## 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(biOps) # for basic image processing
require(doMC) # for parallel code
require(png) # for reading png images
require(gridExtra)
require(reshape2) # for the melt function
## To install EBImage
# source("http://bioconductor.org/biocLite.R")
# biocLite("EBImage")
require(EBImage) # for more image processing
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: 12 March 2015 width: 1440 height: 900 css: ../common/template.css transition: rotate

ETHZ: 227-0966-00L

Advanced Segmentation

Course Outline

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

• 19th February - Introduction and Workflows
• 26th February - Image Enhancement (A. Kaestner)
• 5th March - Basic Segmentation, Discrete Binary Structures
• 12th March - Advanced Segmentation
• 19th March - Applying Graphical Models and Machine Learning (A. Lucchi)
• 26th March - Analyzing Single Objects
• 2nd April - Analyzing Complex Objects
• 16th April - Spatial Distribution
• 23rd April - Statistics and Reproducibility
• 30th April - Dynamic Experiments (K. Mader and A. Patera)
• 7th May - Scaling Up / Big Data
• 21th May - Guest Lecture, Applications in Material Science
• 28th May - Project Presentations

Literature / Useful References

• Jean Claude, Morphometry with R
  • Online through ETHZ
  • Buy it
• John C. Russ, “The Image Processing Handbook”,(Boca Raton, CRC Press)
  • Available online within domain ethz.ch (or proxy.ethz.ch / public VPN)

Advanced Segmentation

• Markov Random Fields for Image Processing Lecture
• Markov Random Fields Chapter
• Fuzzy set theory
• Superpixels

Contouring

• Active Contours / Snakes

Lesson Outline

• Motivation
  • Many Samples
  • Difficult Samples
  • Training / Learning
• Thresholding
  • Automated Methods
  • Hysteresis Method
• Feature Vectors
  • K-Means Clustering
  • Superpixels
  • Probabalistic Models
• Working with Segmented Images
  • Contouring
• Beyond
  • Fuzzy Models
  • Component Labeling

Previously on QBI

• Image acquisition and representations
• Enhancement and noise reduction
• Understanding models and interpreting histograms
• Choosing a threshold
  • Examining more complicated, multivariate data sets
• Improving segementation with morphological operations
  • Filling holes
  • Connecting objects
  • Removing Noise
• Partial Volume Effect

Where segmentation fails: Inconsistent Illumination

With inconsistent or every changing illumination it may not be possible to apply the same threshold to every image.

+/- r Code

cellImage<-im.to.df(jpeg::readJPEG(qbi.file("Cell_Colony.jpg")))
max.il<-2.5
il.vals<-runif(9,min=1/max.il,max=max.il)
im.vals<-ldply(1:length(il.vals),function(il.idx,th.val=0.75)
  cbind(cellImage[,c("x","y")],
        val=cellImage$val*il.vals[il.idx],
        in.thresh=ifelse(cellImage$val*il.vals[il.idx]

+/- r Code

ggplot(subset(im.vals,in.thresh=="Cells"),aes(x=x,y=y))+
  geom_raster(fill="red")+facet_wrap(~il.idx)+
  labs(fill="Phase",title="Cell Phase")+
  theme_bw(20)+coord_equal()

Where segmentation fails: Canaliculi

Here is a bone slice

1. Find the larger cellular structures (osteocyte lacunae)
2. Find the small channels which connect them together

The first task

is easy using a threshold and size criteria (we know how big the cells should be)

The second

is much more difficult because the small channels having radii on the same order of the pixel size are obscured by partial volume effects and noise.

Where segmentation fails: Brain Cortex

• The cortex is barely visible to the human eye
• Tiny structures hint at where cortex is located

+/- r Code

alz.df<-im.to.df(t(png::readPNG("ext-figures/cortex.png")))
ggplot(alz.df,aes(x=x,y=518-y))+
  geom_raster(aes(fill=val))+
  labs(fill="Electron Density",y="y",x="x")+
  coord_equal()+
  theme_bw(20)

• A simple threshold is insufficient to finding the cortical structures
• Other filtering techniques are unlikely to magicially fix this problem
  

  +/- r Code

  ggplot(alz.df,aes(x=x,y=518-y))+
  geom_raster(aes(fill=cut_interval(val,4)))+
  labs(fill="Segmented Phases",y="y",x="x")+
  coord_equal()+
  theme_bw(20)
  

Automated Threshold Selection

Given that applying a threshold is such a common and signficant step, there have been many tools developed to automatically (unsupervised) perform it. A particularly important step in setups where images are rarely consistent such as outdoor imaging which has varying lighting (sun, clouds). The methods are based on several basic principles.

