Computer Science Aggregation operator in image reduction.

1. By
Abu Sadat Mohammed Yasin
Debotosh Dey

2. Aggregation Operator Definition
 An aggregation is a collection, or the gathering of
things together.
 Aggregation operators are mathematical functions.
 A real number y to any n-tuple (x1,x2, …,xn) of real
numbers: y = Aggreg(x1,x2, …,xn)

3. Aggregation Operator Definition
 M. Detyniecki, "Fundamentals On Aggregation
Operators," AGOP, Berkerley, 2001, defines an
aggregation operator as a
 function Aggreg:
 Satisfy the following properties
Aggreg (x) = x Identity when unary
Aggreg (0,…,0) = 0 and Aggreg
(1,…,1) = 1
Boundary conditions
Aggreg (x1,…,xn) ≤Aggreg (y1,…, yn)
if (x1,…, xn) ≤ (y1,…, yn)
Non decreasing

4. Properties of Aggregation Operator
 Properties into two families
 The mathematical properties
 The behavioral properties

5. Properties of Aggregation Operator
 The mathematical properties
 Boundary Conditions:
Aggreg (0, 0,..., 0) = 0
Aggreg (1,1,..., 1 ) = 1
 Monotonicity (non decreasing)
if yi ≥ xi
Than Aggreg(x1, y1, xn) ≥ Aggreg(x1, xi, xn)
 Continuity
 Associativity
Aggreg(x1,x2,x3) = Aggreg(Aggreg (x1,x2),x3)= Aggreg(x1, Aggreg (x2,x3))
 Symmetry
Aggreg(x σ(1), x σ(2),..., x σ(n)) = Aggreg(x1,x2,...,xn)
 Bisymmetry
A(A(x11, x12),A(x21, x22)) = A(A(x11, x21),A(x12, x22))
 Absorbent Element
Aggreg(x1,...,a,....xn) = a

6. Properties of Aggregation Operator
 The mathematical properties
 Neutral Element
Aggreg[n](x1,...,e,....xn-1) = Aggreg[n-1]( x1,..., xn-1)
 Idempotence
Aggreg(x,x,...,x) = x
 Compensation
 Counter balancement
∀t ∈]0,1 [,∀(x1,...,xn )∃ (y1,...,ym)
so that Aggreg(x1,...,xn, y1,...,ym)=t
 Reinforcement
 Stability for a linear function
Aggreg(r.x1+t, r.x2+t,...,r.xn+t) = r.(Aggreg(x1,x2,...,xn))+t
 Invariance
Aggreg(f(x1), f(x2),..., f(xn)) = f(Aggreg(x1, x2,..., xn))

7. Properties of Aggregation Operator
 Behavioral properties
 Decisional behavior
 Interpretability of the parameters
 Weights on the arguments

8. Different Types of Aggregation
Operators
 The arithmetic mean
 The weighted mean
 The Median
 The minimum and the maximum
 The weighted minimum and the weighted maximum

9. Different Types of Aggregation
Operators
 Quasi-arithmetic means
 geometric mean
 harmonic mean
 Aczel J. Defines
 Dujmovic, Dyckhoff Defines: (f:x →xα )

10. Different Types of Aggregation
Operators
 Quasi-arithmetic means
 for α =1, we obtain the arithmetic mean.
 for α = 2, we obtain the quadratic mean (also called the
Euclidean mean).
 for α = -1, we obtain the harmonic mean.
 when α tends to -∞, this formula tends to the maximum
operator.
 when α tends to +∞, this formula tends to the minimum
operator.
 when α tends to 0, this formula tends to the geometric
mean.

