Dn|)× (1 + |Bn|) → [An Bn] rank deficient!
When can we recover x(n) and yDn ?
A. Ortega (USC) Wavelets on Graphs June 2011 17 / 71
Distributed Transforms on Graphs Invertibility Conditions
Invertibility Conditions
Note that
yn = [An Bn] ·
x(n)yDn
yBn
= An ·
[x(n)yDn
]+ Bn · yBn .
Can recover x(n) and yDn if
(i) yBn is decoded before yn
(ii) An is invertible
A. Ortega (USC) Wavelets on Graphs June 2011 18 / 71
Distributed Transforms on Graphs Unidirectional Lifting Transforms
Unidirectional Lifting Transforms
Unidirectional wavelet transforms
Construct using lifting [Sweldens’95]
Split nodes into evens E and odds O, E ∩ O = ∅Predict odd x(n) from x(Nn), Nn ⊂ E
d(n) = x(n)−∑i∈Nn
pn(i)x(i)
Update even x(m) from d(Nm), Nm ⊂ Os(m) = x(m) +
∑j∈Nm
um(j)d(j)
A. Ortega (USC) Wavelets on Graphs June 2011 19 / 71
Distributed Transforms on Graphs Split Designs
Split Designs
Split into even and odd sets
Tree-based splits [Shen’08a]
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(a) Tree-based split with 4 even nodes
Graph-based splits [Jansen’01, Narang:ICASSP’10]
A. Ortega (USC) Wavelets on Graphs June 2011 20 / 71
Distributed Transforms on Graphs Lifting Filter Designs
Prediction Filters
d(n) = x(n)−∑
i∈Nnpn(i)x(i)∑
i∈Nn
pn(i)x(i) ≈ x(n)→ d(n) ≈ 0
Averaging [Shen’08a] : d(n) = x(n)−∑
i∈Nnx(i)/|Nn|
Planar [Wagner’05] and data adaptive [Shen’09b] designs
A. Ortega (USC) Wavelets on Graphs June 2011 21 / 71
Distributed Transforms on Graphs Lifting Filter Designs
Update Filters
s(m) = x(m) +∑
j∈Nmum(j)d(j)
- Provides data smoothing- Improves numerical stability of inverse
Smoothing [Shen’08a]: um(Nm) = 12|Nm|
Mean-preserving [Wagner’05]
Orthogonalizing [Shen’09c]
- Minimizes reconstruction MSE due to quantization [Girod’05]
A. Ortega (USC) Wavelets on Graphs June 2011 22 / 71
Distributed Transforms on Graphs Lifting Filter Designs
Experimental Results
Simulated data with second order AR model (600× 600 grid)
Compare T-DPCM [Shen’09b] with raw data gathering
an : Data adaptive prediction filters [Shen’09b]
Cost model [Wang’02, Heinz’00]
- ET (k ,D) = Eelec · k + εamp · k · D2 Joules- ER (k) = Eelec · k Joules
Encoding
- Quantization: y(n) = Q · round[y(n)/Q]- Entropy Coding: arithmetic coder
Performance measure : distortion (SNR) vs. energy consumption
- SNR(Q) = 10 log10(Ex/MSE (Q))
A. Ortega (USC) Wavelets on Graphs June 2011 23 / 71
Distributed Transforms on Graphs Lifting Filter Designs
Experimental Results
Multi-level Haar-like wavelets w/ and w/o broadcast
0 100 200 300 400 500 6000
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600Transform Structure on SPT (Variable RR)
0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.0555
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Total Energy Consumption (Joules)
SN
R (
dB
)
SNR vs. Energy Consumption (Variable RR)
Haar−like w/ Broad.
