Prediction of Stream Channel Location from Drainage Basin Boundaries
by
Robert J. Fleming Jr.
B.A., Geology (1984)
Harvard University
Submitted to the Department of Earth, Atmospheric and Planetary Sciences
in partial fulfillment of the Requirements for the Degree of
Master of Science in Geosystems
at the Massachusetts Institute of Technology
Ladgre'
February 2001
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
@ Massachusetts Institute of Technology
All rights reserved
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Signature of Author .....
DepartmenYof Earth, Atmospheric, and Planetary Sciences
11 September, 2000
Certified by .........
...................
.............
7/......KelinX.
Whipple
Assistant Professor of Geology
Thesis Supervisor
.........
Ronald G. Prinn
Chairman, Department of Earth, Atmospheric and Planetary Sciences
Accepted by ......................
.....................
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Ilmm
Abstract
Common methods of extracting representations of drainage networks from raster digital
elevation models for hydrological and geomorphological applications are similar to a
class of image processing methods known as grayscale watershed algorithms. These
algorithms partition a field of scalar values into connected regions based on a local
minimum associated with each region. A related class of image processing algorithms,
known as 2-dimensional skeletonization algorithms, reduce a planar shape to a onedimensional, connected, graph-like structure, called a skeleton, that maintains significant
information about the properties of the original shape. The morphological similarity
between the skeleton of a region and a drainage network suggest that skeletonization
algorithms might be used to relate basin shape to the drainage network within the basin.
This idea was examined by applying two 2-dimensional skeletonization algorithms to two
drainage basin boundary shapes extracted from digital elevation models to attempt to
predict stream channel locations within the basin.
The skeletons computed for the two basins studied did not predict the location of
principal channels in the interiors of the basins studied. This is due, at least in part, to the
fact that these two dimensional algorithms only consider symmetry with respect to plan
view basin shape, with no consideration made of relative elevations along basin
boundaries or position of the boundary points with respect to the basin outlet. In convex
outward salients of the upper reaches of the two basins studied, the position and planform
of computed skeletons agree reasonably well with the upper reaches of drainage networks
derived from the digital elevation model. This observation suggests a relationship
between basin boundary shape and the location and form of the channel network, at least
in the neighborhood of the boundary in upper portions of the basins.
A brief review of recent results from computational geometry and image analysis suggest
several possible methods of extending this analysis to incorporate relative elevation along
the boundary and orientation of the boundary with respect to the basin outlet, and
possibly resolving this question.
Prediction of Stream Channel Locations from Drainage Basin Boundaries
Introduction
Digital representations of drainage basins derived from digital elevation models
(DEMs) are widely used in hydrological and geomorphological studies of fluvial process
and landscape evolution. Techniques and algorithms for extraction and manipulation of
drainage basins and drainage networks derived from raster DEMs are well-established.
To a large extent, methods of analysis and modeling of hydrological and
geomorphological processes are dependent on assumptions about the nature and the scale
of the range of physical processes or phenomena being studied in a given problem.
Often, many such assumptions must be applied in order to render a real-world problem
tractable and apply even the simplest models of physical processes.
Assessing the relative importance of spatial and temporal variation in lithology,
uplift rates, bedrock or soil cover, climate, and integrated basin history to a specific
location or basin can be a significant issue in setting up a such a problem or model. One
little-studied aspect of this problem is the relationship of the bounding shape of a basin to
the three-dimensional form of the channel network within the basin. A geometric
description of basin shape is independent of assumptions about physical process and
process rates and offers a means of basin description and characterization that is only
limited by the scale and resolution of the measurements used to delineate the basin.
This study applies a simple concept from mathematical morphology, the skeleton
of a planar shape, to a raster image of a drainage basin. The question addressed is to
what extent the skeleton of basin corresponds to the location of channels within the basin.
If it is possible to predict channel location to some extent by using the basin boundary
alone, then it may be possible, for some applications, to use a basin boundary as a
compact representation of a drainage basin rather than a much larger grid of interpolated
topography. Such a relationship would also offer the possibility of applying a well-
developed body of theory and tools from computational geometry and image processing
to topographic analysis. Further, a quantitative relationship between basin shape and the
form of the drainage network within the basin may offer some insight on the dynamics of
the interactions between basins and drainage basin evolution.
Two 2-dimensional skeletonization algorithms are applied to two drainage basin
boundaries extracted from DEMs, and the location of the resulting skeletons are
qualitatively compared to channel locations in drainage networks derived from the
DEMs. The two basins used in this study, Oat Creek Basin, (Shubrick Peak Quad) and
Juan Creek Basin; (Westport and Lincoln Ridge Quads) are located in the King Range of
northern California.
Elevation maps of the two basins are shown in Figures 1 and 2.
Background
A raster digital elevation model is a regular grid of elevation values representing
topography in a specific geographic location. The USGS Level 2 1:24,000 scale raster
DEMs used in this study were derived by interpolation of digitized contour lines from
topographic maps, which were in turn derived from stereophotogrammetric analysis of
air photos, coupled with ground surveys (Elassal 1983). Horizontal (position) accuracy
is prescribed to 15 m. A DEM produced in this way is, as the name implies, a model, and
does not directly represent a discrete set of measurements.
