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

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1 Microscopy Supervised Segmentation of Domain Boundaries in STM Images of Self-Assembled Molecule Layers Daniel Lander, Rodrigo Rios, Matt Vollmer, Yu (Dan) Zhou Supervised by: Dominique Zosso Department of Mathematics, University of California, Los Angeles (UCLA) 08/07/2013

2 2/31 Overview 1 Introduction 2 Methods 3 K-means 4 Spectral Clustering 5 Markov Random Fields 6 Modeling Data 7 Mixture Modeling 8 Conclusion

3 3/31 Introduction Scanning Tunneling Microscopy (STM) Used to measure topography of surface at nano-scale Produces two different types of images of interest Self-assembled cage molecules on Au{111} [left] Beta-sheets of peptide on graphite [right] [2] [3]

4 4/31 Introduction Cage Molecules Produces SAMs with relatively few defects SAMs made from 1-carboranethiol, 1-adamantanethiol, and 2-adamantanethiol [2] [6]

5 5/31 Introduction Image Segmentation I 0 be the observed image stored as set of pixels and P is a uniformity (homogeneity) predicate Partitioning of I 0 into a set of subregions (S 1, S 2,..., S n ) where n S i = F with S i S j = Ø, i j (1) i=1 P(S i ) = true to S j i and P(S i S j ) = false, when S i is adjacent

6 6/31 Introduction Previous Work Applied principal component analysis, mean diffused orientation Gnorm-TV, and a structure tensor for pre-processing Chan-Vese for contour modeling Data is on beta sheets as opposed to self-assembled cage molecules [3] [3]

7 Introduction Methods K-means Spectral Clustering Markov Random Fields Modeling Data Mixture Modeling Conclusion Introduction Objective Accurately segment and characterize different domains in an STM image with self-assembled cage molecules [2] [6] 7/31

8 8/31 Methods Preprocessing Mexican hat filter separates structure from texture Hole finder to isolate artifacts

9 9/31 Methods Fourier Transform Chunking image 1 Block-wise without overlap 2 Pixel by pixel Apply Gaussian window and Fourier transform Power spectrum is invariant under spatial translation

10 10/31 Methods Fourier Domain K-means Spectral clustering Bayesian Probability in Fourier Domain Markov random fields 1 Spectral clustering initialization 2 Partial labeling Mixture modeling Assessing Correctness Pixel by pixel difference with ground truth

11 11/31 The k-means Algorithm Data: X = {x i i = 1,..., n} and Centroids: C = {c i i = 1,..., k} S j = {x x is a member of a cluster j} n Cost = dist(ω i, c k ), c k = S k i=1 ω i S k ω i

12 12/31 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

13 13/31 k-means pixel by pixel results Original Synthetic 32 pixel size, 0.59

14 14/31 Spectral Clustering D i,i = m 2 j=0 W i,j L = D W Given weight matrix, W. ω(a, B) = i A,j B i A,j B W i,j e Î 0 (i) RatioCut(A 1,..., A k ) = 1 2 k i=1 2 Î 0 (j) 2 2 2σ 2 ω(a i, A i C ) A i Finishing steps: 1 Define matrix U, in R nxk 2 Cluster points in R k using the k-means algorithm

15 (c) σ = log(2.5), 0.78 (d) σ = log(3), /31 Spectral clustering block-wise results Various σ at 32 pixel size Original synthetic (a) σ = log(1.5), 0.69 (b) σ = log(2), 0.54

16 Spectral clustering block-wise results Various pixel sizes at optimal σ = log(2.5) Original synthetic (a) 8 pixel, 0.29 (b) 16 pixel, 0.48 (c) 32 pixel, 0.59 (d) 64 pixel, /31

17 17/31 Spectral clustering pixel by pixel results (a) Original synthetic (b) σ = 0.071, 32 pixel, 0.95 (c) 32 pixel

18 18/31 Markov random fields Markov Random Fields (MRFs) Assume local dependence of pixels Solve optimization problem: P(L) is the clique potential P(I 0 L) is the data term max P(L I 0) P(L)P(I 0 L) L