Haar−like
5/3−like
T−DPCM
Raw Data
Broadcast provides up to 1 dB improvement
A. Ortega (USC) Wavelets on Graphs June 2011 24 / 71
Distributed Transforms on Graphs Conclusions
Conclusions
Unidirectional lifting always invertible
Haar-like transforms give best performance:
(1) At most 1-hop raw data forwarding(2) More broadcasts → more improvements(3) Broadcast more useful for odd leaf nodes
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x(11)
[x(10), d(11)]
x(12)
[d(9), s(10), d(11), s(12)]
(c) No Broadcasts
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x(12)
d(11)
[x(10), d(11)]
x(12)
[d(9), s(10), d(11), s(12)]
(d) With Broadcasts
A. Ortega (USC) Wavelets on Graphs June 2011 25 / 71
Image Compression with Graph-based Transforms
Next Section
1 Introduction
2 Distributed Transforms on Graphs
3 Image Compression with Graph-based Transforms
4 Wavelet Transforms on Arbitrary Graphs
5 Downsampling of Graph Signals
6 Conclusions
A. Ortega (USC) Wavelets on Graphs June 2011 26 / 71
Image Compression with Graph-based Transforms Motivation
Motivation - Edges in Image Coding
Separable wavelets for image compression (JPEG 2000)- Good performance if only vertical / horizontal discontinuities- Complex discontinuities → large HP coeffs → higher bitrate
Even more significant for depth images (no texture)
Taken from [Ding’07].
A. Ortega (USC) Wavelets on Graphs June 2011 27 / 71
Image Compression with Graph-based Transforms Motivation
Edge Adaptive Transforms?
Need edge-adaptive transforms for depth maps for better viewinterpolation
DCT not good for complex edgesGraph representation →edge-adaptive transforms
A. Ortega (USC) Wavelets on Graphs June 2011 28 / 71
Image Compression with Graph-based Transforms Motivation
Related Work
Directional DCT [Zeng’06]
Only efficiently represents blocks with a single linear edgeGood for (a) and (b), but not for (c)
Bandelets [Pennec’05], Directionlets [Velisavljevic’06]
Separable approaches after choosing a locally dominant direction
Platelet-based coding [Morvan’06]
Piecewise planar approximation → fixed approximation error
A. Ortega (USC) Wavelets on Graphs June 2011 29 / 71
Image Compression with Graph-based Transforms Tree Wavelets Based Coding
Wavelets on Trees
Existing transforms find “good” paths for filtering- Path-wise filtering is 1D (e.g., de-generate trees)- Special case of trees
Alternatively : Use lifting on arbitrary trees [Shen’08c, Shen’09c]- Simply avoid discontinuities- Use merges to improve filtering
(a) Toy Image with Edges0
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(b) Horizontal Tree
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(d) Vertical Tree on Odds
A. Ortega (USC) Wavelets on Graphs June 2011 30 / 71
Image Compression with Graph-based Transforms Tree Wavelets Based Coding
Experimental Results
Natural image test: Peppers- Coeffs. coded with SPIHT [Said’96], edge map coded with JBIG
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PS
NR
PSNR vs. bpp
Standard
Tree−based
A. Ortega (USC) Wavelets on Graphs June 2011 31 / 71
Image Compression with Graph-based Transforms Tree Wavelets Based Coding
Experimental Results
Depth map image test: Tsukuba
0 0.2 0.4 0.6 0.8 125
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PS
NR
PSNR vs. bpp
Tree−based (Orth. Update)
Tree−based (Non−orth. Update)
Standard
A. Ortega (USC) Wavelets on Graphs June 2011 32 / 71
Image Compression with Graph-based Transforms Tree Wavelets Based Coding
Experimental Results
Depth map image test: Tsukuba
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PS
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PSNR vs. bpp
Tree−based (Orth. Update)
Tree−based (Non−orth. Update)
Standard
A. Ortega (USC) Wavelets on Graphs June 2011 32 / 71
Image Compression with Graph-based Transforms Tree Wavelets Based Coding
Experimental Results
Subjective comparison at 0.25 bpp
(g) Ours, PSNR = 42.6 dB (h) 9/7, PSNR = 35.8 dB
A. Ortega (USC) Wavelets on Graphs June 2011 33 / 71
Image Compression with Graph-based Transforms Tree Wavelets Based Coding
Conclusions
Proposed transforms give gains
More gain for depth maps
Ortho. updates [Shen’09c] give gains
Improvements in subjective quality
Merges give 0.05 dB improvement
- Path-wise transforms are sufficient- Consistent with [Sanchez’09]
Further work on block-based, graph transform approaches (e.g., [PCS2010]. )
A. Ortega (USC) Wavelets on Graphs June 2011 34 / 71
Wavelet Transforms on Arbitrary Graphs
Next Section
1 Introduction
2 Distributed Transforms on Graphs
3 Image Compression with Graph-based Transforms
4 Wavelet Transforms on Arbitrary Graphs
5 Downsampling of Graph Signals
6 Conclusions
A. Ortega (USC) Wavelets on Graphs June 2011 35 / 71
Wavelet Transforms on Arbitrary Graphs Applications
Graph signals
Many datasets naturally modeled as graph-signals.