Automated Methods

Histogram-based methods

Just like we visually inspect a histogram an algorithm can examine the histogram and find local minimums between two peaks, maximum / minimum entropy and other factors

• Otsu, Isodata, Intermodes, etc

Image based Methods

These look at the statistics of the thresheld image themselves (like entropy) to estimate the threshold

Results-based Methods

These search for a threshold which delivers the desired results in the final objects. For example if you know you have an image of cells and each cell is between 200-10000 pixels the algorithm runs thresholds until the objects are of the desired size

• More specific requirements need to be implemented manually

Histogram Methods

Taking a typical image of a bone slice, we can examine the variations in calcification density in the image

+/- r Code

bone.df<-im.to.df(png::readPNG("ext-figures/bonegfiltslice.png"))
ggplot(bone.df,aes(x=x,y=y))+
  geom_raster(aes(fill=val))+
  labs(fill="Calcification Dens",y="y",x="x")+
  coord_equal()+
  theme_bw(20)

We can see in the histogram that there are two peaks, one at 0 (no absorption / air) and one at 0.5 (stronger absorption / bone)

+/- r Code

bhist<-ggplot(bone.df,aes(x=val))+
  geom_histogram(binwidth=0.05)+
  labs(x="Calcification Density")+
  theme_bw(20)
bhist

Histogram Methods

+/- r Code

bhist

Intermodes

• Take the point between the two modes (peaks) in the histogram

Otsu

Search and minimize intra-class (within) variance $$\sigma^2_w(t)=\omega_{bg}(t)\sigma^2_{bg}(t)+\omega_{fg}(t)\sigma^2_{fg}(t)$$

Isodata

• $\textrm{thresh}= \max(img)+\min(img)/2$
• while the thresh is changing
  • $bg = img<\textrm{thresh}, obj = img>\textrm{thresh}$
  • $\textrm{thresh} = (\textrm{avg}(bg) + \textrm{avg}(obj))/2$

Fiji -> Adjust -> Auto Threshold

There are many methods and they can be complicated to implement yourself. FIJI offers many of them as built in functions so you can automatically try all of them on your image

Pitfalls

While an incredibly useful tool, there are many potential pitfalls to these automated techniques.

Histogram-based

These methods are very sensitive to the distribution of pixels in your image and may work really well on images with equal amounts of each phase but work horribly on images which have very high amounts of one phase compared to the others

Image-based

These methods are sensitive to noise and a large noise content in the image can change statistics like entropy significantly.

Results-based

These methods are inherently biased by the expectations you have. If you want to find objects between 200 and 1000 pixels you will, they just might not be anything meaningful.

Realistic Approaches for Dealing with these Shortcomings

Imaging science rarely represents the ideal world and will never be 100% perfect. At some point we need to write our master's thesis, defend, or publish a paper. These are approaches for more qualitative assessment we will later cover how to do this a bit more robustly with quantitative approaches

Model-based

One approach is to try and simulate everything (including noise) as well as possible and to apply these techniques to many realizations of the same image and qualitatively keep track of how many of the results accurately identify your phase or not. Hint: >95% seems to convince most biologists

Sample-based

Apply the methods to each sample and keep track of which threshold was used for each one. Go back and apply each threshold to each sample in the image and keep track of how many of them are correct enough to be used for further study.

Worst-case Scenario

Come up with the worst-case scenario (noise, misalignment, etc) and assess how inacceptable the results are. Then try to estimate the quartiles range (75% - 25% of images).

Hysteresis Thresholding

For some images a single threshold does not work

• large structures are very clearly defined
• smaller structures are difficult to differentiated (see partial volume effect)

ImageJ Source

+/- r Code

bone.df<-im.to.df(png::readPNG("ext-figures/bonegfiltslice.png"))
ggplot(bone.df,aes(x=x,y=y))+
  geom_raster(aes(fill=val))+
  labs(fill="Calcification Dens",y="y",x="x")+
  coord_equal()+
  theme_bw(20)

+/- r Code

ggplot(bone.df,aes(x=val))+
  geom_histogram(aes(y=..density..),alpha=0.5)+
  geom_density()+scale_y_sqrt()+
  labs(x="Calcification Dens")+
  theme_bw(20)

+/- r Code

thresh.fun<-function(x) {ifelse(x<0.01,"Air",ifelse(x<0.30,"Between","Bone"))}
bone.df$phase<-thresh.fun(bone.df$val)
ggplot(bone.df,aes(x=val))+
  geom_histogram(aes(fill=phase),binwidth=0.02,alpha=0.5)+
  geom_density(aes(y=15000/1.5*..scaled..))+
  labs(x="Calcification Density (au)")+
  scale_y_sqrt()+#(c(0,20))+
  theme_bw(20)

Hysteresis Thresholding

Comparing the original image with the three phases

+/- r Code

ggplot(bone.df,aes(x=x,y=y))+
  geom_raster(aes(fill=val))+
  labs(fill="Calcification Dens",y="y",x="x")+
  coord_equal()+
  theme_bw(20)

+/- r Code

ggplot(bone.df,aes(x=x,y=y))+
  geom_raster(aes(fill=phase))+
  labs(fill="Phase",y="y",x="x")+
  coord_equal()+
  theme_bw(20)

Hysteresis Thresholding: Reducing Pixels

Now we apply two important steps. The first is to remove the objects which are not cells (too small) using an opening operation.

