Graph Learning, Sampling & Filtering for
Image & Signal Estimation
Gene Cheung
York University
Toronto, Canada

March 9, 2021

Acknowledgement• Graph and Image Signal Processing (GISP) Lab (York University, Toronto, Canada)➢ Post-docs: Cheng Yang*, Xue Zhang➢ Grad students: Saghar Bagheri, Fengbo Lan, Huy Vu➢ Visiting researchers: Weng-tai Su (NTHU), Chinthaka Dinesh (SFU), Fen Wang (Xidian)➢ Collaborators➢ Richard Wildes, Michael Brown (York Univ., Canada)➢ Ivan V. Bajic (Simon Fraser Univ., Canada)➢ Antonio Ortega (Univ. of Southern California, USA)
➢ Stanley Chan (Purdue Univ., USA)➢ Wai-Tian Tan (Cisco, USA)
➢ Jiahao Pang, Dong Tian (InterDigital, USA)
➢ Yuji Nakatsukasa (Oxford Univ., UK)➢ Vladimir Stankovic (Univ. of Strathclyde, UK)➢ Wei Hu, Wen Gao (Peking Univ., China)➢ Weng-tai Su, Chia-Wen Lin (NTHU, Taiwan)➢ Yonina C. Eldar (Weizmann Inst. of Science, Israel)* September 2018 to June 2020.
2

Gene Cheung (genec@yorku.ca)

Outline➢ What is Graph Signal Processing?
➢ Graph spectrum➢ Graph Fourier transform (GFT), graph Laplacian regularizer (GLR)➢ Graph Learning➢ Precision / Graph Laplacian Matrix Estimation (w/ eigen-structure constraint)➢ Feature Graph Learning: Gershgorin Disc Perfect Alignment (GDPA)➢ Application: Semi-supervised classifier learning➢ Graph Sampling➢ Gershgorin Disc Alignment Sampling (GDAS)➢ Application: Sampling for matrix completion, 3D point cloud sub-sampling➢ Graph Filtering➢ Signal-dependent GLR, GTV➢ Application: Image denoising
3

Gene Cheung (genec@yorku.ca)

Outline➢ What is Graph Signal Processing?
➢ Graph spectrum➢ Graph Fourier transform (GFT), graph Laplacian regularizer (GLR)➢ Graph Learning➢ Precision / Graph Laplacian Matrix Estimation (w/ eigen-structure constraint)➢ Feature Graph Learning: Gershgorin Disc Perfect Alignment (GDPA)➢ Application: Semi-supervised classifier learning➢ Graph Sampling➢ Gershgorin Disc Alignment Sampling (GDAS)➢ Application: Sampling for matrix completion, 3D point cloud sub-sampling➢ Graph Filtering➢ Signal-dependent GLR, GTV➢ Application: Image denoising
4

Gene Cheung (genec@yorku.ca)

Digital Signal Processing▪ Discrete signals on regular data kernels.
▪ Ex.1: audio on regularly sampled timeline.
▪ Ex.2: image on 2D grid.

▪ Harmonic analysis tools (transforms, wavelets):▪ Compression, restoration, segmentation, etc.

a =xsparsetransform coeff.

desiredsignal

DCT transform

2D DCT basis
5

Gene Cheung (genec@yorku.ca)

Graph Signal Processing▪ Signals on irregular data kernels described by graphs.
▪ Graph: nodes and edges.
▪ Edges reveals node-to-node relationships.
1. Harmonic Analysis of graph signals.
2. Embed pairwise (dis)similarity info into edge weights.
▪ Eigenvectors provide global info aggregated from local info.

[1] A. Ortega, P. Frossard, J. Kovacevic, J. M. F. Moura, and P. Vandergheynst, “Graph signal processing: Overview,
challenges, and applications,” Proceedings of the IEEE, vol. 106, no. 5, pp. 808–828, 2018.
[2] G. Cheung, E. Magli, Y. Tanaka, and M. K. Ng, “Graph spectral image processing,” Proceedings of the IEEE, vol. 106, no.
5, pp. 907–930, 2018.
6

Gene Cheung (genec@yorku.ca)

signal on graph kernel

signal on graph kernelGraph Signal Processing▪ Signals on irregular data kernels described by graphs.
▪ Graph: nodes and edges.
▪ Edges reveals node-to-node relationships.
1. Harmonic Analysis of graph signals.
2. Embed pairwise (dis)similarity info into edge weights.
▪ Eigenvectors provide global info aggregated from local info.

signal on graph kernel

Graph Signal Processing (GSP) provides spectral
analysis tools for signals residing on graphs.
[1] A. Ortega, P. Frossard, J. Kovacevic, J. M. F. Moura, and P. Vandergheynst, “Graph signal processing: Overview,
challenges, and applications,” Proceedings of the IEEE, vol. 106, no. 5, pp. 808–828, 2018.
[2] G. Cheung, E. Magli, Y. Tanaka, and M. K. Ng, “Graph spectral image processing,” Proceedings of the IEEE, vol. 106, no.
5, pp. 907–930, 2018.
6

Gene Cheung (genec@yorku.ca)

signal on graph kernelGraph Fourier Transform (GFT)
Graph Laplacian:undirected graph• Adjacency Matrix W: entry Wi,j has non-negative edge weight wi,jconnecting nodes i and j.
• Degree Matrix D: diagonal matrix w/ entry Di,i being sum of column
entries in row i of W.
𝐷𝑖,𝑖 = ෍ 𝑊𝑖,𝑗𝑖

• Combinatorial Graph Laplacian L: L = D - W▪ L is related to 2nd derivative.

L3,: x = − x2 + 2 x3 − x4
f (x + h ) − 2 f (x ) + f (x − h )f ( x ) = lim
h →0h2▪ L is a differential operator on graph.
*https://en.wikipedia.org/wiki/Second_derivative
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Gene Cheung (genec@yorku.ca)

w1, 2

1

1

2

3

0
𝑤
𝐖 = 1,2
0
0

𝑤1,2
0
1
0

0
1
0
1

0
0
1
0

𝑤1,2
0
𝐃=
0
0

0
𝑤1,2 + 1
0
0

0
0
2
0

0
0
0
1

1

𝑤1,2
−𝑤1,2
𝐋=
0
0

4

−𝑤1,2
0
0
𝑤1,2 + 1 −1 0
−1
2 −1
0
−1 1

Graph Fourier Transform (GFT)
Graph Laplacian:undirected graph• Adjacency Matrix W: entry Wi,j has non-negative edge weight wi,jconnecting nodes i and j.
• Degree Matrix D: diagonal matrix w/ entry Di,i being sum of column
entries in row i of W.
𝐷𝑖,𝑖 = ෍ 𝑊𝑖,𝑗𝑖

• Combinatorial Graph Laplacian L: L = D - W▪ L is related to 2nd derivative.

