Clustering TechniquesK-Means & EM

Mario Haddad

What is Clustering? Cluster: a collection of data objects

Similar to one another within the same clusterDissimilar to the objects in other clusters

Clustering is unsupervised classification: no predefined classes

Typical applications As a stand-alone tool to get insight into data

distribution As a preprocessing step for other algorithms

K-Means

K-Means

Groups data into K clusters and attempts to group data points to minimize the sum of squares distance to their central mean.

Algorithm works by iterating between two stages until the data points converge.

Problem Formulation

Given a data set of {x1,…,xN} which consists of N random instances of a random D-dimensional Euclidean variable x.

Introduce a set of K prototype vectors, µk where k=1,…,K and µk corresponds to the mean of the kth cluster.

Problem Formulation

Goal is to find a grouping of data points and prototype vectors that minimizes the sum of squares distance of each data point.

Problem Formulation (cont.) This can be formalized by introducing

an indicator variable for each data point: rnk is {0,1}, and k=1,…,K

Our objective function becomes:

𝐽=∑𝑛=1

𝑁

∑𝑘=1

𝐾

𝑟𝑛𝑘 ¿|𝑥𝑛−𝜇𝑘|∨¿2¿

How K-Means works

Algorithm initializes the K prototype vectors to K distinct random data points.

Cycles between two stages until convergence is reached.

Convergence: since there are only a finite set of possible assignments.

How K-Means works

1. For each data point, determine rnk where:

2. Update µk :

𝜇𝑘=∑𝑛𝑟𝑛𝑘 𝑥𝑛

∑𝑛𝑟𝑛𝑘

𝑟𝑛𝑘={1𝑖𝑓 𝑘=𝑎𝑟𝑔𝑚𝑖𝑛 𝑗¿|𝑥𝑛−𝜇𝑘|∨¿2 ¿0 h𝑜𝑡 𝑒𝑟𝑤𝑖𝑠𝑒

How K-Means works (cont)

If k (number of clusters) and d (dimension of variables) are fixed, the clustering can be performed in time.

Exponential bounds for plane:

𝑂 (𝑁 𝑑𝑘+1 𝑙𝑜𝑔𝑁 )

K-Means Initialization example Pick K cluster centers (unfortunate choice)

Pick K cluster centers (random choice)

K-Means Initialization example

K-Means++ K-Means with smart initial seeding

Choose one center uniformly at random from among the data points.

For each data point x, compute D(x), the distance between x and the nearest center that has already been chosen.

Choose one new data point at random as a new center, using a weighted probability distribution where a point x is chosen with probability proportional to D(x)2.

Repeat Steps 2 and 3 until k centers have been chosen. Now that the initial centers have been chosen, proceed

using standard k-means.

K-Means++

This seeding method yields considerable improvement in the final error of k-means

Takes more time to initialize Once initialized, K-Means converges

quickly Usually faster than K-Means 1000 times less prone to error than K-

Means

K-Means Example

Cluster black and white intensities: Intensities: 1 3 8 11 Centers c1 = 7, c2=10 Assign 1, 3, 8 to c1, 11 to c2 Update c1 = (1+3+8)/3 = 4, c2 = 11 Assign 1,3 to c1 and 8 and 11 to c2 Update c1 = 2, c2 = 9 ½ Converged

Computer Vision - A Modern ApproachSet: SegmentationSlides by D.A. Forsyth

Image Clusters on intensity

K-Means

K-Means

Original:

After Intensity Clustering:

Computer Vision - A Modern ApproachSet: SegmentationSlides by D.A. Forsyth

K-means using color alone,11 segments.

Computer Vision - A Modern ApproachSet: SegmentationSlides by D.A. Forsyth

K-means using color and position,20 segments.

Pros and Cons of K-Means

Convergence: J may converge to a local minima and not the global minimum. May have to repeat algorithm multiple times.

With a large data set, the Euclidian distance calculations can be slow.

K is an input parameter. If K is inappropriately chosen it may yield poor results.