Models of drainage networks are extracted in a variety of ways from DEMs
(Martz and Garbrecht 1993). Typically they are constructed by an algorithm which
makes one or more passes through the entire field of elevations in the DEM and
associates a flow direction or a flow path for each grid cell. Flow direction may be based
on one of several numerical estimates of gradient using neighboring grid points, or
simply by the compass direction on the grid to the neighbor of lowest elevation. This last
method, called D8 or steepest descent, appears to be widely used, is found in
commercial packages such as Arc/Info and Rivertools, and has the advantage of being
fast and simple. A number of more elaborate methods of flow direction estimation,
involving weighted partitioning of flow from a grid cell to multiple neighbors (Freeman
1991), or vector-based flow routing (Costacabral and Burges 1994), allow for estimation
of flow routing over divergent and planar sections of topography. For any of the
methods, adjustments to topography, such as pit filling or channel routing across flat
areas, may be required to ensure a connected drainage path (Moore, Grayson et al. 1993)
. New views on and techniques for pit filling, drainage enforcement, and calculation of
flow direction still seem to appear regularly, e.g. (Tarboton 1997; Martz and Garbrecht
1999).
The boundaries of drainage basins comprise drainage divides, or watersheds,
where precipitation on the topographic surface is partitioned, flowing into different basins
or subbasins and ultimately to a distinct outlet or point of minimum elevation associated
with each basin. Basin boundaries, derived as described above from a DEM, are a
byproduct of the process of determination of a flow network -- they are simply the
dividing lines between basins that are associated with different elevation minima. Both
drainage basin boundaries and drainage networks can be delineated without explicit
reference to elevation data within the basin by field survey or analysis of imagery.
This process of partitioning a field of scalar values into bounded regions on the
basis of an minimum value connected to interior points, has an analogy in the field of
image processing, a class of algorithms called watershed algorithms. Though initial work
on watershed algorithms was concerned with topographic analysis, watershed methods
have been widely applied in other areas. These algorithms are currently the subject of
active research with applications in such fields as machine vision (Bieniek and Moga
2000), pattern analysis(Li, Benie et al. 1999), medical imaging (Mariano-Goulart, Collet
et al. 1999; Riddell, Brigger et al. 1999), and machine tool control (Barre and Lopez
2000). The concept of a watershed is intuitively clear, but a rigorous computational
implementation of the concept does not appear to be a simple matter (Vincent and Soille
1991). General computational approaches include "arrowing", similar to the D8
approach (Hagyard, Razaz et al. 1996) used in DEM processing, "flooding" algorithms,
(Vincent and Soille 1991), and iterative or sequential skeletonization algorithms.
Another class of problems in image processing, so called skeletonization, or
thinning, problems, appear to deal with what is very nearly the reverse of the above
process. Two-dimensional skeletonization is the transformation of a planar shape to a
one-dimensional, graph-like representation that maintains significant information about
the properties of the original shape.
Skeletonization techniques are also the subject of current active reseach, with a
number of applications in medical imaging (Jacq, Roux et al. 2000; Wu and Bourland
2000), robot path planning (de Leon and Sossa 1998), interpretation of seismic data (Li,
Vasudevan et al. 1997),(Vasudevan, Tozser et al. 1997), and studies of transport in
porous media (Liang, loannidis et al. 2000).
A common thread in the applications cited
above is the representation of a geometric relationships in a large data set by at least a
partially connected subset of lower dimension.
The skeleton of a two-dimensional shape is frequently defined in terms of the
medial axis. The medial axis of a bounded region can be defined as the locus of points
that are minimally equidistant from at least two boundary points (Blanding, Turkiyyah et
al. 2000), or as the set of centers of largest circles tangent to the region boundary at two
or more points. Figure 3a illustrates this schematically. The relation of the medial axis
to the boundary of a shape is often described heuristically in terms of a "grassfire
algorithm" e.g. (Chin 1999). If a fire is started simultaneously along a region boundary,
and allowed to propagate inwards at a uniform rate, those points ("quench points") where
the flame line meets and extinguishes itself define the medial axis. Defined in this way,
the medial axis is unique for a given shape. If distances to the associated boundary points
for each point on the skeleton (or similarly, radii of inscribed circles, or simulated time to
"quenching") are recorded along with the skeleton position, a shape can be reconstructed
exactly from the medial axis.
Although this definition is intuitively clear, computation of a medial axis from a
discrete approximation to a shape is not so straightforward. A very wide range of
computational techniques is used for computing approximations to the medial axis for
discrete data sets. Figure 3b illustrates the sensitivity of the medial axis to small
perturbations in a simple boundary shape. Given a complex boundary shape, small
changes in position of the boundary can result in significant changes in the shape of the
medial axis that are non-local and difficult to predict. In practical applications, data
smoothing, filtering or multiscale computational approaches may be used to deal with
noisy boundary data (Ogniewicz and Kubler 1995; Costa 1999; Gauch 1999). (Pitas and
Cotsaces 2000) have pointed out that boundary propagation-based algorithms are among
the most computationally efficient implementations known for both watershed and
skeletonization algorithms.
Approximations to the medial axis of a polygon are commonly called Voronoi
skeletons, because the medial axis can be directly related to the line Voronoi diagram for
the segments comprising the polygon boundary (Okabe, Boots et al. 2000). The Voronoi
diagram of a 2-dimensional point set is the partitioning of the plane into distinct regions,
each of which comprises the set of all points nearer to one point in the original point set
than to any other. In the simplest case, the Voronoi diagram of a point set is represented
by polygons (variously called Voronoi polygons, Dirichlet polygons, or Theissen
polygons) which mark the region boundaries. A line Voronoi diagram is a similar
partitioning based on relative proximity to members of a set of line segments rather than
points, and may include curved region boundaries.