+/- r Code

air.im<-df.to.im(cbind(bone.df,isair=bone.df$phase=="Air"),"isair")
air.df<-im.to.df(opening(air.im,makeBrush(5,"disc")))
names(air.df)[3]<-"stillair"
nbone.df<-merge(air.df,bone.df)
ggplot(nbone.df,aes(x=x,y=y))+
  geom_raster(aes(fill=phase,alpha=stillair))+
  labs(fill="Phase",y="y",x="x",title="After Opening")+
  coord_equal()+guides(alpha=F)+
  theme_bw(20)

+/- r Code

# if its air make sure its still air otherwise demote it to between, 
# if its not air leave it alone
nbone.df$phase<-ifelse(nbone.df$phase=="Air",
       ifelse(nbone.df$stillair>0,"Air","Between"),
       nbone.df$phase)

The second step to keep the between pixels which are connected (by looking again at a neighborhood $\mathcal{N}$) to the air voxels and ignore the other ones. This goes back to our original supposition that the smaller structures are connected to the larger structures

+/- r Code

# incredibly low performance implementation (please do not copy)
bone.idf<-nbone.df
bet.pts<-nbone.df
# run while there is still new air being created
while(nrow(subset(bet.pts,phase=="Air"))>0) {
  air.pts<-subset(bone.idf,phase=="Air")[,c("x","y")]
  bone.pts<-subset(bone.idf,phase=="Bone")[,c("x","y")]
  bet.pts<-ddply(subset(bone.idf,phase=="Between"),.(x,y),function(in.pixel.lst) {
    in.pixel<-in.pixel.lst[1,]
    data.frame(phase=ifelse(min(with(air.pts,(in.pixel$x-x)^2+(in.pixel$y-y)^2))<=1,
         "Air",
         "Between"))
  })
  bone.idf<-rbind(bet.pts,
                  cbind(air.pts,phase="Air"),
                  cbind(bone.pts,phase="Bone"))
  print(nrow(subset(bet.pts,phase=="Air")))
}
ggplot(bone.idf,aes(x=x,y=y))+
  geom_raster(aes(fill=phase))+
  labs(fill="Phase",y="y",x="x")+
  coord_equal()+
  theme_bw(20)

More Complicated Images

As we briefly covered last time, many measurement techniques produce quite rich data.

• Digital cameras produce 3 channels of color for each pixel (rather than just one intensity)
• MRI produces dozens of pieces of information for every voxel which are used when examining different contrasts in the system.
• Raman-shift imaging produces an entire spectrum for each pixel
• Coherent diffraction techniques produce 2- (sometimes 3) diffraction patterns for each point. $$I(x,y) = \hat{f}(x,y)$$

+/- r Code

nx<-4
ny<-4
n.pi<-4
grad.im<-expand.grid(x=c(-nx:nx)/nx*n.pi*pi,
                     y=c(-ny:ny)/ny*n.pi*pi)

grad.im<-cbind(grad.im,
               col=1.5*with(grad.im,abs(cos(x*y))/(abs(x*y)+(3*pi/nx)))+
                 0.5*runif(nrow(grad.im)),
              a=with(grad.im,sqrt(2/(abs(x)+0.5))),
               b=with(grad.im,0.5*sqrt(abs(x)+1)),
              th=0.5*runif(nrow(grad.im)),
              aiso=1,count=1)

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])                    
}

# normalize vector
tens.im<-ddply(grad.im,.(x,y),deform.ellipse.draw)

ggplot(tens.im,aes(x=x,y=y,group=as.factor(id),fill=col))+
  geom_polygon(color="black")+coord_fixed(ratio=1)+scale_fill_gradient(low="black",high="white")+guides(fill=F)+
  theme_bw(20)

Feature Vectors

A pairing between spatial information (position) and some other kind of information (value). $$\vec{x} \rightarrow \vec{f}$$

We are used to seeing images in a grid format where the position indicates the row and column in the grid and the intensity (absorption, reflection, tip deflection, etc) is shown as a different color

basic.image<-im.to.df(matrix(round(runif(5*5,0,100)),nrow=5))
names(basic.image)[3]<-"Intensity"

+/- r Code

simple.image<-ggplot(basic.image,aes(x,y,fill=Intensity))+
  geom_tile(color="black")+
  geom_text(aes(label=paste("(",x,",",y,")\n",round(Intensity))))+
  coord_equal()+
  theme_bw(15)

simple.image

The alternative form for this image is as a list of positions and a corresponding value

$$\hat{I} = (\vec{x},\vec{f})$$

kable(head(basic.image))

This representation can be called the feature vector and in this case it only has Intensity

Why Feature Vectors

If we use feature vectors to describe our image, we are no longer to worrying about how the images will be displayed, and can focus on the segmentation/thresholding problem from a classification rather than a image-processing stand point.

