1. 04/11/13 Collections of Sets Discrete Structures (CS 173) Derek Hoiem, University of Illinois 1

2. Today’s class Collections of sets: basics Review of midterm 2

3. Midterm grade distribution min23 median81 mean78 max100 pctlgrade 1056 2068 3073 4078 5081 6085 7088 8091 9094 3

4. Example: Image segmentation An image is divided into regions based on color or other appearance features Each region is a set of pixels The segmentation is a collection of regions, or a collection of sets of pixels 4

5. Sets of sets 5

6. 6 Example: the powerset of the set of pixels is the set of all possible image regions (including discontinuous regions)

7. Functions on sets 7

8. 8

9. Partitions 9

10. Partitions with Finite Sets 10 The segmentation is a partition of the image pixels: the regions do not overlap, are non-empty, and their union includes all pixels

11. Partitions with Infinite Sets 11 Example 2: Any elements of the set of real numbers that round to the same integer are in a set. The collection of these sets is a partition on the set of real numbers.

12. Partitions of continuous spaces 12 Represent any points that fall into each cell with an integer Treat all points within a cell as the same Represent points in cell with summary statistics, such as mean Simplifies matching and function fitting

13. Partition of continuous space (R 2 ) 13 Source: http://en.wikipedia.org/wiki/List_of_U.S._states_by_population_densityhttp://en.wikipedia.org/wiki/List_of_U.S._states_by_population_density Population density by state Population distribution over area approximated by population within each state or county Population density by county

14. Things that are not partitions 14

15. Some nit-picky things to remember 15

16. Review of midterm 16

17. Next week Combinations and counting – Dice rolls and poker hands Finite state machines 17