Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

MicroscopySupervised Segmentation of Domain Boundaries in STM Images

of Self-Assembled Molecule Layers

Daniel Lander, Rodrigo Rios, Matt Vollmer, Yu (Dan) ZhouSupervised by: Dominique Zosso

Department of Mathematics, University of California, Los Angeles (UCLA)

08/07/2013

Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Overview

1 Introduction

2 Methods

3 K-means

4 Spectral Clustering

5 Markov Random Fields

6 Modeling Data

7 Mixture Modeling

8 Conclusion

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Introduction

Scanning Tunneling Microscopy (STM)

Used to measure topography of surface at nano-scale

Produces two different types of images of interestSelf-assembled cage molecules on Au111 [left]Beta-sheets of peptide on graphite [right]

[2] [3]

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Introduction

Cage Molecules

Produces SAMs with relatively few defects

SAMs made from 1-carboranethiol, 1-adamantanethiol, and2-adamantanethiol

[2] [6]

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Introduction

Image Segmentation

I0 be the observed image stored as set of pixels and P is auniformity (homogeneity) predicate

Partitioning of I0 into a set of subregions (S1,S2, ...,Sn) wheren⋃

i=1

Si = F with Si ∩ Sj = Ø, i 6= j (1)

P(Si ) = true ∀i and P(Si ∪ Sj) = false, when Si is adjacentto Sj

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Introduction

Previous Work

Applied principal component analysis, mean diffusedorientation Gnorm-TV, and a structure tensor forpre-processing

Chan-Vese for contour modeling

Data is on beta sheets as opposed to self-assembled cagemolecules

[3]

[3]

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Introduction

Objective

Accurately segment and characterize different domains in anSTM image with self-assembled cage molecules

[2] [6]

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Methods

Preprocessing

Mexican hat filter separates structure from texture

Hole finder to isolate artifacts

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Methods

Fourier Transform

”Chunking” image1 Block-wise without overlap2 Pixel by pixel

Apply Gaussian window and Fourier transform

Power spectrum is invariant under spatial translation

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Methods

Fourier Domain

K-means

Spectral clustering

Bayesian Probability in Fourier Domain

Markov random fields1 Spectral clustering initialization2 Partial labeling

Mixture modeling

Assessing Correctness

Pixel by pixel difference with ground truth

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

The k-means Algorithm

Data: X = xi |i = 1, ..., n and Centroids: C = ci |i = 1, ..., k

Sj = x |x is a member of a cluster j

Cost =n∑

i=1

dist(ωi , ck) , ck =

∑ωi∈Sk

ωi

|Sk |

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

k-means block-wise results

Original synthetic

(a) 8 pixel, 0.23 (b) 16 pixel, 0.27

(c) 32 pixel, 0.32 (d) 64 pixel, 0.54

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

k-means pixel by pixel results

Original Synthetic 32 pixel size, 0.59

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Spectral Clustering

Given weight matrix, W.

ω(A,B) =∑

i∈A,j∈BWi ,j

∝∑

i∈A,j∈Be−|||I0(i)|2−|I0(j)|2||

2σ2

2

RatioCut(A1, ...,Ak) =1

2

k∑i=1

ω(Ai ,AiC )

|Ai |

Di ,i =m2∑j=0

Wi ,j

L = D −W

Finishing steps:

1 Define matrix U, inRnxk

2 Cluster points in Rk

using the k-meansalgorithm

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Spectral clustering block-wise results

Original synthetic

Various σ at 32 pixel size

(a) σ = log(1.5), 0.69 (b) σ = log(2), 0.54

(c) σ = log(2.5), 0.78 (d) σ = log(3), 0.7215/31

Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Spectral clustering block-wise results

Original synthetic

Various pixel sizes at optimal σ = log(2.5)

(a) 8 pixel, 0.29 (b) 16 pixel, 0.48

(c) 32 pixel, 0.59 (d) 64 pixel, 0.7616/31

Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Spectral clustering pixel by pixel results

(a) Original synthetic (b) σ = 0.071, 32pixel, 0.95

(c) 32 pixel

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Markov random fields

Markov Random Fields (MRFs)

Assume local dependence of pixels

Solve optimization problem:

maxL

P(L|I0) ∝ P(L)P(I0|L)

P(L) is the clique potential

P(I0|L) is the ”data” term

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Markov random fields

The Tentative Procedure

Initialize mean (µk) and variance (Σk) for spectrum of eachcluster using k-means

Apply Expectation-Maximization (EM) algorithm on

P(L) =∏x∈I0

P(L(x)|LΩ\x

)=∏x∈I0

P(L(x)|LN(x)

