## Loading required package: knitr

## The basic files and libraries needed for most presentations
# creates the libraries and common-functions sections
read_chunk("../common/utility_functions.R")

require(ggplot2) #for plots
require(lattice) # nicer scatter plots
require(plyr) # for processing data.frames
require(grid) # contains the arrow function

require(doMC) # for parallel code
require(png) # for reading png images
require(gridExtra)
require(reshape2) # for the melt function
#if (!require("biOps")) {
#  # for basic image processing
#  devtools::install_github("cran/biOps")
#  library("biOps")
#}
## To install EBImage
if (!require("EBImage")) { # for more image processing
  source("http://bioconductor.org/biocLite.R")
  biocLite("EBImage")
}

used.libraries<-c("ggplot2","lattice","plyr","reshape2","grid","gridExtra","biOps","png","EBImage")

# start parallel environment
registerDoMC()
# functions for converting images back and forth
im.to.df<-function(in.img,out.col="val") {
  out.im<-expand.grid(x=1:nrow(in.img),y=1:ncol(in.img))
  out.im[,out.col]<-as.vector(in.img)
  out.im
}
df.to.im<-function(in.df,val.col="val",inv=F) {
  in.vals<-in.df[[val.col]]
  if(class(in.vals[1])=="logical") in.vals<-as.integer(in.vals*255)
  if(inv) in.vals<-255-in.vals
  out.mat<-matrix(in.vals,nrow=length(unique(in.df$x)),byrow=F)
  attr(out.mat,"type")<-"grey"
  out.mat
}
ddply.cutcols<-function(...,cols=1) {
  # run standard ddply command 
  cur.table<-ddply(...)
  cutlabel.fixer<-function(oVal) {
    sapply(oVal,function(x) {
      cnv<-as.character(x)
      mean(as.numeric(strsplit(substr(cnv,2,nchar(cnv)-1),",")[[1]]))
    })
  }
  cutname.fixer<-function(c.str) {
    s.str<-strsplit(c.str,"(",fixed=T)[[1]]
    t.str<-strsplit(paste(s.str[c(2:length(s.str))],collapse="("),",")[[1]]
    paste(t.str[c(1:length(t.str)-1)],collapse=",")
  }
  for(i in c(1:cols)) {
    cur.table[,i]<-cutlabel.fixer(cur.table[,i])
    names(cur.table)[i]<-cutname.fixer(names(cur.table)[i])
  }
  cur.table
}

show.pngs.as.grid<-function(file.list,title.fun,zoom=1) {
  preparePng<-function(x) rasterGrob(readPNG(x,native=T,info=T),width=unit(zoom,"npc"),interp=F)
  labelPng<-function(x,title="junk") (qplot(1:300, 1:300, geom="blank",xlab=NULL,ylab=NULL,asp=1)+
                                        annotation_custom(preparePng(x))+
                                        labs(title=title)+theme_bw(24)+
                                        theme(axis.text.x = element_blank(),
                                              axis.text.y = element_blank()))
  imgList<-llply(file.list,function(x) labelPng(x,title.fun(x)) )
  do.call(grid.arrange,imgList)
}
## Standard image processing tools which I use for visualizing the examples in the script
commean.fun<-function(in.df) {
  ddply(in.df,.(val), function(c.cell) {
    weight.sum<-sum(c.cell$weight)
    data.frame(xv=mean(c.cell$x),
               yv=mean(c.cell$y),
               xm=with(c.cell,sum(x*weight)/weight.sum),
               ym=with(c.cell,sum(y*weight)/weight.sum)
    )
  })
}

colMeans.df<-function(x,...) as.data.frame(t(colMeans(x,...)))

