Wavelets on Graphs: Theory and Applications 1sti.epfl.ch/files/content/sites/sti/files/shared/sel/pdf/...Distributed Transforms on Graphs Lifting Filter Designs Experimental Results

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


Introduction

Motivation

Wavelets popular tool in many signal processing applications

- Signal analysis- Compression- Storage

Useful properties

- Multiresolution representation- Energy compaction/Sparsity- Computational efficiency


Introduction

Wavelets in 2 slides – 1

(a) 2 Channel Filterbank (b) Tree-structured Filterbank

From Vetterli and Kovacevic, Wavelets and Subband Coding, ’95


Introduction

Wavelets in 2 slides – 2

(a) Separable Transform (b) Example Image

From Vetterli and Kovacevic, [Ding’07]


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


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?


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


Distributed Transforms on Graphs

Next Section

1 Introduction





6 Conclusions


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


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


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)


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]



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


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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?


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 ?


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


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)


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]


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



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]



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))




Multi-level Haar-like wavelets w/ and w/o broadcast

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600Transform Structure on SPT (Variable RR)

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Total Energy Consumption (Joules)

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R (

dB

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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


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


Image Compression with Graph-based Transforms

Next Section

1 Introduction





6 Conclusions


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].



Edge Adaptive Transforms?

Need edge-adaptive transforms for depth maps for better viewinterpolation

DCT not good for complex edgesGraph representation →edge-adaptive transforms



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


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

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Natural image test: Peppers- Coeffs. coded with SPIHT [Said’96], edge map coded with JBIG

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PSNR vs. bpp

Standard

Tree−based




Depth map image test: Tsukuba

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PSNR vs. bpp

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Standard




Depth map image test: Tsukuba

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PSNR vs. bpp

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Standard




Subjective comparison at 0.25 bpp

(g) Ours, PSNR = 42.6 dB (h) 9/7, PSNR = 35.8 dB



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]. )


Wavelet Transforms on Arbitrary Graphs

Next Section

1 Introduction





6 Conclusions


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.


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


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

5 10


5 10


5 10


5 10


eigen-vectors for line-graph


Wavelet Transforms on Arbitrary Graphs Randomly Generated Graph

Randomly Generated Graph with 50 nodes

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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.


Wavelet Transforms on Arbitrary Graphs Randomly Generated Graph

Eigenvectors of graph Laplacian

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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 (+)).


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]


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



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




Lifting Scheme


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




Lifting Scheme


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



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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(c) ”bad” split

Maximum Bipartite Sub-graph Problem:NP-hard problem in general.Iterative Approximation Solution [Narang:APSIPA’09]


Downsampling of Graph Signals

Next Section

1 Introduction





6 Conclusions


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)



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



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)


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 .



Graph based downsampling in Images

Examples:

Horizonal

downsamplingQuincunx

downsampling

Diamond

downsampling



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.



Experiments

Fourier frequency response of Tlow under different downsamplingtechniques:

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(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.



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

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ideal response

approx. response

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approx. response

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Experiments

(c)

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Noisy Samples

(d)

(c) Random graph with N = 670 nodes and, (d) Voronoi plot of Noisy Samples;



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.



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


Conclusions

Next Section

1 Introduction





6 Conclusions


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


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.

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Wavelets on Graphs: Theory and Applications 1sti.epfl.ch/files/content/sites/sti/files/shared/sel/pdf/...Distributed Transforms on Graphs Lifting Filter Designs Experimental Results