Lecture of 29 August 1994

Points and Connectivity

• We say that a point p is k-adjacent to point q if p is in n-k of q.
• A k-path from point p = (x,y) to point q = (s,t) is a sequence of points (x0,y0), ... , (xn,yn) satisfying
  • (x0,y0) = (x,y)
  • (xn,yn) = (s,t)
  • for each i in 0 .. n-1, (xi,yi) is k-adjacent to (xi+1),yi+1).
• If p and q are points in a set S, then p is k-connected to q in S if there is a k-path of points from p to q consisting entirely of points in S.

  Connected Component Labeling

• Review the algorithm for labeling connected components given in section 2.4.3.

  Equivalence Relations and Connectivity

• A binary relation R on a set A is a set of ordered pairs from A. For each pair (a,b) in R, we say a is related to b under R, also written aRb.
• It is often helpful to envision a relation as a directed graph with nodes corresponding to the elements of A and arcs corresponding to the pairs in R.
• A binary relation is
  • reflexive if aRa for all a in A.
  • symmetric if either both of neither of aRb and bRa for all a and b in A.
  • transitive if whenever aRb and bRc, then aRc as well for all a, b, and c in A.
• Any relation satisfying all three of these properties is termed an equivalence relation.
• If you draw the graph of an equivalence relation, you will find that there are a number of cliques, or subgraphs satisfying the property that every node in the subgraph is directly connected to every other node in the subgraph. Those collections of elements of A corresponding to the nodes in a clique in the graph are referred to as the equivalence classes of A with respect to relation R.
• One forms the reflexive closure of a non-reflexive relation R on set A by adding to R the pair (a,a) for each a in A.
• One forms the transitive closure of a non-transitive relation by adding all those pairs and only those pairs that are necessary to cause the relation to be transitive.
• The reflexive-transitive closure of a relation is the result of forming the transitive closure of the reflexive closure of the relation.
• The k-connected relation is not transitive. It is, however, symmetric. If one forms the reflexive-transitive closure of the k-adjacent relation, then one has formed the k-connected relation. Informally verify this by consider the relationship between paths and transitivity.
• If one represents the connections between points with an adjacency matrix, having ones wherever two points are adjacent. Warshall's algorithm uses a method similar to matrix multiplication to form the transitive closure of a relation represented by an adjacency matrix.

  Metrics

• A metric (or distance function) is a binary function D from a set A into the non-negative reals s.t.
  • D(p,q) >= 0 if and only if p<>q,
  • D(p,q) = D(q,p), and
  • D(p,r) <= D(p,q) + D(q,r)
• Know the following distance formulae:
  • Euclidean Distance
  • Manhattan Distance
  • Chessboard Distance

  Imaging Geometry

  Basic Transformations

• Know about the affine transformations presented in this section.