Local Minima

K-Means might not find the best possible assignments and centers.

Consider points 0, 20, 32. K-means can converge to centers at 10,

32. Or to centers at 0, 26.

Heuristic solutions Start with many random starting points

and pick the best solution.

EMExpectation Maximization

Soft Clustering

Clustering typically assumes that each instance is given a “hard” assignment to exactly one cluster.

Does not allow uncertainty in class membership or for an instance to belong to more than one cluster.

Soft clustering gives probabilities that an instance belongs to each of a set of clusters.

Each instance is assigned a probability distribution across a set of discovered categories (probabilities of all categories must sum to 1).

EM

Tends to work better than K-Means. Soft Assignments

A point is partially assigned to all clusters. Use probabilistic formulation

Mixture of Gaussians

g(x; m, σ)

The probability of a point x based on a Gaussian Distribution with mean m and variance σ

Intuition

A mixture of K Gaussians

A distribution generated by randomly selecting one of K Gaussians, then randomly draw a point from that distribution.

Gaussian k with a probability of pk

Goal: find pk, σk, mk that maximize the probability of our data points.

Back to EM

Iterative Algorithm Goal: group some primitives together Chicken and Egg problem:

Items in group -=> Description of the group

Description Of the group -=> Items in group

Brace Yourselves..

EM

Iterative Algorithm: E Step and M Step

E Step: Compute the probability that point n

is generated by distribution k𝑝 (𝑖 ) (𝑘|𝑛)=

𝑝𝑘(𝑖 )𝑔(𝑥𝑛 ;𝑚𝑘

(𝑖 ) ,𝜎𝑘( 𝑖 ))

∑𝑚=1

𝐾

𝑝𝑘(𝑖 )𝑔(𝑥𝑛 ;𝑚𝑘

(𝑖) ,𝜎𝑘(𝑖))

EM M Step:

𝑚𝑘(𝑖+1 )=

∑𝑛=1

𝑁

𝑝(𝑖)(𝑘∨𝑛)𝑥𝑛

∑𝑛=1

𝑁

𝑝(𝑖)(𝑘∨𝑛)

𝜎 𝑘(𝑖+1)=√ 1𝐷 ∑

𝑛=1

𝑁

𝑝 ( 𝑖 ) (𝑘|𝑛)∨¿ 𝑥𝑛−𝑚𝑘(𝑖+1)∨¿2

∑𝑛=1

𝑁

𝑝(𝑖)(𝑘∨𝑛)

𝑝𝑘(𝑖+1)= 1

𝑁 ∑𝑛=1

𝑁

𝑝(𝑖) (𝑘∨𝑛)

EM

Converges to a locally optimal solution

Each step increases the probability of the points given the distributions.

Can get stuck in local optima (less than K-Means)

EM vs K-Means local optima 1D points at 0 20 32 Centers at 10 and 32 A local minima for k-means EM: 20 almost evenly shared

between the two centers. The center at 32 moves closer to 20

and takes over First center shifts to the left.

EM and K-means

Notice the similarity between EM for Normal mixtures and K-means. The expectation step is the assignment. The maximization step is the update of

centres.

K-means is a simplified EM. K-means makes a hard decision while EM makes

a soft decision when updating the parameters of the model.

EM and K-Means

𝑝 (𝑖 ) (𝑘|𝑛)=𝑝𝑘

(𝑖 )𝑔(𝑥𝑛 ;𝑚𝑘(𝑖 ) ,𝜎𝑘

( 𝑖 ))

∑𝑚=1

𝐾

𝑝𝑘(𝑖 )𝑔(𝑥𝑛 ;𝑚𝑘

(𝑖) ,𝜎𝑘(𝑖))