In this study, two skeletonization techniques were applied to Oat Creek and Juan
Creek basins to examine the relationship between the shape of the basins and the form of
the drainage networks within the basin. The first technique uses a discrete
approximation to the Voronoi skeleton of a polygonal representation of the basin
boundary. The second technique is an application of an algorithm that uses a simulated
magnetic field to generate a smooth skeleton from a set of points comprising the basin
boundary.
Voronoi Skeletonization
The Voronoi skeleton for both basins was calculated using Jonathan Shewchuk's
general triangulation code (triangle.c). This code, used for finite element mesh
generation and refinement, and generation of triangulated irregular networks (TINs),
allows input of a set of bounding edges to be included in the Delaunay triangulation of a
point set. The triangulation is then constrained to include the boundary edges, in this
case the set of edges from the polygonal approximation to the basin boundary. The
Voronoi diagram is then derived from the triangulation. The skeleton generated in this
way is not a line Voronoi diagram (region boundaries are all straight), but a reasonable
approximation based on the Voronoi diagram of a point set. Figures 4 and 5 display the
planar Voronoi skeleton, in black, overlain on an elevation map of each basin.
Also shown in figures 4 and 5 is a set of white lines approximating the location of
the drainage network within the basin for comparison. These were generated by applying
a 2D streamline interpolation routine (the MATLABTM stream2 function) to the
numerical gradient of the elevation field from the original DEM. A starting point for the
streamlines slightly inward from the boundary was estimated by removing 2 layers of
pixels from the basin perimeter. These lines are not continuous because no effort was
made to enforce drainage continuity by adjusting elevation or gradient - streamlines are
allowed to simply stop when they reach a local minimum. Because these streamlines are
based strictly on elevation values from the DEM, they represent approximate channel
location only where multiple streamlines have converged and are clearly following
topography. Estimation of the location of specific channels, channel heads, or the
transition from hillslope processes to channel processes in a basin is probably below the
resolution of 30m DEMs (Montgomery and Dietrich 1988; Montgomery and
Foufoulageorgiou 1993). The simple method used here has the advantage that it is fast
and simple to implement, serves as a reasonable basis for a qualitative comparison of
network structure, and does not depend on assumptions of the magnitude or nature of a
critical or threshold support area for portraying network geometry.
In the case of Oat Creek basin, the Voronoi skeleton and the drainage network are
roughly similar in form - groups of converging streamlines and groups of subparallel
streamlines (indicating planar areas of topography in the basin) correspond to similar
groups of skeleton branches. Notable exceptions include the southwest-sweeping arc on
the northwest side of the basin, not represented in the skeleton, and the significant
difference in direction between polygon edges and streamlines at lower elevations in the
basin. The interior lines of the skeleton in these areas tend to remain generally normal to
the boundary while the streamlines trend toward the basin outlet.
For Juan Creek basin, the central, or principal axis of the skeleton passes through
the centerline of the basin. Because of the presence of the southern subbasin and its
significant drainage, the pattern of the skeleton does not resemble that of the calculated
streamlines, though these agree roughly in the eastern part of the basin. A useful
exercise would have been to split Juan Creek into separate basins for this analysis. Given
the basin geometry, one would expect a result similar to Oat Creek Basin.
For both basins, the strongest qualitative similarities between calculated
streamlines and the planar Voronoi skeleton occur in prominent convex outward portions
of the boundaries at upper elevations in the basin.
The sensitivity of the calculated skeletons to boundary position was examined by
adding noise within the spatial uncertainty level to the boundary positions and
recalculating the skeleton. Each boundary data point was shifted by a random distance
uniformly distributed on the interval 0-15 m. in a random direction, also uniformly
distributed. Recalculated skeletons are shown in black in Figure 6 for Oat Creek, and
Figure 7 for Juan Creek. In each figure the skeleton calculated from the original
boundary positions is shown in gray for comparison. As can be seen, the central portion
of the skeleton appears to remain relatively stable with changes occurring in the skeleton
segments connecting with flat or concave outward sections of the boundary.
The range of possible skeleton configurations for the basin boundaries was
examined by iterating the above process. Figures 8 and 9 show the results of 1000
calculations of the Voronoi skeleton following after a random perturbation of boundary
point position, as described above. The skeleton was rendered after each iteration, and
the resulting binary image was summed. This was carried out on a grid of higher
resolution than the original DEM (5x5 grid pixels per DEM pixel for Oat Creek and 3x3
grid pixels for Juan Creek) to improve the spatial resolution of the skeleton. The color
scale in these figures indicates the number of times a pixel was included in a rendered
skeleton. Again, the central axis appears relatively stable with respect to small
perturbation of boundary position, while the greatest variability (low pixel counts) is seen
in areas near concave outward and flat sections of the boundary. Skeleton branches are
consistently persistent in convex outward portions of the boundary in the upper basin.
Constraining the maximum area and/or the minimum angles allowed in triangles
comprising a triangulation (quality meshing) is a technique used to refine finite element
meshes and to dynamically adjust mesh or TIN configurations e.g. (Braun and Sambridge
1997). If a conforming triangulation cannot be generated for a given point set subject to
such constraints, additional points, called Steiner points, are added to the data set until a
successful conforming triangulation is possible. The effect of dynamically added points
was examined in a cursory fashion by imposing a 2 degree minimum angle constraint on
the Delaunay triangulation used to generate the Voronoi skeleton, then generating and
integrating multiple iterations subject to random variation in spatial position as described
earlier. Such a constraint had little effect on the Voronoi skeleton for Oat Creek basin.
For Juan Creek, the central portion of the skeleton following the long axis of the basin
becomes less stable, and is bypassed by subparallel branches nearer the boundaries.