Examples:
Graphs Vertices Links Signals
Sensor networks sensors comm. links sensor-measurements
Social networks Users social ties Users’ attributes
Internet computers internet links traffic measurements
PPI graphs2 proteins interactions chem. props.
2protein-protein interactionA. Ortega (USC) Wavelets on Graphs June 2011 36 / 71
Wavelet Transforms on Arbitrary Graphs Graph Signals
General Graph-Signals
Graph : vertices (nodes) connected via some links (edges)
Graph Signal: set of scalar/vector values defined on the vertices.
Graph-signal
Graph G = (V,E)
Vertex Set V = {v1, v2, ...}
Edge Set E = {(v1, v2), (v1, v3), ...}
Graph Signal x = {x1, x2, ...}
Neighborhood, h-hopNh(i) = {j ∈ V : hop dist(i , j) ≤ h}
Flexible model for representing data in many problems.
A. Ortega (USC) Wavelets on Graphs June 2011 37 / 71
Wavelet Transforms on Arbitrary Graphs Graph Signals
Graph Transforms
Input Signal Transform Output Signal Processing/
Analysis
Can handle arbitrary connectivity encoded in the edge weights.
Other desirable properties
InvertibleCritically sampledOrthogonalLocalized in graph (space) and graph spectrum (frequency)
Local Linear Transform
A. Ortega (USC) Wavelets on Graphs June 2011 38 / 71
Wavelet Transforms on Arbitrary Graphs Basic Theory
Spectrum of Graphs
Graph Laplacian Matrix L = D− A = UΛU′
Eigen-vectors of L : U = {uk}k=1:N
Eigen-values of L : diag{Λ} = λ1 ≤ λ2 ≤ ... ≤ λN
Eigen-pair system {(λk ,uk )} provides Fourier-like interpretation ofgraph signals.
5 10
Line−Graph with 15 Nodes
5 10
Eigen Vector for λ2
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Eigen Vector for λ3
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Eigen Vector for λ8
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Eigen Vector for λ12
5 10
Eigen Vector for λ15
eigen-vectors for line-graph
A. Ortega (USC) Wavelets on Graphs June 2011 39 / 71
Wavelet Transforms on Arbitrary Graphs Randomly Generated Graph
Randomly Generated Graph with 50 nodes
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1
A randomly generated graph with 50 nodes. Nodes are uniformly sampled from a 2Dplane. The probability of connection between a pair of nodes is inversely proportional tosquare of their norm-2 distance.
A. Ortega (USC) Wavelets on Graphs June 2011 40 / 71
Wavelet Transforms on Arbitrary Graphs Randomly Generated Graph
Eigenvectors of graph Laplacian
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λ2: # zero−crossings =60
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λ26: # zero−crossings =150
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λ39: # zero−crossings =168
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λ50: # zero−crossings =189
Voronoi diagrams of the sampled points representing eigen-vectors of Graph Laplacian.The color of the Voronoi cell represent the sign of eigen-vector value on thecorresponding Voronoi point (black for (-) and white for (+)).
A. Ortega (USC) Wavelets on Graphs June 2011 41 / 71
Wavelet Transforms on Arbitrary Graphs Work to date
Related Work
Spatial Transform Designs
Graph Wavelets [Crovella’03]Random Multi-resolution transforms on graphs [Wang’06]Lifting wavelets on Sensor Networks [Wagner’05], [Shen’08a] 2008Lifting wavelets on arbitrary graphs [Narang:APSIPA’09]
Spectral Transform Designs
Diffusion Wavelets [Coifman’06]Spectral Wavelets on Graphs [Hammond’09]
A. Ortega (USC) Wavelets on Graphs June 2011 42 / 71
Wavelet Transforms on Arbitrary Graphs Lifting Wavelet Transform
Distributed Lifting Wavelet Transforms
Lifting Scheme
Originally proposed in[Sweldens’95] for 1-D irregularsignals.
