applied independently to each set of domain points, and the resulting surfaces are
independently interpolated with a polynomial function to generate a visually
smooth, topographically variable manifold/surface.
We suggest that the resulting manifold structures (Figure 8.15) correspond to a
modified “scaling ladder” as conceptualized by Wu (1999), except that in this case,
instead of a single ladder that represents the entire landscape at “constant scales” (i.e.,
the spaces between the rungs of the ladder are constant), we have essentially generated
many geo-object-based ladders, with variable spacing between rungs. Conceptually,
the base of each ladder corresponds to the center (x, y) of the spatially dominant
structures (i.e., patches/image objects) in the scene, while the height of the rungs is
defined by the location of the annihilation points at unique scales (t) within the stack.
These annihilation points signify the spatial extent (i.e., range of influence) of the
initial object through scale. Consequently, coarser scale domains will include fewer
larger objects, each composed of an aggregation of smaller scale objects. Thus the
resulting manifold will be smoother at higher scales (as illustrated in Figure 8.15).
Conceptually, if a manifold were stretched between the center point of all the first
rungs of the many ladders, then another were stretched between rungs 2, 3, . . .
(similar to that described in Section 8.3.1.2), this would result in a multiscale
hierarchical scaling manifold, where the space between each manifold represents
unique scale domains of different size and shape that are defined by the persistence of
the objects they model. We note that while we have modeled “hard” domain
manifolds (i.e., actual threshold-based structures), this is for illustration purposes
only. Conceptually, these structures would have a diffusive (i.e., Gaussian) composition that corresponds to the notion of the “near decomposability of hierarchical
structures,” as outlined in hierarchy theory. Additionally an important result these
domain manifolds visually demonstrate is that there is no single scale of analysis for
assessing the different sized, shaped, and spatially distributed components (i.e.,
patches) within a scene but rather that object-based scales are required (Hay and
Marceau, 2004; Hay et al., 2005).
8.4 CONCLUSION
To better understand how landscape components interact through scale, we need
appropriate theory and methods for generating multiscale representations of a scene,
techniques to automatically define landscape components (i.e., patches or image
objects) of interest in remote sensing imagery, and object topology that facilitates the
ability to link and query these image objects within appropriate hierarchical structures. This is necessary as both nested and unseated hierarchies may contain the same
focal object, but their relationships are completely different; consequently the
information resulting from each type of hierarchy will be very different, which in
turn will affect the methods used for scaling, the questions that can be posed, and the
conclusions that can be drawn (see Section 8.1.2).
What does a scale domain look like and where is it located? In this chapter we
describe a novel integration of scale space and hierarchy theory for automatically
162
VISUALIZING SCALE-DOMAIN MANIFOLDS
independently interpolated with a polynomial function to generate a visually
smooth, topographically variable manifold/surface.
We suggest that the resulting manifold structures (Figure 8.15) correspond to a
modified “scaling ladder” as conceptualized by Wu (1999), except that in this case,
instead of a single ladder that represents the entire landscape at “constant scales” (i.e.,
the spaces between the rungs of the ladder are constant), we have essentially generated
many geo-object-based ladders, with variable spacing between rungs. Conceptually,
the base of each ladder corresponds to the center (x, y) of the spatially dominant
structures (i.e., patches/image objects) in the scene, while the height of the rungs is
defined by the location of the annihilation points at unique scales (t) within the stack.
These annihilation points signify the spatial extent (i.e., range of influence) of the
initial object through scale. Consequently, coarser scale domains will include fewer
larger objects, each composed of an aggregation of smaller scale objects. Thus the
resulting manifold will be smoother at higher scales (as illustrated in Figure 8.15).
Conceptually, if a manifold were stretched between the center point of all the first
rungs of the many ladders, then another were stretched between rungs 2, 3, . . .
(similar to that described in Section 8.3.1.2), this would result in a multiscale
hierarchical scaling manifold, where the space between each manifold represents
unique scale domains of different size and shape that are defined by the persistence of
the objects they model. We note that while we have modeled “hard” domain
manifolds (i.e., actual threshold-based structures), this is for illustration purposes
only. Conceptually, these structures would have a diffusive (i.e., Gaussian) composition that corresponds to the notion of the “near decomposability of hierarchical
structures,” as outlined in hierarchy theory. Additionally an important result these
domain manifolds visually demonstrate is that there is no single scale of analysis for
assessing the different sized, shaped, and spatially distributed components (i.e.,
patches) within a scene but rather that object-based scales are required (Hay and
Marceau, 2004; Hay et al., 2005).
8.4 CONCLUSION
To better understand how landscape components interact through scale, we need
appropriate theory and methods for generating multiscale representations of a scene,
techniques to automatically define landscape components (i.e., patches or image
objects) of interest in remote sensing imagery, and object topology that facilitates the
ability to link and query these image objects within appropriate hierarchical structures. This is necessary as both nested and unseated hierarchies may contain the same
focal object, but their relationships are completely different; consequently the
information resulting from each type of hierarchy will be very different, which in
turn will affect the methods used for scaling, the questions that can be posed, and the
conclusions that can be drawn (see Section 8.1.2).
What does a scale domain look like and where is it located? In this chapter we
describe a novel integration of scale space and hierarchy theory for automatically
162
VISUALIZING SCALE-DOMAIN MANIFOLDS