Adaptive Median Filters
Elements of visual perception
Representing Digital Images
Spatial and Intensity Resolution
cones and rods
Brightness Adaptation
Spatial and Intensity Resolution
Adaptive Median Filters
Elements of visual perception
Representing Digital Images
Spatial and Intensity Resolution
cones and rods
Brightness Adaptation
Spatial and Intensity Resolution
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
In this paper generation of binary sequences derived from chaotic sequences defined over Z4 is proposed.
The six chaotic map equations considered in this paper are Logistic map, Tent Map, Cubic Map, Quadratic
Map and Bernoulli Map. Using these chaotic map equations, sequences over Z4 are generated which are
converted to binary sequences using polynomial mapping. Segments of sequences of different lengths are
tested for cross correlation and linear complexity properties. It is found that some segments of different
length of these sequences have good cross correlation and linear complexity properties. The Bit Error Rate
performance in DS-CDMA communication systems using these binary sequences is found to be better than
Gold sequences and Kasami sequences.
Fixed-Point Code Synthesis for Neural Networksgerogepatton
Over the last few years, neural networks have started penetrating safety critical systems to take decisions in robots, rockets, autonomous driving car, etc. A problem is that these critical systems often have limited computing resources. Often, they use the fixed-point arithmetic for its many advantages (rapidity, compatibility with small memory devices.) In this article, a new technique is introduced to tune the formats (precision) of already trained neural networks using fixed-point arithmetic, which can be implemented using integer operations only. The new optimized neural network computes the output with fixed-point numbers without modifying the accuracy up to a threshold fixed by the user. A fixed-point code is synthesized for the new optimized neural network ensuring the respect of the threshold for any input vector belonging the range [xmin, xmax] determined during the analysis. From a technical point of view, we do a preliminary analysis of our floating neural network to determine the worst cases, then we generate a system of linear constraints among integer variables that we can solve by linear programming. The solution of this system is the new fixed-point format of each neuron. The experimental results obtained show the efficiency of our method which can ensure that the new fixed-point neural network has the same behavior as the initial floating-point neural network.
CYCLIC RESOLVING NUMBER OF GRID AND AUGMENTED GRID GRAPHSijcoa
For an ordered set W = {w1, w2 … wk} V (G) of vertices, we refer to the ordered k-tuple r(v W) = (d(v, w1), d(v, w2) … d(v, wk)) as the (metric) representation of v with respect to W. A set W of a connected graph G is called a resolving set of G if distinct vertices of G have distinct representations with respect to W. A resolving set with minimum cardinality is called a minimum resolving set or a basis. The dimension, dim(G), is the number of vertices in a basis for G. By imposing additional constraints on the resolving set, many resolving parameters are formed. In this paper, we introduce cyclic resolving set and find the cyclic resolving number for a grid graph and augmented grid graph.
High Speed Memory Efficient Multiplier-less 1-D 9/7 Wavelet Filters Based NED...IJERA Editor
Conventional distributed arithmetic (DA) is popular in field programmable gate array (FPGA) design, and it
features on-chip ROM to achieve high speed and regularity. In this paper, we describe high speed area efficient
1-D discrete wavelet transform (DWT) using 9/7 filter based new efficient distributed arithmetic (NEDA)
Technique. Being area efficient architecture free of ROM, multiplication, and subtraction, NEDA can also
expose the redundancy existing in the adder array consisting of entries of 0 and 1. This architecture supports any
size of image pixel value and any level of decomposition. The parallel structure has 100% hardware utilization
efficiency.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
In this paper generation of binary sequences derived from chaotic sequences defined over Z4 is proposed.
The six chaotic map equations considered in this paper are Logistic map, Tent Map, Cubic Map, Quadratic
Map and Bernoulli Map. Using these chaotic map equations, sequences over Z4 are generated which are
converted to binary sequences using polynomial mapping. Segments of sequences of different lengths are
tested for cross correlation and linear complexity properties. It is found that some segments of different
length of these sequences have good cross correlation and linear complexity properties. The Bit Error Rate
performance in DS-CDMA communication systems using these binary sequences is found to be better than
Gold sequences and Kasami sequences.
Fixed-Point Code Synthesis for Neural Networksgerogepatton
Over the last few years, neural networks have started penetrating safety critical systems to take decisions in robots, rockets, autonomous driving car, etc. A problem is that these critical systems often have limited computing resources. Often, they use the fixed-point arithmetic for its many advantages (rapidity, compatibility with small memory devices.) In this article, a new technique is introduced to tune the formats (precision) of already trained neural networks using fixed-point arithmetic, which can be implemented using integer operations only. The new optimized neural network computes the output with fixed-point numbers without modifying the accuracy up to a threshold fixed by the user. A fixed-point code is synthesized for the new optimized neural network ensuring the respect of the threshold for any input vector belonging the range [xmin, xmax] determined during the analysis. From a technical point of view, we do a preliminary analysis of our floating neural network to determine the worst cases, then we generate a system of linear constraints among integer variables that we can solve by linear programming. The solution of this system is the new fixed-point format of each neuron. The experimental results obtained show the efficiency of our method which can ensure that the new fixed-point neural network has the same behavior as the initial floating-point neural network.