Extended to 2-D and 3-D WSNcase in [Wagner’05].
[Narang:APSIPA’09] extendslifting scheme to arbitrarygraphs.
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initial graph
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Wavelet Transforms on Arbitrary Graphs Lifting Wavelet Transform
Lifting Wavelet Transforms
Lifting Scheme
Step 1: Split nodes into even(E)and odd (O) nodes.
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split nodes into �: even and u: odd nodes
A. Ortega (USC) Wavelets on Graphs June 2011 44 / 71
Wavelet Transforms on Arbitrary Graphs Lifting Wavelet Transform
Lifting Wavelet Transforms
Lifting Scheme
Step 1: Split nodes into even(E)and odd (O) nodes.
Step 2: Compute detailcoefficients d(n) at odd nodesusing the data x from their evenneighbors.
d(n) = x(n)−∑
m∈N1(n)∩E
pn,mxm 1
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predict red nodes data using neighboring
blue nodes
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Wavelet Transforms on Arbitrary Graphs Lifting Wavelet Transform
Lifting Wavelet Transforms
Lifting Scheme
Step 1: Split nodes into even(E)and odd (O) nodes.
Step 2: Compute detailcoefficients d(n) at ag. nodesusing the data x from their evenneighbors.
d(n) = x(n)−∑
m∈N1(n)∩E
pn,mxm
Step 3: Compute smoothcoefficients s at even nodesusing data d from their oddneighbors.
s(n) = x(n) +∑
m∈N1(n)∩O
un,mdm
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update blue nodes data using neighboring
red nodes
A. Ortega (USC) Wavelets on Graphs June 2011 46 / 71
Wavelet Transforms on Arbitrary Graphs Lifting Wavelet Transform
Optimal Even/Odd Splitting
By construction, any split will guarantee invertibility.
Goal: Want to split the graph to minimize the number of conflicts.
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(a) Initial Graph
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(b) ”good” split
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(c) ”bad” split
Maximum Bipartite Sub-graph Problem:NP-hard problem in general.Iterative Approximation Solution [Narang:APSIPA’09]
A. Ortega (USC) Wavelets on Graphs June 2011 47 / 71
Downsampling of Graph Signals
Next Section
1 Introduction
2 Distributed Transforms on Graphs
3 Image Compression with Graph-based Transforms
4 Wavelet Transforms on Arbitrary Graphs
5 Downsampling of Graph Signals
6 Conclusions
A. Ortega (USC) Wavelets on Graphs June 2011 48 / 71
Downsampling of Graph Signals Problem Formulation
Problem Formulation: Downsampling in General Sequence
What is downsampling?given a general sequence {x(n)}n∈Vchoose a subset S ⊂ Vdiscard all samples {x(n)}n/∈S
Downsampling function :
βS(n) =
{1 if n ∈ S−1 otherwise
Downsampling a general sequence f with a downsampling function βS(n)
In matrix form Jβ = {diag(β(n))} and
xds =1
2(x + Jβx) (2)
A. Ortega (USC) Wavelets on Graphs June 2011 49 / 71
Downsampling of Graph Signals Problem Formulation
Downsampling in 1D signals
DU by a factor of 2 on a 1-D signal
For 1-D signals β(n) = (−1)n,
Fdu(z) = 1/2(F (z) + F (−z)) (3)
Can we find downsampling function β for graphs with similarproperties?
In general graphs – NOFor k-regular bipartite graphs(k-RBG) – YES
bipartiteevery node degree k
A. Ortega (USC) Wavelets on Graphs June 2011 50 / 71
Downsampling of Graph Signals Problem Formulation
Downsampling in k-RBG [ICASSP 2011]
Properties of k-RBG spectrum:
Symmetrically distributed in range [0 2k].If λ ∈ σ(G ) ⇒ 2k − λ ∈ σ(G )For connected graph {0, 2k} ∈ σ(G ) with multiplicity 1.
Proposed downsampling function for k-RBG
β(n) =
{1 if n ∈ S1
−1 if n ∈ S2
and downsampling matrix Jβ = diag{β(n)} .