Example

So we have an image of a cell and we want to identify the membrane (the ring) from the nucleus (the point in the middle).

ring.image<-mutate(
  expand.grid(x=c(-10:10),y=c(-10:10)),
  Intensity = abs(cos(sqrt(x^2+y^2)/15*6.28))
  )

+/- r Code

simple.image<-ggplot(ring.image,aes(x,y,fill=Intensity))+
  geom_tile(color="black")+
  coord_equal()+
  theme_bw(15)

simple.image

A simple threshold doesn't work because we identify the point in the middle as well. We could try to use morphological tricks to get rid of the point in the middle, or we could better tune our segmentation to the ring structure.

+/- r Code

ggplot(ring.image,aes(x,y,fill=Intensity))+
  geom_tile(color="black")+
  geom_tile(data=subset(ring.image,Intensity>0.75),aes(color="Threshold"),fill="red",alpha=0.5)+
  coord_equal()+
  labs(color="")+
  theme_bw(15)

Adding a new feature

In this case we add a very simple feature to the image, the distance from the center of the image (distance).

ring.image.dist<-ddply(ring.image,.(x,y),function(in.pts) {
  in.pts$Distance<-with(in.pts,sqrt(
      min(
        x^2+y^2
      )
    ))
  in.pts
})

+/- r Code

ggplot(ring.image.dist,aes(x,y,fill=Distance))+
  geom_tile(color="black")+
  coord_equal()+
  scale_fill_gradientn(colours=rainbow(5))+
  theme_bw(15)

kable(head(ring.image.dist))

We now have a more complicated image, which we can't as easily visualize, but we can incorporate these two pieces of information together.

+/- r Code

ggplot(ring.image.dist,aes(y=Distance,x=Intensity))+
  geom_density2d()+
  geom_point(aes(color=Distance,size=Intensity))+
  scale_color_gradientn(colours=rainbow(5))+
  labs(color="Distance",size="Intensity")+
  theme_bw(20)

Applying two criteria

Now instead of trying to find the intensity for the ring, we can combine density and distance to identify it

$$iff (5<\textrm{Distance}<10 \\ \& 0.5<\textrm{Intensity}>1.0)$$

+/- r Code

ggplot(ring.image.dist,aes(y=Distance,x=Intensity))+
  geom_density2d(aes(color="Density"))+
  geom_point()+
  geom_rect(aes(color="Threshold"),xmin=0.5,xmax=1,ymin=5,ymax=10,fill=NA)+
  labs(color="")+
  theme_bw(20)

+/- r Code

ring.image.seg<-mutate(ring.image.dist,
                       Segmented = Intensity>0.5 & Distance>5 & Distance<10)
ggplot(ring.image,aes(x,y,fill=Intensity))+
  geom_tile(color="black")+
  geom_tile(data=subset(ring.image.seg,Segmented),
            aes(color="Threshold"),fill="red",alpha=0.5)+
  coord_equal()+
  labs(color="")+
  theme_bw(15)

Common Features

The distance while illustrative is not a commonly used features, more common various filters applied to the image

• Gaussian Filter (information on the values of the surrounding pixels)
• Sobel / Canny Edge Detection (information on edges in the vicinity)
• Entroy (information on variability in vicinity)

  +/- r Code

  ring.array<-df.to.im(ring.image,val.col="Intensity")
  sobel.image<-im.to.df(imgSobel(ring.array*255)/255,out.col="Sobel")
  gauss.image<-im.to.df(imgStdBlur(ring.array*255,10)/220,out.col="Gaussian")
  merged.ring.image<-merge(
  merge(
      mutate(ring.image.seg,x=x+11,y=y+11,Distance=Distance/10),
      sobel.image),
      gauss.image
      )
  

kable(head(merged.ring.image[,c("x","y","Intensity","Sobel","Gaussian")]),digits=2)

+/- r Code

mfeat.vec<-melt(merged.ring.image,id.vars=c("x","y","Segmented","Distance"))
ggplot(mfeat.vec,
       aes(x=x,y=y,fill=value))+
  geom_tile(size=1,alpha=0.75)+
  scale_fill_gradientn(colours=rainbow(5))+
  coord_equal()+
  labs(fill="")+
  facet_grid(variable~.)+
  theme_bw(20)

Analyzing the feature vector

The distributions of the features appear very different and can thus likely be used for identifying different parts of the images.

+/- r Code

ggplot(mfeat.vec,aes(x=value,color=variable))+
  geom_density()+
  scale_y_sqrt()+
  labs(x="Feature Value",color="Feature")+
  theme_bw(20)

Combine this with our a priori information (called supervised analysis)

+/- r Code

ggplot(mutate(mfeat.vec,Structure=ifelse(Segmented,"Ring","Background")),
       aes(x=value,color=Structure))+
  geom_density()+
  labs(x="Feature Value",color="Structure")+
  facet_grid(variable~.,scales="free")+
  theme_bw(12)

+/- r Code

splom(merged.ring.image[,c("Intensity","Sobel","Gaussian")],groups=merged.ring.image$Segmented,pch=16)

K-Means Clustering / Classification (Unsupervised)

