Use of Topology to Find Connected Areas in Medicine

1
Use of Topology to Find Connected Areas in
Medicine

• Problems of Segmenting Ultrasound Images

prepared by Georg Doblhoff
2
What is Topology

• Many different definitions depending on the
  source
• Differing definitions depending on the
  application
• Geographical
• Computer Sciences
• Medicine
• Mathematics
• .....
• Basic and phenomenological approaches
• Some of these definitions are going to be shown
  in the following slides

3
Topology definitions 1

• The American Heritage Dictionaries1
• Topographic study of a given place (especially
  the history of a region as indicated by its
  topography).
• Computer Science The arrangement in which the
  nodes of a LAN are connected to each other.
• Medicine The anatomical structure of a specific
  area or part of the body.
• Mathematics The study of the properties of
  geometric figures or solids that are not changed
  by homeo-morphisms, such as stretching or
  bending. Donuts and picture frames are
  topologically equivalent for example.

4
Topology and Homeomphism

• Topology as the study of those properties of a
  geometric model, such as connectivity, which are
  not dependent on position i.e. that are not
  changed by homeomorphism
• A graph G is said to be homomorphic to a graph H
  if there is a mapping, called a homomorphism,
  from V(G) to V(H) such that if two vertices are
  adjacent in G then their corresponding vertices
  are adjacent in H.
• Two graphs G and H are said to be isomorphic,
  denoted by G H, if there is a one-to-one
  correspondence, called an isomorphism, between
  the vertices of the graph such that two vertices
  are adjacent in G if and only if their
  corresponding vertices are adjacent in H.

5
Topology versus Morphology

• Due to the definition via the homeomorphism there
  is an ambiguity between topology and morphology.
• Just a few definitions of morphology from
  encyclopedias on the internet
• I) The study of the form of things.
• II) 1. Literally, the study of form. The study
  of structure. 2. The form itself, as of an
  organ or part of the body. www.emedicinehealth.
  com
• To avoid confusion lets return to a more basic
  mathematical approach or definition of topology

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Topology Formal Mathematical Definition

• There is a formal definition for a topology
  defined in terms of set operations. A set X along
  with a collection of subsets T of it is said to
  be a topology if the subsets in T obey the
  following properties
• 1. The (trivial) subsets X and the empty set Ø
  are in .
• 2. Whenever sets A and B are in T , then so is
  A?B .
• 3. Whenever two or more sets are in T , then so
  is their union
• Weisstein, Eric W. "Topology." From MathWorld--A
  Wolfram Web Resource. http//mathworld.wolfram.com
  /Topology.html

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Different Branches of topology

• Algebraic topology (which includes combinatorial
  topology),
• Differential topology
• Low-dimensional topology.
• The low-level language of topology, which is not
  really considered a separate "branch" of
  topology, is known as point-set topology.
• Same source as above

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Algebraic Topology

• Study of intrinsic qualitative aspects of spatial
  objects
• surfaces,
• spheres,
• tori,
• circles,
• knots,
• links,
• configuration spaces, etc.
• Includes combinatorial topology

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Algebraic Topology

• Rubber-sheet geometry"
• Study of disconnectivities.
• Mathematical machinery for studying different
  kinds of hole structures
• Weisstein, Eric W. "Algebraic Topology." From
  MathWorld--A Wolfram Web Resource.
  http//mathworld.wolfram.com/AlgebraicTopology.htm
  l

10
Differential Topology

• Study of smooth (differentiable) manifolds.
• Nonmetrical notions of manifolds
• (while differential geometry deals with metrical
  notions of manifold)
• Weisstein, Eric W. "Differential Topology." From
  MathWorld--A Wolfram Web Resource.
  http//mathworld.wolfram.com/DifferentialTopology.
  html

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Low-Dimensional Topology

• Deals with objects that are two-, three-, or
  four-dimensional in nature.
• Low-dimensional topology should be part of
  differential topology, but
• General machinery of algebraic and differential
  topology gives only limited information
  (particularly noticeable in dimensions three and
  four)
• alternative specialized methods have evolved.

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Point Set Topology

• Point-set topology, also called set-theoretic
  topology or general topology, is the study of the
  general abstract nature of continuity or
  "closeness" on spaces.
• Basic point-set topological notions are ones like
  continuity, dimension, compactness, and
  connectedness.
• Weisstein, Eric W. "Point-Set Topology." From
  MathWorld--A Wolfram Web Resource.
  http//mathworld.wolfram.com/Point-SetTopology.htm
  l

13
Back to the Roots

• The basic definition of topology is clearly true
  for describing a set of connected elelments.
• Two elements of a set are connected if it is
  possible move from one element to the other just
  by advancing from one neighbouring element
  contained in the set to the next.
• A set is connected if any two points of the set
  are connected.
• The first thing that needs to be defined is
  thereforWich elements in a 2D/3D space are
  direct neighbours, i.e. are directly connect?