Proposition : If us is a spectral basis function for a k-RBG witheigen-value λs , then modified basis function Jβus is also a spectralbasis function of the k-RBG with eigenvalue λ2k−s .
Jβus = uN−s (4)
A. Ortega (USC) Wavelets on Graphs June 2011 51 / 71
Downsampling of Graph Signals Local Transform Examples
Downsampling for k-RBG contd.
Let x ∈ RN : a graph-signal defined on a k-RBG G = (S1,S2,E ).
x =N∑
s=1
x(s)us
Let xdu: graph-signal after DU operation with βS1 .
xdu =N∑
s=1
xdu(s)us
From our proposition:
xdu(s) =1
2(x(s) + x(N − s)) (5)
Nyquist-rate Result for k-RBG : A graph-signal f on a k-RBG,G = (S1;S2; E) can be completely described by only half of itssamples in the set S1 or S2 if the spectrum of f is bandlimited byλN/2 = k .
A. Ortega (USC) Wavelets on Graphs June 2011 52 / 71
Downsampling of Graph Signals Local Transform Examples
Graph based downsampling in Images
Examples:
Horizonal
downsamplingQuincunx
downsampling
Diamond
downsampling
A. Ortega (USC) Wavelets on Graphs June 2011 53 / 71
Downsampling of Graph Signals Local Transform Examples
Experiments
Ideal low-pass filter (projection matrix) on k-RBG:
Tlow =∑λi<k
ui uti (6)
projects a graph signal into the space spanned by only low-passeigenvectors (λ < 1).recovers original signal x from the downsampled signal xdu without lossif Nyquist condition is true.
A. Ortega (USC) Wavelets on Graphs June 2011 54 / 71
Downsampling of Graph Signals Local Transform Examples
Experiments
Fourier frequency response of Tlow under different downsamplingtechniques:
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
(a) (c)(b)
(a) horizontal downsampling (b) for quincunx downsampling and (c) for diamondgraph based downsampling.Frequency responses of anti-aliasing low-pass filters for different downsamplingcases. The filters are approximation of ideal filters with a M = 160 orderChebychev polynomial approximation.
A. Ortega (USC) Wavelets on Graphs June 2011 55 / 71
Downsampling of Graph Signals Local Transform Examples
Two approaches
f1even
+
f0even
f0odd
Bi-partition
Block
f0
P
+
+
U
+
-
+
f1odd
To next level decomp
Lifting: Downsample then transform
f1low
DownSampler
f0
TH
f1high
To next level decompTL
General: Transform then downsample
0
0.5
1
ideal response
approx. response
0
(a) low-pass filters
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ideal response
approx. response
0
(b) high-pass filters
A. Ortega (USC) Wavelets on Graphs June 2011 56 / 71
Downsampling of Graph Signals Local Transform Examples
Experiments
(c)
200 400 600 800 1000
Noisy Samples
(d)
(c) Random graph with N = 670 nodes and, (d) Voronoi plot of Noisy Samples;
A. Ortega (USC) Wavelets on Graphs June 2011 57 / 71
Downsampling of Graph Signals Local Transform Examples
Experiments
200 400 600 800 1000
Lowpass Filter Response
(a)
200 400 600 800 1000
Highpass Filter Response
(b)
Voronoi plots of (a) reconstructed signal from only low-pass coeffs and (b) from only
highPass coeffs of the noisy signal in a 2-level general 2-channel filter-bank.
A. Ortega (USC) Wavelets on Graphs June 2011 58 / 71
Downsampling of Graph Signals Local Transform Examples
Summary
Proposed two designs for constructing two-channel filter-banks ongraphs.
Analysis of sampling
Solution for general graphs:
Based on a separable approachSplit graph into bipartite graphsJust posted to Arxiv
A. Ortega (USC) Wavelets on Graphs June 2011 59 / 71
Conclusions
Next Section
1 Introduction
2 Distributed Transforms on Graphs
3 Image Compression with Graph-based Transforms
4 Wavelet Transforms on Arbitrary Graphs
5 Downsampling of Graph Signals
6 Conclusions
A. Ortega (USC) Wavelets on Graphs June 2011 60 / 71
Conclusions
This Talk – Summary
- Some theoretical results
- Lifting transforms- General transforms, invertibility conditions- New directions, downsampling
- Example applications
- Distributed data gathering in sensor network- Image compression- Graph data
A. Ortega (USC) Wavelets on Graphs June 2011 61 / 71
Conclusions References
References I
S.K. Narang and A. Ortega.