Figure 10Oa shows the result of 1000 iterations of the skeleton calculation subject to noise.
Figure 10Ob shows the same information with a log scale to illustrate the nature of the
skeleton structures in the center of the basin.
In this simple examination, no use was made of boundary elevation values, only
the plan-view shape of the basin. As treated above, and as generally used in image
processing, a Voronoi skeleton provides only a descriptor of shape. But a drainage basin
is defined with reference to an outlet, or point of minimum elevation on the basin
boundary. The fact that a basin boundary has an orientation and a range of elevation
values on the boundary suggests that the notion of the Voronoi skeleton could be
extended to include these attributes. Several results from the field of computational
geometry suggest possible approaches. (Aichholzer 1999) suggest a different distance
metric is useful in calculation of Voronoi diagrams in some applications where spatial
relationships are not isotropic. They define a direction-dependent distance metric, a skew
distance, to generate nearest neighbor relationships on a tilted plane. (Sugihara 1992)
defines a structure called the "Voronoi diagram on a river", which establishes proximity
relationships on the basis of an assumed directed flow. (Okabe, Boots et al. 2000)
summarize a number of weighting techniques and distance metrics which might be
applicable to this problem, such as the geodesic distance and the shortest visible path
distance metrics. Neither these approaches nor other generalizations of Voronoi
diagrams were pursued here, though they appear to be applicable, with modification, to
this problem.
Smooth Skeletonization
A second skeletonization method examined in this study is based on an algorithm
by (Shroff and Ben-Arie 1999) for determination of the smooth skeleton of a 2dimensional shape. The algorithm is implemented as follows. The basin plan-view
shape is reduced to a binary image on a white field. The numerical gradient of this image
is calculated, and a simulated magnetic dipole is placed at each non-zero point in the
gradient. The dipoles are assumed to lie in the image plane, aligned in the direction of
the gradient, with strength proportional to the gradient magnitude. Figure 1la shows the
location and orientation of the simulated dipoles with respect to the basin boundary as
defined in the DEM. Figure 1lb shows the shape of the magnetic field lines associated
with each dipole. The magnetic flux density, using an arbitrary unit value for magnetic
permeability of free space, is calculated for each point in the interior of the image by
integrating the contribution of each dipole.
The magnitude of the resulting vector field calculated for the two basins, is shown
with log scaling in Figures 12 and 13. The direction of the resulting field is shown for
each basin in Figures 14 and 15. (Shroff and Ben-Arie 1999) used a local valley finding
algorithm applied to the numerical gradient of the calculated field magnitude to
determine the skeleton. The same approach was taken here, again using the MATLABTM
streamline interpolation function applied to the gradient of the calculated field magnitude,
starting two cells in from the boundary to avoid edge effects. The streamlines generated
trace out a smooth path from points along the boundary to a sink along the central axis.
Plots of these calculated streamlines are shown in black in figures 16 and 17 for both
basins over an elevation map of the basin. Overlain in white for comparison are
streamlines estimated from the original DEMs in the manner described in the previous
section on Voronoi skeletonization.
The collection of streamlines, or skeleton, resulting from this method is smoother
and individual streamlines converge much more slowly than the with the Voronoi
skeleton. This is due to the fact that small contributions from individual dipoles are
integrated over the entire basin interior in this method, in contrast with Voronoi skeleton,
in which skeleton segments are determined principally by a small set of nearest neighbors
on the boundary. In this regard, the algorithm of (Shroff and Ben-Arie 1999) is similar
to other skeletonization algorithms that use electrostatic fields or generalized field
potentials (Ahuja and Chuang 1997; Grigorishin 1998). Otherwise the skeletons
produced are similar to the Voronoi skeletons in that the calculated streamlines in the
upper portions of the basin near prominent convex outward portions of the boundary
correspond reasonably well with streamlines estimated from the DEM.
The (Shroff and Ben-Arie 1999) algorithm simply generates a vector field with
strength inversely proportional to the cube of distance from dipoles on or near a given
boundary point. Because of the geometry of the simulated magnetic field generated by
inward pointing dipoles, field strength is reduced in convex outward areas on the
boundary along an axis roughly normal to the local convexity. The converse is true for
concave areas along the boundary. This method simulates, in a rough fashion, the
geometry of convergent and divergent topography at convex and concave outward
portions of the boundary, respectively.
Again, in this simple treatment, only plan view shape of the boundary was
considered and neither boundary elevation data nor position relative to the basin outlet
were used in generating the skeleton. One cursory attempt made to incorporate direction
into this skeletonization algorithm is shown in figure 18. Here, an additional dipole, of
unit magnitude, oriented toward the basin outlet was placed at each of the original dipole
positions. The principal (central) branches of the resulting skeleton correspond closely
with the principal drainages derived from the DEM in the upper portion of the basin,
rather than following the "smooth symmetric" centerline of the basin. This observation
just suggests that the boundary position relative to the basin outlet in this algorithm may
be locally significant. Other variations, such as weighting dipole magnitudes by relative
elevation or distance from the outlet (which did not result in skeletons significantly
different from the original algorithm). A systematic examination of the effects of such
variations is made difficult by the dependence of this method on the complex geometry
resulting from the interaction of many dipole fields.
Conclusions
Skeletons generated by applying the 2-dimensional skeletonization algorithms in
this study do not closely approximate planform of stream channels in the interior of the
two basins studied. This approach is a simple one and could only be expected to succeed
in the case of principal channels in a drainage located symmetrically with respect to the
basin boundaries. However, in convex outward salients of the upper reaches of the two
basins studied, the position and planform of computed skeletons agree reasonably well
with the upper reaches of drainage networks derived from the digital elevation model.
