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Active Contours and Active Surfaces

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Book cover Guide to Medical Image Analysis

Part of the book series: Advances in Computer Vision and Pattern Recognition ((ACVPR))

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Abstract

Active contours and active surfaces are means of model-driven segmentation. Their use enforces closed and smooth boundaries for each segmentation irrespective of the image content. They are particularly useful if such properties cannot be derived everywhere from the data. We will discuss explicit and implicit active contours, their definition, parameterization, and properties. Different fitting methods for active contours will be presented in detail, since their understanding is necessary to understand parameterization and stability issues. We present methods to extend implicit active contours to multi-label segmentation and we discuss how to deal with intensity variations and inhomogeneous background intensity.

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Notes

  1. 1.

    The procedure is similar to optical flow computation by simultaneous minimization of the error for the Horn–Schunck constraint and the difference between neighboring displacement vectors.

  2. 2.

    The advantage over using a pixel or voxel grid is that unique surface representations for a surface passing the cell exist. This is important for an active contour because it is a surface in 2D or 3D. Using a pixel or voxel grid requires an additional mechanism to resolve ambiguities, which are known, for instance, from the marching cube algorithm on a voxel grid.

  3. 3.

    The chain rule for a function g of several functions f 1, f 2, …, f n is \( \frac{{{\text{d}}\left[ {g\left( {f_{1} \left( t \right),f_{2} \left( t \right), \ldots ,f_{n} \left( t \right)} \right)} \right]}}{{{\text{d}}t}} = \frac{{{\text{d}}g}}{{{\text{d}}f_{1} }}\frac{{{\text{d}}f_{1} }}{{{\text{d}}t}} + \frac{{{\text{d}}g}}{{{\text{d}}f_{2} }}\frac{{{\text{d}}f_{2} }}{{{\text{d}}t}} + \cdots + \frac{{{\text{d}}g}}{{{\text{d}}f_{n} }}\frac{{{\text{d}}f_{n} }}{{{\text{d}}t}} \).

  4. 4.

    It is a solution under some specializing assumptions that are necessary to define a unique optimum because the formulation contains a free parameter. Details are described in Caselles et al. (1997). It does not restrict the use of this formalism to find an optimal active contour.

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  1. Computer Science Department, ISG, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

    Klaus D. Toennies

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  1. Klaus D. Toennies
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Toennies, K.D. (2017). Active Contours and Active Surfaces. In: Guide to Medical Image Analysis. Advances in Computer Vision and Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-4471-7320-5_9

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  • DOI: https://doi.org/10.1007/978-1-4471-7320-5_9

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