![Page 1: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/1.jpg)
Graph Fourier Transform
Prof. Luis Gustavo Nonato
SME/ICMC - USP
August 29, 2017
![Page 2: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/2.jpg)
Graph Fourier Transform
From previous classes we learned that the eigenvectors of agraph Laplacian behave similarly to a Fourier basis, motivatingthe development of graph-based Fourier analysis theory.
A major obstacle to the development of a graph signalprocessing theory is the irregular and coordinate-free nature ofa graph domain.
For instance, signal translation is basic operation for signalprocessing. However, that operation is not naturallyimplemented in graph domains.
![Page 3: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/3.jpg)
Graph Fourier Transform
From previous classes we learned that the eigenvectors of agraph Laplacian behave similarly to a Fourier basis, motivatingthe development of graph-based Fourier analysis theory.
A major obstacle to the development of a graph signalprocessing theory is the irregular and coordinate-free nature ofa graph domain.
For instance, signal translation is basic operation for signalprocessing. However, that operation is not naturallyimplemented in graph domains.
![Page 4: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/4.jpg)
Graph Fourier Transform
From previous classes we learned that the eigenvectors of agraph Laplacian behave similarly to a Fourier basis, motivatingthe development of graph-based Fourier analysis theory.
A major obstacle to the development of a graph signalprocessing theory is the irregular and coordinate-free nature ofa graph domain.
For instance, signal translation is basic operation for signalprocessing. However, that operation is not naturallyimplemented in graph domains.
![Page 5: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/5.jpg)
Graph Fourier Transform
Let G = (V, E) be a weighted graph, L be its correspondinggraph Laplacian, and f : V → R a function defined on thevertices of G.
Graph Fourier TransformThe Graph Fourier Transform of f is defined as
GF [f ](λl) = f (λl) =< f , ul >=n
∑i=1
f (i)ul(i)
Inverse Graph Fourier TransformThe Inverse Graph Fourier Transform is given by
IGF [f ](i) = f (i) =n−1
∑l=0
f (λl)ul(i)
![Page 6: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/6.jpg)
Graph Fourier Transform
Let G = (V, E) be a weighted graph, L be its correspondinggraph Laplacian, and f : V → R a function defined on thevertices of G.
Graph Fourier TransformThe Graph Fourier Transform of f is defined as
GF [f ](λl) = f (λl) =< f , ul >=n
∑i=1
f (i)ul(i)
Inverse Graph Fourier TransformThe Inverse Graph Fourier Transform is given by
IGF [f ](i) = f (i) =n−1
∑l=0
f (λl)ul(i)
![Page 7: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/7.jpg)
Graph Fourier Transform
Let G = (V, E) be a weighted graph, L be its correspondinggraph Laplacian, and f : V → R a function defined on thevertices of G.
Graph Fourier TransformThe Graph Fourier Transform of f is defined as
GF [f ](λl) = f (λl) =< f , ul >=n
∑i=1
f (i)ul(i)
Inverse Graph Fourier TransformThe Inverse Graph Fourier Transform is given by
IGF [f ](i) = f (i) =n−1
∑l=0
f (λl)ul(i)
![Page 8: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/8.jpg)
Graph Fourier Transform
f (λl) =n
∑i=1
f (i)ul(i)
The eigenvalues of L play the role of frequencies and theeigenvectors the Fourier basis.
The graph Fourier transform (and its inverse) gives a way torepresent a signal in two different domains: the vertex domainand the graph spectral domain.
![Page 9: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/9.jpg)
Graph Fourier Transform
f (λl) =n
∑i=1
f (i)ul(i)
The eigenvalues of L play the role of frequencies and theeigenvectors the Fourier basis.
The graph Fourier transform (and its inverse) gives a way torepresent a signal in two different domains: the vertex domainand the graph spectral domain.
![Page 10: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/10.jpg)
Graph Fourier Transform: Synthesizing Signals
f (i) =n−1
∑l=0
f (λl)ul(i)
![Page 11: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/11.jpg)
Graph Fourier Transform: Synthesizing Signals
f (i) =n−1
∑l=0
f (λl) ul(i)⇒ f (i) =n−1
∑l=0
g(λl)ul(i)
We can generate signals in the graph domain by using a kernelin the spectral domain.