11. Different Types of Aggregation
Operators
 T-norms and T-conorms
 The t-norms generalize the conjunctive 'AND' operator.
 The t-conorms generalize the disjunctive 'OR' operator.
 t-norm : function T : [0,1]x[0,1] → [0,1]
 t-conorm : function S : [0,1]x[0,1] → [0,1]
 Properties
 Same properties
 Commutativity T(x,y) = T(y,x), S(x,y) = S(y,x)
 Monotonicity (increasing) T(x,y) ≤ T(u,v), if x ≤ u and y ≤ v
S(x,y) ≤ S(u,v), if x ≤ u and y ≤ v
 Associativity T(x,T(y,z)) = T(T(x,y),z), S(x,S(y,z)) = S(S(x,y),z)
 Common properties but for different element.
 One as a neutral element in T-norm, T(x,1) = x
 Zero as a neutral element in T-conorms, S(x,0) = x

12. Different Types of Aggregation
Operators
 Ordered Weighted Averaging Operators

 And many more
GOWA, Quasi OWA, fuzzy OWA, LOWA, ULOWA,
OWAWA, FGOWAWA

13. Usage of Aggregation Operators
 Reducing a set of numbers into a unique representative (or
meaningful) number.
 Has the purpose the simultaneous use of different pieces of
information in order to come to a conclusion or a decision.
 Basic concerns for all kinds of knowledge based systems,
from image processing to decision making, from pattern
recognition to machine learning.
 Several research groups are directly interested in finding
solutions, among them the multi-criteria community, the
sensor fusion community, the decision-making
community, the data mining community, image processing
community etc

14. Image reduction
 Image reduction is the process of diminishing the
resolution of the image but maintaining as much
information as possible from the original image
 As an example, original multi-megapixel size image
showing on a camera viewfinder, on a computer or
mobile screen

15. Image reduction methods
 Lots of different image reduction method has been
developed. But two methods are used mostly..
 Image to be reduced globally or in a transform domain
 divide the image in pieces and act on each of them
 Last method, is very much efficient in time and keeps
some of the specific properties of the images such as
textures, edges, etc.

16. A Study of Aggregator Operator in
Image reduction
 Construction of image reduction operators using
averaging aggregation functions [Paternain,
Fernandez, Bustince, Mesiar, Beliakov]
 Two objectives:
 design a reduction algorithm that, given an image,
provides a new image of lower dimension that keeps the
intensity properties of the original image.
 design mechanisms to reduce small regions of an image
into a single pixel that represents the intensities of the
region.

17. Image reduction operators
 As an operator from an image (which is a matrix or a
relation) and results in a new reduced image of lower
in dimension.

18. Reduction operators in the literature
 Undersampling/subsampling
 removing a given number of pixels, for example removing odd
rows/columns from the image.
 Fuzzy transform
 a fuzzy partition of a universe into fuzzy subsets (factors, clusters,
granules etc.).
 a function can be associated with a mapping from a set of fuzzy subsets
to the set of obtained average values.
 Image interpolation
 using the information of the pixels of an image to estimate the value of
pixels in unknown locations.
 Nearest neighbor interpolation
 Bilinear interpolation
 Bicubic interpolation

19. Construction of reduction operators from
local reduction operators
 This study provides an algorithm that allows
constructing reduction operators.
 The main idea of the reduction algorithm is to divide
the image in small (non- overlapping) regions, to
reduce each region into a single pixel and to collect all
the pixels in the new reduced image.
 Then, the whole algorithm can be seen as a reduction
operator.

20. Local reduction operators from
aggregation functions (I)
 The reduction operator allows construction of
reduction operators by means of local reduction
operators.
 Here, they studied several examples of local reduction
operators constructed from aggregation functions.
 Then, analyze the effect of these functions in the
reduced image obtained by reduction algorithm.

21. Local reduction operators from
aggregation functions (II)
 Local reduction operators constructed from
aggregation functions
 T-norms and T-conorms
 Quasi-arithmetic means
 OWA operators
 Median
 α-migrative operators

22. Best reduction operator
 For finding the best reduction operator, whole
process divided in to two sub- processes.
1. Reduction and reconstruction of images
2. Image reduction as a preprocessing step in
pattern recognition

23. 1) Reduction and reconstruction of
images
 In the literature, image reduction process is associated
with procedures of reduction and later reconstruction
of the image.
 Given an original image, build several reduced images
using different local reduction operators, by means of
Algorithm or by means of reduction operators given in
the literature.
 Reconstruct all the reduced images using one single
magnification method.
 Compare the reconstructed images with the original one
and decide which is the best reduction operator.

