Application of fuzzy set theory in image analysis

Robin StrandCentre for Image analysisSwedish University of Agricultural SciencesUppsala University

Fuzzy systems

• Image data are rarely of perfect quality• Fuzzy systems are capable of

representing diverse, non-exact, uncertain, and inaccurate knowledge or information.

Fuzzy systemsTwo forms of knowledge:

• Objective knowledge – mathematical knowledge, used in engineering problems.

• Subjective knowledge – exists in linguistic form, often not possible to quantify.

Fuzzy systems can coordinate these two forms of knowledge.

Fuzzy systems can handle numerical data and linguistic knowledge simultaneously.

Fuzzy systems can model inherently imprecisely defined conditions.

Example - Fuzzy set of tall men

The degree of membershipdepends on the height.

Fuzzy set

A fuzzy subset S of a set X is a set ofordered pairs S = {(x, μS(x) |x ∈ X},where the membership function μS(x)∈[0,1]represents the grade of membership of x in S.

Example – small numbers

0 150

1

Reference setx={0,15,13,2,11,7,8,9,3,7,5,10}

x μS(x)

0 1

2 1

3 1

5 1

7 1

7 1

8 0

9 0

10 0

11 0

13 0

15 0

Example – small numbers

Reference setx={0,15,13,2,11,7,8,9,3,7,5,10}

x μS(x)

0 1

2 13/15

3 12/15

5 10/15

7 8/15

7 8/15

8 7/15

9 6/15

10 5/15

11 4/15

13 2/15

15 0

0 150

1

Example – small numbers

Reference setx={0,15,13,2,11,7,8,9,3,7,5,10}

x μS(x)

0 1

2 1

3 1

5 1

7 2/4

7 2/4

8 1/4

9 0

10 0

11 0

13 0

15 0

0 150

1

Membership functions

0 15

0

1μSMALL(x)

μMEDIUM(x)

μLARGE(x)

0 15

0

1

0 15

0

1

Fuzzy set operators

Intersection A∩B:μA∩B(x)=min(μA(x),μB(x))

Union A∪B:μA∪B(x)=max(μA(x),μB(x))

Complement Ac:μAc(x)=1−μA(x)

Fuzzy c-means clustering

• The fuzzy c-means algorithm (FCM) iteratively optimizes an objective function in order to detect its minima, starting from a reasonable initialization.

• Its objective is to partition a collection of numerical data into a series of overlapping clusters. The degrees of belongingness are interpreted as fuzzy membership values.

Fuzzy c-means clusteringExtends K-means, ch. 9.2.5

K-means algorithm minimizes the within-cluster variance:

Where the matrix I (with elements iik) is a k-partition of the data set X={x1, x2, ... , xn}

and vi is the cluster center of class i (1≤i≤K) anddik

2=ǁxk−viǁ2, where ǁ∙ǁ is an inner product norm metric.

K

K-means clusteringExample: four points and three clusters

I=

1 0 00 1 01 0 00 0 1

Point four (row) does notbelong to cluster two (column).

d11 d12 d13d21 d22 d23d31 d32 d33d41 d42 d43

The distance between point fourand center of cluster two.

Fuzzy c-means clustering

The FCM algorithm makes use of iterativeoptimization to approximate minima of an objective function which is a member of a family of fuzzy c- means functionals defined as

where the matrix U (with elements uik) is a fuzzy c-partition of the data set X={x1, x2, ... , xn}

Fuzzy c-means clusteringExample: four points and three clusters

U=

0.8 0.1 0.10.2 0.5 0.30.4 0.3 0.30.1 0.1 0.8

Point four (row) has membership0.1 to cluster two (column).

d11 d12 d13d21 d22 d23d31 d32 d33d41 d42 d43

The distance between point fourand center of cluster two.

Basic steps of FCM• make initial guess for cluster means

• iteratively– use the current means to assign

samples to clusters*

– update means

• until there are no changes

*) in k-means clustering assignment is crisp, to only one (the nearest) cluster; in FCM assignment is fuzzy, based on relative distance to cluster centers

Fuzzy c-means clustering example- background- phantom body- cold and hot lesions

Regions as four fuzzy sets (note white=zero here)

• boundaries between subgroups are not crisp• each element may belong to more than one cluster – its

”overall” membership sums up to one• objective function includes parameter m controlling

degree of fuzziness (suitable values in range [1.5,2.5])

Fuzzy connectedness (7.4)

• Graded composition– heterogeneity of intensity in the object

region due to heterogeneity of object material and blurring caused by the imaging device

• Hanging-togetherness– natural grouping of voxels constituting an

object a human viewer readily sees in a display of the scene in spite of intensity heterogeneity

Fuzzy connectedness (7.4)

• If two regions have about the same grey-level and if they are relatively close to each other, then they likely belong to the same object.

• Group pixels that seem to hang together.

• Spatial relationship between pixels.

• Determine relationship between each pair of pixels in the entire image.

Fuzzy connectedness

Fuzzy connectedness combines– fuzzy adjacency (closeness in space)

– fuzzy affinity (closeness in terms of intensities or other properties)

and assigns a strength of connectedness to each pair of image points determined as the strength of the weakest link of the strongest path between the points

Jayaram K. Udupa, et al.MIPG, University of Pennsylvania, Philadelphia

Fuzzy connectedness

Fuzzy connectedness

Let Pc,d denote the set of all paths between c and d.• The strength of connectedness of the pathP=<c=c1, c2, ..., cn=d> is

• The fuzzy connectedness between c and d is

Fuzzy affinity

Fuzzyadjacency

Fuzzy affinity-homogeneity

based component

Fuzzy affinity-object-feature

based component

Fuzzy affinity

Expected properties of g• Range within [0,1]• Monotonically non-decreasing in both argumentsExamples:

Fuzzy adjacency

• Spatial closeness

• Compare 4- or 8-adjacency in binary 2D images

• For example:

Fuzzy affinityHomogeneity-based component

The degree of local hanging-togetherness due tothe similarity in intensity.

Expected properties of Wφ• Range within [0,1] and Wφ(0)=1• Monotonically non-increasingExamples:The right-hand side of an appropriately scaled box,trapezoid, or Gaussian function.

Fuzzy affinityObject-feature-based componentThe degree of local hanging-togetherness with respect to some given feature, for example intensity distribution.

Expected properties of Wo and Wb• Range within [0,1]Examples: An appropriately scaled and shifted box, trapezoid, or Gaussian function.

Fuzzy affinity

In the computer exercise (and in the book):

An object as a fuzzy connected component

Given one or several seeds:• Compute connectedness map for all possible

paths

• Threshold map• An object is a fuzzy

connected component of a given strength

Iterative relative fuzzy connectedness

P2

P1

Due to weak boundary between object O1 and O2, the cost of path P1 and P2 can be equal. Solution: Path P1 is not allowed to pass through the core of O1.

An example

Segmentation of vascular trees. (a) MIP.(b) Segmentation using absolute fuzzy connectedness.(c) Artery-vein separation using relative fuzzy connectedness.

Algorithm for Computing FuzzyConnectedness (Dijkstra-like)Set all elements of FC to 0 except s which is set to 1 ;

Push s to Q ;

While Q is not empty do

Remove a spel c from Q for which FC(c) is maximal ;

For each spel e such that μκ(c,e) > 0 do

Set fc = min(FC(c), μκ(c,e)) ;

If fc > FC(e) then

Set FC(e) = fc ;

If e is already in Q then

Update e in Q ;

Else

Push e to Q ;

Summary

● Fuzzy system● Fuzzy set operations● Fuzzy c-means● Fuzzy connectedness