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there has been no systematic study of how to extend edge finder algorithms to these other types of properties.

more general, but has not been tested on real images.

In the surface modelling approach, represented by Haralick 1980, 1984) Har-alick, Watson, and Laffey 1983), and Parvin and Medioni 1987), the image intensities 'in each patch of the 'image are modelled. The model for each surface patch is then analyzed to detect the presence of boundaries, e.g. by looking for zero-crossings of the second-differences of the model. The weakness in this approach is, again, the set of models. Surface models in current use can only provide good approximations for patches of image in which the intensities vary smoothly or in which there are only restricted types of boundaries (typically, again, isolated straight step edges). Thus, these approaches also fail on 'intersec-tions, sharp comers, and dense texture. Brooks 1978) discusses how some earlier edge operators can be vewed in terms of surface modelling. The segmentation algorithm of Besl and Jain 1988), the regularization proposal of Torre and Pog-gio 1986), and the corner detector of Noble 1987) represent similar approaches to image description.

In the edge operator approach, some operation (such as taking second dif-ferences) 'is applied to the image to yeld a map of "edge responses." Some test is then applied to dstinguish significant responses from those due to cam-era noise and boundaries are hypothesized to account for significant boundaries.

The Phantom edge finder falls into this class of algorithms. Other recent ex-amples 'include Marr and Hildreth 1980), Hildreth 1983) Canny 1983 186), Pearson and Robinson 1985), Grimson and Pavlidis 1985) (stereo depth data), Watt and Morgan 1985), Huertas and Medioni 1986), Young 1986), Gennert (1986), Boie, Cox and Rehak 1986), Deriche 1987), Spacek 1985), Argyle (1971), Macleod 1972), Nevatia and Babu 1980), Huttenlocher 1988) (cur-vature data), and Lee, Pavlidis, and Huang 1988). These algorithms can be

described in terms of two independent problems: what operator to use and how to distinguish real responses from noise.

Quite a variety of operator shapes have been proposed, most of them close variants of one another. Consider the ID case first. There are two basic shapes of edge operators: first difference and second dfference's Boundaries are hy-pothesized at maxima of first difference responses and at zero-crossings (and occasionally isolated maxima) of second derence responses. On a perfect step edge, the two types of operators behave similarly. First difference operators have the problem of producing spurious responses on ramps, formed by blurred bound-aries and smooth shading. They are also unable to detect isolated maxima of the second difference, known as creases or roof edges. Second difference operators, on the other hand, produce spurious boundaries in staircase patterns. These problem behaviors are shown in Chapter 9 Section 6.

Many of the edge finders lsted above use second difference operators, as the Phantom edge finder does. Those using first derence, or smilar, operators include Canny 1983, 1986), Argyle 1971), Macleod 1972), Spacek 1985), De-riche 1987), Nevatia and Babu 1980), and Gennert (1986). Gabor filters (cf Young 1986) and Difference of Gaussian operators (Marr and Hildreth 1980) are similar in shape to the second difference. Residual operators (Grimson and Pavlidis 1985, Lee, Pavlidis, and Huang 1988, Huang, Lee, and Pavlidis 1987) also seem similar in shape to second differences. The details, however, depend on the type of approximation used and have not been explored in detail. Boie, Cox, and Rehak 1986) use a combination of first and second difference type operators.

There are three methods of extending these operators to 2D.- drectional,

18 Either type may, of course, be combined with smoothing. See below.

oriented, and isotropic, shown in Fgure 34. In the drectional method, the D operator is applied along straight paths through the 2D image. This is the method used by the Phantom edge finder. Oriented operators are formed by taking directional responses from a set of parallel paths and averaging them.

This favors extended straight boundaries. Isotropic operators are created by averaging responses from directional dfferences taken about a common point, but in derent directions. Isotropic and oriented operators both distort the shape of boundaries that are not straight.

4

V

Figure 34. Left to right: directional, oriented, and isotropic methods of taking differences.

When directional or oriented edge operators are used, the results from differ-ent directions must be combined. The Phantom edge finder is unique in having a robust method for combining directional responses. Nevatia and Babu 1980) use a similar method for combining oented first difference operator responses, but it 'is unclear that their later thinning and linking algorithms are robust. Canny (1983, 1986) assumes that the directional first dfferences approximate a linear transformation and summarizes them into a gradient direction and magnitude on this basis. As we saw 'in Section 3 this assumption is not valid near sharp corners and intersections, at which Canny's edge finder performs poorly. Gennert (1986) accepts a directional response at a cell only 'if it is an extremum over all

directions and larger than the response in the perpendicular direction. This has not been thoroughly tested, but it also seems liable to make errors at comers and intersections.

The other main variation in edge operator algorithms is in how they eliminate the effects of camera noise-19 The most popular method of eliminating noise is to smooth the image before applying the operator and then remove responses wth low amplitude. The problem with this technique is that smoothing reduces the resolution of the edge finder output. Because of this, recent work has attempted to reduce the amount of smoothing required by better methods of distinguishing real responses from noise 'in the output of the edge operator. Methods using edge linking (Nevatia and Babu 1980, Persoon 1976) have been proposed, but it is unclear how well they work.

Matching representations from different scales is occasionally suggested as a method of identifying spurious edge finder responses (Marr and Hildreth 1980, Hildreth 1983, Schunck 1987, Bergholm 1987). Other researchers have suggested evaluating responses based on a sum or product of responses from different scales (Watt and Morgan 1985, Rosenfeld 1970, Schunck 1987). While preservation over multiple scales or occurrence at a sufficiently coarse scale may be useful as a measure of the importance of a boundary, neither criterion seems helpful in identifying spurious boundaries. First, many legitimate features in images appear only at the finest scale, because they are simply too small to be detected at any other scale. Secondly, in 'images with qualitatively different representations at different scales, such as the cleaning cloth image discussed in Section and 19Pearson and Robinson 1985) seem to achieve good results with only inor amounts of noise suppression. However, since my re-implementation of their algorithm is sensitive to camera noise, their low-resolution images may have been produced by some type of sub-sampling. Since camera noise 'is primarily high-frequency, sub-sampled 'images contain far less noise.

Chapter 5, Section 5, many legitimate features last only one or two scales.

Two methods for distinguishing real response from camera no'se have recently been proposed, both using image topology 'in addition to response amplitudes.

Blake 1983) and Geman and Geman 1984) use iterative procedures to assem-ble responses into extended boundaries. Although interesting, these techniques have not yet been developed 'Into robust algorithms. Furthermore, they make excessively strong assumptions about the form of boundaries (see Section 7 As discussed 'in Section 4 algorithms similar to Phantom's have also been proposed by Watt and Morgan 1985), Huertas and Medioni 1986), and Huttenlocher (1988) (curvature data). However, these researchers dcuss only the 1D case and, thus far, Phantom's algorithm is the only robust 2D version of this 'idea.

Lee, Pavlidis, and Huang 1988; also Huang, Lee and Pavlidis 1987) propose another 2D version, but the details are unclear and it has not been extensively tested.