• Automatic clustering of multidimensional data into groups based on a distance metric
• Fast and scalable to petabytes of data (Google, Facebook, Twitter, etc. use it regularly to classify customers, advertisements, queries)
• Input = feature vectors, distance metric, number of groups
• Output = a classification for each feature vector to a group

K-Means Example

Input

• Feature vectors ($\vec{v}$)

  +/- r Code

  kable(row.names=T,head(merged.ring.image[,c("x","y","Intensity","Sobel","Gaussian")]),digits=2)
  

• Distance metric $$D_{ij}=||\vec{v}_i-\vec{v}_j||$$

• Group Count ($N=2$)

$$\downarrow$$

Output

• Group 1

  +/- r Code

  kable(row.names=T,merged.ring.image[c(20:24),c("x","y","Intensity","Sobel","Gaussian")],digits=2)
  

• Group 2

  +/- r Code

  kable(row.names=T,merged.ring.image[c(100:104),c("x","y","Intensity","Sobel","Gaussian")],digits=2)
  

K-Means Algorithm

We give as an initial parameter the number of groups we want to find and possible a criteria for removing groups that are too similar

1. Randomly create center points (groups) in vector space
2. Assigns group to data point by the “closest” center
3. Recalculate centers from mean point in each group
4. Go back to step 2 until the groups stop changing

What vector space to we have?

• Sometimes represent physical locations (classify swiss people into cities)
• Can include intensity or color (K-means can be used as a thresholding technique when you give it image intensity as the vector and tell it to find two or more groups)
• Can also include orientation, shape, or in extreme cases full spectra (chemically sensitive imaging)

Note: If you look for N groups you will almost always find N groups with K-Means, whether or not they make any sense

K-Means Example

Continuing with our previous image and applying K-means to the Intensity, Sobel and Gaussian channels looking for 2 groups we find

+/- r Code

sring.image<-merged.ring.image[,c("x","y","Intensity","Sobel","Gaussian")]
sring.image$km.cluster<-kmeans(sring.image[,c("Intensity","Sobel","Gaussian")],2)$cluster
sring.vec<-melt(sring.image,id.vars=c("x","y","km.cluster"))
ggplot(sring.vec,
       aes(x=x,y=y,fill=value,color=as.factor(km.cluster)))+
  geom_tile(size=0.75,alpha=0.75)+
  #scale_fill_gradientn(colours=rainbow(5))+
  coord_equal()+
  labs(fill="",color="Cluster")+
  facet_grid(~variable)+
  theme_bw(10)

+/- r Code

splom(sring.image[,c("Intensity","Sobel","Gaussian")],groups=sring.image$km.cluster,pch=16)

Looking for 5 groups

+/- r Code

sring.image$km.cluster<-kmeans(sring.image[,c("Intensity","Sobel","Gaussian")],5)$cluster
sring.vec<-melt(sring.image,id.vars=c("x","y","km.cluster"))
ggplot(sring.vec,
       aes(x=x,y=y,fill=value,color=as.factor(km.cluster)))+
  geom_tile(size=0.75,alpha=0.75)+
  #scale_fill_gradientn(colours=rainbow(5))+
  coord_equal()+
  labs(fill="",color="Cluster")+
  facet_grid(~variable)+
  theme_bw(10)

+/- r Code

splom(sring.image[,c("Intensity","Sobel","Gaussian")],groups=sring.image$km.cluster,pch=16)

Changing the feature vector

Including the position in the features as well

+/- r Code

sring.image$km.cluster<-kmeans(sring.image[,c("x","y","Intensity","Sobel","Gaussian")],5)$cluster
sring.vec<-melt(sring.image,id.vars=c("x","y","km.cluster"))
ggplot(sring.vec,
       aes(x=x,y=y,fill=value,color=as.factor(km.cluster)))+
  geom_tile(size=0.75,alpha=0.75)+
  #scale_fill_gradientn(colours=rainbow(5))+
  coord_equal()+
  labs(fill="",color="Cluster")+
  facet_grid(variable~.)+
  theme_bw(10)

+/- r Code

splom(sring.image[,c("x","y","Intensity","Sobel","Gaussian")],groups=sring.image$km.cluster,pch=16)

Rescaling components

Since the distance is currently calculated by $||\vec{v}_i-\vec{v}_j||$ and the values for the position is much larger than the values for the Intensity, Sobel or Gaussian they need to be rescaled so they all fit on the same axis $$\vec{v} = \left\{\frac{x}{10}, \frac{y}{10}, \textrm{Intensity},\textrm{Sobel},\textrm{Gaussian}\right\}$$

+/- r Code

sring.image$km.cluster<-kmeans(mutate(
  sring.image[,c("x","y","Intensity","Sobel","Gaussian")],
  x=x/20,y=y/20)
                               ,5)$cluster

sring.vec<-melt(sring.image,id.vars=c("x","y","km.cluster"))
ggplot(sring.vec,
       aes(x=x,y=y,fill=value,color=as.factor(km.cluster)))+
  geom_tile(size=0.75,alpha=0.75)+
  #scale_fill_gradientn(colours=rainbow(5))+
  coord_equal()+
  labs(fill="",color="Cluster")+
  facet_grid(~variable)+
  theme_bw(10)

+/- r Code

splom(sring.image[,c("x","y","Intensity","Sobel","Gaussian")],groups=sring.image$km.cluster,pch=16)

Super-pixels

An approach for simplifying images by performing a clustering and forming super-pixels from groups of similar pixels.