pca.fun<-function(in.df) {
  ddply(in.df,.(val), function(c.cell) {
    c.cell.cov<-cov(c.cell[,c("x","y")])
    c.cell.eigen<-eigen(c.cell.cov)
    
    c.cell.mean<-colMeans.df(c.cell[,c("x","y")])
    out.df<-cbind(c.cell.mean,
                  data.frame(vx=c.cell.eigen$vectors[1,],
                             vy=c.cell.eigen$vectors[2,],
                             vw=sqrt(c.cell.eigen$values),
                             th.off=atan2(c.cell.eigen$vectors[2,],c.cell.eigen$vectors[1,]))
    )
  })
}
vec.to.ellipse<-function(pca.df) {
  ddply(pca.df,.(val),function(cur.pca) {
    # assume there are two vectors now
    create.ellipse.points(x.off=cur.pca[1,"x"],y.off=cur.pca[1,"y"],
                          b=sqrt(5)*cur.pca[1,"vw"],a=sqrt(5)*cur.pca[2,"vw"],
                          th.off=pi/2-atan2(cur.pca[1,"vy"],cur.pca[1,"vx"]),
                          x.cent=cur.pca[1,"x"],y.cent=cur.pca[1,"y"])
  })
}

# test function for ellipse generation
# ggplot(ldply(seq(-pi,pi,length.out=100),function(th) create.ellipse.points(a=1,b=2,th.off=th,th.val=th)),aes(x=x,y=y))+geom_path()+facet_wrap(~th.val)+coord_equal()
create.ellipse.points<-function(x.off=0,y.off=0,a=1,b=NULL,th.off=0,th.max=2*pi,pts=36,...) {
  if (is.null(b)) b<-a
  th<-seq(0,th.max,length.out=pts)
  data.frame(x=a*cos(th.off)*cos(th)+b*sin(th.off)*sin(th)+x.off,
             y=-1*a*sin(th.off)*cos(th)+b*cos(th.off)*sin(th)+y.off,
             id=as.factor(paste(x.off,y.off,a,b,th.off,pts,sep=":")),...)
}
deform.ellipse.draw<-function(c.box) {
  create.ellipse.points(x.off=c.box$x[1],
                        y.off=c.box$y[1],
                        a=c.box$a[1],
                        b=c.box$b[1],
                        th.off=c.box$th[1],
                        col=c.box$col[1])                    
}
bbox.fun<-function(in.df) {
  ddply(in.df,.(val), function(c.cell) {
    c.cell.mean<-colMeans.df(c.cell[,c("x","y")])
    xmn<-emin(c.cell$x)
    xmx<-emax(c.cell$x)
    ymn<-emin(c.cell$y)
    ymx<-emax(c.cell$y)
    out.df<-cbind(c.cell.mean,
                  data.frame(xi=c(xmn,xmn,xmx,xmx,xmn),
                             yi=c(ymn,ymx,ymx,ymn,ymn),
                             xw=xmx-xmn,
                             yw=ymx-ymn
                  ))
  })
}

# since the edge of the pixel is 0.5 away from the middle of the pixel
emin<-function(...) min(...)-0.5
emax<-function(...) max(...)+0.5
extents.fun<-function(in.df) {
  ddply(in.df,.(val), function(c.cell) {
    c.cell.mean<-colMeans.df(c.cell[,c("x","y")])
    out.df<-cbind(c.cell.mean,data.frame(xmin=c(c.cell.mean$x,emin(c.cell$x)),
                                         xmax=c(c.cell.mean$x,emax(c.cell$x)),
                                         ymin=c(emin(c.cell$y),c.cell.mean$y),
                                         ymax=c(emax(c.cell$y),c.cell.mean$y)))
  })
}

common.image.path<-"../common"
qbi.file<-function(file.name) file.path(common.image.path,"figures",file.name)
qbi.data<-function(file.name) file.path(common.image.path,"data",file.name)

th_fillmap.fn<-function(max.val) scale_fill_gradientn(colours=rainbow(10),limits=c(0,max.val))

Quantitative Big Imaging

author: Kevin Mader date: 30 April 2015 width: 1440 height: 900 css: ../common/template.css transition: rotate

ETHZ: 227-0966-00L

Dynamic Experiments

Course Outline

source('../common/schedule.R')

Literature / Useful References

Books


Papers / Sites

Previously on QBI ...

Quantitative "Big" Imaging

The course has covered imaging enough and there have been a few quantitative metrics, but "big" has not really entered.

What does big mean?


So what is "big" imaging

Objectives

  1. What sort of dynamic experiments do we have?
  2. How can we design good dynamic experiments?
  3. How can we track objects between points?
  4. How can we track topology?
  5. How can we track voxels?
  6. How can we assess deformation and strain?
  7. How can assess more general cases?

