Advances in DIP2

7/31/2019 Advances in DIP2

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Advances in DIP( Fuzzy , Artificial Neural Networks,Expert System and Image Segmentation)

Minakshi Kumar

Photogrammetry and Remote Sensing Division ,Indian Institute of Remote Sensing

Dehradun


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Fuzzy

ORIGIN:

Aristotle some 2000 years ago formulated Conventional Logicas the Law of Excluded Middle (ie., Crisp Set) , which hasdominated Western Logic ever since..

But the era was over..

Lotfi Zadeh, Professor, University of California at Berkley,had

first conceived the idea of Fuzzy Logic during 1960s as a way ofprocessing data by allowing partial set membership rather thancrisp set membership.


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WHAT IS FUZZY LOGIC (FL).?

FL provides a simple way to arrive at adefinite conclusion based upon vague,

ambiguous, imprecise, noisy, or missing inputinformation.

FL is basically a multi-valued logic that allows

intermediate values to be defined betweenconventional evaluations like yes/no,

true/false, black/white etc. Notions like ratherwarm or pretty cold can be formulated

mathematically and processed by computers


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Fuzzy Set Basics

Classical (crisp) sets: Membership in a set is all or nothing

Membership function cS: Universe {0, 1}

cS(x) = 1 iff x S

Fuzzy sets: Membership in a set is a degree

membership function cS

: Universe [0, 1]

rangeof real

numbers


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Linguistic Characterizations of Degree of

Membership Consider the set of cold days in Dehradun

in December 2005. Was December 31 cold? It might have

been called one of:

very cold

sort of cold

not cold

The answer depends on the observer, time,

etc.


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Sounds similar to probability, but isnt

Probability deals with uncertainty, likelihood

Fuzzy logic deals with ambiguity, vagueness


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Fuzzy Membership


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Membership Functions

A membership function (MF) is a curve that defineshow each point in the input space is mapped to amembership value (or degree of membership)between 0 and 1.

The input space is sometimes referred to as theuniverse of discourse, a fancy name for a simpleconcept.

It associates a weighting with each of the inputs thatare processed, define functional overlap between

inputs, and ultimately determines an output response. The rules use the input membership values as

weighting factors to determine their influence on the

fuzzy output sets of the final output conclusion.


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Membership function plots

is a pile

# grains of sand

1

0

10 100 1000 10000 100000


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Crisp vs. Fuzzy

Membership Functionsis a pile

grains of sand

fuzzycrisp

1

0

10 100 1000 10000 100000


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Set of Tall people


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More Crisp vs. Fuzzy

Membership Functions1

0

1

0

grade ofmembershipCrisp

universe of

possibilities

Fuzzy

1

0

1

0

Note: Universe can be continuous or discrete,

ordered or unordered.


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Alternative Notation


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Fuzzy-Set Operationsexpressed using membership functions

1

0

A B

Fuzzy OR (union) Fuzzy AND (intersection)

cA U B(x) = max(cA (x), cB(x)) cA B(x) = max(cA (x), cB(x))

1

0

A U B1

0

A B


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Fuzzy Complement

(not the only possible model)

cA(x) = 1 - cA(x).

1

0A

A


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Fuzzy Classification

Method

In traditional classification, information is

represented in a one-pixel -one - class method andits classification result is definite, i.e. a pixel belongsto a class.

In fuzzy classification, a pixel is considered havingdifferent membership grades obtained from fuzzyclassification indicate the area proportion of everycover classes in a mixed pixel.

As the traditional method, the fuzzy classificationcan also be divided into fuzzy supervised and fuzzyunsupervised classification.


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Fuzzy Supervised Classification

Sample pixels should be chosen for

estimating fuzzy parameters beforeclassification, different to conventional

method, chosen sample need not be

sufficiently homogeneous. Fuzzy mean canbe expressed as


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where n is the total number of sample pixels,

fc is membership of a pixel to class c, xi ispixel value of sample pixels. Fuzzy

covariance matrix can be expressed as

f f


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After the fuzzy parameters are determined, everypixel is classified according to its spectral value.

To perform a fuzzy classification, the membershipfunction must be defined for each class,

In this work, the membership functions are defined

based on maximum likelihood classificationalgorithm with fuzzy and fuzzy covariance matrixreplacing the conventional mean and covariancematrix.

The following is the definition of membershipfunction for cover class c:

N is the pixel

vector dimension,m is number of

predefined

classes.


