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Chapter 2 ■ Edge-Detection Techniques 39

2.3 Edge Models: The Marr-Hildreth
Edge Detector

In the late 1970s, David Marr attempted to combine what was known about
biological vision into a model that could be used for machine vision. According
to Marr, ‘‘ ... the purpose of early visual processing is to construct a primitive but
rich description of the image that is to be used to determine the reflectance and
illumination of the visible surfaces, and their orientation and distance relative to the
viewer’’ [Marr 1980]. He called the lowest level description the primal sketch,a
major component of which are the edges.
Marr studied the literature on mammalian visual systems and summarized
these in five major points:
1. In natural images, features of interest occur at a variety of scales. No single
operator can function at all of these scales, so the result of operators at
each of many scales should be combined.

2. A natural scene does not appear to consist of diffraction patterns or other
wave-like effects, and so some form of local averaging (smoothing)must
take place.
3. The optimal smoothing filter that matches the observed requirements of
biological vision (smooth and localized in the spatial domain and smooth
and band-limited in the frequency domain) is the Gaussian.
4. When a change in intensity (an edge) occurs there is an extreme value in
the first derivative or intensity. This corresponds to a zero crossing in the
second derivative.
5. The orientation independent differential operator of lowest order is the
Laplacian.

Each of these points is either supported by the observation of vision systems
or derived mathematically, but the overall grounding of the resulting edge
detector is still a little loose. However, based on the preceding five points, an
edge-detection algorithm can be stated as follows:
1. Convolve the image I with a two-dimensional Gaussian function.
2. Compute the Laplacian of the convolved image; call this L.
3. Find the edge pixels — those for which there is a zero crossing in L.

The results of convolutions with Gaussians having a variety of standard
deviations are combined to form a single edge image. Standard deviation is a
measure of scale in this instance.