Outline


Motivation


We can say that it looks like, but many pieces of quantitative information are difficult to extract

Scientific Goals

Rheology

Understanding the flow of liquids and mixtures is important for many processes


Deformation

Deformation is similarly important since it plays a significant role in the following scenarios

Experiments

The first step of any of these analyses is proper experimental design. Since there is always


There are always trade-offs to be made between getting the best possible high-resolution nanoscale dynamics and capturing the system level behavior.

Simulation

In many cases, experimental data is inherited and little can be done about the design, but when there is still the opportunity, simulations provide a powerful tool for tuning and balancing a large number parameters

Simulations also provide the ability to pair post-processing to the experiments and determine the limits of tracking.

What do we start with?

Going back to our original cell image

  1. We have been able to get rid of the noise in the image and find all the cells (lecture 2-4)
  2. We have analyzed the shape of the cells using the shape tensor (lecture 5)
  3. We even separated cells joined together using Watershed (lecture 6)
  4. We have created even more metrics characterizing the distribution (lecture 7)

We have at least a few samples (or different regions), large number of metrics and an almost as large number of parameters to tune

How do we do something meaningful with it?

Basic Simulation

We start with a starting image

# Fill Image code
# ... is for extra columns in the data set
fill.img.fn<-function(in.img,step.size=1,...) {
  xr<-range(in.img$x)
  yr<-range(in.img$y)
  ddply(expand.grid(x=seq(xr[1],xr[2],step.size),
                  y=seq(yr[1],yr[2],step.size)),
        .(x,y),
      function(c.pos) {
        ix<-c.pos$x[1]
        iy<-c.pos$y[1]
        nset<-subset(in.img,x==ix & y==iy)
        if(nrow(nset)<1) nset<-data.frame(x=ix,y=iy,val=0,...)
        nset
        })
}
make.spheres<-function(sph.list,base.gr=seq(-1,1,length.out=40)) {
  start.image<-expand.grid(x=base.gr,y=base.gr)
  start.image$val<-c(0)
  for(i in 1:nrow(sph.list)) {
    start.image$val<-with(start.image,
                          val + (
                            ((x-sph.list[i,"x"])^2+(y-sph.list[i,"y"])^2)<
                              sph.list[i,"r"]^2)
      )
  }
  start.image$phase<-with(start.image,ifelse(val>0,TRUE,FALSE))
  start.image
}
rand.list<-function(n.pts,r=0.15,min=-1,max=1) data.frame(x=runif(n.pts,min=-1,max=1),y=runif(n.pts,min=-1,max=1),r=r)
grid.list<-function(n.pts,r=0.15,min=-1,max=1) cbind(expand.grid(x=seq(min,max,length.out=n.pts),y=seq(min,max,length.out=n.pts)),r=r)


Analysis

Describing Motion

\[ \vec{v}(\vec{x})=\langle 0,0.1 \rangle \]

ggplot(subset(make.spheres(test.grid),phase),aes(x,y))+
  geom_raster(fill="gray50",alpha=0.75)+
  geom_segment(data=cbind(test.grid,xv=0,yv=0.1),
               aes(xend=x+xv,yend=y+yv),arrow=arrow(length = unit(0.3,"cm")))+
  coord_equal()+
  theme_bw(20)
plot of chunk unnamed-chunk-3


\[ \vec{v}(\vec{x})=0.3\frac{\vec{x}}{||\vec{x}||}\times \langle 0,0,1 \rangle \]

test.grid$xyr<-with(test.grid,sqrt(x^2+y^2))
ggplot(subset(make.spheres(test.grid),phase),aes(x,y))+
  geom_raster(fill="gray50",alpha=0.75)+
  geom_segment(data=with(test.grid,cbind(test.grid,xv=-0.3*y/xyr,yv=0.3*x/xyr)),
               aes(xend=x+xv,yend=y+yv),arrow=arrow(length = unit(0.3,"cm")))+
  coord_equal()+
  theme_bw(20)
plot of chunk unnamed-chunk-4