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Fuzzy Unsupervised Classification

The fuzzy unsupervised classification can be

performed in many methods, in this work, k-

means clustering algorithm is adopted and isrun using iteration as following:

determine classification number k, 1

give a initial membership matrix


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Fuzzy Unsupervised Classification

Cont calculate fuzzy clustering where centers of

every class

calculate new membership matrix U(t+1)

using formula

if the criterion is satisfied, stop the iteration.Otherwise, repeat process 3 to 5. Thecriterion of ending iteration can choose


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Artificial neural Networks

Introduction of Neural Network:Introduction of Neural Network: Brain is a highly complex, nonlinear, and parallel computer. It

has structural constituents, known as neurons for computations.

Definition of Neural Network:Definition of Neural Network:

A neural network is a massively parallel distributed processormade up of simple processing units, which has a naturalpropensity for storing experimental knowledge and making itavailable for use. It resembles the brain in two respects;

Knowledge is acquired by the network from its environmentthrough a learning process.

Inter-neuron connection strengths, known as synaptic weights,are used to store the acquired knowledge.

The produre used to perform the learning process is called alearning algorithm, the function of which is to modify the synapticweights of the network in an orderly fashion to attain a desireddesign objective.

Th NTh NB i f NB i t f N


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The Neuron:The Neuron:

The basic neuron consists of;

Synapses: are connectionsbetween neurons they are not

physical connections, but

miniscule gaps that allow

electric signals to jump across

from neuron to neuron.

Soma: these electrical signals

are then passed across to the

soma which performs some

operation and sends out itsown electrical signal to the

axon.

Axon: the axon then

distributes this signal to

dendrites.

Dendrites: carry the signals

out to the various synapses,and

the cycle repeats

Basic components of Neuron:Basic components of Neuron:


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Artificial Neuron:Artificial Neuron:

Each neuron has a certain number of inputs.Each of which have a weight assigned to them.

The weights simply are an indication of how important the

coming signal for that input is.

The net value of the neuron is then calculated.

The net is simply the weighted sum, the sum of all the inputsmultiplied by their specific weight.

Each neuron has its own unique threshold value, and if the

net is greater than the threshold, the neuron fires(or output a

1), otherwise it stays quiet(outputs a 0).

The output is then fed into all the neurons it is connected to.


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Architecture of Neural Network:Architecture of Neural Network:

The network consists of several layers of neurons

9An input layer

9Hidden layer

9Output layer

Hidden layer-because the user cannot see the inputs or outputs for those

layers.

Ups and downs of neural network:Ups and downs of neural network:

They are excellent as pattern classification/recognizers.

Neural Network can handle exceptions and abnormal input data.

The power of neural network lies in their ability to process

information in a parallel fashion but machines today are serial they

only execute on instruction at a time.

General Mechanism of Neural Network:General Mechanism of Neural Network:


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Schematic representation of an neural network:

An artificial neuron is a model whose components have direct analogs to the


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An artificial neuron is a model whose components have direct analogs to the

components of actual neuron.

The input signal are represented by x0, x1, ..xn.

Each of these inputs is modified by a weight (synaptic weight) whose

function is analogous to that of the synaptic junction in a biological neuron.

These weights can be either positive or negative.

The processing element consists of two parts

9The first part simply sums the weighted input resulting in a quantity I.

9The second part is effectively a non-linear filter, usually called the activation

function through which the combined signal flows.

Activation function be;

It can be threshold function that passes information, when the output of the

first part of the artificial neuron exceeds the threshold T.

(I) =1, I>T

0, I


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Perceptron:Perceptron:

Perceptron is the most basic term used in ANN.

It can be single layered or multi-layered structure.

Single perceptron:Single perceptron:

The structure of a single perceptron is very simple.

There are two inputs, a bias and an output

In the perceptron, the most common form of learning is byadjusting the weights by the difference between the desired output

and the actual output.

i= x

i

= desired output actual output.


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Linear associator neural network:Linear associator neural network:

The most elementary neural network is a linearassociator

The fundamental neuron sums the weighted inputs and

then subjects this sum to a nonlinear activation function tokeep the output with in reasonable range.

The architecture of a linear associator is a set of input

neurons that are connected to a set of output neurons. (i.e.,

a two =layer neural network)

When the activation functions of the neurons are linear,

this network is a linear associator

It is not very accurate, if two many items are stored in the

associators.

Learning:Learning:


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Learning:Learning:

This talk about weights and thresholds.

Which leads to an obvious question, how are all these set?

Most popular training methods are;

Back-propagation

The delta rule

Kohonen learning

Learning rules can be categorized into two areas:

Supervised

Un-supervised

Supervised learning rules require a teacher.

Unsupervised learning do not require teacher because they produce their

own output, which is then further evaluated.

Learning and Recall:Learning and Recall:


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e g d ec :g

Learning is the process of adapting the connection weights in an artificial

network to produce the desired output vector presented to the inputbuffer.