)(2)

∝∏x∈I0

e−λH where H =∑

y∈N(x)

1− 1[L(x)=L(y)] (3)

P(I0|L) =∏x∈I0

N (PS |µk ,Σk) where PS = |I0(x)|2 (4)

Maximize label configuration L in double product forP(L)P(I0|L) across k clusters

Update expected new mean and variance parameters ofspectral clusters

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Markov random fields with spectral clustering initialization

Original synthetic

(a) Initial, 0.95 (b) λ = 0.05,0.87, 25 iterations

(c) λ = 0.05,0.80, 50 iterations

(d) λ = 0.05,0.74, 75 iterations

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Markov random fields with user input

(a) Original synthetic (b) Initial, 0.60 (c) λ = 0.0244, 0.84,50 iterations

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Markov random fields results with partial labeling

(a) Original synthetic (b) λ = 1, 32 pixel,0.86

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Applying algorithms on data

(a)1-adamantanethiol

(b) Spec clust,σ = 0.071, 32pixel, 0.52

(c) User input,λ = 0.0244, 32pixel, 0.61

(d) Partiallabeling, λ = 1,32 pixel, 0.64

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Applying algorithms on data

(a)2-adamantanethiol

(b) Spec clust,σ = 0.071, 32pixel, 0.69

(c) User input,λ = 0.0244, 32pixel, 0.85

(d) Partiallabeling, λ = 1,32 pixel, 0.89

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Mixture modeling

Mixture models

Assigns mixture vector u(x) = (u1(x), ..., uk(x)) to pixel x ofbeing in k clusters

New clique potential with spatial gradient:

P(u) =∏x∈I0

P(u(x)) =∏x∈I0

N(∇u(x)|µ = 0, σ2

g

)(5)

Probability vector used as weights for product of normaldistributions in data term:

P(I0|u) =∏x∈I0

(k∏

i=1

N(PS |µi , σ2

)ui (x)

)(6)

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Mixture modeling

Mixture models

New maximization problem:

maxu

∏x∈I0

(k∏

i=1

N(∇u(x)|µ = 0, σ2

)N(PS |µi , σ2

g

)ui (x)

)After a −log transformation:

minu,λ

∑x∈I0

k∑i=1

ui (x)2 ||(PS − µi )||2 + λ ||∇u||2

subject tok∑

i=1

ui (x) = 1 and ui (x) ≥ 0 ∀x ∈ I0

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

Conclusions & Outlook

1 Block by block is not as good as pixel by pixel

2 Spectral clustering works best on synthetic images

3 Markov random fields with k-means manual initialization andpartial labeling works best on microscopy data

4 Optimize Markov random fields to be interactive

5 Attempt mixture modeling

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

References I

Julian Besag.On the statistical analysis of dirty pictures.Journal of the Royal Statistical Society. Series B(Methodological), pages 259–302, 1986.

Arrelaine A. Dameron, Lyndon F. Charles, and Paul S. Weiss.Structures and displacement of 1-adamantanethiolself-assembled monolayers on au 111.Journal of the American Chemical Society,127(24):8697–8704, 2005.

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

References II

Huynh Nen Lim Tawny Meyer Travis Dragomiretskiy, JonathanSiegel Konstantin and Joseph Woodworth.Image analysis and classification in scanning tunnelingmicroscopy.2012.

Rafael C. Gonzalez and R.E. Woods.Digital image processing, 2008.

John A. Hartigan and Manchek A. Wong.Algorithm as 136: A k-means clustering algorithm.Journal of the Royal Statistical Society. Series C (AppliedStatistics), 28(1):100–108, 1979.

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

References III

J. Nathan Hohman, Shelley A. Claridge, Moonhee Kim, andPaul S. Weiss.Cage molecules for self-assembly.Materials Science and Engineering: R: Reports,70(3):188–208, 2010.

Anil K Jain, M. Narasimha Murty, and Patrick J Flynn.Data clustering: a review.ACM computing surveys (CSUR), 31(3):264–323, 1999.

Todd K. Moon.The expectation-maximization algorithm.Signal processing magazine, IEEE, 13(6):47–60, 1996.

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Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion

References IV

Ulrike Von Luxburg.A tutorial on spectral clustering.Statistics and computing, 17(4):395–416, 2007.

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