   Next: Using Regularization to Find Edges Up: Edges Previous: Using Mathematical Morphology

## Energy-Based Methods

In  and  Venkatesh, Owens et.\ al. describe the approach of using ``local energy'' (in the frequency domain) to find features. The local energy is found from quadrature pairs of image functions, such as the image function and its Hilbert transform. Fourier transforms of the image were initially used to find the required functions, at large computational cost. In a more ``realistic'' implementation also described, processing is performed using a function pair very similar to a first and a second derivative; this is shown to be equivalent to the method proposed originally.

This approach has the advantage of being able to find a wider variety of edge types than that typically found using antisymmetric linear filters, however, this is at the expense of optimizing the signal to noise ratio. (In  Canny makes a similar point, when discussing the symmetric and antisymmetric components of a filter.)

In  the concept of filter idempotency is also discussed, that is, the property of making no changes to the initially processed image when further iterations are applied.

A similar approach to that of Venkatesh, Owens et. al. is described in , where Perona and Malik use pairs of filters to detect ``composite edges''. The emphasis here is on edges which are formed from combinations of different simple types, rather than on a variety of separate simple edge types. The different filters which are combined to give the ``composite edge filter'' can work at different spatial scales, to match the expected scales of the various edge components. A set of filters is used to find the responses at different orientations. Typically, the second derivative of the Gaussian is used for the symmetric filter, and the Hilbert transform of the image function is used for the antisymmetric filter. The method appears to give edges which are quite fragmented.

In  Rosenthaler et. al. use a similar method; a set of oriented Gabor-based filters is used to find local energy. The oriented energy map is used to find edges and two dimensional features (see later). The results presented are fairly good, but junction connectivity is poor.   Next: Using Regularization to Find Edges Up: Edges Previous: Using Mathematical Morphology

© 1997 Stephen M Smith. LaTeX2HTML conversion by Steve Smith (steve@fmrib.ox.ac.uk)