CYCLIC RESOLVING NUMBER OF GRID AND AUGMENTED GRID GRAPHSijcoa
For an ordered set W = {w1, w2 … wk} V (G) of vertices, we refer to the ordered k-tuple r(v W) = (d(v, w1), d(v, w2) … d(v, wk)) as the (metric) representation of v with respect to W. A set W of a connected graph G is called a resolving set of G if distinct vertices of G have distinct representations with respect to W. A resolving set with minimum cardinality is called a minimum resolving set or a basis. The dimension, dim(G), is the number of vertices in a basis for G. By imposing additional constraints on the resolving set, many resolving parameters are formed. In this paper, we introduce cyclic resolving set and find the cyclic resolving number for a grid graph and augmented grid graph.
High Speed Memory Efficient Multiplier-less 1-D 9/7 Wavelet Filters Based NED...IJERA Editor
Conventional distributed arithmetic (DA) is popular in field programmable gate array (FPGA) design, and it
features on-chip ROM to achieve high speed and regularity. In this paper, we describe high speed area efficient
1-D discrete wavelet transform (DWT) using 9/7 filter based new efficient distributed arithmetic (NEDA)
Technique. Being area efficient architecture free of ROM, multiplication, and subtraction, NEDA can also
expose the redundancy existing in the adder array consisting of entries of 0 and 1. This architecture supports any
size of image pixel value and any level of decomposition. The parallel structure has 100% hardware utilization
efficiency.
the slides are aimed to give a brief introductory base to Neural Networks and its architectures. it covers logistic regression, shallow neural networks and deep neural networks. the slides were presented in Deep Learning IndabaX Sudan.
1. Fuzzy Rule Based Networks
Alexander Gegov
University of Portsmouth, UK
alexander.gegov@port.ac.uk
2. Presentation Outline
Part 1 - Theory: Introduction
Types of Fuzzy Systems
Formal Models for Fuzzy Networks
Basic Operations in Fuzzy Networks
Structural Properties of Basic Operations
Advanced Operations in Fuzzy Networks
Part 2 - Applications: Feedforward Fuzzy Networks
Feedback Fuzzy Networks
Evaluation of Fuzzy Networks
Fuzzy Network Toolbox
Conclusion
References
2
3. Introduction
Modelling Aspects of Systemic Complexity
• non-linearity (input-output functional relationships)
• uncertainty (incomplete and imprecise data)
• dimensionality (large number of inputs and outputs)
• structure (interacting subsystems)
Complexity Management by Fuzzy Systems
• model feasibility (achievable in the case of non-linearity)
• model accuracy (achievable in the case of uncertainty)
• model efficiency (problematic in the case of dimensionality)
• model transparency (problematic in the case of structure)
3
4. Types of Fuzzy Systems
Fuzzification
• triangular membership functions
• trapezoidal membership functions
Inference
• truncation of rule membership functions by firing strengths
• scaling of rule membership functions by firing strengths
Defuzzification
• centre of area of rule base membership function
• weighted average of rule base membership function
4
5. Types of Fuzzy Systems
Logical Connections
• disjunctive antecedents and conjunctive rules
• conjunctive antecedents and conjunctive rules
• disjunctive antecedents and disjunctive rules
• conjunctive antecedents and disjunctive rules
Inputs and Outputs
• single-input-single-output
• single-input-multiple-output
• multiple-input-single-output