This observation suggests a relationship between boundary shape and the location and
form of the channel network, at least in the neighborhood of the boundary in the upper
reaches of the basins.
Intuitively, convergent topography (and the initiation of channel heads) is to be
expected in the neighborhood of convex outward portions of the boundary, from simple
geometric considerations - inward directed path normals to the boundary must converge
to some degree in these areas. Methods in general use to compute digital representations
of basin boundaries and channel networks suggest a close geometric relationship between
basin boundaries and the drainage networks they contain. The extent to which this
relationship extends into the basin interior was not made clear by this study.
The question of whether a more general notion of boundary shape can describe or
predict channel locations in three dimensions within a drainage basin is also left open, as
3-dimensional methods were not attempted. The brief review of recent results from
computational geometry and image analysis made during this study suggest that moving
from 2- to a 3-dimensional description and analysis of shape and form introduces a
number of theoretical and computational problems which bear directly on this problem.
Some of these results might be useful in addressing this more general question.
This study could have been improved in several ways. As is common in
hydrological and geomorphological applications, both basin boundaries and the drainage
network were derived from a numerical treatment of the same set of numbers, elevations
from the basin DEMs. Knowledge of the drainage network as derived from the DEM
was necessary in this case to delineate the position of basin boundary. This dependency
is not necessary. A basin boundary delineated from a precision field survey or from
photographic image analysis would provide an independent basis for comparison with the
drainage network derived from a DEM. Incorporating the elevation of basin boundaries
or the position of boundary data points relative to the basin outlet would have provided a
more complete assessment of skeletonization in this context. Comparison of generated
skeletons with complete (connected) drainage networks derived from DEMs using
standard methods (rather that the simple streamline interpolation used in this study)
would allow quantitative comparison of the spatial position, length, and other atttributes
of the drainage network. Juan Creek Basin, with the substantial subbasin in the southern
portion, should probably have been separated into two drainages for comparison with the
main basin.
List of Figures
Figure 1 - Oat Creek DEM elevation map - 30 m pixel size
elevation in meters. source: USGS 1:24,000 DEM Shubrick Peak Quad,
CA
Figure 2 - Juan Creek DEM elevation map - 30 m pixel size
elevation in meters. source USGS 1:24000 DEM Westport, Lincoln Ridge
Quads, CA.
Figure 3
a. medial axis of a rectangle, showing maximal circles
b. sensitivity of medial axis to perturbations in boundary position
from (Kalmar, Marczell et al. 1999)
Figure 4 - Voronoi skeleton compared with channel position estimated from DEM - Oat
Creek. Black lines indicate the Voronoi skeleton. White lines indicate estimate of actual
channel position derived from DEM by streamline interpolation, as described in the text.
Color scale indicates elevation in meters from the original DEM.
Figure 5 - Voronoi skeleton and channel position estimated from DEM - Juan Creek.
Black lines indicate the Voronoi skeleton. White lines indicate estimate of actual channel
position derived from DEM by streamline interpolation, as described in the text. Color
scale indicates elevation in meters from the original DEM.
Figure 6 - Sensitivity of skeleton to small changes in boundary position - four examples
of random perturbations of boundary position - Oat Creek. Blue lines indicate the
Voronoi skeleton computed from the original basin boundary derived from the digital
elevation model. Black lines indicate the Voronoi skeleton computed from perturbations
to the boundary positions as described in the text.
Figure 7 - Sensitivity of skeleton to small changes in boundary position - two examples
of random perturbations of boundary position - Juan Creek. Blue lines indicate the
Voronoi skeleton computed from the original basin boundary derived from the digital
elevation model. Black lines indicate the Voronoi skeleton computed from perturbations
to the boundary positions as described in the text.
Figure 8 - Sensitivity of Voronoi skeleton to position of boundary locations. - Oat Creek
Sum of Voronoi skeletons computed for 1000 perturbations (realizations) to boundary
positions. Color scale indicates number of times a pixel was included in a skeleton.
Pixel size used in the computation was 6 m.
Figure 9 - Sensitivity of Voronoi skeleton to position of boundary locations. - Juan
Creek Sum of Voronoi skeletons computed for 1000 perturbations (realizations) to
boundary positions. Color scale indicates number of times a pixel was included in a
skeleton. Pixel size used in the computation was 10 m.
Figure 10
a - effect of a small angle constraint on Voronoi Skeleton, Juan Creek.
calculations and color scale as in Figure 9.
b - same data as 10a, with log scaling to show interior of shape
Figure 11
a - position and layout of simulated magnetic dipoles used in smooth
skeletonization algorithm in relation to basin boundary.
b - shape of magnetic dipole field used in smooth skeleton algorithm
(from (Shroff and Ben-Arie 1999))
Figure 12 - Calculated field magnitude from smooth skeleton algorithm (log scale) - Oat
Creek
Figure 13 - Calculated field magnitude from smooth skeleton algorithm (log scale) Juan Creek
Figure 14 - Calculated field direction from smooth skeleton algorithm (log scale) - Oat
Creek. Color scale indicates direction in radians, O=East.
Figure 15 - Calculated field direction from smooth skeleton algorithm (log scale) - Juan
Creek. Color scale indicates direction in radians, 0=East.
Figure 16 - Skeleton (black lines) generated as described in the text from smooth
skeleton algorithm and comparison with streamlines from DEM - Oat Creek. White lines
indicate estimate of actual channel position derived from DEM by streamline
interpolation, as in earlier figures.