Recall is the process of accepting an input stimulus and producing an

output response in accordance with the network weight structure.

Recall is an integral part of the learning process since a desired response

to the network must be compared to the actual output to create an error

function.

In supervised learning, the artificial neural network is trained to give the

desired response to a specific input stimulus.

In graded learning, the output is graded as good or bad on a numerical

scale, and the connection weights are adjusted in accordance with thegrade.

In unsupervised learning there is no specific response sought, but rather

the response is based on the networks ability to organize itself.

BackBackPropagation:Propagation:


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Back-propagation is a systematic method for training multiple (three or more) layer

artificial neural networks.

The problem with the perception is that it cannot express non-linear decisions.

The perceptron is basically a linear threshold device, which returns a certain value, 1

for example , if the dot product of the input vector and the associated weight vector plus

bias surpasses the threshold, and another value, -1 for example, if the threshold is not

reached.

For example:

If the dot product of the input vector and the associated weight plus the bias, is graphed it

is obviously linear.

More over, all the input vectors that will give a value greater than the threshold are

separated into one space, and those that will not, will be separated into another.

+ +

-

-

+

+-

-

Non-linear plotLinear plot

Multi-layer feed forward network normally consist of three or


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Multi layer feed forward network normally consist of three or

four layers; with one output and minimum one hidden layer.

The raw values of input layer are fed to hidden layer.

Once the neurons for the hidden layer are computed, their

activations are then fed downstream to the next layer, until all

the activations eventually reach the output layer, in which each

output layer neuron is associated with a specific classification

category.

Thus in computing the values of each neuron in the hidden and

output layers one must first take the sum of the weighted sums

and the bias, and then apply sigmoid function to calculate the

neurons activation.

Then how does the network learn the problem? The solution is

by modifying all the weights.

How to modify the weights;How to modify the weights;


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ow o od y e we g s;y g ;

Take partial derivative of the error of the network with

respect to each weight, we will learn a little about thedirection the error of the network is moving.

If the derivative is positive, this tells us that the error is

increasing when the weight is increasing, then add a negative

value to the weight and vice versa if the derivative is

negative.

Because the taking of these partial derivatives and then

applying them to each of the weights takes place starting

from the output layer to hidden layer weights, then the

hidden layer to input layer weights.

This algorithm has been called the Back Propagation

Algorithm.

Basic steps in backBasic steps in back propagation training for a multipropagation training for a multi layer neural network:layer neural network:


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Basic steps in backBasic steps in back--propagation training for a multipropagation training for a multi--layer neural network:layer neural network:

1. Randomize the weights to small random values (both positive &

negative) to ensure that the network is not saturated by largevalues of weights.(if all weights start at equal values, and desired

performance requires unequal weights, the network would not

train at all)

2. Select a training pair from the training set.

3. Apply the input vector to network input.

4. Calculate the network output.

5. Calculate the error, the difference between the network output

and the desired output.

6. Adjust the weights of the network in a way that minimizes thiserror.

7. Repeat steps 2 6 for each pair of input output vectors in the

training set until the error fro the entire system is acceptably low.

Neural Networks


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A NN is a machine learning approach inspired by the way in

which the brain performs a particular learning task:

Knowledge about the learning task is given in the form of

examples.

Inter neuron connection strengths (weights) are used to

store the acquired information (the training examples).

During the learning process the weights are modified in

order to model the particular learning task correctly on the

training examples.

Learning


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Learning

Supervised Learning

Recognizing hand-written digits, pattern recognition,

regression.

Labeled examples(input , desired output)

Neural Network models: perceptron, feed-forward, radial

basis function, support vector machine.

Unsupervised Learning

Find similar groups of documents in the web, content

addressable memory, clustering.

Unlabeled examples(different realizations of the input alone)

Neural Network models: self organizing maps, Hopfield

networks.

Network architectures


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Three different classes of network architectures

single-layer feed-forward neurons are organized

multi-layer feed-forward in acyclic layers

recurrent

The architecture of a neural network is linked with the learning

algorithm used to train

Single Layer Feed-forward


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g y

Input layer

of

source nodes

Output layer

of

neurons


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Recurrent network


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Recurrent network

Recurrent Network with hidden neuron(s): unit

delay operatorz-1 implies dynamic system

z-1

z-1

z-1

input

hidden

output

Neural Network Architectures


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The Neuron


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The neuron is the basic information processing unit of a NN.It consists of:

1 A set ofsynapses orconnecting links, each link

characterized by a weight:

W1, W2, , Wm

2 An adderfunction (linear combiner) which computes

the weighted sum of the inputs:

3 Activation function (squashing function) for limiting

the amplitude of the output of theneuron.