![Page 12: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/12.jpg)
Graph Fourier Transform: Synthesizing Signals
f (i) =n−1
∑l=0
f (λl) ul(i)⇒ f (i) =n−1
∑l=0
g(λl)ul(i)
We can generate signals in the graph domain by using a kernelin the spectral domain.
![Page 13: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/13.jpg)
Graph Fourier Transform: Filtering
In classical signal processing, frequency filtering is given byamplifying or attenuating the contributions of some FourierBasis.
L[f ] = f ∗ h→ F [f ∗ h] = f h
Taking the IGFT
L[f ](i) =n−1
∑l=0
f (λl)h(λl)ul(i) (1)
Suppose h(λl) = ∑Kk=0 akλk
l (polynomial on the spectral domain)
Replacing h(λl) and the definition of f (λl) in Eq. (1) we get
L[f ](i) =n
∑j=1
f (j)K
∑k=0
ak
n
∑l=1
λkl ul(j)ul(i)
![Page 14: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/14.jpg)
Graph Fourier Transform: Filtering
In classical signal processing, frequency filtering is given byamplifying or attenuating the contributions of some FourierBasis.
L[f ] = f ∗ h→ F [f ∗ h] = f h
Taking the IGFT
L[f ](i) =n−1
∑l=0
f (λl)h(λl)ul(i) (1)
Suppose h(λl) = ∑Kk=0 akλk
l (polynomial on the spectral domain)
Replacing h(λl) and the definition of f (λl) in Eq. (1) we get
L[f ](i) =n
∑j=1
f (j)K
∑k=0
ak
n
∑l=1
λkl ul(j)ul(i)
![Page 15: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/15.jpg)
Graph Fourier Transform: Filtering
In classical signal processing, frequency filtering is given byamplifying or attenuating the contributions of some FourierBasis.
L[f ] = f ∗ h→ F [f ∗ h] = f h
Taking the IGFT
L[f ](i) =n−1
∑l=0
f (λl)h(λl)ul(i) (1)
Suppose h(λl) = ∑Kk=0 akλk
l (polynomial on the spectral domain)
Replacing h(λl) and the definition of f (λl) in Eq. (1) we get
L[f ](i) =n
∑j=1
f (j)K
∑k=0
ak
n
∑l=1
λkl ul(j)ul(i)
![Page 16: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/16.jpg)
Graph Fourier Transform: Filtering
L[f ](i) =n
∑j=1
f (j)K
∑k=0
ak
n
∑l=1
λkl ul(j)ul(i)
(Lk)ij =n
∑l=1
λkl ul(j)ul(i)
It can be shown that (Lk)ij = 0 when the shortest distancebetween nodes i and j is greater than k.Defining
bij =K
∑k=0
ak Lkij
L[f ](i) =n
∑j∈Nk(i)
bijf (j)
Therefore, when the filter is a polynomial in the spectral domain, the filteredsignal in each node i is a linear combination of the original signal in theneighborhood of i.
![Page 17: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/17.jpg)
Graph Fourier Transform: Filtering
L[f ](i) =n
∑j=1
f (j)K
∑k=0
ak
n
∑l=1
λkl ul(j)ul(i)
(Lk)ij =n
∑l=1
λkl ul(j)ul(i)
It can be shown that (Lk)ij = 0 when the shortest distancebetween nodes i and j is greater than k.Defining
bij =K
∑k=0
ak Lkij
L[f ](i) =n
∑j∈Nk(i)
bijf (j)
Therefore, when the filter is a polynomial in the spectral domain, the filteredsignal in each node i is a linear combination of the original signal in theneighborhood of i.