L3,: x = − x2 + 2 x3 − x4
f (x + h ) − 2 f (x ) + f (x − h )f ( x ) = lim
h →0h2▪ L is a differential operator on graph.
*https://en.wikipedia.org/wiki/Second_derivative
7

Gene Cheung (genec@yorku.ca)

w1, 2

1

1

2

3

0
𝑤
𝐖 = 1,2
0
0

𝑤1,2
0
1
0

0
1
0
1

0
0
1
0

𝑤1,2
0
𝐃=
0
0

0
𝑤1,2 + 1
0
0

0
0
2
0

0
0
0
1

1

𝑤1,2
−𝑤1,2
𝐋=
0
0

4

−𝑤1,2
0
0
𝑤1,2 + 1 −1 0
−1
2 −1
0
−1 1

Graph Spectrum from GFT
Graph Fourier Transform (GFT) is eigen-matrix of graph Laplacian L.
eigenvalues along diagonal

L= VVT

GFT

eigenvectors in columns~
x = VT x
GFT coefficients

1. Eigenvectors aggregate info from edge weights.
▪ Constant 1st eigenvector is DC.
▪ # zero-crossings increases as λ increases.
2. Eigenvalues (≥ 0) as graph frequencies.
GFT defaults to DCT for un-weighted connected line.
GFT defaults to DFT for un-weighted connected circle.

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Gene Cheung (genec@yorku.ca)

1

w1, 2

1
2

1

3

2nd eigenvector

4

…

8

Graph Spectrum from GFT
Graph Fourier Transform (GFT) is eigen-matrix of graph Laplacian L.
eigenvalues along diagonal

L= VVT

GFT

eigenvectors in columns~
x = VT x
GFT coefficients

1. Eigenvectors aggregate info from edge weights.
▪ Constant 1st eigenvector is DC.
▪ # zero-crossings increases as λ increases.
2. Eigenvalues (≥ 0) as graph frequencies.
GFT defaults to DCT for un-weighted connected line.
GFT defaults to DFT for un-weighted connected circle.

8

Gene Cheung (genec@yorku.ca)

1

w1, 2

1
2

1

3

2nd eigenvector

4

…

8

Graph Frequency Examples (US Temperature)
Weather stations from 100 most populated cities.
Graph connections from Delaunay Triangulation*.
Edge weights inverse proportional to distance.

location diff.
 − l −l 2 
 i j 2wi , j = exp
2

 




Edge weights

V1: DC component*https://en.wikipedia.org/wiki/Delaunay triangulation
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Gene Cheung (genec@yorku.ca)

Graph Frequency Examples (US Temperature)
Weather stations from 100 most populated cities.
Graph connections from Delaunay Triangulation*.
Edge weights inverse proportional to distance.

location diff.
 − l −l 2 
 i j 2wi , j = exp
2

 




Edge weights

V2: 1st AC component

10

Gene Cheung (genec@yorku.ca)

Graph Frequency Examples (US Temperature)
Weather stations from 100 most populated cities.
Graph connections from Delaunay Triangulation*.
Edge weights inverse proportional to distance.

location diff.
 − l −l 2 
 i j 2wi , j = exp
2

 




Edge weights

V3: 2nd AC component

11

Gene Cheung (genec@yorku.ca)

Graph Frequency Examples (US Temperature)
Weather stations from 100 most populated cities.
Graph connections from Delaunay Triangulation*.
Edge weights inverse proportional to distance.

location diff.
 − l −l 2 
 i j 2wi , j = exp
2

 




Edge weights

V4: 9th AC component

12

Gene Cheung (genec@yorku.ca)

GSP and Graph-related Research
GSP: SP framework that unifies concepts from multiple fields.
Partial Differential
Eq’ns

MachineLearning

Laplace equation,
Diffusiongraphical model,
manifold learning,classifier learning

LaplaceBeltrami
operator

Graph Signal
Processing* (GSP)DSP

Max cut, graphtransformation

Combinatorial
Graph Theory
13

Computer Graphics

Gene Cheung (genec@yorku.ca)

spectral Computerclustering

Visioneigen-analysis ofgraph Laplacian,adjacency matrices

SpectralGraph Theory

Outline➢ What is Graph Signal Processing?
➢ Graph spectrum➢ Graph Fourier transform (GFT), graph Laplacian regularizer (GLR)➢ Graph Learning➢ Precision / Graph Laplacian Matrix Estimation (w/ eigen-structure constraint)➢ Feature Graph Learning: Gershgorin Disc Perfect Alignment (GDPA)➢ Application: Semi-supervised classifier learning➢ Graph Sampling➢ Gershgorin Disc Alignment Sampling (GDAS)➢ Application: Sampling for matrix completion, 3D point cloud sub-sampling➢ Graph Filtering➢ Signal-dependent GLR, GTV➢ Application: Image denoising
14

Gene Cheung (genec@yorku.ca)

What is a good graph?
• Graph Signal Processing (GSP) provides spectral analysis
tools for signals on fixed graphs.
• Graph captures pairwise relationships.
• Correlations.
• Feature distance.

signal on line kernel• Goal:1. Learn inverse covariance matrix from limited data.
2. Learn metric to determine feature distance.
signal on graph kernel[1] X. Dong et al., Learning graphs from data: A signal representation perspective," IEEE SPM, vol. 36, no. 3, pp. 44-63, 2019.
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Gene Cheung (genec@yorku.ca)

Sparse Precision Matrix Estimation: GLASSO• Given empirical covariance matrix Σ, Graphical Lasso
computes positive-definite (PD) precision matrix Θ:• 1st and 2nd terms are likelihood.
• 3rd term promotes sparsity.
• Solved via block-coordinate descent (BCD) algorithm.

