Spectral Graph Theory

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Spectral Graph Theory
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Outline

• Definitions and different spectra
• Physical analogy
• Description of bisection algorithm
• Relationship of spectrum to graph structure
• My own recent work on graphical images

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I. Definition and Different Spectra

• Spectrum The set of eigenvalues corresponding to
  a matrix
• Two major kinds of graph matricies used in this
  context Adjacency and Laplacian

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Adjacency Matrix
If graph is undirected, A is symmetric. This
means that its eigenvectors are real and
orthogonal
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Laplacian Matrix
Let D be the matrix composed of I multiplied by a
vector containing the degree of each vertex
Then,
Sometimes normalized by
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Spectra Intuition

• For each vertex, assign a number so that the
  number is proportional to the sum of all its
  neighbors numbers Proportions are eigenvalues
  of A
• For each vertex, assign a number so that the
  number is proportional to the difference between
  of all its neighbors numbers and itself
  multiplied by its degree Proportions are
  eigenvalues of L

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Formally
Laplacian Matrix
In matrix notation
Proportional to the differences
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II. Physical Analogies

• Harmonic modes of a vibrating string
• Chemistry Hückel theory showed that the spectra
  of a molecules adjacency matrix is related to
  the energy of the corresponding molecular orbitals

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Modes of a string
x3
x2
x1
x5
x4
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Seeking solutions to the differential equation of
the form
where a and x0 are a scalar and vector
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Therefore eigenvalues of M are
With eigenvectors
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Modes of a string concluded

• Therefore the eigenvectors of the Laplacian form
  the harmonic basis set with coefficients equal to
  the eigenvalues
• Demmel suggests that harmonic analogy works in
  2D, giving fault lines in the surface

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III. Spectral bisection algorithm

• Builds L, finds the eigenvector corresponding to
  the second smallest eigenvalue (Fiedler value)
  and partitions nodes based on a cut point in the
  corresponding eigenvector (Fiedler vector)

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Rationale behind method

• Since rows/colums all add to zero in L, the first
  eigenvector will be all ones and the first
  eigenvalue will be zero
• Second eigenvalue/eigenvector known as the
  Fiedler vector/value

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Back to modes in a string
First harmonic

Second harmonic

-
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Possible choices of a cut-point

• Median cut Use median value in eigenvector
• Ratio cut Use point which gives the best ratio
  of vertices separated to edges cut
• Sign cut Cut-point equals zero
• Gap cut Choose value at largest gap in the
  sorted list of Fiedler vector components

Demmel endorses sign cut, most applied
researchers seem to favor median cut, while
mathematicians (and Shi/Malik) favor ratio cut
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Q. How to approximate eigenvalues/vectors of a
sparse, symmetric matrix?
A. The Lanczos method
Lanczos method takes an n x n sparse, symmetric
matrix A and computes a k x k tridiagonal matrix
T whose eigenvalues/vectors are good
approximations of those in A
Even with k much smaller than n, the
approximation is fairly good. Fortunately, the
values which converge first are the largest and
smallest, including the Fiedler values
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Lanczos overview
Choose an arbitrary starting vector r b(0)
norm(r) i 0 while not converged i v(i) r
/ b(i-1) r Av(i) r r - b(i-1)v(i-1) a(i)
dotproduct(v(i), r) r r -a(i)v(i) b(i)
norm(r) end
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Lanzcos matrix at step i
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Problem Lanczos method too slow
Solution Multilevel method

• Coarsen using MIS
• Find eigenvectors of coarsened graph using
  Rayleigh Quotient Iteration (RQI)
• Project eigenvectors back to original graph
  using coarsened values to seed RQI

Although approximation is quick and dirty, when
youre only concerned with the sign (or median
or...), a rough approximation is okay
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IV. Beyond partitioning Structural relationships

• Knowing the spectrum of a graph can tell you
  certain things about the graphs structure and
  visa versa

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Isomorphisms

• Cospectral graphs are not necessarily isomorphic,
  but isomorphic graphs are always cospectral
• Spectrum is invariant to nonsingular
  transformations (e.g. permutations, affine
  transforms, etc.)

