Many Objects and Distributions

ETHZ: 227-0966-00L

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- 25th February - Introduction and Workflows
- 3rd March - Image Enhancement (A. Kaestner)
- 10th March - Basic Segmentation, Discrete Binary Structures
- 17th March - Advanced Segmentation
- 24th March - Analyzing Single Objects
- 7th April - Analyzing Complex Objects
- 14th April - Many Objects and Distributions
- 21st April - Statistics and Reproducibility
- 28th April - Dynamic Experiments
- 12th May - Scaling Up / Big Data
- 19th May - Guest Lecture - High Content Screening
- 26th May - Guest Lecture - Machine Learning / Deep Learning and More Advanced Approaches
- 2nd June - Project Presentations

- Voronoi Tesselations
- Ghosh, S. (1997). Tessellation-based computational methods for the characterization and analysis of heterogeneous microstructures. Composites Science and Technology, 57(9-10), 1187–1210
- Self-Avoiding / Nearest Neighbor
- Schwarz, H., & Exner, H. E. (1983). The characterization of the arrangement of feature centroids in planes and volumes. Journal of Microscopy, 129(2), 155–169.
Kubitscheck, U. et al. (1996). Single nuclear pores visualized by confocal microscopy and image processing. Biophysical Journal, 70(5), 2067–77.

- Alignment / Distribution Tensor
- Mader, K. et al (2013). A quantitative framework for the 3D characterization of the osteocyte lacunar system. Bone, 57(1), 142–154
- Aubouy, M., et al. (2003). A texture tensor to quantify deformations. Granular Matter, 5, 67–70. Retrieved from http://arxiv.org/abs/cond-mat/0301018
- Two point correlation
Dinis, L., et. al. (2007). Analysis of 3D solids using the natural neighbour radial point interpolation method. Computer Methods in Applied Mechanics and Engineering, 196(13-16)

- Image Enhancment
- Highlighting the contrast of interest in images
- Minimizing Noise
- Understanding image histograms
- Automatic Methods
- Component Labeling
- Single Shape Analysis
- Complicated Shapes

- Local Environment
- Neighbors
- Voronoi Tesselation
- Distribution Tensor

- Alignment
- Self-Avoidance
- Two Point Correlation Function

We examine a number of different metrics in this lecture and additionally to classifying them as Local and Global we can define them as point and voxel-based operations.

- Nearest Neighbor
- Delaunay Triangulation
- Distribution Tensor
- Point (Center of Volume)-based Voronoi Tesselation
- Alignment

x | y | z |
---|---|---|

2 | 3 | 4 |

1 | 1 | 3 |

1 | 0 | 4 |

0 | 0 | 4 |

- Voronoi Tesselation
- Neighbor Counting
- 2-point (N-point) correlation functions

Going back to our original cell image

- We have been able to get rid of the noise in the image and find all the cells (lecture 2-4)
- We have analyzed the shape of the cells using the shape tensor (lecture 5)
- We even separated cells joined together using Watershed (lecture 6)

We can characterize the sample and the average and standard deviations of volume, orientation, surface area, and other metrics

With all of these images, the first step is always to understand exactly what we are trying to learn from our images.

- We want to know how many cells are alive

- Maybe small cells are dead and larger cells are alive \(\rightarrow\) examine the volume distribution
- Maybe living cells are round and dead cells are really spiky and pointy \(\rightarrow\) examine anisotropy

- We want to know where the cells are alive or most densely packed

- We can visually inspect the sample (maybe even color by volume)
- We can examine the raw positions (x,y,z) but what does that really tell us?
- We can make boxes and count the cells inside each one
- How do we compare two regions in the same sample or even two samples?

- We want to know how the cells are communicating

- Maybe physically connected cells (touching) are communicating \(\rightarrow\) watershed
- Maybe cells oriented the same direction are communicating \(\rightarrow\)
*average?*orientation - Maybe cells which are close
**enough**are communicating \(\rightarrow\)**?** - Maybe cells form hub and spoke networks \(\rightarrow\)
**?**

- We want to know how the cells are nourished

- Maybe closely packed cells are better nourished \(\rightarrow\) count cells in a box__?__
- Maybe cells are oriented around canals which supply them \(\rightarrow\)
**?**

- A way for counting cells in a region and estimating density without creating arbitrary boxes
- A way for finding out how many cells are
*near*a given cell, it’s nearest neighbors - A way for quantifying how far apart cells are and then comparing different regions within a sample
- A way for quantifying and comparing orientations

A tool which could be adapted to answering a large variety of problems - multiple types of structures - multiple phases

With most imaging techniques and sample types, the task of measurement itself impacts the sample. - Even techniques like X-ray tomography which *claim* to be non-destructive still impart significant to lethal doses of X-ray radition for high resolution imaging - Electron microscopy, auto-tome-based methods, histology are all markedly more destructive and make longitudinal studies impossible - Even when such measurements are possible - Registration can be a difficult task and introduce artifacts

- techniques which allow us to compare different samples of the same type.
- are sensitive to common transformations
- Sample B after the treatment looks like Sample A stretched to be 2x larger
- The volume fraction at the center is higher than the edges but organization remains the same

\[ \downarrow \]

x | y | vx | vy |
---|---|---|---|

20.19 | 10.69 | -0.95 | -0.30 |

20.19 | 10.69 | 0.30 | -0.95 |

293.08 | 13.18 | -0.50 | 0.86 |

293.08 | 13.18 | -0.86 | -0.50 |

243.81 | 14.23 | 0.68 | 0.74 |

243.81 | 14.23 | -0.74 | 0.68 |

\[ \cdots \]

So if we want to know the the mean or standard deviations of the position or orientations we can analyze them easily.

Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. | |
---|---|---|---|---|---|---|

x | 6.90 | 215.70 | 280.50 | 258.20 | 339.00 | 406.50 |

y | 10.69 | 111.60 | 221.00 | 208.60 | 312.50 | 395.20 |

Length | 1.06 | 1.57 | 1.95 | 2.08 | 2.41 | 4.33 |

vx | -1.00 | -0.94 | -0.70 | -0.42 | 0.07 | 0.71 |

vy | -1.00 | -0.70 | 0.02 | 0.04 | 0.71 | 1.00 |

Theta | -180.00 | -134.10 | -0.50 | -4.67 | 130.60 | 177.70 |

- But what if we want more or other information?

When given a group of data, it is common to take a mean value since this is easy. The mean bone thickness is 0.3mm. This is particularly relevant for groups with many samples because the mean is much smaller than all of the individual points.

- the mean of 0\(^\circ\) and 180\(^\circ\) = 90\(^\circ\)
- the distance between -180\(^\circ\) and 179\(^\circ\) is 359\(^\circ\)
- since we have not defined a tip or head, 0\(^\circ\) and 180\(^\circ\) are actually the same