Resel Processing

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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

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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.

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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

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U.S. Counties
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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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Modifying the Center Cell in the Case of
PixelsNeighborhood operators are frequently used
in image processing
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First and Second Order Neighbors of
KansasNeighborhood operators can also be used
with resels
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Census tract data from a small Midwestern city
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The same Data Shown As a Conventional
(Choropleth) MapWith Shading Proportional to
Population Density.
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The Same Data Shown As a Bivariate HistogramWith
Volumes Proportional to Population
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Linearly Modified (Smoothed) VersionsPopulation
by Census Tracts
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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

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3090 Communities. 3.6 km. average resolution
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Migration Turbulence in the Alps. 3090 units -
3.6 km resolution
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184 Districts. 14.7 km. average resolution
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Less of the Fine Detail. 184 units - 14.7 km
resolution
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26 Cantons. 39.2 km. average resolution
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The Broad Pattern Only. 26 units - 39.2 km
resolutionChanging the resolution has the effect
of a spatial filter.
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Three levels of administrative units and three
levels of migration resolution all at
once.Communities
Districts
Cantons
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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

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Boundary Polygons and Their Density Values32 of
48 states are on the boundary. 16 state values to
be estimated
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US interior state populationOne Actual, One
EstimatedFrom boundary state values using
Laplaces equation with a Dirichlet condition
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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.

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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.

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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.

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Population Density in KansasBy CountyCourtesy
of T. Slocum

• A piecewise continuous surface

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Population Density in Kansasby CountyEach
County Still Contains the Same Number of People

• A smooth continuous surface, with population
  pycnophylactically redistributed

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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.

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Mass Preserving Reallocation Using Areal Data

• W. Tobler, 1979, Smooth Pycnophylactic
  Interpolation for Geographical Regions, J. Am.
  Stat. Assn., 74(367)519-536.

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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

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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.

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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.

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Left to Right1. Data Polygons 2. Rasterized
3. Smoothed
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Colored Clay Before Smoothing
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Five Iterations
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Ten Iterations
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Fifteen Iterations
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Twenty Iterations
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The Smoothed Surface
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Finite Elements Also Work Courtesy of Wolf Rase
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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

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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.

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An Image Processing ExampleA 20 by 14 Image
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Quadrupled to 80 by 56but with the same total
mass
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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

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Laplacian Biharmonic Smoothing Dirichlet
Boundary Condition
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Laplacian Biharmonic Smoothing Neumann
Boundary Condition
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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

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