04/11/13

Collections of Sets

Discrete Structures (CS 173)Derek Hoiem, University of Illinois 1

Today’s class

• Collections of sets: basics

• Review of midterm

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Midterm grade distributionmin 23median 81mean 78max 100

pctl grade 10 5620 6830 7340 7850 8160 8570 8880 9190 94

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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

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Sets of sets

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The powerset of set is the set of all possible subsets of

Example: the powerset of is

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

If we have pixels, how many possible regions are there?

Functions on setsPowersets most often used for defining the domain or co-domain of a function that operates on sets

Example 1: function to compute the mean intensity of pixels in a region

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is a set of pixel indices is an element of (an image index) is the pixel intensity

Functions on setsPowersets most often used for defining the domain or co-domain of a function that operates on sets

Example 2: function to return indices of red pixels

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is an element of (an image index) is the intensity in red channel

Partitions

There are 10 kinds of people: those who understand binary and those who don’t.

A partition of is a collection of non-empty subsets of , such that each element of is contained by exactly one subset

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Partitions with Finite SetsA collection of sets is a partition of (1) : (2) No overlap within elements of : if (3) No element of is empty

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

Partitions with Infinite SetsA collection of sets is a partition of if

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Example 1: The set of natural numbers can be partitioned into those between 0 and 9, 10 and 99, 100 and 999, and -1, etc. (elements of each set have the same number of digits)

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.

Partitions of continuous spacesQuantization: map to a partition of

– Clustering– Decision trees

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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

Partition of continuous space (R2)

13Source: http://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

Things that are not partitions

is not a partition of

is not a partition of

, is not a partition of the natural numbers

, is not a partition of the natural numbers

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Some nit-picky things to remember

• Important to distinguish between a one-element set and an element: – Common programming error (e.g., function is expecting a

set but gets a number)

• Every powerset contains the empty set as an element

• An element of a collection (or set of sets) is a set

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Review of midterm

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Next week• Combinations and counting

– Dice rolls and poker hands• Finite state machines

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