• multiple-input-multiple-output
5
6. Types of Fuzzy Systems
Operation Mode
• Mamdani (conventional fuzzy system)
• Sugeno (modern fuzzy system)
• Tsukamoto (hybrid fuzzy system)
Rule Base
• single rule base (standard fuzzy system)
• multiple rule bases (hierarchical fuzzy system)
• networked rule bases (fuzzy network)
6
10. Formal Models for Fuzzy Networks
Node Modelling by If-then Rules
Rule 1: If x is low, then y is small
Rule 2: If x is average, then y is medium
Rule 3: If x is high, then y is big
Node Modelling by Integer Tables
Linguistic terms for x Linguistic terms for y
1 (low) 1 (small)
2 (average) 2 (medium)
3 (high) 3 (big)
10
23. Advanced Operations in Fuzzy Networks
Node Transformation in Input Augmentation
[N] (x | y) ⇒ [NAI] (x , xAI | y)
Node Transformation in Output Permutation
PO
[N] (x | y1, y2) ⇒ [N ] (x | y2, y1)
Node Transformation in Feedback Equivalence
[N] (x, z | y, z) ⇒ [NEF] (x, xEF | y, yEF)
23
24. Advanced Operations in Fuzzy Networks
Node Identification in Horizontal Merging
A*U =C
A, C – known nodes, U – non-unique unknown node
U*B=C
B, C – known nodes, U – non-unique unknown node
24
25. Advanced Operations in Fuzzy Networks
A*U*B=C
A, B, C – known nodes, U – non-unique unknown node
A*U=D
D*B=C
D – non-unique unknown node
U*B=E
A*E=C
E – non-unique unknown node
25
26. Advanced Operations in Fuzzy Networks
Node Identification in Vertical Merging
A+U=C
A, C – known nodes, U – unique unknown node
U+B=C
B, C – known nodes, U – unique unknown node
26
27. Advanced Operations in Fuzzy Networks
A+U+B=C
A, B, C – known nodes, U – unique unknown node
A+U=D
D+B=C
D – unique unknown node
U+B=E
A+E=C
E – unique unknown node
27
28. Advanced Operations in Fuzzy Networks
Node Identification in Output Merging
A;U =C
A, C – known nodes, U – unique unknown node
U;B=C
B, C – known nodes, U – unique unknown node
28
29. Advanced Operations in Fuzzy Networks
A;U;B=C
A, B, C – known nodes, U – unique unknown node
A;U=D
D;B=C
D – unique unknown node
U;B=E
A;E=C
E – unique unknown node
29
33. Feedforward Fuzzy Networks
Network with Multiple Levels and Multiple Layers
• grid of ‘m by n’ fuzzy systems
1,1 … 1,n
… … …
m,1 … m,n
33
34. Feedback Fuzzy Networks
Network with Single Local Feedback
• one node embraced by feedback
N(F) N N N(F)
N N N N
N N N N
N(F) N N N(F)
N(F) – node embraced by feedback
N – nodes with no feedback
34
35. Feedback Fuzzy Networks
Network with Multiple Local Feedback
• at least two nodes embraced by separate feedback
N(F1) N(F2) N N N(F1) N
N N N(F1) N(F2) N N(F2)
N(F1) N N N(F1) N N(F1)
N(F2) N N N(F2) N(F2) N
N(F1), N(F2) – nodes embraced by separate feedback
N – nodes with no feedback
35
36. Feedback Fuzzy Networks
Network with Single Global Feedback
• one set of at least two adjacent nodes embraced by feedback
N1(F) N2(F) N N
N N N1(F) N2(F)
N1(F) N N N1(F)
N2(F) N N N2(F)
{N1(F), N2(F)} – set of nodes embraced by feedback
N – nodes with no feedback
36
37. Feedback Fuzzy Networks
Network with Multiple Global Feedback
• at least two sets of nodes embraced by separate feedback with
at least two adjacent nodes in each set
N1(F1) N2(F1) N1(F1) N1(F2)
N1(F2) N2(F2) N2(F1) N2(F2)
{N1(F1), N2(F1)}, {N1(F2), N2(F2)} – sets of nodes embraced
by separate feedback
37
38. Evaluation of Fuzzy Networks
Linguistic Evaluation Metrics
• composition of hierarchical into standard fuzzy systems
• decomposition of standard into hierarchical fuzzy systems
Composition Formula
x–1 x–1
NE,x = *p=1 (N1,p + +q=p+1 Iq,p)
