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Edge Linking-Digital Image Processing-Lecture 19 Slides Slides-Electrical and Computer Engineering, Slides of Digital Image Processing

University of Alabama (UA)Digital Image Processing

Edge Linking, Graph Theoretic Techniques, Global, Processing, Theoretic, Graph Theory, Edge Element, Costs, Minimum Cost, Path, Heuristic, Goal Node, Dijkstra, Digital Image Processing, Lecture Slides, Dr D J Jackson, Department of Electrical and Computer Engineering, University of Alabama, United States of America.

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Dr. D. J. Jackson Lecture 19-1Electrical & Computer Engineering
Computer Vision &
Digital Image Processing
Edge Linking Via Graph Theoretic
Techniques
Dr. D. J. Jackson Lecture 19-2Electrical & Computer Engineering
Global Processing via Graph-Theoretic
Techniques
•The previous method for edge-linking discussed is based on
obtaining a set of edge points through a gradient operation.
•As the gradient is a derivative, the operation is seldom
suitable as a preprocessing step in situations characterized
by high noise content.
•Here, we discuss a global approach based on representing
edge segments in the form of a graph and searching the
graph for low-cost paths that correspond to significant
edges.
–This representation provides a rugged approach that performs well in
the presence of noise.
–As might be expected, the procedure is considerably more
complicated and requires more processing time than the methods
discussed so far.
pf3
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Download Edge Linking-Digital Image Processing-Lecture 19 Slides Slides-Electrical and Computer Engineering and more Slides Digital Image Processing in PDF only on Docsity!

Electrical & Computer Engineering Dr. D. J. Jackson Lecture 19-

Computer Vision &

Digital Image Processing

Edge Linking Via Graph Theoretic

Techniques

Global Processing via Graph-Theoretic

Techniques

  • The previous method for edge-linking discussed is based on

obtaining a set of edge points through a gradient operation.

  • As the gradient is a derivative, the operation is seldom

suitable as a preprocessing step in situations characterized

by high noise content.

  • Here, we discuss a global approach based on representing

edge segments in the form of a graph and searching the

graph for low-cost paths that correspond to significant

edges.

  • This representation provides a rugged approach that performs well in the presence of noise.
  • As might be expected, the procedure is considerably more complicated and requires more processing time than the methods discussed so far.

Electrical & Computer Engineering Dr. D. J. Jackson Lecture 19-

Basic Graph Theory

  • We begin the development with some basic definitions.
  • A graph G = ( N, A ) is a finite, nonempty set of nodes N ,

together with a set A of unordered pairs of distinct elements

of N.

  • Each pair ( ni , nj ) of A is called an arc.
  • A graph in which the arcs are directed is called a directed

graph.

  • If an arc is directed from node ni to node nj , then nj is said to

be a successor of its parent node ni.

  • The process of identifying the successors of a node is called

expansion of the node.

Basic Graph Theory (continued)

Ni

Nj

Ai,j =( ni , nj )

Ni

Nk Nj Nl

Successors of parent node N i

Electrical & Computer Engineering Dr. D. J. Jackson Lecture 19-

Edge Elements

  • An edge element is the boundary between two

pixels p and q, such that p and q are 4-neighbors

  • In this context, an edge is a sequence of edge

elements.

p q

Edge Elements

  • We illustrate how the foregoing concepts apply to

edge detection with the 3 x 3 image shown below:

  • The outer numbers are pixel coordinates and the

numbers in parentheses represent intensity.

(7) (2) (2)

(5) (7) (2)

(5) (1) (0)

Electrical & Computer Engineering Dr. D. J. Jackson Lecture 19-

Edge Element Costs

  • Each edge element defined by pixels p and q has

an associated cost, defined as

  • where H is the highest intensity value in the image

(7 in this case), f(p) is the intensity value of p, and

f(q) is the intensity value of q. As indicated earlier, p

and q are 4-neighbors.

c ( p , q )= H −[ f ( p )− f ( q )]

Edge Element Cost Graph

(7) (2) (2)

(5) (7) (2)

(5) (1) (0)

Start

Electrical & Computer Engineering Dr. D. J. Jackson Lecture 19-

Approximating a Minimum Cost Path

  • In general, the problem of finding a minimum-cost path is not

trivial in terms of computation.

  • Typically, the approach is to sacrifice optimality for the sake

of speed, and the following algorithm represents a class of

procedures that use heuristics in order to reduce the search

effort.

  • Let r(n) be an estimate of the cost of a minimum-cost path from the start node s to a goal node, where the path is constrained to go through n.
  • This cost can be expressed as the estimate of the cost of a minimum- cost path from s to n plus an estimate of the cost of that path from n to a goal node; that is,

r(n) = g(n) + h(n)

Approximating a Minimum Cost Path

  • Here, g(n) can be chosen as the lowest cost path

from s to n found so far, and h(n) is obtained by

using any available heuristic information (such as

expanding only certain nodes based on previous

costs in getting to that node).

  • An algorithm that uses r(n) as the basis for

performing a graph search is as follows.

Electrical & Computer Engineering Dr. D. J. Jackson Lecture 19-

Approximating a Minimum Cost Path

  • Step 1: Mark the start node OPEN and set g(s) = 0.
  • Step 2: If no node is OPEN exit with failure; otherwise, continue.
  • Step 3: Mark CLOSED the OPEN node n whose estimate r(n) computed from r(n)=g(n)+h(n) is smallest. (Ties for minimum r values are resolved arbitrarily, but always in favor of a goal node.)
  • Step 4: If n is a goal node, exit with the solution path obtained by tracing back through the pointers; otherwise, continue.
  • Step 5: Expand node n , generating all of its successors. (If there are no successors go to step 2.)
  • Step 6: If a successor n (^) i is not marked, set r ( n (^) i ) = g ( n ) + c ( n, n (^) i ) mark it OPEN, and direct pointers from it back to n.
  • Step 7 : If a successor n (^) j is marked CLOSED or OPEN, update its value by letting g' ( n (^) i ) = min[ g ( n (^) i ), g ( n ) + c ( n , n (^) i )]. Mark OPEN those CLOSED successors whose g' values were thus lowered and redirect to n the pointers from all nodes whose g' values were lowered. Go to step 2.

Approximating a Minimum Cost Path

  • In general, this algorithm does not guarantee a

minimum-cost path; its advantage is speed via the

use of heuristics.

  • However, if h ( n ) is a lower bound on the cost of the

minimal-cost path from node n to a goal node, the

procedure indeed yields an optimal path to a goal

(Hart, Nilsson, and Raphael [1968]).

  • If no heuristic information is available (that is, h =0),

the procedure reduces to the uniform-cost algorithm

of Dijkstra [1959].