Chapter 3 responses to real features are dstinguished from those due to camera noise based on the sum of second difference responses over a maximal star-convex neighborhood of each cell. This constitutes the main use of topological structure in the edge finder.
Sections 23 present the algorithm for describing second dfference responses.
Since the noise suppression algorithm does not interact with the rest of the edge finding process, 'it is presented separately, 'in Section 4 Sections 57 discuss how the clean responses are classified and boundary locations hypothesized and Section discusses problems of combining information from derent scales. In Section 9 I present results of the edge finder on a range of images and Section 0 compares the Phantom edge finder to previous edge finding. algorithms.
any second derences larger than this bound must indicate the presence of a boundary.
Boundaries in images can be classified on the basis of the shape of the inten-sities in a straight D path across the boundary. Figure shows several common intensity shapes and their second differences. In all of these patterns intensity varies continuously. Because images are represented only to finite resolution and the space of ntensities is connected, we can never observe a pattern of intensities in a digitized image that could not represent a continuous function. However, in the patterns representing boundaries, the second difference is significantly different from zero.
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Figure 1. Common patterns of intensity values along straight ID paths in an image. Left to right: no change, smooth variation 'in lighting, step edge, roof
edge, thin bar. The top row shows the image intensities, the middle row shows
their first differences, and the bottom row shows their second differences. The righthand three cases indicate the presence of boundaries, whereas the lefthand two cases do not.
We are more accustomed to thinking of abrupt changes in properties in terms the applications discussed below, all differences are taken using a consistent spacing, so this point can be finessed.
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*M.-of hgh first differences. For example, if I place a white cup in front *M.-of a dark background, first differences taken across this object boundary will be very hgh.
However, in analyzing camera images, high first dfferences are not reliable evi-dence of a boundary. Smooth variations in light intensity and smooth shading can create high first dfferences even within a region whose physical properties (material, surface color, and so forth) are homogeneous. Imposing a bound on first differences would cause spurious boundaries to be reported in regions with variations in shading. Such markings would be intuitively unreasonable and un-stable under changes in viewpoint and lghting.
Analysis of 'image intensities is not unusual 'in having usable bounds on sec-ond differences but not on first dfferences. This pattern occurs also 'in reasoning about changes in physical properties over time, because processes of change, such as boiling water or moving objects, often create high first differences (see Chap-ter for discussion). When a textured surface is seen at an angle, perspective distortion causes the size of regions composing the texture to change rapidly across the visual field. In all of these cases, hgh second differences or changes in first difference sign reliably indicate the presence of a boundary and high first differences do not.
In analyzing camera images, or other real input, we do not have access to the underlying function values, but only to dgitized versions of these values.
Intensity values are smoothed before sampling, to avoid the aliasing effects dis-cussed in Chapter 2 3 The second difference values depend on the amount of smoothing and the density at which the image has been sampled. However, the Gaussian-like smoothing used in most computer vision systems consistently
de-3 More or less. I have occasionally seen aliasing in video camera images, so apparently the smoothing is not exactly the right shape to accomplish this goal or perhaps some of it is applied after, rather than before, smoothing.
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creases differences taken between any two points 'in the 'image. Thus, any high difference detected 'in an image must reflect a high difference in the underlying continuous intensities. The converse, naturally, does not hold.
The second differences used in my edge finder are taken along a straight five-cell path. Intensities at the five-cells along the path are added together, weighted by the values [- 1, 0 2 0 - ].' All other things being equal, differences should be computed using cells as close together as possible, to provide the most de-tailed representation. However, the narrowest second difference, using three-cell paths wth weights 1 2 -11, detects artifacts due to the interlacing used in most video cameras. The dfferences are taken along straight paths, because the processes responsible for high first differences in images produce differences that are constant along straight paths, at least locally.
Readers familiar with recent research in computer vision may notice that I have been very cautious in making assertions about the real world. It is currently the fashion for theoretical analyses of computer vision algorithms to build very precise models of reality. Unfortunately, these more specific models are typically unverifiable or, in some cases, incorrect. For example, it 'is often stated that phys-ical properties change dscontinuously across boundaries. First of all, not even the physicists have any solid evidence about the dfferential structure of space and an algorithm whose input is dgitized can hardly depend on structure finer than its digitization. Secondly, at a macroscopic level, most physical processes change in a way that seems continuous, 'if viewed at a high enough resolution.
4The usual definition of the second difference is the negative of this mask. I have 'inverted the mask so that lighter regions of the 'image produce positive values, because that seems intuitively more natural.