Many Frames

\[ \vec{v}(\vec{x})=\langle 0,0.1 \rangle \]

many.frames<-ldply(seq(0,2,length.out=9),function(in.offset) {
  cbind(
    make.spheres(data.frame(x=test.grid$x,y=test.grid$y+in.offset,r=test.grid$r)),
    offset=in.offset
    )
})

ggplot(subset(many.frames,phase),aes(x,y))+
  geom_raster(fill="gray50",alpha=0.75)+
  coord_equal()+
  facet_wrap(~offset)+
  labs(title="Different Frames in Linear Flow Image")+
  theme_bw(20)
plot of chunk unnamed-chunk-5


\[ \vec{v}(\vec{x})=0.3\frac{\vec{x}}{||\vec{x}||}\times \langle 0,0,1 \rangle \]

many.frames<-cbind(make.spheres(test.grid),offset=0)
last.frame<-test.grid
for(in.offset in seq(0,2,length.out=9)) {
  last.frame$xyr<-with(last.frame,sqrt(x^2+y^2))
  last.frame$xyr[which(last.frame$xyr==0)]<-1
  last.frame<-with(last.frame,data.frame(x=x-0.05*y/xyr,y=y+0.05*x/xyr,r=r))
  
  many.frames<-rbind(many.frames,
                     cbind(make.spheres(last.frame),offset=in.offset)
                     )
}
ggplot(subset(many.frames,phase),aes(x,y))+
  geom_raster(fill="gray50",alpha=0.75)+
  coord_equal()+
  facet_wrap(~offset)+
  labs(title="Different Frames in Spiral Flow")+
  theme_bw(20)
plot of chunk unnamed-chunk-6

Random Appearance / Disappearance

Under perfect imaging and experimental conditions objects should not appear and reappear but due to

It is common for objects to appear and vanish regularly in an experiment.


many.frames<-ldply(seq(0,1.,length.out=9),function(in.offset) {
  c.grid<-test.grid[sample(nrow(test.grid), 18), ]
  cbind(
    make.spheres(data.frame(x=c.grid$x,y=c.grid$y+in.offset,r=c.grid$r)),
    offset=in.offset
    )
})
ggplot(subset(many.frames,phase),aes(x,y))+
  geom_raster(fill="gray50",alpha=0.75)+
  coord_equal()+
  facet_wrap(~offset)+
  labs(title="Different Frames in Linear Flow Image")+
  theme_bw(20)
plot of chunk unnamed-chunk-7

Jitter / Motion Noise

Even perfect spherical objects do not move in a straight line. The jitter can be seen as a stochastic variable with a random magnitude (\(a\)) and angle (\(b\)). This is then sampled at every point in the field

\[ \vec{v}(\vec{x})=\vec{v}_L(\vec{x})+||a||\measuredangle b \]

ggplot(subset(make.spheres(test.grid),phase),aes(x,y))+
  geom_raster(fill="gray50",alpha=0.75)+
  geom_segment(data=cbind(test.grid,
                          xv=0+runif(nrow(test.grid),min=-0.05,max=0.05),
                          yv=0.1+runif(nrow(test.grid),min=-0.05,max=0.05)),
               aes(xend=x+xv,yend=y+yv),arrow=arrow(length = unit(0.3,"cm")))+
  coord_equal()+
  theme_bw(20)
plot of chunk unnamed-chunk-8

Jitter (Continued)

Over many frames this can change the path significantly

last.frame<-test.grid[,c("x","y","r")]
last.frame$id<-1:nrow(last.frame)
many.frames<-cbind(make.spheres(last.frame),offset=0)
many.grids<-cbind(last.frame,offset=0)

for(in.offset in cumsum(rep(0.2,8))) {
  last.frame<-with(last.frame,
                   data.frame(x=x+runif(nrow(last.frame),min=-0.1,max=0.1),
                              y=y+0.2+runif(nrow(last.frame),min=-0.1,max=0.1),
                              r=r)
                   )
  
  last.frame$id<-1:nrow(last.frame)
  
  many.frames<-rbind(many.frames,
                     cbind(make.spheres(last.frame),offset=in.offset)
                     )
  
  many.grids<-rbind(many.grids,
                    cbind(last.frame,offset=in.offset)
                    )
}
ggplot(subset(many.frames,phase),aes(x,y))+
  geom_raster(fill="gray50",alpha=0.75)+
  coord_equal()+
  facet_wrap(~offset)+
  labs(title="Different Frames in Linear Flow Image")+
  theme_bw(20)
plot of chunk unnamed-chunk-9


