Application of fuzzy set theory in image analysis Robin Strand Centre for Image analysis Swedish University of Agricultural Sciences Uppsala University
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