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Graphs, Grids and Connectivity (definitions 1 )

• General graph
• Hexagonal graph3-connect
• Hexagonal cell
• 6-connect
• 4-connect graph
• Orthogonal Grid
• 8-connect graph

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Graphs, Grids and Connectivity 3D-Graphs

• 6-connect graph (corresponds to the 4-connect
  graph) cubic lattice 6 faces of the cube
• 12-connect graph (corresponds to hexagonal graph
  in 2D) hexagonal densest or cubic-space-centred
  lattice
• 18-connect graph (corresponds to the 8-connect
  graph) cubic lattice 6 faces 12 edges
• 26-connect graph (corresponds to an extended
  8-connect graph) cubic lattice 6 faces 8
  corners 12 edges

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Image Segmentation (definition )

• Image segmentation is the division of an image
  into different regions.
• In a segmented image the elementary picture
  elements no longer are the pixels (or voxals) but
  different connected sets of pixels. All pixels of
  an individual connected set belong to the same
  region.
• After segmentation measurements can be performed
  on each region. Results may be used for
  individual evaluation or further segmentation.

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Boundaries in Euclidian Space

• Boundary of discrete open and closed sets
• the closed set includes the boundary
• the open set does not include the boundary
• In Euclidian space boundaries maybe
  infinitesimally narrow and thus two open sets may
  be adjacent to each other

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Discrete Boundaries

• Formally the following concept was followed
• In discrete space the boundaries will have a
  width of at least one pixel. ?
• At least one of two neighbouring sets must
  include the boundary between the two

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Internal and Extrnal Discrete Boundaries

• Internal boundaries of a four connect set of
  pixels. The internal boundary has at least one
  pixel from a neighbouring region as its direct
  neighbour.
• External boundaries of a four connect set of
  pixels are not part of the region itself but have
  at least one neighbour pixel from the region
• The general four connect boundary combines the
  internal and the external boundary to a common
  line.

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Connectivity of Boundaries

• A 4-connect region will produce a boundary (
  ), which in itself is not 4-connected, but
  8-connected.
• 8-connected areas will produce boundaries, which
  are 4-connected, thus not the thinnest
  8-connected line possible. Elements diogonally
  connected to the neighbour ( ) are added
• Possible work-around Use the boundary region of
  both regions ( and )and combine them.
  This will produce a 4-connected graph, that may
  however be thicker than necessary.

Region 2
Region 1
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Introducing Cracks as Boundaries between Discrete
Sets

• If the boundaries other than the discrete
  elements are acceptable one may define cracks
  i.e. the line between two discrete neighbouring
  elements as a borderline, instead of using a
  pixel as description of a boundrary
• Introducing cracks as boundaries allows
• Use of two open sets for two adjacent regions
• Avoiding the awkward situation of 8-connect
  boundaries for 4-connect regions and vice versa
  (see last slide)

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Skeletons

• Skeletons may be used as the basic elements that
  contain compressed information of a complete
  segmented region.
• Especially oblong elements are predestined for
  this kind of data reduction.
• The skeleton itself (i.e. the neighbourhood
  relations of each skeleton pixel) contains the
  information of direction.
• Using adequate reduction tools the total
  connected length may contain important
  information of object length.
• The number of depletions necessary to reduce to
  the skeleton pixel will contain information on
  transverse dimensions of the object and thus
  allow its reconstruction.

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Moving from Boundaries to Skeletons

• We cannot use erosion and delation to obtain an
  addequate skeleton
• Simple erosion (i.e. reducing the image by its
  boundary pixels) may led to disconnected areas
• Erosion and delation are not inverse transforms.
• Using maximal balls
• Maximal balls are fitted into the contour of the
  object
• The line of center points contains the
  information of the radii and thus allows
  reconstruction of the origin
• Points may never the less get disconected
• Defining that skeleton end points need to have
  0-radus will allow to retain connectivity and
  total skeleton length

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Moving from Boundaries to Skeletons

• Ultimate erosion
• keeps the information of all residui before they
  disapear is thus similar to the maximum ball
  method.
• Using the distance transform the original image
  can be reconstructed
• Segmenation of a region intosubsets can be
  performed due to the distance information with
  respect to the border. Regions with identical
  distance to the border are defined.

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Quality Assurance in Ultrasound

• Basics of ultrasonic beam
• Backreflection from the body
• Scanning beam image (B-Image and volume images)
• Images with Speckle Patterns of the same size
  as beam diameter
• Reduction of SNR by filling of small cystic
  objects, due to beam diameter and side lobes
• Using SNR of nonreflecting structures as
  segmentation and quality cryteria

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Using Topology for Medical Imaging

• It is obvious, that organs or other anatomically
  interesting regions constitute multiple connected
  areas.
• As connected areas are represented by topologies,
  the basic theorems of topologies may be used for
  handling the medical morphology manipulations.
• The first important task that has to be solved,
  when using medical images, is the segmentation.
• Finding robust segmentation criteria may be quite
  difficult due to high variance in overall image
  amplitude and low signal to noise ratios.
• The signal to noise ratio (SNR) may in some cases
  be a useful segmentation criterum.

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References

• Topology." The American Heritage Dictionary of
  the English Language, Fourth Edition. Houghton
  Mifflin Company, 2004. Answers.com 05 Jan. 2008.
  http//www.answers.com/topic/network-topology
• Morphological image analysis P.Soille Springer
  2004 ISBN 3-540-42988-3
• H.Heijmans. Mathematical morphology A
  mathematical approach in image processing based
  on algebra and geometry.
• Image Processing, Analysis and Machine vision,
  Milan Slonka et al. Thompson 2008 (3rd Editon)
  ISBN 978-0-495-08252-1