Lifting based wavelet transforms on graphs.In Proc. of Asia Pacific Signal and Information Processing Association (APSIPA), October 2009.
S. Narang, G. Shen, and A. Ortega.
Unidirectional graph-based wavelet transforms for efficient data gathering in sensor networks.In In Proc. of ICASSP’10.
S. Narang and A. Ortega.
Downsampling Graphs using Spectral TheoryIn In Proc. of ICASSP’11.
G. Shen and A. Ortega.
Transform-based Distributed Data Gathering.To Appear in IEEE Transactions on Signal Processing.
G. Shen, S. Pattem, and A. Ortega.
Energy-efficient graph-based wavelets for distributed coding in wireless sensor networks.In Proc. of ICASSP’09, April 2009.
G. Shen, S. Narang, and A. Ortega.
Adaptive distributed transforms for irregularly sampled wireless sensor networks.In Proc. of ICASSP’09, April 2009.
G. Shen and A. Ortega.
Tree-based wavelets for image coding: Orthogonalization and tree selection.In Proc. of PCS’09, May 2009.
A. Ortega (USC) Wavelets on Graphs June 2011 62 / 71
Conclusions References
References II
I.F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci.
A survey on sensor networks.IEEE Communication Magazine, 40(8):102–114, August 2002.
R. Baraniuk, A. Cohen, and R. Wagner.
Approximation and compression of scattered data by meshless multiscale decompositions.Applied Computational Harmonic Analysis, 25(2):133–147, September 2008.
C.L. Chang and B. Girod.
Direction-adaptive discrete wavelet transform for image compression.IEEE Transactions on Image Processing, 16(5):1289–1302, May 2007.
C. Chong and S. P. Kumar.
Sensor networks: Evolution, opportunities, and challenges.Proceedings of the IEEE, 91(8):1247–1256, August 2003.
R.R. Coifman and M. Maggioni.
Diffusion wavelets.Applied Computational Harmonic Analysis, 21(1):53–94, 2006.
R. Cristescu, B. Beferull-Lozaon, and M. Vetterli.
Networked Slepian-Wolf: Theory, algorithms, and scaling laws.IEEE Transactions on Information Theory, 51(12):4057–4073, December 2005.
M. Crovella and E. Kolaczyk.
Graph wavelets for spatial traffic analysis.In IEEE INFOCOMM, 2003.
A. Ortega (USC) Wavelets on Graphs June 2011 63 / 71
Conclusions References
References III
TinyOS-2.
Collection tree protocol.http://www.tinyos.net/tinyos-2.x/doc/.
I. Daubechies, I. Guskov, P. Schroder, and W. Sweldens.
Wavelets on irregular point sets.Phil. Trans. R. Soc. Lond. A, 357(1760):2397–2413, September 1999.
W. Ding, F. Wu, X. Wu, S. Li, and H. Li.
Adaptive directional lifting-based wavelet transform for image coding.IEEE Transactions on Image Processing, 16(2):416–427, February 2007.
M. Gastpar, P. Dragotti, and M. Vetterli.
The distributed Karhunen-Loeve transform.IEEE Transactions on Information Theory, 52(12):5177–5196, December 2006.
B. Girod and S. Han.
Optimum update for motion-compensated lifting.IEEE Signal Processing Letters, 12(2):150–153, February 2005.
V.K. Goyal.
Theoretical foundations of transform coding.IEEE Signal Processing Magazine, 18(5):9–21, September 2001.
S. Haykin.
Adaptive Filter Theory.Prentice Hall, 4th edition, 2004.
A. Ortega (USC) Wavelets on Graphs June 2011 64 / 71
Conclusions References
References IV
W. R. Heinzelman, A. Chandrakasan, and H. Balakrishnan.
Energy-efficient routing protocols for wireless microsensor networks.In Proc. of Hawaii Intl. Conf. on Sys. Sciences, January 2000.