24. 1) Reduction and reconstruction of images
(Operators)
 6 reduction operators :
 Minimum,
 Geometric mean,
 Arithmetic mean,
 Median,
 Centered OWA
 Maximum
 4 reduction operators from the literature
 Nearest neighbor interpolation,
 Bilinear interpolation,
 The fuzzy transform
 Subsampling

25. 1) Reduction and reconstruction of images
(Results)
 Worst results are obtained with minimum and
maximum.
 Arithmetic mean, geometric mean, median and
centered OWA give better result.
 With PSNR(peak signal to noise ratio) the best is
achieved by arithmetic mean.
 With SSIM(structural similarity) the best is obtained
by centered OWA operator.

26. 1) Reduction and reconstruction of images
(Reaction to noise )
Input images with noise, the reduction operator
act in different ways.
To check the reaction to different types of noise,
original images are modified with two types of
noise
 impulsive noise (salt and pepper noise)
 Gaussian noise.

27. 1) Reduction and reconstruction of images (Reaction to
noise ) - Experiments
 10% of pixels corrupted by impulsive noise
 Best result: median.
 Centered OWA gives very good result.
 Signification increment of impulsive noise
 Centered OWA is giving worse results.
 Pixels corrupted by Gaussian noise
 Best result: arithmetic mean.
 Centered OWA is also good.

28. 2) Image reduction as a preprocessing step in
pattern recognition
 The experiment is carried on from 13 images each of 15
different persons
 All of the original images are reduced to 48 × 36 pixels to
avoid the high running time.
 Original images are reduced using the same reduction
operators as before,
Minimum, Geometric mean,
Arithmetic mean, Median,
Centered OWA, Maximum
 Result is compared with the measurement obtained using
the imresize function from Matlab.

29. 2) Image reduction as a preprocessing step in
pattern recognition
 The results of the reduction operators are very
competitive.
 Best result is obtained by means of the reduction
operator based on the minimum.
 Similar experiment been performed, but by reducing
the dimension of the images to 36 × 27 pixels.
 Again the minimum provides the best results.

30. Study Summary
 There is not a single operator that works well in every
perspective.
 In reduction and reconstruction of images
 For better, PSNR: arithmetic mean
 For better, SSIM: centered OWA operator.
 For impulsive noise: median.
 For Gaussian noise: arithmetic mean.
 The centered OWA, provides good result for both kind of
noise in the image.
 Image reduction as a preprocessing step in pattern
recognition: minimum

31. Image reduction in Machine
Learning
 Dimensionality Reduction
 Process of reducing the number of random variables
from a set of data.
 Combination of
 Principal component analysis (PCA)
 A powerful tool for data analysis and pattern recognition
 Frequently used in signal and image processing.
 Linear discriminant analysis (LDA)
 Canonical correlation analysis (CCA)

32. The PCA Theory
(the Karhunen–Loève theorem )
 PCA – data samples x =[x1,x2, ...xn] T
 Compute the mean
 Computer the covariance:
 Compute the eigenvalues
and eigenvectors of the data matrix.
 Order them by magnitude
 PCA reduces the dimension by keeping direction
such that

33. PCA Use for Image Compression(I)
 An image can be expressed as a weighted sum of
three colour components R, G, B according to
relation
 Images of size MxN saved in 3D matrix with size
MxNx3
 PCA theory applied and 3-dimension vector
reconstructed

34. PCA Use for Image Compression(II)
 Only the first - largest eigenvalue was used for its
definition
 This theory implies that the image obtained by
reconstruction contains the majority of information so
this image should have the maximum contrast.

35. Conclusion
 Aimed to specify an overview of aggregator operators
in image reduction.
 Described aggregation operators.
 Described image reduction.
 Described a study related to aggregator operator in
image reduction.
 Image reduction in machine learning.