[1] Friedman J, Hastie T, Tibshirani R. “Sparse inverse covariance estimation with the graphical lasso,” Biostatistics. 2008; 9(3): 432-441.
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Gene Cheung (genec@yorku.ca)

Sparse Precision Matrix Estimation: GLASSO• Given empirical covariance matrix Σ, Graphical Lasso
computes positive-definite (PD) precision matrix Θ:• 1st and 2nd terms are likelihood.
• 3rd term promotes sparsity.
• Solved via block-coordinate descent (BCD) algorithm.

[1] Friedman J, Hastie T, Tibshirani R. “Sparse inverse covariance estimation with the graphical lasso,” Biostatistics. 2008; 9(3): 432-441.
16

Gene Cheung (genec@yorku.ca)

α-incoherence
condition

Graph Laplacian Estimation• Assume precision matrix is:• Generalized graph Laplacian (GGLs),• Diagonally dominant generalized graph Laplacian (DDGLs), or• Combinatorial graph Laplacian (CGLs).
• Given empirical covariance matrix S, computes Laplacian Θ:• K = S + H, H is regularization matrix.
• Lg(A) ensures Θ is GGL.
• Solved via block-coordinate descent (BCD) algorithm.
[1] H. E. Egilmez, E. Pavez and A. Ortega, "Graph Learning From Data Under Laplacian and Structural Constraints," in IEEE Journal of
Selected Topics in Signal Processing, vol. 11, no. 6, pp. 825-841, Sept. 201717

Gene Cheung (genec@yorku.ca)

Graph Laplacian Estimation w/ Eigen-Structure Constraint• Assume graph Laplacian matrix L has:Pre-determined first K eigenvectors.
• Define convex coneof PSD matrices sharing same K eigenvectors.
• Design projection operator toinspired by Gram-Schmidt procedure.
• Given empirical covariance matrix S, computes Laplacian L:• Solve via alternating BCD and projection algorithm.
[1] S. Bagheri, G. Cheung, A. Ortega, F. Wang, "Learning Sparse Graph Laplacian with K Eigenvector Prior via Iterative GLASSO and
Projection," accepted to IEEE International Conference on Acoustics, Speech and Signal Processing, Toronto, Canada, June 2021.
18

Gene Cheung (genec@yorku.ca)

Graph Laplacian Estimation w/ Eigen-Structure Constraint• Assume graph Laplacian matrix L has:Pre-determined first K eigenvectors.

Ex:
1. 1st e-vector is constant for image coding.
2. 1st e-vector is PWC for voting in Senate.
3. Sparse first K e-vectors for transform coding.

• Define convex coneof PSD matrices sharing same K eigenvectors.
• Design projection operator toinspired by Gram-Schmidt procedure.
• Given empirical covariance matrix S, computes Laplacian L:• Solve via alternating BCD and projection algorithm.
[1] S. Bagheri, G. Cheung, A. Ortega, F. Wang, "Learning Sparse Graph Laplacian with K Eigenvector Prior via Iterative GLASSO and
Projection," accepted to IEEE International Conference on Acoustics, Speech and Signal Processing, Toronto, Canada, June 2021.
18

Gene Cheung (genec@yorku.ca)

Graph Laplacian Estimation: results• Randomly located 20 nodes in 2D space. Use the Erdos-Renyi model [23] to determine connectivity
with probability 0.6. Compute edge weights using a Gaussian kernel. Remove weights <0.75. Flipsign of each edge with probability 0.5. K=1.
• (a) Ground Truth Laplacian L , (b) Proposed Proj-Lasso with K = 1, (c) GLASSO, (d) DDGL and (e)GL-SigRep .

[1] S. Bagheri, G. Cheung, A. Ortega, F. Wang, "Learning Sparse Graph Laplacian with K Eigenvector Prior via Iterative GLASSO and
Projection," accepted to IEEE International Conference on Acoustics, Speech and Signal Processing, Toronto, Canada, June 2021.
19

Gene Cheung (genec@yorku.ca)

Metric Learning for Graph Construction• Construct graph when ≤ 1 signal observation, but
Each node has K-dimension feature vector.
• Example: semi-supervised graph classifier• Each node i has feature vector• Use PSD metric matrix M, establish Mahalanobis
distance:• Compute positive edge weight using exp:

[1] C. Yang, G. Cheung, W. Hu, "Signed Graph Metric Learning via Gershgorin Disc Alignment," submitted to IEEE Transactions on
Pattern Analysis and Machine Intelligence, June 2020.
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Gene Cheung (genec@yorku.ca)

signal on graph kernel

Signal Reconstruction using GLR▪ Signal Model:

observation

sampling matrix
desired signal

𝐲 = 𝐇𝐱 + 𝐯

noise▪ Signal prior is graph Laplacian regularizer (GLR):𝐱 𝑇 Lx =12
෍ 𝑤𝑖,𝑗 𝑥𝑖 − 𝑥𝑗 = ෍ 𝜆𝑘 𝑥෤𝑘2
2
𝑖,𝑗

𝑘

signal smooth w.r.t. graph▪ MAP Formulation:
signal priorfidelity term

1

2

1
4

3

Sample set {2, 4}
𝐇=signal contains
mostly low graph freq.

1

1

0 1 0
0 0 0

0
0
𝐇T𝐇 =
0
0

0
1
0
0

0
0
0
0

0
1
0
0
0
1

min 𝐲 − 𝐇𝐱 22 + 𝜇 𝐱 𝑇 𝐋𝐱
𝐱

𝐇𝑇𝐇 + 𝜇𝐋 𝐱 ∗ = 𝐲linear system of eqn’s solved using conjugate gradient[1] J. Pang, G. Cheung, "Graph Laplacian Regularization for Image Denoising: Analysis in the Continuous Domain," IEEE TIP, vol. 26, no.4, pp.1770-1785, April 2017.

[2] C. Yang, G. Cheung, V. Stankovic, "Alternating Binary Classifier and Graph Learning from Partial Labels," APSIPA ASC 2018, Hawaii, USA, November 2018.
21

Gene Cheung (genec@yorku.ca)

Metric Learning for Graph Construction• Optimal metric matrix M:

for convex, differentiable Q(M).
• For example, Graph Laplacian Regularizer (GLR):[1] C. Yang, G. Cheung, W. Hu, "Signed Graph Metric Learning via Gershgorin Disc Alignment," submitted to IEEE Transactions on
Pattern Analysis and Machine Intelligence, June 2020.
22

Gene Cheung (genec@yorku.ca)

Metric Learning for Graph Construction• Optimal metric matrix M:
PSD cone constraint is hard!

for convex, differentiable Q(M).

