Toward a Statistical Approach to Automatic Particle ...chuanhai/teaching/598M/Docs/detection.pdf˜...

Data and Data ReductionInitial Estimation of Particle Locations

Particle DetectionDiscussion

Data (a.eps and noise.eps generated by GIMP@linux)Background noiseData reduction via blocking

Toward a Statistical Approach to AutomaticParticle Detection in Electron Microscopy

Chuanhai Liu

Chuanhai Liu Toward a Statistical Approach to Automatic Particle Detection




Image size = 2048 × 2048.

The data in the rectangular

region

(700, 300) : (1100, 700) is

used to study the

distribution of the

background noise.





A block of background noise

The R-code generating this image:png(”noise.png”)par(mai=rep(0,4))image(700:1100, 300:700, X[700:1100,300:700])

dev.off()





The marginal distribution

For alternative approximations, try, for example,

qqnorm(X[700:1100,300:700] 0̂.6)?

The R-code for “QQ-fit”:z= sort(c(X[700:1100,300:700]))qqfit.nbinom = function (y = z) {˜ n = length(y); p = ((1:n)-.5)/n˜ f = function(par){˜ ˜ par=exp(par)˜ ˜ size = par[1]; prob = par[2]/(1+par[2])˜ ˜ x = qnbinom(p, size=size, prob=prob)˜ ˜ return(mean(abs(lm(y x)$res)))˜ }˜ out = optim(par=c(log(200), 0), fn=f)˜ par=exp(out$par)˜ size = par[1]; prob = par[2]/(1+par[2])˜ return(list(size = size, prob = prob))}˜The noise distribution is a mixture of Poissondistributions with Gamma distributed means:Z ∼ Gamma(shape = size)*(1-prob)/prob;Y ∼ Poisson(Z)˜

EVEV+VE

≈ 2.2% =⇒ model the noise by Gamma

distributions (for simplicity?)





The (spatial) dependence

Ignore the spatial trend for the selected block of noise and computethe (homogeneous/stationary) spatial correlation coefficients.

2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

distance

corr

elat

ion

coef

ficie

nt

Do the pattern of the correlation coefficients,

(1.00, 0.47, 0.10, 0.02, 0.02, 0.02, 0.01, 0.01, ...),

suggest a moving-average model?





Specification of the (joint) sampling distribution

To do when it is needed....





Due to the small signal-to-ratio and the high image resolution with respect to particlesof interest, data reduction for computational simplicity is possible with much loss ofinformation for particle detection. Consider square blocks of size 2m for some integerm; see R-functions named ParticleDetection/.RData/block and block.png.

Blocks of size 24 = 16 are too large!





R-functions block and block.png

block=function (X, block.size=8, plot.it=TRUE){˜ m = block.size; n = dim(X) %/% m˜ x = as.list(1:(m*m))˜ for(i in 1:m){ for(j in 1:m){˜ ˜ x[[(i-1)*m+j]] = X[(0:(n[1]-1))*m + i, (0:(n[2]-1))*m + j]˜ }}˜ A = 0; ˜ for(i in 1:length(x)) A = A + x[[i]]; ˜ A = A/(m*m)˜ V = 0; ˜ for(i in 1:length(x)) V = V + (x[[i]]-A) 2̂; ˜ v = V/(m*m-1)˜ if(plot.it){; ˜ par(mfrow=c(1,2))˜ ˜ image(A, main = paste(”Average over every ”,m,”x”,m, ” blocks”, sep=””))˜ ˜ image(log(v), main = ”Associated variances in log scale”)˜ }˜ return(list(a=A, v=v))}˜block.png = function (X, block.size, file, height, width) {˜ png(file, width=width, height=height)˜ b=block(X, block.size=block.size)˜ dev.off()}





Blocks of size 24 = 16 look satisfactory — a formal statisticalapproach needs to be considered!




Reduced data and assumptionsGradientsInitial estimation of particle boundary locations

In what follows, we use the reduced data via blocking with blocksize 23 = 8, using the averages and ignoring the (within-block)variances. Thus, we have

◮ The reduced image size is 256 × 256.

◮ The (background) noise distribution is approximately that ofwhite (Gaussian) noise.





The definition

Notice that the background noise intensities, especially, those nearthe boundaries of particles of interest are higher than those ofparticle images. Let

Ii ,j (1 ≤ i , j ≤ 256)

be the intensity at the pixel location (i , j). Then, the differencevectors, called gradients,

Gi ,j =(

Imin(i+1,n1),j − Imax(i−1,1),j , Ii ,min(j+1,n2) − Ii ,max(j−1,1)

)

provide useful information for finding particle locations, where(n1, n2) = (256, 256).