Figure 17 - Skeleton (black lines) generated from smooth skeleton algorithm and
comparison with streamlines from DEM - Oat Creek. White lines indicate estimate of
actual channel position derived from DEM by streamline interpolation, as in earlier
figures.
Figure 18 - Skeleton (black lines) for Oat Creek from modification to smooth skeleton
algorithm. In this case an additional "dipole" oriented toward basin outlet was added at
each grid position with a non-zero gradient value, as described in the text.
References
Ahuja, N. and J. H. Chuang (1997). "Shape representation using a generalized potential
field model." IEEE Transactions On Pattern Analysis and Machine Intelligence
19(2): 169-176.
Aichholzer, O., F. Aurenhammer, D.Z. Chen, and D.T.Lee (1999). "Skew Voronoi
Diagrams." International Journal of Computational Geometry and Applications
9(3): 235-247.
Barre, F. and J. Lopez (2000). "Watershed lines and catchment basins: a new 3D-motif
method." International Journal of Machine Tools & Manufacture 40(8): 11711184.
Bieniek, A. and A. Moga (2000). "An efficient watershed algorithm based on connected
components." Pattern Recognition 33(6): 907-916.
Blanding, R. L., G. M. Turkiyyah, et al. (2000). "Skeleton-based three-dimensional
geometric morphing." Computational Geometry-Theory and Applications 15(13): 129-148.
Braun, J. and M. Sambridge (1997). "Modelling landscape evolution on geological time
scales: a new method based on irregular spatial discretization." Basin Research 9:
27-52.
Chin, F. J. S., and C.A. Wing (1999). "Finding the medial axis of a simple polygon in
linear time." Discrete and Computational Geometry 21: 405-420.
Costa, L. D. (1999). "Particle systems analysis by using skeletonization and exact
dilations." Particle & Particle Systems Characterization 16(6): 273-277.
Costacabral, M. C. and S. J. Burges (1994). "Digital Elevation Model Networks (Demon)
- a Model of Flow Over Hillslopes For Computation of Contributing and
Dispersal Areas." Water Resources Research 30(6): 1681-1692.
de Leon, J. L. D. and J. H. Sossa (1998). "Automatic path planning for a mobile robot
among obstacles of arbitrary shape." IEEE Transactions On Systems Man and
Cybernetics Part B- Cybernetics 28(3): 467-472.
Elassal, A. A., and Caruso, V.M. (1983). USGS Digital Cartographic Data Standards:
Digital Elevation Models. U.S. Geological Survey Circular 895-B, U.S.
Geological Survey: 40.
Freeman, T. G. (1991). "Calculating Catchment-Area With Divergent Flow Based On a
Regular Grid." Computers & Geosciences 17(3): 413-422.
Gauch, J. M. (1999). "Image segmentation and analysis via multiscale gradient watershed
hierarchies." IEEE Transactions On Image Processing 8(1): 69-79.
Grigorishin, T., G Abdel-Hamid, Y.H. Yang (1998). "Skeletonization: an electrostatic
field-based approach." Pattern Analysis and Applications 1(3): 163-177.
Hagyard, P., M. Razaz, et al. (1996). Analysis of watershed algorithms for greyscale
images. International Conference on Image Processing, IEEE.
Jacq, J. J., C. Roux, et al. (2000). "Analytical surface recognition in three-dimensional
(3D) medical images using genetic matching: Application to the extraction of
spheroidal articular surfaces in 3D computed tomography data sets." International
Journal of Imaging Systems and Technology 11(1): 30-43.
Kalmar, Z., Z. Marczell, et al. (1999). "Parallel and robust skeletonization built on selforganizing elements." Neural Networks 12(1): 163-173.
Li, Q., K. Vasudevan, et al. (1997). "Seismic skeletonization: A new approach to
interpretation of seismic reflection data." Journal of Geophysical Research-Solid
Earth 102(B4): 8427-8445.
Li, W., G. B. Benie, et al. (1999). "Watershed-based hierarchical SAR image
segmentation." International Journal of Remote Sensing 20(17): 3377-3390.
Liang, Z., M. A. Ioannidis, et al. (2000). "Geometric and topological analysis of threedimensional porous media: Pore space partitioning based on morphological
skeletonization." Journal of Colloid and Interface Science 221(1): 13-24.
Mariano-Goulart, D., H. Collet, et al. (1999). "Routine measurements of right and left
ejection fractions thanks to the segmentation of gated blood pool emission
tomographic images by a watershed algorithm." European Journal of Nuclear
Medicine 26(9): PS103.
Martz, L. W. and J. Garbrecht (1993). "Automated Extraction of Drainage Network and
Watershed Data From Digital Elevation Models." Water Resources Bulletin
29(6): 901-908.
Martz, L. W. and J. Garbrecht (1999). "An outlet breaching algorithm for the treatment of
closed depressions in a raster DEM." Computers & Geosciences 25(7): 835-844.
Montgomery, D. R. and W. E. Dietrich (1988). "Where do channels begin?" Nature
336(6196): 232-234.
Montgomery, D. R. and E. Foufoulageorgiou (1993). "Channel Network Source
Representation Using Digital Elevation Models." Water Resources Research
29(12): 3925-3934.
Moore, I. D., R. B. Grayson, et al. (1993). Digital terrain modelling: A review of
hydrological, geomorphological, and biological applications. Terrain Analysis and
Distributed Modelling in Hydrology. K. J. Beven and I. D. Moore. Norfolk, John
Wiley & Sons: 249.