Naïve Approach:• Gradient descent via• Projection to PSD cone.
• Repeat.

Our Approach:• For example, Graph Laplacian Regularizer (GLR): • Convert PSD cone to K adaptive
linear constraints via Gershgorin
Disc Alignment (GDA).
• Min Q(M) w/ linear constraints.
• Repeat.

[1] C. Yang, G. Cheung, W. Hu, "Signed Graph Metric Learning via Gershgorin Disc Alignment," submitted to IEEE Transactions on
Pattern Analysis and Machine Intelligence, June 2020.
22

Gene Cheung (genec@yorku.ca)

Gershgorin Circle Theorem
Gershgorin Circle Theorem:▪ Row i of M maps to a Gershgorin disc w/ centre Mii
and radius Ri
𝑅𝑖 = ෍ 𝑀𝑖𝑗𝑗≠𝑖▪ λmin is lower-bounded by smallest disc left-end:
𝜆−min 𝐌 ≜ min 𝑀𝑖,𝑖 − 𝑅𝑖 ≤ 𝜆min𝑖

▪ To ensure PSDness, apply linear constr’s
𝑀𝑖,𝑖 − ෍ 𝑀𝑖𝑗 ≥ 0
𝑗≠𝑖[1] R. S. Varga, Gershgorin and His Circles, Springer, Dec 2004.
23

Gene Cheung (genec@yorku.ca)𝜆−minGershgorin Circle Theorem
Gershgorin Circle Theorem:▪ Row i of M maps to a Gershgorin disc w/ centre Mii
and radius Ri
𝑅𝑖 = ෍ 𝑀𝑖𝑗𝑗≠𝑖▪ λmin is lower-bounded by smallest disc left-end:
𝜆−min 𝐌 ≜ min 𝑀𝑖,𝑖 − 𝑅𝑖 ≤ 𝜆min𝑖

▪ To ensure PSDness, apply linear constr’s
𝑀𝑖,𝑖 − ෍ 𝑀𝑖𝑗 ≥ 0
𝑗≠𝑖[1] R. S. Varga, Gershgorin and His Circles, Springer, Dec 2004.
23

Gene Cheung (genec@yorku.ca)𝜆−minGershgorin Disc Perfect Alignment (GDPA)• Consider similarity transform of M (same eigenvalues!):

𝐁 = 𝐒 𝐌 𝐒 −1

similarity transformdiagonal matrix w/ scale
factors 𝑠𝑖−• Different S’s induce different lower bounds 𝜆min 𝐁[1] C. Yang, G. Cheung, W. Hu, "Signed Graph Metric Learning via Gershgorin Disc Alignment," submitted to IEEE Transactions on
Pattern Analysis and Machine Intelligence, June 2020.
24

Gene Cheung (genec@yorku.ca)

Gershgorin Disc Perfect Alignment (GDPA)• Consider similarity transform of M (same eigenvalues!):

𝐁 = 𝐒 𝐌 𝐒 −1

similarity transformdiagonal matrix w/ scale
factors 𝑠𝑖−• Different S’s induce different lower bounds 𝜆min 𝐁

Theorem 1: Let M be a generalized graph Laplacian matrix corresponding to an irreducible,
positive graph G. Denote by v the first eigenvector of M corresponding to the smallest
1eigenvalue 𝜆min . Then by computing scalars 𝑠𝑖 = , ∀𝑖, all Gershgorin disc left-ends of𝑣𝑖

𝐁 = 𝐒 𝐌 𝐒−1 , 𝐒 = 𝑑𝑖𝑎𝑔 𝑠1 , … , 𝑠𝑁 , are aligned at 𝜆min .
[1] C. Yang, G. Cheung, W. Hu, "Signed Graph Metric Learning via Gershgorin Disc Alignment," submitted to IEEE Transactions on
Pattern Analysis and Machine Intelligence, June 2020.
24

Gene Cheung (genec@yorku.ca)

Metric Optimization via GDPA
• Original diagonal opt w/ PSD cone constraint:
original metric optimization• Revised diagonal opt w/ linear constraints:[1] C. Yang, G. Cheung, W. Hu, "Signed Graph Metric Learning via Gershgorin Disc Alignment," submitted to IEEE Transactions on
Pattern Analysis and Machine Intelligence, June 2020.
25

Gene Cheung (genec@yorku.ca)

Metric Optimization via GDPA
• Original diagonal opt w/ PSD cone constraint:
original metric optimization• Revised diagonal opt w/ linear constraints:

scalars si computed from 1st e-vectorof last sol’n M[1] C. Yang, G. Cheung, W. Hu, "Signed Graph Metric Learning via Gershgorin Disc Alignment," submitted to IEEE Transactions on
Pattern Analysis and Machine Intelligence, June 2020.
25

Gene Cheung (genec@yorku.ca)

Metric Learning Results (speed)• Running time comparison against PD-cone and HBNB1, for different metrics,
using Madelon dataset.

[1] W. Hu, X. Gao, G. Cheung, and Z. Guo, “Feature graph learning for 3D point cloud denoising," IEEE TSP, vol. 68, pp. 2841-2856, 2020.
26

Gene Cheung (genec@yorku.ca)

Metric Learning Results (accuracy)• Using a GLR objective, SGML achieved the best classification results in 7 outof 14 datasets and remained competitive for 12 out of 14 datasets.