M. Jansen, G. Nason, and B. Silverman.
Scattered data smoothing by empirical Bayesian shrinkage of second generation wavelet coefficients.In Wavelets: Applications in Signal and Image Processing IX, Proc. of SPIE, 2001.
D. Jungnickel.
Graphs, Networks and Algorithms.Springer-Verlag Press, 2nd edition, 2004.
M. Maitre and M. N. Do,
“Shape-adaptive wavelet encoding of depth maps,”In Proc. of PCS’09, 2009.
K. Mechitov, W. Kim, G. Agha, and T. Nagayama.
High-frequency distributed sensing for structure monitoring.In In Proc. First Intl. Workshop on Networked Sensing Systems (INSS), 2004.
Y. Morvan, P.H.N. de With, and D. Farin,
“Platelet-based coding of depth maps for the transmission of multiview images,”2006, vol. 6055, SPIE.
S. Pattem, B. Krishnamachari, and R. Govindan.
The impact of spatial correlation on routing with compression in wireless sensor networks.ACM Transactions on Sensor Networks, 4(4):60–66, August 2008.
A. Ortega (USC) Wavelets on Graphs June 2011 65 / 71
Conclusions References
References V
E. Le Pennec and S. Mallat.
Sparse geometric image representations with bandelets.IEEE Transactions on Image Processing, 14(4):423– 438, April 2005.
J.G. Proakis, E.M. Sozer, J.A. Rice, and M. Stojanovic.
Shallow water acoustic networks.IEEE Communications Magazine, 39(11):114–119, 2001.
P. Rickenbach and R. Wattenhofer.
Gathering correlated data in sensor networks.In Proceedings of the 2004 Joint Workshop on Foundations of Mobile Computing, October 2004.
A. Said and W.A. Pearlman.
A New, Fast, and Efficient Image Codec Based on Set Partitioning in Hierarchical Trees.IEEE Transactions on Circuits and Systems for Video Technology, 6(3):243– 250, June 1996.
A. Sanchez, G. Shen, and A. Ortega,
“Edge-preserving depth-map coding using graph-based wavelets,”In Proc. of Asilomar’09, 2009.
G. Shen and A. Ortega.
Optimized distributed 2D transforms for irregularly sampled sensor network grids using wavelet lifting.In Proc. of ICASSP’08, April 2008.
G. Shen and A. Ortega.
Joint routing and 2D transform optimization for irregular sensor network grids using wavelet lifting.In IPSN ’08, April 2008.
A. Ortega (USC) Wavelets on Graphs June 2011 66 / 71
Conclusions References
References VI
G. Shen and A. Ortega.
Compact image representation using wavelet lifting along arbitrary trees.In Proc. of ICIP’08, October 2008.
G. Shen, S. Pattem, and A. Ortega.
Energy-efficient graph-based wavelets for distributed coding in wireless sensor networks.In Proc. of ICASSP’09, April 2009.
G. Shen, S. Narang, and A. Ortega.
Adaptive distributed transforms for irregularly sampled wireless sensor networks.In Proc. of ICASSP’09, April 2009.
G. Shen and A. Ortega.
Tree-based wavelets for image coding: Orthogonalization and tree selection.In Proc. of PCS’09, May 2009.
G. Shen, W.-S. Kim, A. Ortega, J. Lee and H.C. Wey.
Edge-aware Intra Prediction for Depth Map Coding.Submitted to Proc. of ICIP’10.
G. Shen and A. Ortega.
Transform-based Distributed Data Gathering.To Appear in IEEE Transactions on Signal Processing.
G. Strang.
Linear Algebra and its Applications.Thomson Learning, 3rd edition, 1988.
A. Ortega (USC) Wavelets on Graphs June 2011 67 / 71
Conclusions References
References VII
W. Sweldens.
The lifting scheme: A construction of second generation wavelets.Tech. report 1995:6, Industrial Mathematics Initiative, Department of Mathematics, University of South Carolina, 1995.
M. Tanimoto, T. Fujii, and K. Suzuki,
“View synthesis algorithm in view synthesis reference software 2.0(VSRS2.0),” ISO/IEC JTC1/SC29/WG11, Feb. 2009.
G. Valiente.
Algorithms on Trees and Graphs.Springer, 1st edition, 2002.
V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P.L. Dragotti.