The simulation can be represented in a more clear fashion by using single lines to represent each spheroid

ggplot(many.grids,aes(x,y,))+
  geom_path(aes(color=id,group=id))+
  coord_equal()+
  labs(title="Different Paths in Linear Jittered Flow Image")+
  scale_color_gradientn(colours=rainbow(10))+
  theme_bw(20)
plot of chunk unnamed-chunk-10

Limits of Tracking

We see that visually tracking samples can be difficult and there are a number of parameters which affect the ability for us to clearly see the tracking.


We thus try to quantify the limits of these parameters for different tracking methods in order to design experiments better.

Acquisition-based Parameters

Experimental Parameters

Tracking: Nearest Neighbor

While there exist a number of different methods and complicated approaches for tracking, for experimental design it is best to start with the simplist, easiest understood method. The limits of this can be found and components added as needed until it is possible to realize the experiment

We then return to nearest neighbor which means we track a point (\(\vec{P}_0\)) from an image (\(I_0\)) at \(t_0\) to a point (\(\vec{P}_1\)) in image (\(I_1\)) at \(t_1\) by

\[ \vec{P}_1=\textrm{argmin}(||\vec{P}_0-\vec{y}|| \forall \vec{y}\in I_1) \]

Scoring Tracking

In the following examples we will use simple metrics for scoring fits where the objects are matched and the number of misses is counted.

There are a number of more sensitive scoring metrics which can be used, by finding the best submatch for a given particle since the number of matches and particles does not always correspond. See the papers at the beginning for more information

source('~/Dropbox/tipl_maven/snippets/R/trackingCode.R')
source('../common/data/simFlow.R')

Basic Simulations

Input flow from simulation

\[ \vec{v}(\vec{x})=\langle 0,0,0.05 \rangle+||0.01||\measuredangle b \]

# Generate a simple synthetic dataset
in.frames<-generate.frames(base.rand=0.01,crop.size=c(-10,10),n.objects=20,n.frames=10,flow.rate=0.05,force.2d=T) 

all.tracks<-track.frames(in.frames,offset=c(0,0,0.1),run.offset=T,run.adaptive=T,maxVolPenalty=NA)

ggplot(do.call(rbind,in.frames),
       aes(x=POS_X,y=POS_Z,color=as.factor(REAL_LACUNA_NUMBER)))+
  geom_path(aes(linetype="Original"))+
  labs(x="X Position",y="Z Position",color="Object ID",title="Flow Simulation Results")+
  theme_bw(20)
plot of chunk unnamed-chunk-11


Nearest Neighbor Tracking

ggplot(do.call(rbind,in.frames),
       aes(x=POS_X,y=POS_Z,color=as.factor(REAL_LACUNA_NUMBER)))+
  #geom_path(aes(linetype="Original"))+
  geom_path(data=subset(all.tracks,Matching=="No Offset"),aes(linetype="Tracked"),size=2,alpha=0.5)+
  facet_wrap(~Matching)+
  labs(x="X Position",y="Z Position",color="Object ID",title="Tracking Results")+
  theme_bw(20)
plot of chunk unnamed-chunk-12

More Complicated Flows

Input flow from simulation

\[ \vec{v}(\vec{x})=\langle 0,0,0.01 \rangle+||0.05||\measuredangle b \]

# Generate a simple synthetic dataset
in.frames<-generate.frames(base.rand=0.1,crop.size=c(-10,10),n.objects=20,n.frames=20,flow.rate=0.01,
                           force.2d=T,rand.fun=function(x,min=0,max=1) rnorm(x,(min+max)/2,(max-min)/2)) 
all.tracks<-track.frames(in.frames,offset=c(0,0,0.1),run.offset=T,run.adaptive=T,maxVolPenalty=NA)

ggplot(do.call(rbind,in.frames),
       aes(x=POS_X,y=POS_Z,color=as.factor(REAL_LACUNA_NUMBER)))+
  geom_path(aes(linetype="Original"))+
  labs(x="X Position",y="Z Position",color="Object ID",title="Flow Simulation Results")+
  theme_bw(20)