Ogniewicz, R. L. and O. Kubler (1995). "Heirarchical Voronoi Skeletons." Pattern
Recognition 28(3): 343-359.
Okabe, A., B. Boots, et al. (2000). Spatial Tesselations: Concepts and Applications of
Voronoi Diagrams. New York, John Wiley.
Pitas, I. and C. I. Cotsaces (2000). "Memory efficient propagation-based watershed and
influence zone algorithms for large images." IEEE Transactions On Image
Processing 9(7): 1185-1199.
Riddell, C., P. Brigger, et al. (1999). "The watershed algorithm: A method to segment
noisy PET transmission images." IEEE Transactions On Nuclear Science 46(3):
713-719.
Shroff, H. and J. Ben-Arie (1999). "Finding shape axes using magnetic fields." IEEE
Transactions On Image Processing 8(10): 1388-1394.
Sugihara, K. (1992). "Voronoi diagrams in a river." International Journal of
Computational Geometry and Applications 2: 29-48.
Tarboton, D. G. (1997). "A new method for the determination of flow directions and
upslope areas in grid digital elevation models." Water Resources Research 33(2):
309-319.
Vasudevan, K., P. Tozser, et al. (1997). "Delineation of surfaces by skeletonization:
Application to interpretation of 3D seismic reflection and georadar data volumes."
Geophysical Research Letters 24(7): 767-770.
Vincent, L. and P. Soille (1991). "Watersheds in digital spaces: an efficient algorithm
based on immersion simulations." IEEE Transactions on Pattern Analysis and
Machine Intelligence 13(6): 583-598.
Wu, Q. J. and J. D. Bourland (2000). "Three-dimensional skeletonization for computerassisted treatment planning in radiosurgery." Computerized Medical Imaging and
Graphics 24(4): 243-251.
.. . ....
.. .
.
I
3000
700
2500
600
2000
500
400
c
.E 1500
0
300
1000
200
500
100
0
0
I
I
500
1000
I
1500
easting (m)
I
I
I
2000
2500
3000
Figure 1 - Oat Creek DEM elevation map - 30 m pixel size
elevation in meters. source: USGS 1:24,000 DEM Shubrick Peak Quad,
CA
0
. ................
4
im iN
IM"
700
4000
-
3500 -
600
500
3000
E 2500 0)400
E 2000
300
o
1500 -
200
1000
100
500
0
0
I
1000
I
2000
I
3000
I
I
5000
4000
easting (m)
I
6000
Figure 2 - Juan Creek DEM elevation map - 30 m pixel size
II0
7000
8000
elevation in meters. source USGS 1:24000 DEM Westport, Lincoln Ridge
Quads, CA.
1
Fig. 1. Rectangles illustrating Marr's criticism. The figure illustrates a
stability problem of traditional skeletonization. Let us consider a rectangle
(top left) and its skeleton (bottom left). The inclusion of a wedge-shaped
perturbation into a side of the rectangle (top right) induces topological
changes in its skeleton (bottom right) and thus inhibits the use of noisy
boundary data. Note that the deformation of the skeleton is sustained even if
the dimensions of the perturbation go to zero.
Figure 3
a. medial axis of a rectangle, showing maximal circles
b. sensitivity of medial axis to perturbations in boundary position
from (Kalmar, Marczell et al. 1.999)
...
SI
...
...
....
...
..
...
..
..
....
I
3000
700
2500
600
.
1500-
400
0
300
1000
200
500100
0
0
500
1000
1500
2000
easting (m)
2500
3000
Figure 4 - Voronoi skeleton compared with channel position estimated from DEM - Oat
Creek. Black lines indicate the Voronoi skeleton. White lines indicate estimate of actual
channel position derived from DEM by streamline interpolation, as described in the text.
Color scale indicates elevation in meters from the original DEM.
0
-- r.
4000
700
3500
600
3000 -
500
E 2500 0)
400
2000
C
300
1500
1000
200
500 0
0
100
1000
2000
3000
4000
5000
easting (m)
6000
7000
8000
Figure 5 - Voronoi skeleton and channel position estimated from DEM - Juan Creek.
Black lines indicate the Voronoi skeleton. White lines indicate estimate of actual channel
position derived from DEM by streamline interpolation, as described in the text. Color
scale indicates elevation in meters from the original DEM.
0
I
I
II
3000:
3000
2500
2500
2000
2000
1500
1500
1000
1000
500
0
0
500
1000
2000
0
3000
3000
0
1000
2000
3000
0
1000
2000
3000
3000
2500
2500
2000
2000
1500
1500
1000
,1000
500
500
0
0
1000
'0
2000
3000
Figure 6 - Sensitivity of skeleton to small changes
in boundary position - four examples
of random perturbations of boundary position
- Oat Creek. Blue lines indicate the
Voronoi skeleton computed from the original
basin boundary derived from the digital
elevation model. Black lines indicate the Voronoi
skeleton computed from perturbations
to the boundary positions as described in the
text.
E 2500
" 2000
€-
0
1000
2000
3000
4000
5000
6000
7000
8000
6000
7000
8000
eastina (m)
E 2500
" 2000
C
0
1000
2000
3000
4000
5000
eastinq (m)
Figure 7 - Sensitivity of skeleton to small changes in boundary position - two examples
of random perturbations of boundary position - Juan Creek. Blue lines indicate the
Voronoi skeleton computed from the original basin boundary derived from the digital
elevation model. Black lines indicate the Voronoi skeleton computed from perturbations
to the boundary positions as described in the text.
....
....
...
.
SI
.....