==

m

1jjxwu

j

)(uy b+=

The Neuron


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Input

signal

Synaptic

weights

Summing

function

Bias

b

ActivationfunctionLocal

Field

vOutput

y

x1

x2

xm

w2

wm

w1

M M

)(

Bias of a Neuron


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Bias b has the effect of applying an affine transformation to u

v = u + b

vis the induced field of the neuron

v

u

==

m

1jjxwu

j

Bias as extra input


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Input

signal

Synaptic

weights

Summing

function

Activation

functionLocal

Fieldv

Output

y

x1

x2

xm

w2

wm

w1

M M

)(

w0x0= +1

Bias is an external parameter of the neuron. Can bemodeled by adding an extra input.

bw

xwv jm

j

j

=

= =

0

0

Segmentation


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Segmentation

is the process of dividing an image into regions of relatedcontent.

is a process of partitioning an image into some non-overlappingmeaningful homogeneous regions

process of subdividing or partitioning of an image intohomogeneous (with respect to some criterion of homogeneity)

and disjoint (non-overlapping, non uniform) regions of differentstatistical behaviour, called image segments

Segmentation


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g

9Grouping together pixels that have similarproperties such as color, texture, motion, etc

9 Each pixels can be treated as a data point in thefeature space

Segmentation


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g

These image regions or segments may be laterassociated with information labels, but thesegmentation process simply gives each region ageneric label (region 1, region 2, etc.).

In the context of earth remote sensing, these labelswould generally be a ground cover type or land use

category.The shape of an object can be described in terms of:

Its boundary requires image edge detection

The region it occupies requires image segmentation in

homogeneous regions, Image regions generally havehomogeneous characteristics (e.g. intensity, texture)

Segmentation Algorithms


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Segmentation Algorithms

Segmentation algorithms are based on one of two basicproperties of intensity values - discontinuity and similarity.

First category is to partition an image based on abrupt changesin intensity, such as edges in an image.

Second category are based on partitioning an image into regionsthat are similar according to a predefined criteria. Histogram

Thresholding approach falls under this category.

Segmentation Approaches


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g pp

Thresholding

Boundary Detection Region Growing

Thresholding


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g

Suppose that the gray-level histogram corresponds to animage, f(x,y), composed of dark objects in a lightbackground, in such a way that object and backgroundpixels have gray levels grouped into two dominant modes.

One obvious way to extract the objects from the backgroundis to select a threshold T that separates these modes.

Then any point (x,y) for which f(x,y) > T is called an object

point, otherwise, the point is called a background point.

Gray Scale Image - bimodal


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y g

Image of rice with black background Image histogram of rice

Segmented Image


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Segmented Image

Image after segmentation


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Region Growing


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A simple approach to image segmentation is tostart from some pixels (seeds) representing

distinct image regions and to grow them, until

they cover the entire image

For region growing we need a rule describing a

growth mechanism and a rule checking the

homogeneity of the regions after each growth

step

Split / Merge


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The opposite approach to region growing isregion shrinking ( splitting ).

It is a top-down approach and it starts with the

assumption that the entire image ishomogeneous

If this is not true , the image is split into four sub

images

This splitting procedure is repeated recursively

until we split the image into homogeneousregions

Split / Merge


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If the original image is square N x N, having dimensions that

are powers of 2(N = 2n):

All regions produced but the splitting algorithm are squares

having dimensions M x M , where M is a power of 2 as well(M=2m,M


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Quadtree

R1R0

R0

R3

R1R2

R00 R01 R02 R04

Split / Merge


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Splitting techniques disadvantage, they

create regions that may be adjacent and

homogeneous, but not merged.

Split and Merge method It is an iterative

algorithm that includes both splitting andmerging at each iteration:

Boundary Based Segmentation


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Methods based on Edge Detection

Filters

Sobel

Roberts

Kirsh

Laplace

Expert Systems


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Expert Systems

Knowledge Based Information System(KBIS)

Expert System (ES):

A KBIS that uses its knowledge about a

specific area to act as an expert consultant

to the end user

Expert Systems


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Expert Systems

Expert Systems and DSS:ES are built into a DSS to improve the

quality of decision making; helps add

structure to unstructured problems.

Support a specific problem domain.

Uses a knowledge database to support

decision making.

Characteristics of an Expert


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System

Can explain their reasoning or suggesteddecisions.

Can display intelligent behavior.Can draw conclusions from complex

relationships.

Can deal with uncertainty.

Components of


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Expert Systems

[Figure 11.7]

Advantages of ES


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Advantages of ES

Serve in the absence of a real expert.

Provide consistent advice, unaffected bystress and time constraints.

Knowledge can be transferred andreproduced.

May be less expensive than an actual

expert, especially if the knowledge is used

over and over.

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