+/- r Code

shale.array<-readTiff(qbi.file("shale-slice.tiff"))
shale.image<-im.to.df(shale.array[,,1],out.col="Intensity")
ggplot(shale.image,aes(x,y,fill=Intensity))+
  geom_tile()+
  coord_equal()+
  labs(fill="")+
  theme_bw(10)

pos.scale<-5
shale.kmeans.data<-mutate(
  shale.image[,c("x","y","Intensity")],
  x=x/pos.scale,y=y/pos.scale)
# evenly spaced grid
grid.size<-40
grid.center<-round(grid.size/2)
shale.superpixels.centers<-subset(shale.kmeans.data, 
                                  x%%grid.size==grid.center & x%%grid.size==grid.center)
# not the real super pixels algorithm but close enough, see paper and websites
# http://www.kev-smith.com/papers/SLIC_Superpixels.pdf
# http://ivrg.epfl.ch/research/superpixels
shale.kmeans<-kmeans(shale.kmeans.data,
                               centers=shale.superpixels.centers,
                               iter.max=50
  )
shale.image$km.cluster<-shale.kmeans$cluster

+/- r Code

ggplot(shale.image,
       aes(x=x,y=y,alpha=Intensity))+
  geom_tile(aes(fill=as.factor(km.cluster)),size=0)+#,alpha=0.75)+
  scale_alpha_continuous(range=c(0.5,1))+
  guides(fill=F)+
  coord_equal()+
  labs(fill="")+
  theme_bw(10)

Why use superpixels

Drastically reduced data size, serves as an initial segmentation showing spatially meaningful groups

+/- r Code

shale.centers<-mutate(as.data.frame(shale.kmeans$centers),x=pos.scale*x,y=pos.scale*y)
ggplot(shale.centers,
       aes(x,y,color=Intensity))+
  geom_point(size=3)+
  scale_color_gradientn(colours=rainbow(5))+
  labs(x="X",y="Y")+
  coord_equal()+
  theme_bw(20)

+/- r Code

ggplot(shale.centers,
       aes(x=Intensity))+
  geom_vline(aes(xintercept=Intensity),alpha=0.02)+
  geom_density(aes(y=..count..))+
  geom_vline(aes(color="Threshold"),color="red",xintercept=c(130,180))+
  labs(y="Number of Superpixels")+
  theme_bw(20)

Segment the superpixels and apply them to the whole image (only a fraction of the data and much smaller datasets)

shale.centers<-mutate(shale.centers,
                      cluster=1:nrow(shale.centers),
                      group=as.factor(ifelse(Intensity<130,"Dark",
                                   ifelse(Intensity<180,"Medium","Light")))
                      )
grp.image<-merge(shale.image,shale.centers[,c("cluster","group")],by.x="km.cluster",by.y="cluster")

+/- r Code

ggplot(grp.image,
       aes(x=x,y=y,alpha=Intensity))+
  geom_tile(aes(fill=as.factor(group)),size=0)+#,alpha=0.75)+
  scale_alpha_continuous(range=c(0.5,1))+
  coord_equal()+
  labs(fill="")+
  theme_bw(10)

Superpixels vs Standard Segmentation

Superpixels (0.06% of the original)

+/- r Code

ggplot(grp.image,
       aes(x=x,y=y,alpha=Intensity))+
  geom_tile(aes(fill=as.factor(group)),size=0)+#,alpha=0.75)+
  scale_alpha_continuous(range=c(0.5,1))+
  coord_equal()+
  labs(fill="")+
  theme_bw(10)

Original

+/- r Code

ggplot(
  mutate(shale.image,group=as.factor(ifelse(Intensity<130,"Dark",
                                   ifelse(Intensity<180,"Medium","Light")))),
       aes(x=x,y=y,alpha=Intensity))+
  geom_tile(aes(fill=as.factor(group)),size=0)+#,alpha=0.75)+
  scale_alpha_continuous(range=c(0.5,1))+
  coord_equal()+
  labs(fill="")+
  theme_bw(10)

Probabilistic Models of Segmentation

A more general approach is to use a probabilistic model to segmentation. We start with our image $I(\vec{x}) \forall \vec{x}\in \mathbb{R}^N$ and we classify it into two phases $\alpha$ and $\beta$

$$P(\{\vec{x} , I(\vec{x})\} | \alpha) \propto P(\alpha) + P(I(\vec{x}) | \alpha)+ P(\sum_{x^{\prime} \in \mathcal{N}} I(\vec{x^{\prime}}) | \alpha)$$

• $P(\{\vec{x} , f(\vec{x})\} | \alpha)$ the probability a given pixel is in phase $\alpha$ given we know it's position and value (what we are trying to estimate)
• $P(\alpha)$ probability of any pixel in an image being part of the phase (expected volume fraction of that phase)
• $P(I(\vec{x}) | \alpha)$ probability adjustment based on knowing the value of $I$ at the given point (standard threshold)
• $P(f(\vec{x^{\prime}}) | \alpha)$ are the collective probability adjustments based on knowing the value of a pixels neighbors (very simple version of Markov Random Field approaches)

Contouring

Expanding on the hole filling issues examined before, a general problem in imaging is identifying regions of interest with in an image.

