1. 1. By Abu Sadat Mohammed Yasin Debotosh Dey
2. 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. 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. 4. Properties of Aggregation Operator Properties into two families The mathematical properties The behavioral properties
5. 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. 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. 7. Properties of Aggregation Operator Behavioral properties Decisional behavior Interpretability of the parameters Weights on the arguments
8. 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. 9. Different Types of Aggregation Operators Quasi-arithmetic means geometric mean harmonic mean Aczel J. Defines Dujmovic, Dyckhoff Defines: (f:x x )
10. 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. 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. 12. Different Types of Aggregation Operators Ordered Weighted Averaging Operators And many more GOWA, Quasi OWA, fuzzy OWA, LOWA, ULOWA, OWAWA, FGOWAWA
13. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.