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Next: Optimal Path Finding Up: Extracting Curvilinear Features from Previous: Introduction

Minimising Cost Functions

There are a wide range of approaches to the problem of curvilinear feature extraction depending on the type of images (noisy or clean, man-made or natural) and the type of features involved (broad or narrow, straight or twisting).

Although many domain dependent algorithms have been published to tackle these problems, it is often possible to pose the problem as one of finding a minimum cost path through a network. For example, the problem of finding a narrow dark line in a light image, given end points on the line, involves finding a path between the two points which minimises


\begin{displaymath}\sum_{\underline{x} \in S} I_{\underline {x}}
\end{displaymath}

where $I_{\underline{x}}$ is the image intensity at pixel position $\underline{x}$ and S is the set of pixels along the path.

If relatively straight features are to be extracted, the curvature along the path could be minimised simultaneously by adding a term of the form


\begin{displaymath}k \sum_{\underline{x} \in S} c_{\underline {x}}
\end{displaymath}

where $c_{\underline{x}}$ is an estimate of the curvature of the path at pixel position $\underline{x}$ and k is a constant which determines the relative strength of the term.

If edges are to be extracted rather than lines, for example along the boundaries of a broad curvilinear feature such as a river estuary in an aerial image, the minimum cost path approach may be applied to the gradient magnitude image rather than the raw image.

If a path with target intensity It is required, an appropriate cost function would be


\begin{displaymath}\sum_{\underline{x} \in S} \vert I_t-I_{\underline {x}}\vert
\end{displaymath}

The use of path searching approaches for line and edge following in images was proposed many years ago. A distinction may be drawn between exhaustive optimal search algorithms which find the global minimum cost path and heuristic search algorithms which use heuristics to reduce the search space. For example, Montanari [] introduced dynamic programming for optimal curve detection with respect to a particular figure of merit and Martelli [,] introduced heuristic search to detect edges and contours.

Since these early papers, many authors have applied heuristic search techniques to the problem of boundary and line detection, for example [,]. Optimal search times have been significantly less attractive than heuristic search when powerful heuristics are available, and their computationally intensive nature has resulted in their lack of popularity. More recently, the cost minimisation approach to feature detection has been reworked in the idea of snakes [] which find a local solution to the minimum cost path problem given an initial path by the user. If the local minimum is not correct, additional terms may be added to the cost function interactively to push or pull the snake towards the correct position.


next up previous
Next: Optimal Path Finding Up: Extracting Curvilinear Features from Previous: Introduction
Mark Dobie
1999-03-04