• For samples like brains it is done to identify different regions of the brain which are responsible for different functions.

• In material science it might be done to identify a portion of the sample being heated or understress.

• There are a number of approaches depending on the clarity of the data and the

Convex Hull Approach

takes all of the points in a given slice or volume and finds the smallest convex 2D area or 3D volume (respectively) which encloses all of those points.

+/- r Code

test.im<-function(n) mutate(
  data.frame(th=c(0:200)/200.0*2*pi),
  r=cos(th*n),
  x=r*cos(th),
  y=r*sin(th)
  )
pthull<-function(pts,x.col="x",y.col="y") {
  pts.hull<-chull(pts[,c(x.col,y.col)])
  pts[c(pts.hull,pts.hull[1]),c(x.col,y.col)]
}
test.imgs<-ldply(c(1:6),function(n) {
  tdata<-test.im(n)[,c("x","y")]
  cbind(rbind(cbind(tdata,src="pts"),
        cbind(pthull(tdata),src="hull")),n=n)
})

ggplot(subset(test.imgs,src=="pts"),aes(x,y))+
  geom_polygon(data=subset(test.imgs,src=="hull"),aes(fill="convex\nhull"),alpha=0.5)+
  geom_point()+
  labs(fill="")+
  facet_wrap(~n)+
  theme_bw(20)

Depending on the type of sample the convex hull can make sense for filling in the gaps and defining the boundaries for a sample.

+/- r Code

test.img<-subset(test.imgs,n==2 & src=="pts")[,c("x","y")]
ggplot(test.img,aes(x,y))+
  geom_polygon(data=pthull(test.img),aes(fill="convex\nhull"),alpha=0.5)+
  geom_point()+
  labs(fill="")+
  theme_bw(20)

The critical short coming is it is very sensitive to single outlier points.

+/- r Code

test.img<-rbind(subset(test.imgs,n==2 & src=="pts")[,c("x","y")],data.frame(x=-1,y=-1))
ggplot(test.img,aes(x,y))+
  geom_polygon(data=pthull(test.img),aes(fill="convex\nhull"),alpha=0.5)+
  geom_point()+
  labs(fill="")+
  theme_bw(20)

Convex Hull Example

The convex hull very closely matches the area we would define as 'bone' without requiring any parameter adjustment, resolution specific adjustments, or extensive image-processing, for such a sample a convex hull is usually sufficient.

+/- r Code

cortbone.im<-imagedata(t(png::readPNG(qbi.file("bone-section.png"))[,,1]),"grey")
cortbone.df<-im.to.df(cortbone.im)
# calculate convex hull
cortbone.chull<-chull(subset(cortbone.df,val>0.5)[,c("x","y")])
cortbone.chull<-c(cortbone.chull,cortbone.chull[1])
cortbone.chull<-subset(cortbone.df,val>0.5)[cortbone.chull,c("x","y")]
ggplot(subset(cortbone.df,val>0.5),aes(x=x,y=518-y))+
  geom_polygon(data=cortbone.chull,aes(fill="convex hull"),alpha=0.5)+
  geom_raster(aes(fill="original"),alpha=0.8)+
  labs(fill="Image",y="y",x="x",title="Convex Hull Creation")+
  coord_equal()+
  theme_bw(20)

Here is an example of the convex hull applied to a region of a cortical bone sample. The green shows the bone and the red shows the convex hull. Compared to a visual inspection, the convex hull overestimates the bone area as we probably would not associate the region where the bone curves to the right with 'bone area'

+/- r Code

cortbone.im<-imagedata(t(png::readPNG(qbi.file("bone.png"))),"grey")
cortbone.df<-im.to.df(cortbone.im)
# calculate convex hull
cortbone.chull<-chull(subset(cortbone.df,val<1)[,c("x","y")])
cortbone.chull<-c(cortbone.chull,cortbone.chull[1])
cortbone.chull<-subset(cortbone.df,val<1)[cortbone.chull,c("x","y")]
ggplot(subset(cortbone.df,val<1),aes(x=x,y=518-y))+
  geom_polygon(data=cortbone.chull,aes(fill="convex hull"),alpha=0.5)+
  geom_raster(aes(fill="original"),alpha=0.8)+
  labs(fill="Image",y="y",x="x",title="Convex Hull Creation")+
  coord_equal()+
  theme_bw(20)