R-function get.gradients

get.gradients = function (I=block(X,block.size=8)$a){˜ n = dim(I)˜ Gy = cbind(I[,2]-I[,1], I[,3:n[2]]-I[,1:(n[2]-2)], I[,n[2]]-I[,n[2]-1])˜ Gx = rbind(I[2,]-I[1,], I[3:n[1],]-I[1:(n[1]-2),], I[n[1],]-I[n[1]-1,])˜ return(list(r = sqrt(Gx*Gx+Gy*Gy), theta=atan2(Gy,Gx)))}





The distribution

Most of pixel values are noise. Thus, at most pixel locations

◮ The gradient directions and lengths are independent.

◮ The gradient directions are uniform.

◮ The gradient lengths have the marginal distribution

‖ Gi ,j ‖2∼ Exp(σ2)

for some σ2 > 0, i.e., a scaled chi-square with 2 degrees offreedom, with small spatial correlation (zero long-range(distance > 1) correlation — independence).





The histograms

Histogram of g$r^(2/3)

g$r^(2/3)

Den

sity

0 5 10 15 20

0.00

0.05

0.10

0.15

0.20

Histogram of g$theta

g$theta

Fre

quen

cy

−3 −2 −1 0 1 2 3

010

0030

0050

0070

00

The fitted density curve and histogram of g$r2/3 indicate thepresence of outliers, as expected.





ML estimation of the null distribution from truncated data

ml.tchi = function (x, df=2, par=log(3.8)) {˜ x=sort(as.numeric(x))˜ b=max(x)˜ n = length(x)˜ fn = function(par){˜ ˜ scale = exp(par)˜ ˜ y = x/scale˜ ˜ P = pchisq(b/scale,df=df)˜ ˜ lf = sum(dchisq(y, df=df, log=TRUE)) - n*log(P*scale)˜ ˜ return(-lf)˜ }˜ out = optim(par=par, fn=fn)˜ scale = out$par = exp(out$par)˜ hist(x, prob=TRUE, main=”ML fit of truncated chi-square”,˜ ˜ breaks = seq(0, max(x), length=13),˜ ˜ xlab=”truncated observations”)˜ lines(x, dchisq(x/scale,df=df)/pchisq(b/scale,df=df)/scale, col=”green”)˜ return(out)}





ML estimation of the null distribution from truncated data

50 60 70 80 90 100

8090

100

110

120

cutting point (percentile)

estim

ate

of s

cale

σ2 ≈ 81., i.e., ‖ Gi ,j ‖2 /81

·∼ Exp(1)





R function fdr

fdr = function (v=g$rˆ2, df=2, scale=81, p=.60) {˜ P = pchisq(as.numeric(v)/scale,df=df); dim(P) = dim(v)˜ a = sort(as.numeric(v)); x = aˆ(1/3)˜ hist(x,breaks=60, prob=TRUE, xlab=”(g$rˆ2)ˆ(1/3)”)˜ g = density(x)˜ k = length(g$x[g$x <= quantile(x,p)]); x.p = g$x[k]˜ lambda = p/pchisq(x.pˆ3/scale,df=df)˜ Z = lambda*dchisq(g$xˆ3/scale,df=df)*3*g$xˆ2/scale˜ y = g$y[(k+1):length(Z)]; ˜ x = g$x[k:length(Z)]; ˜ z = Z[(k+1):length(Z)]˜ Sy = rev(cumsum(rev(diff(x)*y)))˜ Sz = rev(cumsum(rev(diff(x)*z)))˜ FDR = data.frame(x=x[-length(x)]ˆ3,fdr = Sz/Sy)˜ polygon(c(x[-1],rev(x[-1])),c(y,rev(z)),col=”red”)˜ abline(v=x.p, col=”green”,lwd=2)˜ lines(g$x, g$y, col=”blue”, lwd=2, lty=2)˜ lines(g$x, Z, col=”green”, lwd=2)˜ text(9.2,0.028,paste(round((1-lambda)*100),”% outliers”,sep=””),˜ ˜ srt=-40,cex=1.2, col=”white”)˜ return(list(scale=scale, df=df, P=P, fdr=FDR,˜ ˜ outliers=data.frame(x=x[-1],cdf=Sy-Sz)))}





Outlier identification

Histogram of x

(g$r^2)^(1/3)

Den

sity

0 5 10 15 20

0.00

0.05

0.10

0.15

0.20

10% outliers





False discovery rate (FDR)

6 8 10 12 14 16 18 20

0.0

0.2

0.4

0.6

(g$r^2)^(1/3)

fdr





Boundary locations with fdr ≤ 5%





A comment: modeling outliers (if necessary)

−2 −1 0 1 2

1.8

2.0

2.2

2.4

2.6

2.8

ln−normal for the outliers?

normal quantile

ln(o

utlie

rs.q

uant

ile) Question: Does it make sense

to consider a mixture ofexponential and log-normalfor ‖ Gi ,j ‖

2?