𝑟𝑛𝑘={1𝑖𝑓 𝑘=𝑎𝑟𝑔𝑚𝑖𝑛 𝑗¿|𝑥𝑛−𝜇𝑘|∨¿2 ¿0 h𝑜𝑡 𝑒𝑟𝑤𝑖𝑠𝑒K-Means

EM

EM and K-Means

𝑚𝑘(𝑖+1 )=

∑𝑛=1

𝑁

𝑝(𝑖)(𝑘∨𝑛)𝑥𝑛

∑𝑛=1

𝑁

𝑝(𝑖)(𝑘∨𝑛)

𝜇𝑘=∑𝑛𝑟𝑛𝑘 𝑥𝑛

∑𝑛𝑟𝑛𝑘

K-Means

EM

Fast Image Segmentation Based on K-Means Clustering with Histograms in HSV Color

Space

HSVHue-Saturation-Value

Overview

Histogram Generation

Motivation

Gray and color histograms in HSV color space for K-Means clustering

Cluster number automatically set by “Maximin” initialization

Fast and efficient to extract regions with different colors in images.

Segmentation results are close to human perceptions.

Maximin Initialization and Parameter Estimation Use Maximin to initialize number of

clusters and centroid positions:

Step A: From the color histogram bins and gray histogram bins, find the bin which has the maximum number of pixels to be the first centroid.

Maximin Initialization and Parameter Estimation Step B: For each remaining histogram bin,

calculate the min distance, which is the distance between it and its nearest centroid. Then the bin which has the maximum value of min distance is chosen as the next centroid.

Step C: Repeat the process until the number of centroid equals to KMax or the maximum value of the distance in Step B is smaller than a predefined threshold ThM.

Maximin Initialization and Parameter Estimation Kmax is set to 10.

There should be no more than 10 dominant colors in one image for high level image segmentation.

ThM is set to 25. According to human perception of

different colors in HSV color space.

K-Means Clustering in HSV Color Space Step 1: Estimate the parameters of K-

Means.

Step 2: Two kinds of histogram bins will be clustered together in this step. For color histogram bins, since the hue dimension is circular (e.g. 0◦ = 360◦), the numerical boundary should be considered in the distance measurement and the process of centroid calculations.

K-Means Clustering in HSV Color Space For gray histogram bins, there is no

hue information inside. which means that the saturation values of gray histogram bins are all considered as zero, and the hue values can be arbitrary.

K-Means Clustering in HSV Color Space

K-Means Clustering in HSV Color Space Step 3: Recalculate and update K

cluster centroids. Again, since the hue dimension is circular, the indices in the hue dimension should be considered not absolutely but relatively.

K-Means Clustering in HSV Color Space Step 4: Check if the clustering process

is converged according to the total distortion measurement, which is the summation of distances between each histogram bin and its nearest cluster centroid

When the difference of total distortion measurement is smaller than a predefined threshold or max iterations reached, terminate. Else, go to step 2

K-Means Clustering in HSV Color Space Step 4 intuition: G(v) represents the number of pixels

in the gray histogram bin with parameter v

B(h, s, v) represents the number of pixel in the color histogram bin with parameters (h, s, v)

K-Means Clustering in HSV Color Space Step 5: Image pixels are labeled with

the index of nearest centroid of their corresponding histogram bins. A labeled image is obtained in this step, and K-Means clustering is finished.

K-Means Clustering in HSV Color Space Eliminate noise and unnecessary

details of labeled images Statistical filter Windows over pixels The purpose of this filter is to replace

the pixel in the labeled image with the label with maximum number in a window.

Areas smaller than a certain threshold are merged with biggest neighboring region to avoid over segmentation

K-Means Clustering in HSV Color Space

Summary

Clustering K-Means K-Means++ initialization EM EM as a general K-Means Fast Image Segmentation Based on K-

Means Clustering with Histograms in HSV Color Space

Thank You For Listening

References

Fast Image Segmentation Based on K-Means Clustering with Histograms in HSV Color Space Tse-Wei Chen 1, Yi-Ling Chen 2, Shao-Yi Chien

K-Means and EM - David Jacobs D. Forsyth Expectation-Maximization Algorithm and Image

Segmentation - Daozheng Chen