[1] C. Yang, G. Cheung, W. Hu, "Signed Graph Metric Learning via Gershgorin Disc Alignment," submitted to IEEE Transactions on
Pattern Analysis and Machine Intelligence, June 2020.
27

Gene Cheung (genec@yorku.ca)

Outline➢ What is Graph Signal Processing?
➢ Graph spectrum➢ Graph Fourier transform (GFT), graph Laplacian regularizer (GLR)➢ Graph Learning➢ Precision / Graph Laplacian Matrix Estimation (w/ eigen-structure constraint)➢ Feature Graph Learning: Gershgorin Disc Perfect Alignment (GDPA)➢ Application: Semi-supervised classifier learning➢ Graph Sampling➢ Gershgorin Disc Alignment Sampling (GDAS)➢ Application: Sampling for matrix completion, 3D point cloud sub-sampling➢ Graph Filtering➢ Signal-dependent GLR, GTV➢ Application: Image denoising
28

Gene Cheung (genec@yorku.ca)

Graph Sampling (with and without noise)
Q: How to choose best samples for graph-based reconstruction?
• Existing graph sampling strategies extend Nyquist samplingto graph data kernels:• Assume bandlimited signal.
• Greedily select most “informative” samples by computing
extreme eigenvectors of sub-matrix.
• Computation-expensive.
[1] A. Anis, A. Gadde, and A. Ortega, “Efficient sampling set selection for bandlimited graph signals using graph spectral proxies,” IEEE Transactions on Signal Processing,vol. 64, no. 14, pp. 3775–3789, 2016.
[2] Y. Tanaka, Y. C. Eldar, A. Ortega, G. Cheung, "Sampling on Graphs: From Theory to Applications," IEEE Signal Processing Magazine, vol. 37, no.6, pp.14-30,
November 2020.
29

Gene Cheung (genec@yorku.ca)

Signal Reconstruction using GLR▪ Signal Model:

observation

sampling matrix
desired signal

𝐲 = 𝐇𝐱 + 𝐯

noise

𝑇

𝑖,𝑗

𝑘

signal smooth w.r.t. graph

0 1 0
0 0 0
1

1

▪ Signal prior is graph Laplacian regularizer (GLR):
1
2𝐱 Lx = ෍ 𝑤𝑖,𝑗 𝑥𝑖 − 𝑥𝑗 = ෍ 𝜆𝑘 𝑥෤𝑘2
2

𝐇=

1

2

0
1

1
3

4

Sample set {2, 4}
signal contains
mostly low graph freq.

▪ MAP Formulation:
signal priorfidelity term

min 𝐲 − 𝐇𝐱 22 + 𝜇 𝐱 𝑇 𝐋𝐱
𝐱

𝐇𝑇𝐇 + 𝜇𝐋 𝐱 ∗ = 𝐲linear system of eqn’s solved using conjugate gradient[1] J. Pang, G. Cheung, "Graph Laplacian Regularization for Image Denoising: Analysis in the Continuous Domain," IEEE TIP, vol. 26, no.4, pp.1770-1785, April 2017.
30

Gene Cheung (genec@yorku.ca)

Stability of Linear System• Examine solution’s linear system:

1

𝐇𝑇𝐇 + 𝜇𝐋 𝐱 ∗ = 𝐲coefficient matrix B• Stability depends on condition number (λmax/ λmin) of B.
• λmax is upper-bounded by 1+μ2*dmax.
Goal: select H to maximize 𝜆min 𝐁 (w/o computing eigen-pairs)!
Also minimizes worst-case MSE:𝐱ො − 𝐱 2 ≤ 𝜇31

1𝜆𝑚𝑖𝑛 𝐁

1

1

෥
𝐋 𝐱+𝐧

2+

2

Gene Cheung (genec@yorku.ca)

෥ 2
𝐧

2

1
4

3

1 −1 00−1 2 −1 0𝐋=
0 −1 2 −1
0
0 −1 1
0
0
𝐇T𝐇 =
0
0

0
1
0
0

0
0
0
0

0
0
0
1

Sample set {2, 4}

Gershgorin Circle Theorem
11Gershgorin Circle Theorem:▪ Row i of L maps to a Gershgorin disc w/ centre Lii
and radius Ri

𝑅𝑖 = ෍ 𝐿𝑖𝑗

2

1

1
4

3

1 −1 00−1 2 −1 0𝐋=
0 −1 2 −1
0
0 −1 1𝑗≠𝑖▪ λmin is lower-bounded by smallest left-ends ofGershgorin discs:

min 𝐿𝑖,𝑖 − 𝑅𝑖 ≤ 𝜆min𝑖

Graph Laplacian L has all Gershgorin disc left-ends at 0
→ L is PSD.
32

Gene Cheung (genec@yorku.ca)

1

2

3

4

Gershgorin Disc Alignment Sampling (GDAS)

𝐁 = 𝐇𝑇 𝐇 + 𝜇 𝐋

coeff. matrix• Sample node → shift disc.

1

1

Main Idea: Select samples to max smallest disc left-end ofcoefficient matrix B:

1

2

1

3

4

1 −1 00−1 2 −1 0𝐁=0 −1 2 −1
0
0 −1 1• Consider similarity transform of B (same eigenvalues!):

𝐂 = 𝐒 𝐁 𝐒 −1

similarity transform1

diagonal matrix w/ scale
factors• Scale row → expand disc radius.
→ shrink neighbors’ disc radius.

Gene Cheung (genec@yorku.ca)

34

Sample set { }Scale factor {1,1,1,1}[1] Y. Bai, F. Wang, G. Cheung, Y. Nakatsukasa, W. Gao, "Fast Graph Sampling Set Selection Using Gershgorin Disc Alignment," vol.
68, pp. 2419-2434, IEEE Transactions on Signal Processing, March 2020.
33

2

Gershgorin Disc Alignment Sampling (GDAS)

𝐁 = 𝐇𝑇 𝐇 + 𝜇 𝐋

coeff. matrix• Sample node → shift disc.

1

1

Main Idea: Select samples to max smallest disc left-end ofcoefficient matrix B:

1

2

1

1 −1
−1 3
𝐁=
0 −1
0
0

4

3

0
0
−1 0
2 −1
−1 1

• Consider similarity transform of B (same eigenvalues!):

𝐂 = 𝐒 𝐁 𝐒 −1

similarity transform1

diagonal matrix w/ scale
factors• Scale row → expand disc radius.
→ shrink neighbors’ disc radius.

Gene Cheung (genec@yorku.ca)

3

4

5

Sample set {2}Scale factor {1,1,1,1}[1] Y. Bai, F. Wang, G. Cheung, Y. Nakatsukasa, W. Gao, "Fast Graph Sampling Set Selection Using Gershgorin Disc Alignment," vol.
68, pp. 2419-2434, IEEE Transactions on Signal Processing, March 2020.
33

2

Gershgorin Disc Alignment Sampling (GDAS)

𝐁 = 𝐇𝑇 𝐇 + 𝜇 𝐋

coeff. matrix• Sample node → shift disc.

