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Resel Processing

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Title: Resel Processing


1
Resel Processing
  • Presented to the Department of Electrical and
    Computer Engineering
  • 25 May 2001
  • Waldo Tobler
  • Professor emeritus
  • Geography department
  • University of California
  • Santa Barbara, CA 93106-4060
  • http//www.geog.ucsb.edu/tobler

2
Abstract
  • Image processing techniques are of interest to
    geographers because they are used to analyze and
    manipulate two-dimensional spatial information.
    Beyond the obvious application to remotely sensed
    imagery these techniques can also be applied to
    other geographic phenomena. U.S. Census
    population data by county can serve as an
    example. The 3141 counties, as resolution
    elements (resels), vary in size, shape, and
    orientation. Thus methods developed for pixels
    must be generalized. Several examples of such
    extensions are given.

3
Geographic Modeling
  • Some geographers use theoretical models based on
    lattice geometries.
  • Examples include
  • Monte Carlo simulation of the geographic spread
    of ideas, or of disease, or
  • Cellular automata for simulating spatial growth
    of cities.

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From Cellular Models to Real Models
  • To be more realistic the models must be based on
    administrative units for which data are available
  • Examples of such units would include counties
    that vary in size, shape, and orientation

6
U.S. Counties
7
The US County System gives us data at a
Variable Resolution
  • If you received a piece of film with a
    resolution that varied as much as this you would
    send it back to the manufacturer.
  • The sizes generally increase from East to
    West, a function of transportation, history, and
    physical environment.
  • A great deal of societal information is
    collected using these units. Does that affect
    what we think?

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9
Modifying the Center Cell in the Case of
PixelsNeighborhood operators are frequently used
in image processing
10
First and Second Order Neighbors of
KansasNeighborhood operators can also be used
with resels
11
Census tract data from a small Midwestern city
12
The same Data Shown As a Conventional
(Choropleth) MapWith Shading Proportional to
Population Density.
13
The Same Data Shown As a Bivariate HistogramWith
Volumes Proportional to Population
14
Linearly Modified (Smoothed) VersionsPopulation
by Census Tracts
15
Social data are often made available in a
hierarchy of administrative units.
  • Moving through the hierarchy changes the
    resolution and this acts as a spatial filter.
  • This is shown by migration vector fields at
    several levels of resolution for Switzerland.
  • 3.6 km resolution (3090 Gemeinde)
  • 14.7 km resolution (184 Bezirke)
  • 39.2 km resolution (26 Kantone )
  • Maps by Guido Dorigo, University of Z?rich

16
3090 Communities. 3.6 km. average resolution
17
Migration Turbulence in the Alps. 3090 units -
3.6 km resolution
18
184 Districts. 14.7 km. average resolution
19
Less of the Fine Detail. 184 units - 14.7 km
resolution
20
26 Cantons. 39.2 km. average resolution
21
The Broad Pattern Only. 26 units - 39.2 km
resolutionChanging the resolution has the effect
of a spatial filter.
22
Three levels of administrative units and three
levels of migration resolution all at
once.Communities
Districts
Cantons
23
The Dirichlet Problem
  • Given values along the boundary of a region
    determine the interior values
  • The classic example is the distribution of heat
    in a region when the values at the edges is known
  • This is normally solved analytically, or by
    finite differences, or by finite elements
  • Suppose the boundary is given by resels, as in
    the next image

24
Boundary Polygons and Their Density Values32 of
48 states are on the boundary. 16 state values to
be estimated
25
US interior state populationOne Actual, One
EstimatedFrom boundary state values using
Laplaces equation with a Dirichlet condition
26
Often We Have Observations Assembled by Areal
UnitsCensus Tracts, School Districts, and the
Like
  • We would like to convert these to continuous
    densities.
  • It is incorrect, in my opinion, to assign these
    observations to points (centroids).
  • One criterion to be satisfied is that the
    resultant maintain the data values within each
    unit.
  • This is why I invented pycnophylactic
    reallocation.

27
Pycnophylactic Reallocation
  • Allows the production of density or contour maps
    to be made from areal data.
  • It is reallocation - and somewhat of a
    disaggregation operator. My assertion is that it
    may actually improve the data.
  • It is also important for the conversion of data
    from one set of statistical units to another, as
    from census tracts to school districts.

28
An Example Population Density by County
  • Observe the discontinuities at the county
    boundaries.
  • We would like a smooth map of population density,
    in order to draw contours.
  • The usual interpolation procedure will not work
    unless we use centroids and this fiction could
    allow people to be moved from one county to
    another.

