A “fuzzy” boundary

Fuzzy Boundaries

Incorporates the uncertainty or the probability associated with “set membership.” For example, “Are you in the forest or not?” This is not a boolean condition, but a highly complex boundary that changes with scale and time. The image below on the left is the boolean condition (“crisp” they say), whereas the Fuzzy Boundary (a probabilistic one) is on the right. Here the likelihood of being in the forest or not changes gradually over space.

image from Duff, D. and Guesgen, H. (2002). An evaluation of buffering algorithms in fuzzy GISs. In Proc. International Conference on Geographic Information Science (GIScience-02), pages 80–92, Boulder, Colorado. Published as Lecture Notes in Computer Science 2478 by Springer, Berlin.

These types of boundaries can be expert-based rules, where the transition from one condition to another is governed by prescribed or known relationships between conditions or states (in the buffer/not in the buffer, etc). This examples shows a rule-based fuzzy set for correction of a thermostat temperature.


There are many ways to determine the shape of the transition. A linear transition is shown below in (a) or a more Gaussian ones in b-d.

Membership functions for (a) “flat” and “gentle” slope, (b) “close to roads”, (c) “close to town”, and (d) “suitability”.
from “The Enhancement of ArcGIS with Fuzzy Set Theory” by Tahsin A. Yanar and Zuhal Akyurek — ESRI International User Conference, 2004

Explore this rule-based “fuzzy overlay” methodology in ArcGIS which uses “fuzzy membership” and “fuzzy overlay” to analyze the contribution of multiple 0-1 criteria to evaluate set membership. These can be Gaussian (left) or linear (right).

figures from the ESRI Help files. Left is FuzzyGaussian and right is two FuzzyLinear objects (blue: min = 30, max = 80 and red: min =80, max= 30).

Open the project from the “buffer” demo folder, and let’s try “fuzzy membership” on the dist2stream layer in map 2.

Try FuzzyGaussian or FuzzyNear objects (with “zero” as the “crisp value” to center on. What should the “spread” be?).
Try FuzzyLinear with min < max (show show nearness to stream).

Fuzzy boundaries based on data

The degree of “fuzziness” can also be based on data rather than a expert decision. We’ll make our own version. Open your copy of demo\dig_trial\dig_trial project and follow these instructions. We’ll make our own fuzzy boundary to show the applicability to GIS.

  1. Use the empty shapefile to digitize a new polygon
  2. Use the “Edit” tab and “create feature” tool
  3. digitize the extent of the law school as a polygon
  4. make sure the ID attribute of the feature is a 1 after you digitize it (use attribute button on the edit menu bar)
  5. convert it to raster (using your initials as the filename)
    – see Conversion Tools toolbox/ To Raster/ Feature to Raster 
    – set the cell size and extent to that of the photograph in the Environments of the tool
  6. use the reclass tool to change the NoData to zero (with the filename “your_initials_R”)
  7. right-click on the reclassed layer and choose “Data” and “Export data” to the folder \\geodata\vol1\Courses\Geol260\dig_trial.