![Page 18: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/18.jpg)
Graph Fourier Transform: Filtering
L[f ](i) =n
∑j=1
f (j)K
∑k=0
ak
n
∑l=1
λkl ul(j)ul(i)
(Lk)ij =n
∑l=1
λkl ul(j)ul(i)
It can be shown that (Lk)ij = 0 when the shortest distancebetween nodes i and j is greater than k.
Defining
bij =K
∑k=0
ak Lkij
L[f ](i) =n
∑j∈Nk(i)
bijf (j)
Therefore, when the filter is a polynomial in the spectral domain, the filteredsignal in each node i is a linear combination of the original signal in theneighborhood of i.
![Page 19: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/19.jpg)
Graph Fourier Transform: Filtering
L[f ](i) =n
∑j=1
f (j)K
∑k=0
ak
n
∑l=1
λkl ul(j)ul(i)
(Lk)ij =n
∑l=1
λkl ul(j)ul(i)
It can be shown that (Lk)ij = 0 when the shortest distancebetween nodes i and j is greater than k.Defining
bij =K
∑k=0
ak Lkij
L[f ](i) =n
∑j∈Nk(i)
bijf (j)
Therefore, when the filter is a polynomial in the spectral domain, the filteredsignal in each node i is a linear combination of the original signal in theneighborhood of i.
![Page 20: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/20.jpg)
Graph Fourier Transform: Filtering
L[f ](i) =n
∑j=1
f (j)K
∑k=0
ak
n
∑l=1
λkl ul(j)ul(i)
(Lk)ij =n
∑l=1
λkl ul(j)ul(i)
It can be shown that (Lk)ij = 0 when the shortest distancebetween nodes i and j is greater than k.Defining
bij =K
∑k=0
ak Lkij
L[f ](i) =n
∑j∈Nk(i)
bijf (j)
Therefore, when the filter is a polynomial in the spectral domain, the filteredsignal in each node i is a linear combination of the original signal in theneighborhood of i.
![Page 21: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/21.jpg)
Graph Fourier Transform: Filtering
L[f ](i) =n
∑j=1
f (j)K
∑k=0
ak
n
∑l=1
λkl ul(j)ul(i)
(Lk)ij =n
∑l=1
λkl ul(j)ul(i)
It can be shown that (Lk)ij = 0 when the shortest distancebetween nodes i and j is greater than k.Defining
bij =K
∑k=0
ak Lkij
L[f ](i) =n
∑j∈Nk(i)
bijf (j)
Therefore, when the filter is a polynomial in the spectral domain, the filteredsignal in each node i is a linear combination of the original signal in theneighborhood of i.
![Page 22: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/22.jpg)
Graph Fourier Transform: Convolution
The definition of convolution between two functions f and g is
(f ∗ g)(x) =∫ ∞
−∞f (x− t)g(t)dt
Such definition can not be directly applied because the shiftf (x− t) (or singal translation) is not defined in the context ofgraphs.
However, convolution can be defined as:
(f ∗ g) = IGF [GF [(f ∗ g)]]
(f ∗ g)(i) =n
∑l=1
f (λl)g(λl)ul(i)
![Page 23: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/23.jpg)
Graph Fourier Transform: Convolution
The definition of convolution between two functions f and g is
(f ∗ g)(x) =∫ ∞
−∞f (x− t)g(t)dt
Such definition can not be directly applied because the shiftf (x− t) (or singal translation) is not defined in the context ofgraphs.
However, convolution can be defined as:
(f ∗ g) = IGF [GF [(f ∗ g)]]
(f ∗ g)(i) =n
∑l=1
f (λl)g(λl)ul(i)
![Page 24: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/24.jpg)
Graph Fourier Transform: Convolution
The definition of convolution between two functions f and g is
(f ∗ g)(x) =∫ ∞
−∞f (x− t)g(t)dt
Such definition can not be directly applied because the shiftf (x− t) (or singal translation) is not defined in the context ofgraphs.
However, convolution can be defined as:
(f ∗ g) = IGF [GF [(f ∗ g)]]
(f ∗ g)(i) =n
∑l=1
f (λl)g(λl)ul(i)
![Page 25: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/25.jpg)
Graph Fourier Transform: Convolution
The definition of convolution between two functions f and g is
(f ∗ g)(x) =∫ ∞
−∞f (x− t)g(t)dt
Such definition can not be directly applied because the shiftf (x− t) (or singal translation) is not defined in the context ofgraphs.