29
Population Density in KansasBy CountyCourtesy
of T. Slocum
  • A piecewise continuous surface

30
Population Density in Kansasby CountyEach
County Still Contains the Same Number of People
  • A smooth continuous surface, with population
    pycnophylactically redistributed

31
Another example
  • Migration from Illinois to other states.
  • Shown first as a piecewise continuous bivariate
    histogram, based on state outlines with volumes
    according to Illinois outmigration.
  • Then pycnophylactically interpolated.
  • The smoothed surface can be partitioned to yield
    estimated migration by arbitrary regions - the
    Great Lakes Basin for example.

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Another Example
  • This time using population data by Federal
    Planning Regions for Germany.
  • First the data are represented in a perspective
    view of a bivariate histogram.
  • This is followed by a similar view of the
    continuous population density distribution.
  • Courtesy of Wolf Rase in Bonn.

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How Pycnophylactic Reallocation Works
  • Philosophically it is based on the notion that
    people are gregarious, influence each other, are
    mobile, and tend to congregate.
  • This leads to neighboring and adjacent places
    being similar.
  • Mathematically this translates into a smoothness
    criterion (with small partial derivatives).
  • This applies to any data with spatial
    autocorrelation.

38
Mass Preserving Reallocation Using Areal Data
  • W. Tobler, 1979, Smooth Pycnophylactic
    Interpolation for Geographical Regions, J. Am.
    Stat. Assn., 74(367)519-536.

39
The Minimum of the Integral
  • The solution of the integral smoothness equation
    is given by the Laplace equation
  • ?2x ?2y 0
  • This says that the neighboring locations have
    similar values - or in a raster, that the central
    value is the average of those surrounding it.
    This immediately yields a computational algorithm

40
What the Mathematics Means
  • Imagine that each unit is built up of colored
    clay, with a different color for each unit.
  • The volume of clay represents the number of
    people, say, and the height represents the
    density.
  • In order to obtain smooth densities a spatula
    is used, but no clay is allowed to move from one
    unit into another. Color mixing is not allowed.

41
The Smoothing Is Done Using an Iterative Process
  • The first step is to rasterize the region. Then
    the smoothing is done on this raster, all the
    while maintaining the population
  • The number of iteration steps depends on the size
    of the largest region, in raster units
  • That is because the smoothing must cross from
    edge to edge of the largest region. The finer the
    raster, the higher the resolution and the longer
    the iteration time.

42
Left to Right1. Data Polygons 2. Rasterized
3. Smoothed
43
Colored Clay Before Smoothing
44
Five Iterations
45
Ten Iterations
46
Fifteen Iterations
47
Twenty Iterations
48
The Smoothed Surface
49
Finite Elements Also Work Courtesy of Wolf Rase
50
An Important Advantage of Mass Preserving
Reallocation
  • A frequent problem is the reassignment of
    observations from one set of collection units to
    a different set, when the two sets are not nested
    nor compatible. For example converting the number
    of children observed by census tract to a count
    by school district. Boundaries also change over
    time, requiring reallocation for compatibility.
  • The density values obtained using the smooth
    pycnophylactic method allow an estimate to be
    made rather simply. A cookie cutter can cut the
    continuous clay surface into the new zones with
    subsequent addition (summation) to get the count

51
Pycnophylactic reallocation also works for data
assembled within individual cells of a lattice or
grid although this was not the design objective.
  • For example, data given within pixels.
  • Not between pixels which results in a different
    effect.
  • But values in neighboring pixels are taken into
    account within a pixel by the smoothness criteria.

52
An Image Processing ExampleA 20 by 14 Image
53
Quadrupled to 80 by 56but with the same total
mass
54
Smoothing Boundary Conditions
  • The procedure can use different smoothing
    criteria. There is a choice between Laplacian and
    biharmonic smoothing
  • As the solution to a partial differential
    equation it is also necessary to specify boundary
    conditions
  • The Dirichlet condition specifies the value at
    the boundary. The Neumann condition specifies the
    gradient at the boundary

55
Laplacian Biharmonic Smoothing Dirichlet
Boundary Condition
56
Laplacian Biharmonic Smoothing Neumann
Boundary Condition
57
Does It Make a Difference?
  • As far as I know there has been only one
    comparison of mass preserving areal reallocation
    and point based interpolation
  • The following table compares the mass
    preservation property of several point based
    interpolators, based on the German data
  • This was done by Wolf Rase in his dissertation

58
Comparing Volume Preservation Using Different
Interpolations (W. Rase)
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60
U.S. CountiesThink of these the next time you
apply a neighborhood operator
61
I Appreciate Your Attention and Thank You
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