Local smoothingLinear spatial trend removalSegmentationParticle Identification

The four-step procedure

1. (Feature-protected) local smoothing

2. (Linear) spatial trend removal

3. (Adaptive) segmentation: intensity thresholding withthresholds selected based smoothed intensities at initialestimates of particle locations

4. Particle selection (convex shapes and sensible sizes)◮ Trimming?◮ Soft-thresholding — using difference quantiles?◮ Fine tuning by using the original raw 2048× 2048 image?





The method

◮ Specifying neighbors {Ni ,j} for each pixel (i , j).

◮ Estimating the “mean” µi ,j of xi ,j as the smoothed value ofxi ,j based on a simple, both conceptually and computationally,model.

We consider the model

xi,j ∼ N(µi,j , σ2) and xk ∼ N(µi,j , σ

2 + δ2) (k ∈ Ni,j \ {(i , j)}).

We call this local smoothing method Hanning and also consider

Hanning on smoothed values for further smoothing.

The use of mixture of two normals can be interesting, but

computationally can be too expensive. Consider other alternative

models?





An implementation in R

Hanning = function (x, sigmasq) {˜ Z = get.neighbors(x)˜ Z0 = Z[[9]]˜ m = (Z[[1]]+Z[[2]]+Z[[3]]+Z[[4]]+Z[[5]]+Z[[6]]+Z[[7]]+Z[[8]])/8˜ v = ( (Z[[1]]-m)ˆ2+(Z[[2]]-m)ˆ2+(Z[[3]]-m)ˆ2 +(Z[[4]]-m)ˆ2˜ ˜ ˜ +(Z[[5]]-m)ˆ2+(Z[[6]]-m)ˆ2 +(Z[[7]]-m)ˆ2+(Z[[8]]-m)ˆ2)/7˜ if(missing(sigmasq)) sigmasq = median(v)˜ v[v¡=sigmasq] = sigmasq˜ w = sigmasq*8/v˜ a = (Z0+w*m)/(1+w)˜ return(a)}





Performance





The method

◮ Find the boundary locations using a FDR (e.g., 5%) threshold.

◮ Remove isolated single pixels.

◮ Compute the sample mean, µ, and variance, σ2, of pixelintensities at the selected boundary pixels.

◮ Estimate the trend using row and column α-quantiles of pixelintensities, with the comment proportion/probability α chosenby estimates of the number of noise pixels (?).





An implementation in R

remove.trend = function(Z, mu, sigma, alpha = 0.5) {˜ A = Z > mu+alpha*sigma˜ a = as.numeric(A%*%rep(1/n[2],n[2]))˜ b = as.numeric(rep(1/n[1],n[1])%*%A)˜ p = mean(c(a,b))˜ a = as.numeric(apply(Z,1,function(x){return(quantile(x,prob=p))}))˜ b = as.numeric(apply(Z,2,function(x){return(quantile(x,prob=p))}))˜ data = data.frame(y=c(a,b), col=c(1:n[1], rep(0,n[2])),˜ ˜ ˜ ˜ row=c(rep(0,n[1]),1:n[2]))˜ out = lm(y∼col+row, data=data)˜ y.hat = out$fitted; y.hat = y.hat-mean(y.hat)˜ x.hat = y.hat[1:n[1]]; y.hat = y.hat[n[1]+(1:n[2])]˜ Z = Z - matrix(x.hat,n[1],n[2]) - matrix(y.hat,n[1],n[2], byrow=TRUE)˜ return(Z)

}





Performance

Images of pixel values that are less than µi + 0.5 ∗ σi for i = “Before” and “After” (2

hangings), respectively. Notice that a trend from Top-Right to Bottom-Left is

apparent before trend removal.





Thresholding

◮ Compute a normal reference distribution based on smoothedpixel intensities at estimated particle boundary locations, thelocations with large gradients for specification of thresholds.Denote this reference distribution by N(µ, σ2).

◮ Specify a sequence of thresholds, e.g.,

µ + zkσ (zk = .5, .4, ...)





Classification: shapes and sizes of particle (2D) projections

◮




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