However, convolution can be defined as:
(f ∗ g) = IGF [GF [(f ∗ g)]]
(f ∗ g)(i) =n
∑l=1
f (λl)g(λl)ul(i)
![Page 26: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/26.jpg)
Graph Fourier Transform: Convolution
The definition of convolution between two functions f and g is
(f ∗ g)(x) =∫ ∞
−∞f (x− t)g(t)dt
Such definition can not be directly applied because the shiftf (x− t) (or singal translation) is not defined in the context ofgraphs.
However, convolution can be defined as:
(f ∗ g) = IGF [GF [(f ∗ g)]]
(f ∗ g)(i) =n
∑l=1
f (λl)g(λl)ul(i)
![Page 27: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/27.jpg)
Graph Fourier Transform: Convolution
Some properties of the convolution:
f ∗ g = f gf ∗ g = g ∗ ff ∗ (g + h) = f ∗ g + f ∗ h
∑ni=1(f ∗ g)(i) =
√n f (0)g(0)
![Page 28: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/28.jpg)
Graph Fourier Transform: Convolution
Some properties of the convolution:
f ∗ g = f g
f ∗ g = g ∗ ff ∗ (g + h) = f ∗ g + f ∗ h
∑ni=1(f ∗ g)(i) =
√n f (0)g(0)
![Page 29: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/29.jpg)
Graph Fourier Transform: Convolution
Some properties of the convolution:
f ∗ g = f gf ∗ g = g ∗ f
f ∗ (g + h) = f ∗ g + f ∗ h
∑ni=1(f ∗ g)(i) =
√n f (0)g(0)
![Page 30: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/30.jpg)
Graph Fourier Transform: Convolution
Some properties of the convolution:
f ∗ g = f gf ∗ g = g ∗ ff ∗ (g + h) = f ∗ g + f ∗ h
∑ni=1(f ∗ g)(i) =
√n f (0)g(0)
![Page 31: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/31.jpg)
Graph Fourier Transform: Convolution
Some properties of the convolution:
f ∗ g = f gf ∗ g = g ∗ ff ∗ (g + h) = f ∗ g + f ∗ h
∑ni=1(f ∗ g)(i) =
√n f (0)g(0)
![Page 32: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/32.jpg)
Graph Fourier Transform: Translation
The classical translation
(Tt f )(x) = f (x− t)
can not be directly generalized to the graph setting.
However, translation can be seen as a convolution with thedelta function
(Tt f )(x) = (f ∗ δt)(x)
Recalling that F [δ(t− a)](λ) = e−i2πλa, we define translation toa vertex j as
(Tj f )(i) =√
n(f ∗ δj)(i) =√
nn−1
∑l=0
f (λl)ul(j)ul(i)
The normalizing constant√
n ensures that the translationoperator preserves the mean of a signal, i.e.,∑n
i=1(Tj f )(i) = ∑ni=1 f (i)
![Page 33: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/33.jpg)
Graph Fourier Transform: TranslationThe classical translation
(Tt f )(x) = f (x− t)
can not be directly generalized to the graph setting.
However, translation can be seen as a convolution with thedelta function
(Tt f )(x) = (f ∗ δt)(x)
Recalling that F [δ(t− a)](λ) = e−i2πλa, we define translation toa vertex j as
(Tj f )(i) =√
n(f ∗ δj)(i) =√
nn−1
∑l=0
f (λl)ul(j)ul(i)
The normalizing constant√
n ensures that the translationoperator preserves the mean of a signal, i.e.,∑n
i=1(Tj f )(i) = ∑ni=1 f (i)
![Page 34: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/34.jpg)
Graph Fourier Transform: TranslationThe classical translation
(Tt f )(x) = f (x− t)
can not be directly generalized to the graph setting.