1

1

Main Idea: Select samples to max smallest disc left-end ofcoefficient matrix B:

1

2

1

4

3

1 −1 00−1 3 −1 0𝐁=0 −1 2 −1
0
0 −1 1• Consider similarity transform of B (same eigenvalues!):

𝐂 = 𝐒 𝐁 𝐒 −1

similarity transform1

diagonal matrix w/ scale
factors• Scale row → expand disc radius.
→ shrink neighbors’ disc radius.

Gene Cheung (genec@yorku.ca)

3

4

5

Sample set {2}Scale factor {1,s2,1,1}[1] Y. Bai, F. Wang, G. Cheung, Y. Nakatsukasa, W. Gao, "Fast Graph Sampling Set Selection Using Gershgorin Disc Alignment," vol.
68, pp. 2419-2434, IEEE Transactions on Signal Processing, March 2020.
33

2

Gershgorin Disc Alignment Sampling (GDAS)

𝐁 = 𝐇𝑇 𝐇 + 𝜇 𝐋

coeff. matrix• Sample node → shift disc.

1

1

Main Idea: Select samples to max smallest disc left-end ofcoefficient matrix B:

1

2

1

1
−1
𝐁=
0
0

4

3

−1 0
0
3 −1 0−1 2 −1
0 −1 1• Consider similarity transform of B (same eigenvalues!):

𝐂 = 𝐒 𝐁 𝐒 −1

similarity transform1

diagonal matrix w/ scale
factors• Scale row → expand disc radius.
→ shrink neighbors’ disc radius.

Gene Cheung (genec@yorku.ca)

3

4

5

Sample set {2}Scale factor {1,s2,1,1}[1] Y. Bai, F. Wang, G. Cheung, Y. Nakatsukasa, W. Gao, "Fast Graph Sampling Set Selection Using Gershgorin Disc Alignment," vol.
68, pp. 2419-2434, IEEE Transactions on Signal Processing, March 2020.
33

2

Solving Dual Sampling Problem: align discs @ T
Breadth First Iterative Sampling (BFIS):▪ Given initial node set, threshold T.
1. Sample chosen node i (shift disc)2. Scale row i (expand disc radius i to T )3. If disc left-end of connected node j >T,
Scale row j (expand disc radius j to T )Else,
Add node j to node set.
4. Goto step 1 if node set not empty.
5. Output sample set and count K.

[1] Y. Bai, G. Cheung, F. Wang, X. Liu, W. Gao, "Reconstruction-Cognizant Graph Sampling Using Gershgorin Disc Alignment," IEEE
International Conference on Acoustics, Speech and Signal Processing, Brighton, UK, May 2019.
34

Gene Cheung (genec@yorku.ca)

Disc-based Sampling (Intuition)
Analogy: throw pebbles into a pond.
Disc Shifting: throw pebble at sample node i.
Disc Scaling: ripple to neighbors of node i.

Goal: Select min # of samples so ripple at each node is at least T.

35

Gene Cheung (genec@yorku.ca)

Disc-based Sampling (Intuition)
Analogy: throw pebbles into a pond.
Disc Shifting: throw pebble at sample node i.
Disc Scaling: ripple to neighbors of node i.

Goal: Select min # of samples so ripple at each node is at least T.

Takeaway Message: roughly linear time graphsampling algorithm minimizing a global error obj.

35

Gene Cheung (genec@yorku.ca)

Graph Sampling Results: speed• Running time comparisons on two different graphs.
(a) Random sensor raph. (b) Community graph.

36

Gene Cheung (genec@yorku.ca)

Graph Sampling Results: community graph• Visualization of selected nodes on the community graph (N = 500,K = 11). Black circles denotesampled nodes. (a) Original graph. (b) Random [28].(c) E-optimal [25]. (d) SP [16]. (e) MFN [23].
(f) MIA [20]. (g) Ed-free [9]. (h) The proposed BS-GDA.

37

Gene Cheung (genec@yorku.ca)

Graph Sampling Results: matrix completion• Pre-select a subset of matrix entries for sampling to maximize matrix completion fidelity.
• Challenge: select sampling set Ω to maximize λmin of• RMSE of different sampling methods for MC on Synthetic Netflix. The matrix was completed using
the double graph smoothness based method.

[1] F. Wang, Y. Wang, G. Cheung, C. Yang, "Graph Sampling for Matrix Completion Using Recurrent Gershgorin Disc Shift," vol. 68, pp.
1814-2829, IEEE Transactions on Signal Processing, April 2020.
38

Gene Cheung (genec@yorku.ca)

Graph Sampling Results: matrix completion• Pre-select a subset of matrix entries for sampling to maximize matrix completion fidelity.
• Challenge: select sampling set Ω to maximize λmin ofgraph Laplacians for row / column graphs• RMSE of different sampling methods for MC on Synthetic Netflix. The matrix was completed using
the double graph smoothness based method.

[1] F. Wang, Y. Wang, G. Cheung, C. Yang, "Graph Sampling for Matrix Completion Using Recurrent Gershgorin Disc Shift," vol. 68, pp.
1814-2829, IEEE Transactions on Signal Processing, April 2020.
38

Gene Cheung (genec@yorku.ca)

Graph Sampling Results: 3D point cloud sub-sampling• Reduce 3D point cloud size by sub-sampling while preserving the overall object shape.
• Challenge: select sampling matrix H to maximize λmin of• SR reconstruction results from diff. methods of sub-sampled Bunny under 0.2 sub-sampling ratio.

[1] C. Dinesh, G. Cheung, I. V. Bajic, "Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment," submitted to IEEE
Transactions on Pattern Analysis and Machine Intelligence, January 2021.
39

Gene Cheung (genec@yorku.ca)

Graph Sampling Results: 3D point cloud sub-sampling• Reduce 3D point cloud size by sub-sampling while preserving the overall object shape.
• Challenge: select sampling matrix H to maximize λmin ofgeneralized graph Laplacian• SR reconstruction results from diff. methods of sub-sampled Bunny under 0.2 sub-sampling ratio.

