CS654: Digital Image Analysis Lecture 26: Image segmentation.

CS654: Digital Image Analysis

Lecture 26: Image segmentation

Recap of Lecture 25

• Line detection

• Least square, RANSAC

• Hough space/ parametric space

• Voting

• Circle detection

• GHT

Outline of Lecture 26

• Thresholding

• Otsu’s method

• Region based segmentation

• Region growing

• Split and Merge

Introduction

• Detected boundaries between regions

• Pixel distribution was not exploited

• Can the regions be segmented directly?

• How to exploit the top-down/ bottom-up cues?

• Threshold

• Region-based segmentation

Thresholding

𝑔 (𝑥 , 𝑦 )={1 𝐼𝑓 𝑓 (𝑥 , 𝑦 )>𝑇0 𝐼𝑓 𝑓 (𝑥 , 𝑦 )≤𝑇 𝑔 (𝑥 , 𝑦 )={ 𝒂 ; 𝐼𝑓 𝑓 (𝑥 , 𝑦 )>𝑇2

𝒃 ; 𝐼𝑓 𝑇1< 𝑓 (𝑥 , 𝑦 )≤𝑇 2

𝒄 ; 𝐼𝑓 𝑓 (𝑥 , 𝑦 )≤𝑇 1

Global thresholding, local thresholding, adaptive thresholding

Multiple thresholding

Key factors

• Separation between peaks

• The noise content in the

image

• The relative size of the object

and the background

• Uniformity of the illumination

source

• Uniformity of the reflectance

property of the imageAutomatic threshold selection

Iterative threshold selection

1. Select an initial estimate of the threshold . A good initial value is the average intensity of the image

2. Calculate the mean grey values and of the partitions,

3. Partition the image into two groups, , using the threshold

4. Select a new threshold:

5. Repeat steps 2-4 until the mean values and in successive iterations do not change.

𝑇=12

(𝜇1+𝜇2 )

Optimal thresholding

Definition: Methods based on approximation of the histogram of an image using a weighted sum of two or more probability densities with normal distribution.

Example

Input image Histogram Segmented image

Otsu’s Image Segmentation Algorithm

• Find the threshold that minimizes the weighted within-class variance.

• Equivalent to: maximizing the between-class variance.

• Operates directly on the gray level histogram [e.g. 256 numbers, P(i)]

• It is fast (once the histogram is computed).

Otsu: Assumptions

• Histogram (and the image) are bimodal.

• No use of spatial coherence, nor any other notion of object structure.

• Assumes stationary statistics, but can be modified to be locally adaptive

• Assumes uniform illumination (implicitly), so the bimodal brightness behavior arises from object appearance differences only.

The weighted within-class variance is:

w2 (t) q1(t)1

2 (t) q2 (t) 22(t)

Where the class probabilities are estimated as:

q1(t) P(i)i1

t

L

ti

iPtq1

2 )()(

t

i

iiPtq

t11

1 )()(

1)(

L

ti

iiPtq

t12

2 )()(

1)(

And the class means are given by:

Otsu’s method: Formulation

Finally, the individual class variances are:

12(t) [i 1(t)]

2 P(i)

q1(t)i1

t

L

ti tq

iPtit

1 2

22

22 )(

)()]([)(

Run through the full range of t values and pick the value that minimizes

Is this algorithm first enough?

w2 (t)

Otsu’s method: Formulation

w2 (t) q1(t)1

2 (t) q2 (t) 22(t)

Between/ Within/ Total Variance

• The total variance does not depend on threshold (obviously).

• For any given threshold, the total variance is the weighted sum of the within-class variances

• The between class variance, which is the sum of weighted squared distances between the class means and the global mean.

The total variance can be expressed as

2 w2 (t) q1(t)[1 q1 (t)][1(t) 2 (t)]

2

Within-class, from before Between-class,

Minimizing the within-class variance is the same as maximizing the between-class variance.

compute the quantities in recursively as we run through the range of t values.

Total variance

B2 (t)

B2 (t)

Recursive algorithm

q1(t 1)q1(t) P(t 1)

1(t 1) q1(t)1 (t) (t 1)P(t 1)

q1(t 1)

q1(1) P(1) 1(0) 0;

2(t 1) q1(t 1)1(t 1)

1 q1(t 1)

Initialization...

Recursion...

Example

Input image Histogram

Global thresholding

Otsu’s method

Example (in presence of noise)

Input image Histogram Global threshold

Smoothened image Histogram Otsu’s method

Image partitioningInput image Histogram Global threshold

Global Otsu’s Method Image partitioning Local Otsu’s method

Thresholding (non-uniform background)

Input image Global thresholding using Otsu’s method

Local thresholding with moving average

Region based segmentation

• Goal: find coherent (homogeneous) regions in the image

• Coherent regions contain pixel which share some similar property

• Advantages: Better for noisy images

• Disadvantages: Oversegmented (too many regions), Undersegmented (too few regions)

• Can’t find objects that span multiple disconnected regions

Types of segmentations

Oversegmentation Undersegmentation

Multiple Segmentations

Input

Region Segmentation: Criteria

A segmentation is a partition of an imageinto a set of regions satisfying:

1. Si = S Partition covers the whole image.

2. Si Sj = , i j No regions intersect.

3. Si, P(Si) = true

4. P(Si Sj) = false, i j, Si adjacent SjHomogeneity predicate is satisfied by each region.

Union of adjacent regions does not satisfy it.

Define and implement the similarity predicate.

Methods of Region Segmentation

• Region growing

• Split and merge

• Clustering

Region Growing

• It start with one pixel of a potential region

• Try to grow it by adding adjacent pixels till the pixels being compared are too dissimilar

• The first pixel selected can be • The first unlabelled pixel in the image • A set of seed pixels can be chosen from the image.

• Usually a statistical test is used to decide which pixels can be added to a region

Example

Input image

Outputimage

Histogram Initial Seed image

Final seeds

Threshold 1 Threshold 2 Region growing

Split and Merge

• Split into four disjoint coordinates any region for which =false

• When no further splitting is possible, merge adjacent region for which = true

• Stop when no further merging is possible

Example

R1 R2

R4R3

R21 R22

R23 R24

R31 R32

R33 R34

R1

R4

Input image

Split Split Merge

Quadtree representation

R

R1 R2 R3 R4

R21 R22 R23 R24 R31 R32 R33 R34

R21 R22

R23 R24

R31 R32

R33 R34

R1

R4

Clustering

• Task of grouping a set of objects

• Objects in the same group (called a cluster) are more similar (in some sense or another) to each other

• Object of one cluster is different from an object of the another cluster

• Connectivity model, centroid model, distribution model, density model, graph based model, hard clustering, soft-clustering, …

Feature Space

Source: K. Grauman

Centroid model

• Computational time is short

• User have to decide the number of clusters before starting classifying data

• The concept of centroid

• One of the famous method: K-means Method

Partitional Clustering

• K-mean algorithm :

Decide the number of the final classified result with N.

Numbers of cluster: N

we now assume N=3



Randomly choose N point for the centroids of cluster.

(N=3)

Numbers of cluster: N



Find the nearest point for every centroid of cluster. Classify the point into the cluster.

Notice the definition of th nearest!



Calculate the new centroid of every cluster.

Notice the definition of the centroid!



Repeat step1~ step4 until all the point are classified.



Data clustering completed

For N=3

Example

Image Clusters on intensity Clusters on color

Slides by D.A. Forsyth

ExampleInput image Segmentation using K-means

Thank youNext Lecture: Image segmentation

CS654: Digital Image Analysis Lecture 26: Image segmentation.

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