However, translation can be seen as a convolution with thedelta function
(Tt f )(x) = (f ∗ δt)(x)
Recalling that F [δ(t− a)](λ) = e−i2πλa, we define translation toa vertex j as
(Tj f )(i) =√
n(f ∗ δj)(i) =√
nn−1
∑l=0
f (λl)ul(j)ul(i)
The normalizing constant√
n ensures that the translationoperator preserves the mean of a signal, i.e.,∑n
i=1(Tj f )(i) = ∑ni=1 f (i)
![Page 35: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/35.jpg)
Graph Fourier Transform: TranslationThe classical translation
(Tt f )(x) = f (x− t)
can not be directly generalized to the graph setting.
However, translation can be seen as a convolution with thedelta function
(Tt f )(x) = (f ∗ δt)(x)
Recalling that F [δ(t− a)](λ) = e−i2πλa, we define translation toa vertex j as
(Tj f )(i) =√
n(f ∗ δj)(i) =√
nn−1
∑l=0
f (λl)ul(j)ul(i)
The normalizing constant√
n ensures that the translationoperator preserves the mean of a signal, i.e.,∑n
i=1(Tj f )(i) = ∑ni=1 f (i)
![Page 36: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/36.jpg)
Graph Fourier Transform: TranslationThe classical translation
(Tt f )(x) = f (x− t)
can not be directly generalized to the graph setting.
However, translation can be seen as a convolution with thedelta function
(Tt f )(x) = (f ∗ δt)(x)
Recalling that F [δ(t− a)](λ) = e−i2πλa, we define translation toa vertex j as
(Tj f )(i) =√
n(f ∗ δj)(i) =√
nn−1
∑l=0
f (λl)ul(j)ul(i)
The normalizing constant√
n ensures that the translationoperator preserves the mean of a signal, i.e.,∑n
i=1(Tj f )(i) = ∑ni=1 f (i)
![Page 37: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/37.jpg)
Graph Fourier Transform: Translation
Some properties of translationTj(f ∗ g) = (Tj f ) ∗ g = f ∗ (Tj g)
Tj Tk f = Tk Tj f
∑ni=1(Tj f )(i) =
√n f (0) = ∑n
i=1 f (i)‖Tj f‖ 6= ‖f‖ (the energy of the signal is not preserved)
![Page 38: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/38.jpg)
Graph Fourier Transform: Translation
Some properties of translationTj(f ∗ g) = (Tj f ) ∗ g = f ∗ (Tj g)Tj Tk f = Tk Tj f
∑ni=1(Tj f )(i) =
√n f (0) = ∑n
i=1 f (i)‖Tj f‖ 6= ‖f‖ (the energy of the signal is not preserved)
![Page 39: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/39.jpg)
Graph Fourier Transform: Translation
Some properties of translationTj(f ∗ g) = (Tj f ) ∗ g = f ∗ (Tj g)Tj Tk f = Tk Tj f
∑ni=1(Tj f )(i) =
√n f (0) = ∑n
i=1 f (i)
‖Tj f‖ 6= ‖f‖ (the energy of the signal is not preserved)
![Page 40: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/40.jpg)
Graph Fourier Transform: Translation
Some properties of translationTj(f ∗ g) = (Tj f ) ∗ g = f ∗ (Tj g)Tj Tk f = Tk Tj f
∑ni=1(Tj f )(i) =
√n f (0) = ∑n
i=1 f (i)‖Tj f‖ 6= ‖f‖ (the energy of the signal is not preserved)
![Page 41: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/41.jpg)
Graph Fourier Transform: Dilation
(Ds f )(x) =1s
f (1s) =⇒ (Ds f )(λ) = f (sλ)
in the graph setting, ts is not well defined, impairing the direct
generalization of dilation in the graph domain.
An alternative is to define dilation via GFT:
(Ds f )(i) =n−1
∑l=0
f (s · λl)ul(i)
Notice that f (s · λl) might not be in the interval [0, λn].Therefore, dilation can only be used when f is generated from akernel defined in the whole spectral domain.