[1] C. Dinesh, G. Cheung, I. V. Bajic, "Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment," submitted to IEEE
Transactions on Pattern Analysis and Machine Intelligence, January 2021.
39

Gene Cheung (genec@yorku.ca)

Outline➢ What is Graph Signal Processing?
➢ Graph spectrum➢ Graph Fourier transform (GFT), graph Laplacian regularizer (GLR)➢ Graph Learning➢ Precision / Graph Laplacian Matrix Estimation (w/ eigen-structure constraint)➢ Feature Graph Learning: Gershgorin Disc Perfect Alignment (GDPA)➢ Application: Semi-supervised classifier learning➢ Graph Sampling➢ Gershgorin Disc Alignment Sampling (GDAS)➢ Application: Sampling for matrix completion, 3D point cloud sub-sampling➢ Graph Filtering➢ Signal-dependent GLR, GTV➢ Application: Image denoising
40

Gene Cheung (genec@yorku.ca)

GLR for Image Denoising: motivation• Graph Laplacian Regularizer (GLR) 𝐱 𝑇 𝐋𝐱 is a smoothness measure.
• Denoising has simplest formation model 𝐲 = 𝐱 + 𝐳, thus formulation

min 𝐲 − 𝐱 22 + 𝜇 𝐱 𝑇 𝐋𝐱𝐱

𝐈 + 𝜇𝐋 𝐱 ∗ = 𝐲• To promote Piecewise Smoothness (PWS), 𝐋 𝐱 is signal-dependent:• Fix L and solve unconstrained QP each iteration.

min 𝐲 − 𝐱 22 + 𝜇 𝐱 𝑇 𝐋 𝐱 𝐱
𝐱[1] J. Pang, G. Cheung, "Graph Laplacian Regularization for Image Denoising: Analysis in the Continuous Domain," IEEE TIP, vol. 26, no.4, pp.1770-1785, April 2017.
[2] C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” IEEE ICCV, 1998.
41

Gene Cheung (genec@yorku.ca)

GLR for Image Denoising: motivation• Graph Laplacian Regularizer (GLR) 𝐱 𝑇 𝐋𝐱 is a smoothness measure.
• Denoising has simplest formation model 𝐲 = 𝐱 + 𝐳, thus formulation

min 𝐲 − 𝐱 22 + 𝜇 𝐱 𝑇 𝐋𝐱𝐱

𝐈 + 𝜇𝐋 𝐱 ∗ = 𝐲pixel intensity diff.

pixel location diff.

 − x − x 2   − l −l 2 
 i j 2  i j 2wi , j = exp
exp
2
2


 1
  2   filter weights
Bilateral• To promote Piecewise Smoothness (PWS), 𝐋 𝐱 is signal-dependent:• Fix L and solve unconstrained QP each iteration.

min 𝐲 − 𝐱 22 + 𝜇 𝐱 𝑇 𝐋 𝐱 𝐱
𝐱[1] J. Pang, G. Cheung, "Graph Laplacian Regularization for Image Denoising: Analysis in the Continuous Domain," IEEE TIP, vol. 26, no.4, pp.1770-1785, April 2017.
[2] C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” IEEE ICCV, 1998.
41

Gene Cheung (genec@yorku.ca)

GLR for Image Denoising: motivation• Graph Laplacian Regularizer (GLR) 𝐱 𝑇 𝐋𝐱 is a smoothness measure.
• Denoising has simplest formation model 𝐲 = 𝐱 + 𝐳, thus formulation

min 𝐲 − 𝐱 22 + 𝜇 𝐱 𝑇 𝐋𝐱𝐱

𝐈 + 𝜇𝐋 𝐱 ∗ = 𝐲pixel intensity diff.

pixel location diff.

 − x − x 2   − l −l 2 
 i j 2  i j 2wi , j = exp
exp
2
2


 1
  2   filter weights
Bilateral• To promote Piecewise Smoothness (PWS), 𝐋 𝐱 is signal-dependent:• Fix L and solve unconstrained QP each iteration.

min 𝐲 − 𝐱 22 + 𝜇 𝐱 𝑇 𝐋 𝐱 𝐱𝐱

Signal-dependent GLR[1] J. Pang, G. Cheung, "Graph Laplacian Regularization for Image Denoising: Analysis in the Continuous Domain," IEEE TIP, vol. 26, no.4, pp.1770-1785, April 2017.
[2] C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” IEEE ICCV, 1998.
41

Gene Cheung (genec@yorku.ca)

OGLR Denoising Results: visual comparison• Subjective comparisons (  I = 40 )Original

Noisy, 16.48 dB

K-SVD, 26.84 dB

BM3D, 27.99 dB

PLOW, 28.11 dB

OGLR, 28.35 dB[1] J. Pang, G. Cheung, "Graph Laplacian Regularization for Image Denoising: Analysis in the Continuous Domain," IEEE TIP, vol. 26, no.4, pp.1770-1785, April 2017.
42

Gene Cheung (genec@yorku.ca)

OGLR Denoising Results: visual comparison• Subjective comparisons (  I = 30 )Original

Noisy, 18.66 dB BM3D, 33.26 dB NLGBT, 33.41dB OGLR, 34.32 dB[1] J. Pang, G. Cheung, "Graph Laplacian Regularization for Image Denoising: Analysis in the Continuous Domain," IEEE TIP, vol. 26, no.4, pp.1770-1785, April 2017.
43

Gene Cheung (genec@yorku.ca)

Deep GLR: motivation• Recall MAP formulation of denoising w/ GLR:

min y − x 2 +  x T L x
2x
fidelity termsmoothness prior• Solution is system of linear equations:
Sparse PD

𝐈 + μ𝐋 𝐱 ∗ = 𝐲𝐱∗ =• Interpretable filter.

[1] C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” IEEE ICCV, 1998.
44

Gene Cheung (genec@yorku.ca)

LP graph filter
𝐈 + μ𝐋 −𝟏 𝐲

Deep GLR: motivation• Recall MAP formulation of denoising w/ GLR:

min y − x 2 +  x T L x
2x
fidelity termsmoothness prior• Solution is system of linear equations:
Sparse PD

𝐈 + μ𝐋 𝐱 ∗ = 𝐲𝐱∗ =• Interpretable filter.
Q: what is the “most appropriate” graph?
[1] C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” IEEE ICCV, 1998.
44

Gene Cheung (genec@yorku.ca)

LP graph filter
𝐈 + μ𝐋 −𝟏 𝐲

Deep GLR: motivation• Recall MAP formulation of denoising w/ GLR:

min y − x 2 +  x T L x
2x
fidelity termsmoothness prior• Solution is system of linear equations:
Sparse PD

𝐈 + μ𝐋 𝐱 ∗ = 𝐲𝐱∗ =LP graph filter
𝐈 + μ𝐋 −𝟏 𝐲

• Interpretable filter.
Bilateral weights:

Q: what is the “most appropriate” graph?