.............
....
....
....
.....
..
..
.....................................
..
.....
..
...............
.........
I
I
500
I
3000
450
2500-
400
350
2000
300
250
"
"E1500 -
200
1000-
150
100
50050
0
0
I
500
I0
1000
1500
2000
2500
3000
northing (m)
Figure 8 - Sensitivity of Voronoi skeleton to position of boundary locations. - Oat Creek
Sum of Voronoi skeletons computed for 1000 perturbations (realizations) to boundary
positions. Color scale indicates number of times a pixel was included in a skeleton.
Pixel size used in the computation was 6 m.
-
... .. ... .. . .. ....
P
SI
11
.........................
- kin"Rom INIM..........
.....................
........................
I
4000
I
I
I
I
I
700
3500
600
3000-
500
E 2500
400
)400
2000
0
300
1500
200
1000
500
100
0
1000
2000
3000
4000
5000
northing (m)
6000
7000
8000
Figure 9 - Sensitivity of Voronoi skeleton to position of boundary locations. - Juan
Creek Sum of Voronoi skeletons computed for 1000 perturbations (realizations) to
boundary positions. Color scale indicates number of times a pixel was included in a
skeleton. Pixel size used in the computation was 10 m.
-: ..................................
.
4000
800
3500 -
700
3000
600
E 2500 -
500
o
S2000 -
400
1500-
300
1000
200
500
100
0
0
1000
2000
3000
I
I
4000
5000
northing (m)
I
6000
i
7000
I
8000
0
4000
6
3500
5
3000
E 2500
4
C:
2000 -
500
3
0
1000
2000
3000
4000
5000
northing (m)
6000
7000
8000
b.
Figure 10
a - effect of a small angle constraint on Voronoi Skeleton, Juan Creek.
calculations and color scale as in Figure 9.
b - same data as 10a, with log scaling to show interior of shape
outside
non-zero gradient values
(ball - dipole position)
line, marks gradient
direction and magnitude
T
\\I
i/
\
T
I //
/ /T
//
S//
t
-p
S\\
original perimeter
(gray squares)
//
f/
Inside
........ ....
. .. ..
:
""
..
-'.
S'.
... .. . . . . .
Fig. 2.
.:
.
,;
i ..
.- .-
.,
o.
.
. .
.
. .
. .
. .
Theoretical magnetic field lines of magnetic dipole.
Figure 11
a - position and layout of simulated magnetic dipoles used in smooth
skeletonization algorithm in relation to basin boundary.
b - shape of magnetic dipole field used in smooth skeleton algorithm
(from (Shroff and Ben-Arie 1999))
...........................................
...
.................
.....
3000
-1
2500
-2
2000
-3
E
•
-4
1500
S-5
-6
1000
-7
500
-8
-9
0
0
500
1000
1500
2000
easting (m)
2500
3000
Figure 12 - Calculated field magnitude from smooth skeleton algorithm (log scale) - Oat
Creek
~
...........
4000
MONO
WIL_
0
3500
-2
3000
-4
E 2500
1 2000
-6
-
-6
1500
-8
1000
500
-10
0
0
1000
2000
3000
4000
5000
easting (m)
6000
7000
8000
Figure 13 - Calculated field magnitude from smooth skeleton algorithm (log scale) Juan Creek
..
...
...
....
......
.....
...
...............................
3000
3
2500
2
2000
E
C:
0
rC 1500
c-1
1000
-1
500
-2
0
-3
0
500
1000
1500
2000
easting (m)
2500
3000
Figure 14 - Calculated field direction from smooth skeleton algorithm (log scale) - Oat
Creek. Color scale indicates direction in radians, O=East.
~.
. 1111..
.
......
.
WEIMmum---- 0
3
4000
3500
3000
E 2500
-1
1000
-2
500
-3
0
0
1000
2000
3000
5000
4000
easting (m)
6000
7000
8000
Figure 15 - Calculated field direction from smooth skeleton algorithm (log scale) - Juan
Creek. Color scale indicates direction in radians, O=East.
. .......
. ....
..
.........
ANN
I
'
'
'
'
'
3000
700
2500
600
2000
500
C
!E 1500
400
300
1000
200
500
100
0
500
1000
2000
1500
easting (m)
2500
3000
Figure 16 - Skeleton (black lines) generated as described in the text from smooth
skeleton algorithm and comparison with streamlines from DEM - Oat Creek. White lines
indicate estimate of actual channel position derived from DEM by streamline
interpolation, as in earlier figures.
.......
.............
..
.......
..
.
........
.........
..
......
...................
..........
....
. ......
4000
700
3500
600
3000-
-
500
E 2500 400
2000 1500
200
1000
100
500 -
0
1000
2000
3000
4000
5000
easting (m)
6000
7000
8000
Figure 17 - Skeleton (black lines) generated from smooth skeleton algorithm and
comparison with streamlines from DEM - Oat Creek. White lines indicate estimate of
actual channel position derived from DEM by streamline interpolation, as in earlier
figures.
..
....
....
....
.....
....
....
....
.....
....
....
.......
....
.. :.::.....
....
.
...
..
....
..::
:: ::.:::: :r ....
.._
3000
700
2500
600
2000.
500
0)
C
: 1500
400
300
1000
200
500
100
0
0
I
I
500
1000
I
I
1500
2000
easting (m)
I
2500
3000
Figure 18 - Skeleton (black lines) for Oat Creek from modification to smooth skeleton
algorithm. In this case an additional "dipole" oriented toward basin outlet was added at
each grid position with a non-zero gradient value, as described in the text.