Directionlets: Anisotropic multidirectional representation with separable filtering.IEEE Transactions on Image Processing, 15(7):1916– 1933, July 2006.
R. Wagner, H. Choi, R. Baraniuk, and V. Delouille.
Distributed wavelet transform for irregular sensor network grids.In IEEE Stat. Sig. Proc. Workshop (SSP), July 2005.
R. Wagner, R. Baraniuk, S. Du, D.B. Johnson, and A. Cohen.
An architecture for distributed wavelet analysis and processing in sensor networks.In IPSN ’06, April 2006.
A. Wang and A. Chandraksan.
Energy-efficient DSPs for wireless sensor networks.IEEE Signal Processing Magazine, 19(4):68–78, July 2002.
A. Ortega (USC) Wavelets on Graphs June 2011 68 / 71
Conclusions References
References VIII
Y. Zhu, K. Sundaresan, and R. Sivakumar.
Practical limits on achievable energy improvements and useable delay tolerance in correlation aware data gathering inwireless sensor networks.In IEEE SECON’05, September 2005.
S.K. Narang, G. Shen and A. Ortega,
“Unidirectional Graph-based Wavelet Transforms for Efficient Data Gathering in Sensor Networks”.pp.2902-2905, ICASSP’10, Dallas, April 2010.
S.K. Narang and A. Ortega,
“Local Two-Channel Critically Sampled Filter-Banks On Graphs”,Intl. Conf. on Image Proc. (2010),
R. R. Coifman and M. Maggioni,
“Diffusion Wavelets,”Appl. Comp. Harm. Anal., vol. 21 no. 1 (2006), pp. 53–94
D. K. Hammond, P. Vandergheynst, and R. Gribonval,
“Wavelets on graphs via spectral graph theory,”Tech. Rep. arXiv:0912.3848, Dec 2009.
M. Crovella and E. Kolaczyk,
“Graph wavelets for spatial traffic analysis,”in INFOCOM 2003, Mar 2003, vol. 3, pp. 1848–1857.
G. Shen and A. Ortega,
“Optimized distributed 2D transforms for irregularly sampled sensor network grids using wavelet lifting,”in ICASSP’08, April 2008, pp. 2513–2516.
A. Ortega (USC) Wavelets on Graphs June 2011 69 / 71
Conclusions References
References IX
W. Wang and K. Ramchandran,
“Random multiresolution representations for arbitrary sensor network graphs,”in ICASSP, May 2006, vol. 4, pp. IV–IV.
R. Wagner, H. Choi, R. Baraniuk, and V. Delouille.
Distributed wavelet transform for irregular sensor network grids.In IEEE Stat. Sig. Proc. Workshop (SSP), July 2005.
S. K. Narang and A. Ortega,
“Lifting based wavelet transforms on graphs,”(APSIPA ASC’ 09), 2009.
B. Zeng and J. Fu,
“Directional discrete cosine transforms for image coding,”in Proc. of ICME 2006, 2006.
E. Le Pennec and S. Mallat,
“Sparse geometric image representations with bandelets,”IEEE Trans. Image Proc., vol. 14, no. 4, pp. 423–438, Apr. 2005.
M. Vetterli V. Velisavljevic, B. Beferull-Lozano and P.L. Dragotti,
“Directionlets: Anisotropic multidirectional representation with separable filtering,”IEEE Trans. Image Proc., vol. 15, no. 7, pp. 1916–1933, Jul. 2006.
P.H.N. de With Y. Morvan and D. Farin,
“Platelet-based coding of depth maps for the transmission of multiview images,”in In Proceedings of SPIE: Stereoscopic Displays and Applications, 2006, vol. 6055.
A. Ortega (USC) Wavelets on Graphs June 2011 70 / 71
Conclusions References
References X
M. Tanimoto, T. Fujii, and K. Suzuki,
“View synthesis algorithm in view synthesis reference software 2.0 (VSRS2.0),”Tech. Rep. Document M16090, ISO/IEC JTC1/SC29/WG11, Feb. 2009.
Gilbert Strang,
“The discrete cosine transform,”SIAM Review, vol. 41, no. 1, pp. 135–147, 1999.
A. Ortega (USC) Wavelets on Graphs June 2011 71 / 71