Rubber Band

## the rubber band function to fit a boundary around the curve
rubber.band.data<-function(raw.df,binning.pts=10,eval.fun=max) {
  in.df<-raw.df
  # calculate center of mass
  com.val<-colMeans(in.df)
  # add polar coordinates
  in.df$r<-with(in.df,sqrt((x-com.val["x"])^2+(y-com.val["y"])^2))
  in.df$theta<-with(in.df,atan2(y-com.val["y"],x-com.val["x"]))
  # create a maximum path
  outer.path<-ddply.cutcols(in.df,.(cut_interval(theta,binning.pts)),function(c.section) data.frame(r=eval.fun(c.section$r)))
  outer.path$x<-with(outer.path,r*cos(theta)+com.val["x"])
  outer.path$y<-with(outer.path,r*sin(theta)+com.val["y"])
  outer.path
}

1. Calculates the center of mass.
2. Transforms sample into Polar Coordinates
3. Calculates a piecewise linear fit $r=f(\theta)$

+/- r Code

ggplot(subset(cortbone.df,val<1),aes(x=x,y=518-y))+
  geom_polygon(data=rubber.band.data(subset(cortbone.df,val<1),9),aes(fill="rubber band   9pts"),alpha=0.5)+
  geom_polygon(data=rubber.band.data(subset(cortbone.df,val<1),36),aes(fill="rubber band  36pts"),alpha=0.5)+
  geom_raster(aes(fill="original"),alpha=0.8)+
  labs(fill="Image",y="y",x="x")+
  coord_equal()+
  theme_bw(20)

+/- r Code

ggplot(subset(cortbone.df,val<1),aes(x=x,y=518-y))+
  geom_polygon(data=rubber.band.data(subset(cortbone.df,val<1),90),aes(fill="rubber band  90pts"),alpha=0.5)+
  geom_polygon(data=rubber.band.data(subset(cortbone.df,val<1),180),aes(fill="rubber band 180pts"),alpha=0.5)+
  geom_raster(aes(fill="original"),alpha=0.8)+
  labs(fill="Image",y="y",x="x")+
  coord_equal()+
  theme_bw(20)

+/- r Code

ggplot(subset(cortbone.df,val<1),aes(x=x,y=518-y))+
  geom_polygon(data=rubber.band.data(subset(cortbone.df,val<1),180),aes(fill="rubber band 180pts"),alpha=0.5)+
  geom_polygon(data=rubber.band.data(subset(cortbone.df,val<1),720),aes(fill="rubber band 720pts"),alpha=0.5)+
  geom_raster(aes(fill="original"),alpha=0.8)+
  labs(fill="Image",y="y",x="x")+
  coord_equal()+xlim(50,150)+ylim(-100,50)+
  theme_bw(20)

Rubber Band: More flexible constraints

If we use quartiles or the average instead of the maximum value we can make the method less sensitive to outlier pixels

+/- r Code

ggplot(subset(cortbone.df,val<1),aes(x=x,y=518-y))+
  geom_raster(aes(fill="original"),alpha=0.8)+
  geom_polygon(data=rubber.band.data(subset(cortbone.df,val<1),90,eval.fun=mean),
               aes(fill="rubber band mean value"),alpha=0.5)+
  geom_polygon(data=rubber.band.data(subset(cortbone.df,val<1),90,
                                     eval.fun=function(x) quantile(x,0.99)),
               aes(fill="rubber band upper 99%"),alpha=0.5)+

  labs(fill="Image",y="y",x="x")+
  coord_equal()+
  theme_bw(20)

Contouring: Manual - Guided Methods

Many forms of guided methods exist, the most popular is known simply as the Magnetic Lasso in Adobe Photoshop (video).

The basic principal behind many of these methods is to optimize a set of user given points based on local edge-like information in the image. In the brain cortex example, this is the small gradients in the gray values which our eyes naturally seperate out as an edge but which have many gaps and discontinuities.

Active Contours / Snakes

Beyond

• A multitude of other techniques exist for classifying groups and courses in Data Science and Artificial Intelligence go into much greater details.
• These techniques are generally underused because they are complicated to explain and robustly test and can arouse suspicion from reviewers.
  • Because of their added complexity it is easier to manipulate these methods to get desired results from almost any dataset
  • But if the approach is based on a physical model of the images and the underyling system it is acceptable
• Additionally they usually require some degree of implementation (coding).

Fuzzy Classification

Fuzzy classification based on Fuzzy logic and Fuzzy set theory and is a general catagory for multi-value logic instead of simply true and false and can be used to build IF and THEN statements from our probabilistic models.

Instead of

$$P(\{\vec{x} , I(\vec{x})\} | \alpha) \propto P(\alpha) + P(I(\vec{x}) | \alpha)+$$ $$P(\sum_{x^{\prime} \in \mathcal{N}} I(\vec{x^{\prime}}) | \alpha)$$

Clear simple rules

which encompass aspects of filtering, thresholding, and morphological operations

• IF the intensity if dark ($<100$)
  • AND a majority of the neighborhood ($\mathcal{N}$) values are dark ($<100$)
• THEN it is a cell