# Segmentation of Polycrystalline Images Using Voronoi Diagrams Segmentation of Polycrystalline Images

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Segmentation of Polycrystalline Images Using Voronoi Diagrams

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 and marine engineering Graphene: Applications in medicine and electronics Aerogels: Environmental applications

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 2 / 36

Introduction & Motivation

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

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

Material properties

conductivity strength hardness corrosion resistance ...

Material microstrucure properties

Grain size Grain boundary distribution Grain deformations Chemical composition ...

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 3 / 36

Introduction & Motivation

Determining a materials properties through analysis of grain boundary structure is crucial to the development of new materials and subsequent advancement of engineering.

Obtaining accurate measurements through imaging can be expensive and tedious.

Electron Backscatter Diffraction: Equipment and technician costs Light optical microscopy: Requires preprocessing of the material and may affect measurement accuracy

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 4 / 36

Problem Statement

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

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

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 5 / 36

Voronoi Diagrams

We can model grain boundary structure by using Voronoi Diagrams.

Given a set of generating points P = {p1, p2, ..., pn}, a plane is partitioned 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 from pi to q will be shorter than the Euclidean distance from pj to q ∀j 6= i

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 6 / 36

Voronoi Diagrams

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 7 / 36

Voronoi Diagrams

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 8 / 36

Voronoi Diagrams

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 9 / 36

Mean Curvature

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

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

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

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 10 / 36

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 the generating points P = {p1, p2, ..., pn} and start to view some dynamics of the Voronoi diagrams.

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 11 / 36

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

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 12 / 36

Model

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

E = ∑ i , j s.t.

vivj=sk∈S

|vi − vj |+ ∑ i

∫ Ri

(f (x , y)− ci )2dxdy

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 13 / 36

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 ?

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 14 / 36

Calculating Gradients: First Term

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 15 / 36

Calculating Gradients: First Term

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 16 / 36

Calculating Gradients: First Term

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 17 / 36

Calculating Gradients: First Term

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

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

= [w1 w2][s1 s2] −1

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 18 / 36

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 comes from the part of the region that is changed.

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 19 / 36

Calculating Gradients: Second Term

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 20 / 36

Calculating Gradients: Second Term

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 21 / 36

Calculating Gradients: Second Term

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 22 / 36

Calculating Gradients: Second Term

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

n̂ + rL ∂vj ∂p1

n̂

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 23 / 36

Gradient Descent Video

play GD Collision.mp4

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

Handling Topological Events

Vertex collisions (handled well by the algorithm)

Center collisions

Vertices escape the boundary

Center regions collapsed

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 25 / 36

Handling Topological Events - Center Collisions

Repulsion:

R(pi , pj) = R(dij) = e −1

θ2(r−dij )2

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 26 / 36

Handling Topological Events - Boundary Event

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 27 / 36

Handling Topological Events - Boundary Event

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 28 / 36

Handling Topological Events - Boundary Event

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 29 / 36

Handling Topological Events - Collapsed Regions

Area(Ri ) ≤ τ =⇒ Removal of pi

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 30 / 36

Gradient Descent Video

play GD RP.mp4

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 31 / 36

Image Segmentation Result

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 32 / 36

Image Segmentation Result

play Sample Segment.mp4

Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi DiagramsApril 24, 2019 33 / 36

Conclusion and Future Work

Our Model / Algorithm was able to properly handle a pr

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