Segmentation of Polycrystalline Images Using Voronoi Diagrams Uzziel Cortez April 24, 2019 Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 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 Diagrams April 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 Diagrams April 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 Diagrams April 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 Diagrams April 24, 2019 5 / 36
Voronoi Diagrams We can model grain boundary structure by using Voronoi Diagrams. Given a set of generating points P = { p 1 , p 2 , ..., p n } , a plane is partitioned into n regions, { R 1 , R 2 , ..., R n } , such that: Each point p i lies in exactly one region R i . For any point q / ∈ P that lies in region R i , the Euclidean distance from p i to q will be shorter than the Euclidean distance from p j to q ∀ j � = i Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 6 / 36
Voronoi Diagrams Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 7 / 36
Voronoi Diagrams Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 8 / 36
Voronoi Diagrams Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 9 / 36
Mean Curvature Mean-Curvature Flow of Voronoi Diagrams - Matt Elsey & Dejan Slepˇ cev, 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 Diagrams April 24, 2019 10 / 36
Voronoi Diagram Energy Given a Voronoi Diagram with: Generating points P = { p 1 , p 2 , ..., p n } Edges S = { s 1 , s 2 , ..., s k } Vertices V = { v 1 , v 2 , ..., v m } . We define the energy of the Voronoi Diagram as: � � E = Length ( s k ) = | v i − v j | i , j s . t . s k ∈ S v i v j = s k ∈ S Using this definition of energy, we can apply gradient descent on the generating points P = { p 1 , p 2 , ..., p n } and start to view some dynamics of the Voronoi diagrams. Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 11 / 36
Piece-wise Constant Mumford-Shah model � � ( f ( x , y ) − c i ) 2 dxdy R i i f(x,y): the target image’s grayscale value at the pixel (x,y) c i : the average pixel value in region R i computed from f Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 12 / 36
Model Using the energy we defined earlier and the piece-wise constant Mumford-Shah we get: � � � ( f ( x , y ) − c i ) 2 dxdy E = | v i − v j | + R i i , j s . t . i v i v j = s k ∈ S Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 13 / 36
Calculating Gradients: First Term Edges: S = { s 1 , s 2 , ..., s k } Vertices: V = { v 1 , v 2 , ..., v m } ∂ Length ( s i ) ∂ v i ∂ E � � ∂ p i = ∂ v i ∂ p i s i ∈ S v i ∈ vertex ( s i ) = [ v i ( x ) − v j ( x ) Length ( s i ) , v i ( y ) − v j ( y ) ∂ length ( s i ) Length ( s i ) ] ∂ v i How to calculate ∂ v i ∂ p i ? Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 14 / 36
Calculating Gradients: First Term Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 15 / 36
Calculating Gradients: First Term Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 16 / 36
Calculating Gradients: First Term Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 17 / 36
Calculating Gradients: First Term We now have a way to calculate changes in v i for perturbations of p i along two specific directions. Using a change of basis, we can get the gradient in terms of the standard basis ∂ p 1 = [ w 1 w 2 ][ s 1 s 2 ] − 1 ∂ v Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 18 / 36
Calculating Gradients: Second Term g ( x , y ) = ( f ( x , y ) − c i ) 2 � � ∂ g ( x , y ) dxdy ∂ p i R i i 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 Diagrams April 24, 2019 19 / 36
Calculating Gradients: Second Term Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 20 / 36
Calculating Gradients: Second Term Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 21 / 36
Calculating Gradients: Second Term Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 22 / 36
Calculating Gradients: Second Term g ( x , y ) = ( f ( x , y ) − c i ) 2 � � ∂ � � g ( s ) v ( s ) ⊥ ds g ( x , y ) dxdy = ∂ p i R i ∂ R i i i N � � � g ( s ) v k ( s ) ⊥ ∆ s ≈ R i k =1 s i ∈ edges ( R i ) v ( s ) ⊥ = L − r ∂ v j ∂ v i n + r ∂ p 1 ˆ ∂ p 1 ˆ n L L Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 23 / 36
Gradient Descent Video play GD Collision.mp4 Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 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 Diagrams April 24, 2019 25 / 36
Handling Topological Events - Center Collisions Repulsion: − 1 θ 2( r − dij )2 R ( p i , p j ) = R ( d ij ) = e Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 26 / 36
Handling Topological Events - Boundary Event Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 27 / 36
Handling Topological Events - Boundary Event Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 28 / 36
Handling Topological Events - Boundary Event Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 29 / 36
Handling Topological Events - Collapsed Regions Area( R i ) ≤ τ = ⇒ Removal of p i Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 30 / 36
Gradient Descent Video play GD RP.mp4 Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 31 / 36
Image Segmentation Result Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 32 / 36
Image Segmentation Result play Sample Segment.mp4 Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 33 / 36
Conclusion and Future Work Our Model / Algorithm was able to properly handle a preliminary test case segmentation Future Challenges include Non-Uniform Grain Colors Non-Distinct Grain Colors Generating Point Initialization - Location and Number of points Uzziel Cortez Segmentation of Polycrystalline Images Using Voronoi Diagrams April 24, 2019 34 / 36
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