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