![Page 42: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/42.jpg)
Graph Fourier Transform: Dilation
(Ds f )(x) =1s
f (1s) =⇒ (Ds f )(λ) = f (sλ)
in the graph setting, ts is not well defined, impairing the direct
generalization of dilation in the graph domain.
An alternative is to define dilation via GFT:
(Ds f )(i) =n−1
∑l=0
f (s · λl)ul(i)
Notice that f (s · λl) might not be in the interval [0, λn].Therefore, dilation can only be used when f is generated from akernel defined in the whole spectral domain.
![Page 43: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/43.jpg)
Graph Fourier Transform: Dilation
(Ds f )(x) =1s
f (1s) =⇒ (Ds f )(λ) = f (sλ)
in the graph setting, ts is not well defined, impairing the direct
generalization of dilation in the graph domain.
An alternative is to define dilation via GFT:
(Ds f )(i) =n−1
∑l=0
f (s · λl)ul(i)
Notice that f (s · λl) might not be in the interval [0, λn].Therefore, dilation can only be used when f is generated from akernel defined in the whole spectral domain.
![Page 44: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/44.jpg)
Graph Fourier Transform: Dilation
(Ds f )(x) =1s
f (1s) =⇒ (Ds f )(λ) = f (sλ)
in the graph setting, ts is not well defined, impairing the direct
generalization of dilation in the graph domain.
An alternative is to define dilation via GFT:
(Ds f )(i) =n−1
∑l=0
f (s · λl)ul(i)
Notice that f (s · λl) might not be in the interval [0, λn].
Therefore, dilation can only be used when f is generated from akernel defined in the whole spectral domain.
![Page 45: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/45.jpg)
Graph Fourier Transform: Dilation
(Ds f )(x) =1s
f (1s) =⇒ (Ds f )(λ) = f (sλ)
in the graph setting, ts is not well defined, impairing the direct
generalization of dilation in the graph domain.
An alternative is to define dilation via GFT:
(Ds f )(i) =n−1
∑l=0
f (s · λl)ul(i)
Notice that f (s · λl) might not be in the interval [0, λn].Therefore, dilation can only be used when f is generated from akernel defined in the whole spectral domain.
![Page 46: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/46.jpg)
Graph Fourier Transform: Modulation
Continous case:Mλ f (t) = e−(2πiλt) f (t)
In the graph setting we can define
Mλl f (i) =√
n ul(i)f (i)
In contrast to the continous case, modulation in the graphsetting does not correspond to a translation in the spectraldomain.However, if f is generated from a kernel localized in 0, thanMλl f (i) is concentrated in λl.
![Page 47: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/47.jpg)
Graph Fourier Transform: Modulation
Continous case:Mλ f (t) = e−(2πiλt) f (t)
In the graph setting we can define
Mλl f (i) =√
n ul(i)f (i)
In contrast to the continous case, modulation in the graphsetting does not correspond to a translation in the spectraldomain.However, if f is generated from a kernel localized in 0, thanMλl f (i) is concentrated in λl.
![Page 48: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/48.jpg)
Graph Fourier Transform: Modulation
Continous case:Mλ f (t) = e−(2πiλt) f (t)
In the graph setting we can define
Mλl f (i) =√
n ul(i)f (i)
In contrast to the continous case, modulation in the graphsetting does not correspond to a translation in the spectraldomain.
However, if f is generated from a kernel localized in 0, thanMλl f (i) is concentrated in λl.
![Page 49: Graph Fourier TransformGraph Fourier Transform From previous classes we learned that the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development](https://reader030.vdocument.in/reader030/viewer/2022040401/5e7ad8089b6c870ffa3151d4/html5/thumbnails/49.jpg)
Graph Fourier Transform: Modulation
Continous case:Mλ f (t) = e−(2πiλt) f (t)
In the graph setting we can define
Mλl f (i) =√
n ul(i)f (i)
In contrast to the continous case, modulation in the graphsetting does not correspond to a translation in the spectraldomain.However, if f is generated from a kernel localized in 0, thanMλl f (i) is concentrated in λl.