19 19/31 Markov random fields The Tentative Procedure Initialize mean (µ k ) and variance (Σ k ) for spectrum of each cluster using k-means Apply Expectation-Maximization (EM) algorithm on P(L) = P ( ) ( ) L(x) L Ω\x = L(x) LN(x) (2) x I 0 x I 0 P x I 0 e λh where H = y N(x) 1 1 [L(x)=L(y)] (3) P(I 0 L) = x I 0 N (PS µ k, Σ k ) where PS = Î 0 (x) 2 (4) Maximize label configuration L in double product for P(L)P(I 0 L) across k clusters Update expected new mean and variance parameters of spectral clusters

20 20/31 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

21 21/31 Markov random fields with user input (a) Original synthetic (b) Initial, 0.60 (c) λ = , 0.84, 50 iterations

22 22/31 Markov random fields results with partial labeling (a) Original synthetic (b) λ = 1, 32 pixel, 0.86

23 Applying algorithms on data (a) 1-adamantanethiol (b) Spec clust, σ = 0.071, 32 pixel, 0.52 (c) User input, λ = , 32 pixel, 0.61 (d) Partial labeling, λ = 1, 32 pixel, /31

24 Applying algorithms on data (a) 2-adamantanethiol (b) Spec clust, σ = 0.071, 32 pixel, 0.69 (c) User input, λ = , 32 pixel, 0.85 (d) Partial labeling, λ = 1, 32 pixel, /31

25 25/31 Mixture modeling Mixture models Assigns mixture vector u(x) = (u 1 (x),..., u k (x)) to pixel x of being in k clusters New clique potential with spatial gradient: P(u) = P(u(x)) = N ( u(x) µ = 0, σ 2 ) g x I 0 x I 0 (5) Probability vector used as weights for product of normal distributions in data term: P(I 0 u) = ( k N ( ) PS µ i, σ 2) u i (x) x I 0 i=1 (6)

26 26/31 Mixture modeling Mixture models New maximization problem: ( k x I0 i=1 max u N ( u(x) µ = 0, σ 2) N ( PS µ i, σg 2 ) ui (x) ) After a log transformation: min u,λ x I 0 i=1 subject to k u i (x) 2 (PS µ i ) 2 + λ u 2 k u i (x) = 1 and u i (x) 0 x I 0 i=1

27 27/31 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 and partial labeling works best on microscopy data 4 Optimize Markov random fields to be interactive 5 Attempt mixture modeling

28 28/31 References I Julian Besag. On the statistical analysis of dirty pictures. Journal of the Royal Statistical Society. Series B (Methodological), pages , Arrelaine A. Dameron, Lyndon F. Charles, and Paul S. Weiss. Structures and displacement of 1-adamantanethiol self-assembled monolayers on au {111}. Journal of the American Chemical Society, 127(24): , 2005.

29 29/31 References II Huynh Nen Lim Tawny Meyer Travis Dragomiretskiy, Jonathan Siegel Konstantin and Joseph Woodworth. Image analysis and classification in scanning tunneling microscopy Rafael C. Gonzalez and R.E. Woods. Digital image processing, John A. Hartigan and Manchek A. Wong. Algorithm as 136: A k-means clustering algorithm. Journal of the Royal Statistical Society. Series C (Applied Statistics), 28(1): , 1979.

30 30/31 References III J. Nathan Hohman, Shelley A. Claridge, Moonhee Kim, and Paul S. Weiss. Cage molecules for self-assembly. Materials Science and Engineering: R: Reports, 70(3): , Anil K Jain, M. Narasimha Murty, and Patrick J Flynn. Data clustering: a review. ACM computing surveys (CSUR), 31(3): , Todd K. Moon. The expectation-maximization algorithm. Signal processing magazine, IEEE, 13(6):47 60, 1996.

31 31/31 References IV Ulrike Von Luxburg. A tutorial on spectral clustering. Statistics and computing, 17(4): , 2007.