NE,x – equivalent node for a fuzzy network with x inputs
38
39. Evaluation of Fuzzy Networks
Decomposition Algorithm
1. Find N1,1 from the first two inputs and the output.
2. If x=2, go to end.
3. Set k=3.
4. While k≤x, do steps 5-7.
5. Find NE,k from the first k inputs and the output.
6. Derive N1,k–1 from the formula for NE,k, if possible.
7. Set k=k+1.
8. Endwhile.
NE,k – equivalent node for a fuzzy subnetwork with first k inputs
39
40. Evaluation of Fuzzy Networks
Functional Evaluation Metrics
• model performance indicators
• applications to case studies
Feasibility Index (FI)
n
FI = sum i=1 (pi / n)
n – number of non-identity nodes
p i – number of inputs to the i-th non-identity node
lower FI ⇒ better feasibility
40
41. Evaluation of Fuzzy Networks
Accuracy Index (AI)
AI = sum i=1nl sum j=1qil sum k=1vji (|yjik – djik| / vij)
nl – number of nodes in the last layer
qil – number of outputs from the i-th node in the last layer
vji – number of discrete values for the j-th output from the i-th
node in the last layer
yjik , djik – simulated and measured k-th discrete value for the j-th
output from the i-th node in the last layer
lower AI ⇒ better accuracy
41
42. Evaluation of Fuzzy Networks
Efficiency Index (EI)
EI = sum i=1n (qiFID . riFID)
n – number of non-identity nodes
FID – fuzzification-inference-defuzzification
FID
qi – number of outputs from the i-th non-identity node with an
associated FID sequence
riFID – number of rules for the i-th non-identity node with an
associated FID sequence
lower EI ⇒ better efficiency
42
43. Evaluation of Fuzzy Networks
Transparency Index (TI)
TI = (p + q) / (n + m)
p – overall number of inputs
q – overall number of outputs
n – number of non-identity nodes
m – number of non-identity connections
lower TI ⇒ better transparency
43
44. Evaluation of Fuzzy Networks
Case Study 1 (Ore Flotation)
Inputs:
x1 – copper concentration in ore pulp
x2 – iron concentration in ore pulp
x3 – debit of ore pulp
Output:
y – new copper concentration in ore pulp
44
45. Evaluation of Fuzzy Networks
Figure 4: Standard fuzzy system (SFS) for case study 1
45
46. Evaluation of Fuzzy Networks
Figure 5: Hierarchical fuzzy system (HFS) for case study 1
46
47. Evaluation of Fuzzy Networks
Figure 6: Fuzzy network (FN) for case study 1
47
48. Evaluation of Fuzzy Networks
Table 1: Models performance for case study 1
Index / Model Standard Hierarchical Fuzzy
Fuzzy System Fuzzy System Network
Feasibility 3 2 2
Accuracy 4.35 4.76 4.60
Efficiency 343 126 343
Transparency 4 1.33 1.33
FI – FN better than SFS and equal to HFS
AI – FN worse than SFS and better than HFS
EI – FN equal to SFS and worse than HFS
TI – FN better than SFS and equal to HFS
48
49. Evaluation of Fuzzy Networks
Case Study 2 (Retail Pricing)
Inputs:
x1 – expected selling price for retail product
x2 – difference between selling price and cost for retail product
x3 – expected percentage to be sold from retail product
Output:
y – maximum cost for retail product
49
50. Evaluation of Fuzzy Networks
100
90
80
70
60
output
50
40
30
20
10
0
0 20 40 60 80 100 120 140
input permutations
Figure 7: Standard fuzzy system (SFS) for case study 2
50
51. Evaluation of Fuzzy Networks
100
80
60
output
40
20
0
0 20 40 60 80 100 120 140
input permutations
Figure 8: Hierarchical fuzzy system (HFS) for case study 2
51
52. Evaluation of Fuzzy Networks
100
80
60
output
40
20
0
0 20 40 60 80 100 120 140
input permutations
Figure 9: Fuzzy network (FN) for case study 2
52
53. Evaluation of Fuzzy Networks
Table 2: Models performance for case study 2
Index / Model Standard Hierarchical Fuzzy
Fuzzy System Fuzzy System Network
Feasibility 3 2 2
Accuracy 2.86 5.57 3.64
Efficiency 125 80 125
Transparency 4 1.33 1.33
FI – FN better than SFS and equal to HFS
AI – FN worse than SFS and better than HFS
EI – FN equal to SFS and worse than HFS
TI – FN better than SFS and equal to HFS
53
54. Evaluation of Fuzzy Networks
Composition of Hierarchical into Standard Fuzzy System
• full preservation of model feasibility
• maximal improvement of model accuracy
• fixed loss of model efficiency
• full preservation of model transparency
Decomposition of Standard into Hierarchical Fuzzy System
• no change of model feasibility
• minimal loss of model accuracy
• fixed improvement of model efficiency
• no change of model transparency
54
55. Fuzzy Network Toolbox
Toolbox Details
• developed by the PhD student Nedyalko Petrov
• implemented in the Matlab environment
• to be uploaded on the Mathworks web site
• to be uploaded on the Springer web site
Toolbox Structure
• Matlab files for basic operations on nodes
• Matlab files for advanced operations on nodes
• Matlab files for auxiliary operations
• Word files with illustration examples
55
56. Conclusion
Theoretical Significance of Fuzzy Networks
• novel application of discrete mathematics and control theory
• detailed validation by test examples and case studies
Methodological Impact of Fuzzy Networks
• extension of standard and hierarchical fuzzy systems
• bridge between standard and hierarchical fuzzy systems
• novel framework for non-fuzzy rule based systems
Application Areas of Fuzzy Networks
• modelling and simulation in the mining industry
• modelling and simulation in the retail industry
56