 − x − x 2   − l −l 2 
i
j 2 

 i j 2wi , j = expexp
  2

 12

 

2

 


[1] C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” IEEE ICCV, 1998.
44

Gene Cheung (genec@yorku.ca)

Deep GLR: unrolling• Deep GLR:
1.
2.
3.
4.

Learn features f’s using CNN.
Compute distance from features.
Compute edge weights using Gaussian kernel.
Construct graph, solve QP.
featurespre-filter

weight para

[1] K. Gregor and Y. LeCun, “Learning fast approximations of sparse coding,” in Proc. 27th Int. Conf. Machine Learning, 2010..
45

Gene Cheung (genec@yorku.ca)

Deep GLR: CNN implementation[1] J. Zeng et al., “Deep Graph Laplacian Regularization for Robust Denoising of Images,” NTIRE Workshop, CVPR 2019.
46

Gene Cheung (genec@yorku.ca)

Deep GLR: unrolling• Model guarantees numerical stability of solution:(I+ L ) x * = y• Thm 1: condition number κ of matrix satisfies [1]:maximum node degree• Observation: Restricting CNN search space → achieve robust learning.
[1] J. Zeng et al., “Deep Graph Laplacian Regularization for Robust Denoising of Images,” NTIRE Workshop, CVPR 2019.
47

Gene Cheung (genec@yorku.ca)

Deep GLR: numerical comparison• Trained on AWGN on 5 images, patches of size 26-by-26.
• Batch size is 4, model is trained for 200 epochs.
• Trained for both known and blind noise variance.

[1] Kai Zhang et al, “Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising,” TIP 2017.
[2] Marc Lebrun et al, “The noise clinic: a blind image denoising algorithm,” IPOL 2015.
48

Gene Cheung (genec@yorku.ca)

Deep GLR: numerical comparison••••

Cross-domain generalization.
Trained on Gaussian noise, tested on low-light images in (RENOIR).
Competing methods: DnCNN [1], noise clinic [2].
Outperformed DnCNN by 5.74 dB, and noise clinic by 1.87 dB.

[1] Kai Zhang et al, “Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising,” TIP 2017.
[2] Marc Lebrun et al, “The noise clinic: a blind image denoising algorithm,” IPOL 2015.
49

Gene Cheung (genec@yorku.ca)

Deep GLR: visual comparison• Trained on Gaussian noise, tested on low-light images in (RENOIR).
• Competing methods: DnCNN [1], noise clinic [2].
• Outperformed DnCNN by 5.74 dB, and noise clinic by 1.87 dB.

Noise Clinic

CDnCNN[1] Kai Zhang et al, “Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising,” TIP 2017.
[2] Marc Lebrun et al, “The noise clinic: a blind image denoising algorithm,” IPOL 2015.
50

Gene Cheung (genec@yorku.ca)

DeepGLRDeep GTV: motivation• GTV promotes PWS faster than GLR.
min 𝐲 − 𝐱 22 + 𝜇 𝐱 𝐺𝑇𝑉𝑥

𝜇 𝐱 𝐺𝑇𝑉 = ෍ 𝑤𝑖,𝑗 𝑥𝑖 − 𝑥𝑗𝑖,𝑗• Solve as QP via L1-Laplacian:min 𝐲 − 𝐱 22 + 𝜇 𝐱 𝑇 𝐋Г 𝐱
𝑥

• Still interpretable LP graph filter.
[1] Y. Bai, G. Cheung, X. Liu, W. Gao, "Graph-Based Blind Image Deblurring from a Single Photograph," IEEE TIP, vol. 28, no.3, pp.1404-1418, March 2019.
[2] H. Vu, G. Cheung, Y. C. Eldar, "Unrolling of Deep Graph Total Variation for Image Denoising," accepted to IEEE ICASSP, Toronto, Canada, June 2021.
51

Gene Cheung (genec@yorku.ca)

Deep GTV: algorithm• Learn feature via CNN for graph construction.
• Obtain graph filter response:• Fast filter implementation via Lanczos approx.:
1. Compute tri-diagonal matrix
2. Compute approx. filter:𝑔 𝐋 𝐲 ≈ 𝐲 2 𝐕𝑀 𝑔 𝐇𝑀 𝐞1
where• Interpretable graph filter → fast implementation.
[1] J. Zeng et al., “Deep Graph Laplacian Regularization for Robust Denoising of Images,” NTIRE Workshop, CVPR 2019.
[2] A. Susnjara, N. Perraudin, D. Kressner1, and P. Vandergheynst, “Accelerated filtering on graphs using Lanczos method,” in unpublished, arXiv:1509.04537, 2015.
52

Gene Cheung (genec@yorku.ca)

Deep GTV: experimental comparison• Train on Gaussian (𝜎=50) and test on captured noise

save ≥ 80% parameters!

53

Gene Cheung (genec@yorku.ca)

Conclusion• Graph is flexible abstraction to convey pairwise similarities.
• Similarity defined as correlation or feature distance.
• Graph frequencies contains global notions.
• Graph is an expression of domain knowledge.

• GSP leverages on mature understanding in SP and linear
algebra.
• GSP tools are excellent for building hybrid model-based /data-driven systems.

Applications:
Image coding,denoising, deblurring,interpolation, contrast
enhancement, lightfield image coding, 3D
point cloud denoising,enhancement, subsampling, superresolution, inpainting,matrix completion,
semi-supervisedclassifier learning,
video summarization[1] X. Dong*, D. Thanou*, L. Toni, M. Bronstein, P. Frossard, “Graph signal processing for machine learning: A review and new perspectives,” IEEE Signal Processing
Magazine, vol.37, no.6, pp.117-127, Nov., 2020.
54

Gene Cheung (genec@yorku.ca)

Contact Info
• Homepage:https://www.eecs.yorku.ca/~genec/index.html• E-mail:genec@yorku.ca• Forthcoming book:
G. Cheung, E. Magli, (edited) Graph Spectral
Image Processing, ISTE/Wiley, June 2021.
55

Gene Cheung (genec@yorku.ca)