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Connectivity

• Eigenvalues of L(G) are nonnegative, in
  particular

The number of connected components of G is equal
to the number of ?i 0 In particular, ?2 ? 0
iff G is connected
Feidler value ?2 algebraic connectivity
Let S be a subset of G i.e. with the same nodes
and a subset of edges, so that S is less
connected than G, then
The number of spanning trees of a graph G is
given by
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Bipartite Graphs

• A graph containing at least one edge is bipartite
  iff the spectrum of A is symmetric with respect
  to zero

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Cheeger constants
For a subset of the vertices S, let
Define the Cheeger constant as
Then the Fiedler value is bounded by
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Regularity
A graph is regular with degree r iff
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Conclusion

• Graph spectra have many curious and surprising
  relationships to graph structure
• Many more theorems related to graph spectra
• Most work focuses either on applications or
  proving various bounds on the values

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Current Work in Image Processing
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Build a graph-based IP environment

• Problem definition
• Data structure
• Resolved issues
• Current IP routines
• Future directions

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Problem Liberate IP from pixels
Solution Formulate IP on graphs
Advantages Space variant vision possible
Processing on fewer components in same domain
Graph algorithms are fast
Goals Space variant applications Choosing
nodes based on content Use graph theory
algorithms to novel ends Logonoid simulations,
etc.
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Data structure

• Two classes Graph and ImgGraph
• Graph - Only a vertex and edge list
• ImgGraph inherits Graph, plus adds fields for
  Heckbert precomputations and RGB intensity values
  so that images may be imported onto it

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Whats not in the structure

• Neighbor list (i.e. flowers)
• Edge weights
• Face list (more on this later)

Why?

• All of these fields arent necessary for many
  methods (i.e. face list is only necessary if
  visualizing), leading to possibly unneeded
• Precomputation time
• Storage space
• Updating (e.g. when deleting nodes/edges)

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Current methods - Graph

• Get/set - Basic OOP methods
• Neighborhood - Compute neighbor list and
  distances to each neighbor
• Removeedge - Removes an edge
• Removenode - Remove a node

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Current methods - ImgGraph
Get/set - Basic OOP methods Adjacency - Computes
the adjacency matrix Laplacian - Computes the
Laplacian matrix Impotimg - Imports an image
centered at location fovea Edgegraph - Computes
the edge map using 1st derivative Makeweights -
Computes edge weights Neighborhood - Compute
neighbor list and distances to each
neighbor Removenode - Removes a node
list Removeisolated - Finds and removes nodes of
degree zero Threshcut - Segmentation by intensity
thresholding Showstruct -Displays graph structure
without image data Showmesh - Displays graph by
interpolating across enclosed polygons Showgraph
- Displays graph as a traditional stick-and-ball,
where balls are colored to reflect RGB values at
the node Findfaces - Generates a list of enclosed
polygons that can be fed to patch in the
Showmesh call
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Problems to solve

• Importing images
• Assigning edge weights
• Visualization

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Importing images

• Use Heckberts masters thesis to precompute
  domain pixels and weights. Weights are
  normalized to sum to unity
• Vertices in own coordinate system centered around
  the origin. All precomputations are made
  relative the this system (i.e. is independent of
  image)
• The origin is placed by the user in the image
  when importing (i.e. user chooses foveal point)
• Pixels called for outside of the image are
  assigned to zero

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Assigning edge weights
Given two parameters (A, B), I define the weight
between nodes i and j as
Where d represents the (Euclidean) distance
between nodes and c represents the RGB difference
(L1)
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Visualization

• Very important so that the results of IP
  algorithms may be visually assessed
• Data structure supports three visualizations

• Structure - Quick and dirty connectivity display
• Graph - Traditional stick-and-ball, balls
  reflect RGB values of nodes
• Mesh - Displays graph faces as filled-in polygons

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Structure
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Original (Cartesian) image
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Graph visualization
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Mesh visualization
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Current IP algorithms

• Edge detection with 1st derivative
• Segmentation by gray level thresholding
• Mesh partitioning toolbox (Gilbert Teng)

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Edge detection
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Intensity segmentation
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Graph partitioning toolbox

• Failed to produce interesting segmentations for a
  retinal graph
• Algorithms work to produce good load balancing,
  and therefore will cut the retinal disk in half
  through the center of the fovea, regardless of
  the image (although the angle of the cut depends
  on the image)
• Algorithms may still prove useful for image
  guided graphs (i.e. graphs determined by image
  content)

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Future directions

• Coarsening/pyramid algorithms
• More serious segmentation
• Image guided graphs
• Graph matching (i.e. object recognition)
• Hardware implementation
• Fourier domain? FEM? Space/time graphs?