Segmentation of Polycrystalline Images Using Voronoi Diagrams · Segmentation of Polycrystalline Images Using Voronoi Diagrams Uzziel Cortez April 24, 2019 Uzziel Cortez Segmentation

Segmentation of Polycrystalline Images Using VoronoiDiagrams

Uzziel Cortez

April 24, 2019

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 1 / 36

Introduction & Motivation

Material development is essential to solving problems our world faces.

Superalloys: Industrial engineering applications such as aerospace andmarine engineeringGraphene: Applications in medicine and electronicsAerogels: Environmental applications



Polycrystalline materials are composed of many crystalline parts thatare randomly oriented with respect to each other.

The material’s properties are largely dependent on its microstructure.

Material properties

conductivitystrengthhardnesscorrosion resistance ...

Material microstrucure properties

Grain sizeGrain boundary distributionGrain deformationsChemical composition ...



Determining a materials properties through analysis of grain boundarystructure is crucial to the development of new materials andsubsequent advancement of engineering.

Obtaining accurate measurements through imaging can be expensiveand tedious.

Electron Backscatter Diffraction: Equipment and technician costsLight optical microscopy: Requires preprocessing of the material andmay affect measurement accuracy


Problem Statement

Given an image of a polycrystalline material, can we implement analgorithm that will produce an accurate segmentation?

i.e. Can we produce a binary image that accurately represents thegrain boundary structure?


Voronoi Diagrams

We can model grain boundary structure by using Voronoi Diagrams.

Given a set of generating points P = {p1, p2, ..., pn}, a plane ispartitioned into n regions, {R1,R2, ...,Rn}, such that:

Each point pi lies in exactly one region Ri .For any point q /∈ P that lies in region Ri , the Euclidean distance frompi to q will be shorter than the Euclidean distance from pj to q ∀j 6= i


Voronoi Diagrams


Voronoi Diagrams


Voronoi Diagrams


Mean Curvature

Mean-Curvature Flow of Voronoi Diagrams - Matt Elsey & DejanSlepcev, 2014

They were interested in gradient flow of Voronoi diagrams andproving universal bounds on coarsening rates.

We followed a similar method while relaxing some constraints such asperiodic boundary conditions. (more later)


Voronoi Diagram Energy

Given a Voronoi Diagram with:

Generating points P = {p1, p2, ..., pn}Edges S = {s1, s2, ..., sk}Vertices V = {v1, v2, ..., vm}.

We define the energy of the Voronoi Diagram as:

E =∑sk∈S

Length(sk) =∑i , j s.t.

vivj=sk∈S

|vi − vj |

Using this definition of energy, we can apply gradient descent on thegenerating points P = {p1, p2, ..., pn} and start to view somedynamics of the Voronoi diagrams.


Piece-wise Constant Mumford-Shah model

∑i

∫Ri

(f (x , y)− ci )2dxdy

f(x,y): the target image’s grayscale value at the pixel (x,y)

ci : the average pixel value in region Ri computed from f


Model

Using the energy we defined earlier and the piece-wise constantMumford-Shah we get:

E =∑i , j s.t.

vivj=sk∈S

|vi − vj |+∑i

∫Ri

(f (x , y)− ci )2dxdy


Calculating Gradients: First Term

Edges: S = {s1, s2, ..., sk}Vertices: V = {v1, v2, ..., vm}∂E∂pi

=∑si∈S

∑vi∈vertex(si )

∂Length(si )

∂vi

∂vi∂pi

∂length(si )∂vi

= [vi (x)−vj (x)Length(si )

,vi (y)−vj (y)Length(si )

]

How to calculate ∂vi∂pi

?









We now have a way to calculate changes in vi for perturbations of pialong two specific directions.

Using a change of basis, we can get the gradient in terms of thestandard basis∂v∂p1

= [w1 w2][s1 s2]−1


Calculating Gradients: Second Term

g(x , y) = (f (x , y)− ci )2

∂∂pi

∑i

∫Ri

g(x , y)dxdy

note that after perturbing a center, the change in the integral comesfrom the part of the region that is changed.









g(x , y) = (f (x , y)− ci )2

∂∂pi

∑i

∫Ri

g(x , y)dxdy =∑i

∫∂Ri

g(s)v(s)⊥ds

≈∑Ri

∑si∈edges(Ri )

N∑k=1

g(s)vk(s)⊥∆s

v(s)⊥ = L−rL

∂vi∂p1

n + rL∂vj∂p1

n


Gradient Descent Video

play GD Collision.mp4


Handling Topological Events

Vertex collisions (handled well by the algorithm)

Center collisions

Vertices escape the boundary

Center regions collapsed


Handling Topological Events - Center Collisions

Repulsion:

R(pi , pj) = R(dij) = e−1

θ2(r−dij )2


Handling Topological Events - Boundary Event






Handling Topological Events - Collapsed Regions

Area(Ri ) ≤ τ =⇒ Removal of pi


Gradient Descent Video

play GD RP.mp4


Image Segmentation Result


Image Segmentation Result

play Sample Segment.mp4


Conclusion and Future Work

Our Model / Algorithm was able to properly handle a preliminary testcase segmentation

Future Challenges include

Non-Uniform Grain ColorsNon-Distinct Grain ColorsGenerating Point Initialization - Location and Number of points


References

1 Elsey, Matt, and Dejan Slepcev. ”Mean-curvature flow of Voronoidiagrams.” Journal of Nonlinear Science 25.1 (2015): 59-85.

2 Trimby, P., et al. ”Is fast mapping good mapping? A review of thebenefits of high-speed orientation mapping using electron backscatterdiffraction.” Journal of microscopy 205.3 (2002): 259-269.

3 Tai, Xue-cheng, and Chang-hui Yao. ”Image segmentation bypiecewise constant Mumford-Shah model without Estimating theconstants.” Journal of Computational Mathematics, vol. 24, no. 3,2006, pp. 435-443. JSTOR, www.jstor.org/stable/43693303.


Acknowledgments

Special Thanks to:Selim Esedoglu, Tiago Salvador, Yi WenRackham Graduate School for funding

Everyone in the Math Department for their support


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Segmentation of Polycrystalline Images Using Voronoi Diagrams · Segmentation of Polycrystalline Images Using Voronoi Diagrams Uzziel Cortez April 24, 2019 Uzziel Cortez Segmentation