Computer Vision

Lecture 5.

Clustering: Why and How

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 2

This Lecture

• What is segmentation? 14.1• Segmentation and human vision 14.2• Background detection, scene changes 14.3• Segmentation through clustering 14.4

– Digital topology– Split and merge – k-means

• Graph theoretic segmentation 14.5• Hough transform 15.1

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 3

What is Segmentation?

• Examples– Scene change detection– Part search– Identification of humans– Search for buildings in satellite images– Search for images in databases

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 4

Formal Models of Segmentation

• Sub-processes– Image region generation– Curve segment detection– Point correspondence between two images

• Segmentation as clustering– Splitting– Merging

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 5

Gestalt Theories and Segmentation

• Gestalt theory: psychological theory of vision emphasizing how relations between visual objects affects their perception

• Examples of gestalt groupings:

Basic image

NearnessColor similarityShape similarityCommon behavior

Common region

Region grouping is stronger than nearness

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 6

Factors Affecting Grouping

Parallel lines Symmetric Lines

ClosureContinuity

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 7

Shape Decides Figure-Ground Choice

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 8

Figure: MuseumBackground: Sky plus treesGoal: Identify background

Segmentation by Histogram

Strategy: Use red to identify trees,use blue to identify sky

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 9

Edit Detection

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 10

Clustering

• Preliminary processing produces binary image

• Identify connected groups of pixels in the image

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 11

Topological Clustering

• Given: Array with values 0 and -1– Cells with 0 are unmarked (background), cells with -1 are marked

(objects)

• Goal: Assign value 1 for all pixels in first object, value 2 in second object, etc.

• Topological principle: If a pixel is in object k, and another pixel is adjacent to it and is marked, then it is also in object k.

0

-1

21

pixel values

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 12

Topological Clustering Algorithm

topological_clusters {k = 1;p = coordinates_of_marked_pixelwhile (p != -1) {

label_connected_pixels( p, k)k = k+1;p = coordinates_of_marked_pixel

}}coordinages_of_marked_pixel {

for i = 1 to last {if val[i] == -1

return i; }

return -1; }

}

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 13

Connected Pixel Labeling

label_connected_pixels( p, k) { // label all marked // pixels connected to p with kval(p) = k;change = 1;while ( change == 1 ) {

change = 0;for i = 1 to last {

if (val[i] == -1) & (pixel_adjacent_to(i) == k) {

val[i] = k;change = 1;

}}

))

pixel i

pixel adjacentto i

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 14

More General Clustering

• Given: Set of points

• Goal: Set of clusters

• Method (split and merge)

• While (stopping criterion not met) {– If (clusters to loose tight)

• split clusters

– If (clusters too small)• merge clusters

• }

• Many ideas, many algorithms

• Quality measures– Good if small number of clusters– Good if clusters are tight and/or well shaped

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 15

Adjacency Matrix MergeScatter

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20

1

6

5

4

3

2

7

0.0 3.2 5.4 11.0 11.4 15.7 16.43.2 0.0 7.4 13.9 14.2 18.1 19.65.4 7.4 0.0 7.1 7.2 10.8 14.1

11.0 13.9 7.1 0.0 1.0 5.4 7.711.4 14.2 7.2 1.0 0.0 4.6 8.315.7 18.1 10.8 5.4 4.6 0.0 9.916.4 19.6 14.1 7.7 8.3 9.9 0.0

Adjacency Matrix

1 1_2 1_2_3 1_2_3_4_5_6234 4_5 4_5_6567

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 16

Merge Rules

• Find adjacency matrix d(i, j) = ||pi–pj||

• Merge: – Find smallest d(i, j)– Merge these two clusters. Call new cluster

k– For all l ≠ i, j, d(l, k) = min(d(l, i), d(l, j))

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 17

k – Mean Clustering

• Assign data to k clusters

• Iteratively– Compute ci , means of clusters for i = 1 ... k

– Re-assign data to the cluster associated with the closest mean

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 18

Graph-Based Segmentation

• Split image into parts

• Assign a graph vertex to each part

• Assign edges that measure similarity between parts

• Partition graph into components that are similar within component and different among components.

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 19

Advantages and Disadvantages

• Similarity measures can be chosen for the problem at hand– Similarity according to intensity– Similarity in color– Similarity in texture– Similarity in geometry

• No good method of choosing similarity measures and merging criteria

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 20

Model-Based Segmentation

• Method of segmentation should be chosen on the basis of model of objects being segmented

• We will consider clustering of straight lines from edge detection data

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 21

Hough Transform

• Coordinates of a line on a plane are

• The transform

€

x cosθ + ysinθ = r

€

H (r,θ ) = f (x cosθ − l sinθ ,∫ xsinθ + l cosθ )dl

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 22

Hough Coordinates

x

y

r

q

Image of one point

Hough transform

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 23

Line Detection

Computer Vision, Lecture 5 Oleh Tretiak © 2005 Slide 24

Lines With and Without Noise