Genetic algorithms: A Stochastic Approach for Improving the Current

Cadastre AccuraciesAnna ShnaidmanUri Shoshani Yerach Doytsher

Mapping and Geo-Information Engineering Technion – Israel Institute of Technology

2

Introduction

The problem at hand

The proposed algorithm

Implementation of GAs – cadastral analogy

Results

Summary

Future work

Outline

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Introduction

The problem at hand

The proposed algorithm

Implementation of GAs – cadastral analogy

Results

Summary

Future work

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Cadastral measurement is a continuous process of recording the redefinition and updates of boundaries

The cadastre in Israel is of an analogical nature

Different measurements of inconsistent accuracy stored mostly on paper

One of the main objectives on the SOI agenda is transition to a coordinate based (digital) cadastre

Introduction

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Digital cadastre is one of the concrete topics being discussed and researched in many countries:

Digital cadastre accuracies Transformation of graphical data into digital Improvement of dataset points accuracy using GNNS technology Global adjustment model

Most customary solutions are based mainly on the deterministic Least Square (LS) method

Introduction cont.

6

Introduction

The problem at hand

The proposed algorithm

Implementation of GAs – cadastral analogy

Results

Summary

Future work

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Due to population growth an automated and reliable land management system is urgently required

The current cadastre precludes: efficient and computerized management of real estate faster planning of development projectsminimizing border conflictskeeping up with modern customary high work standards

The transition to an analytical cadastre is both crucial and inevitable

The problem at hand

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Introduction

The problem at hand

The proposed algorithm

Implementation of GAs – cadastral analogy

Results

Summary

Future work

9

The proposed algorithm

The conventional methods are mainly analytical and straightforward

The proposed method is based on biological optimizations and is known as Genetic Algorithms (GAs)

Characteristics: stochastic methodfounded on evolutionary ideas and Darwin's principles of selection and survival of the fittest a natural selection which operates on a population of solutions – chromosomes (individuals)

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The proposed algorithm cont.

The generic framework of GAs: Create the initial population of n vectors

Evaluate (grade) the individuals by assigning a fitness value Create the new population by applying variation- inducing operators: selection, crossover and mutation

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The proposed algorithm cont .

Genetic operators - selection:

Two parent chromosomes are selected from a population according to their fitness Guiding principle – selection of the fittest

Superior individuals are of higher probability to be selected (survive)

Selection method – roulette wheel selection Roulette slots’ size is determined by the fitness value

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The proposed algorithm cont .

example

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The proposed algorithm cont .

Genetic operators - crossover: Two offspring are created using single point crossover

Parents chromosomes children chromosomes

Genetic operators - mutation: The new offspring are changed randomly to ensure

diversity

14

Introduction

The problem at hand

The proposed algorithm

Implementation of GAs – cadastral analogy

Results

Summary

Future work

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Implementation of GAs – cadastral analogy A generation of individuals - vectors of turning points coordinates of parcels

Each individual - a set of block coordinates stored in an array (vector) structure

Parcel areas and lines - provide the cadastral and geometrical constraints

An objective function - minimizes the differences between the legal (registered) coordinates and those provided by the solution under the specified conditions

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Implementation of GAs – cadastral analogy cont.

With each generation the vectors are altered according to the best solution provided

Every individual may assumed to be a set of coordinates, representing acceptable observations received from different sources

The GAs method was evaluated using synthetic data

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Implementation of GAs – cadastral analogy cont.

Definitions: The preliminary population of n vectors is produced by randomly altering an "ideal" cadastral block

A registered area criteria, straight, parallel and perpendicular lines were chosen for analyzing the GAs' competence and effectiveness in the cadastral domain

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Cadastral conditions: Objective function employs Cartesian area calculation and a desirable MSE of parcel coordinates Fitness function considers parcel size to determine its weight

Geometrical conditions: Objective function uses turning point angles, line segment lengths and the perpendicular distances Fitness function computes line weight using the number of points in both lines and the total line lengths

Total grade

Implementation of GAs – cadastral analogy cont .

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Implementation of GAs – cadastral analogy cont. – Objective function

1 1

( ) min( )

1 ( )2

f i i ideal calculated

calculated i i i

T u S S S S

S Y X X

1/100 2 2 2 1/100 21 1

( ) min( )

& : :

( )( ) ( )

f

ij i j i j i i i i

T u Delta

paralel perpendicular lines straight lines

Delta l l d d Delta l l d

Areas

Lines

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Implementation of GAs – cadastral analogy cont. – Fitness function

Areas

Lines

Total grade( ) ( ) ( ) ( ) ...b p l lGrade u f u f u f u

2 2 2

( )

100

[( ) ( ) ]XY

ip i i i

i

i ii

i i

i i

i i

Sf u u p pS

A SuA T

S ST mY X

1.5

1.5( ) ( )

( )( ) ( )

100

i j i ji i il

i j i j

i ii i

i

n n l lf u u w w

n n l lA Su S DeltaA

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Implementation of GAs – cadastral analogy cont.

Iterations – creation of successive generation: Parent selection - for each parcel and line/lines in the block two parents are selected according to the tournament method Crossover - a single point crossover is performed Process repetition - until the original population size (N) is reached

Averaging - mean coordinates are calculated

Mutation

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The proposed algorithm - graphical illustration

……Set 1

…Set 2

…Set N

Parcel 1 Parcel 1 Parcel 1 Lines 1 Lines 1 Lines 1

Parents selection

Single point crossover

Offspring 1 Offspring 2

next generation –Averaging coordinates, adding mutation, creation of new sets

…… New set 1

…New set 2

…New set N

Parent 1

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Introduction

The problem at hand

The proposed algorithm

Implementation of GAs – cadastral analogy

Results

Summary

Future work

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Simulation results

The proposed method's quality and accuracy were examined by performing simulations on the synthetic data

The main purpose of these simulations is to test the ability of the GAs to converge to the initial theoretical state - an ideal, errorless solution

For comparison a Least Square iterative adjustment was applied as well

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Simulation results cont.

A characteristic set of synthetic data

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Simulation results cont.

A characteristic example of the solution accuracy (meters)

Param- eters

Min dX

Min dY

Max dX

Max dY

Mean X

Mean Y

X Y

Initial-0.619-0.7040.6300.497-0.0120.0100.2160.224GAs-0.168-0.1900.1540.134-0.0030.0030.0440.043LS-0.619-0.7040.5690.456-0.0120.0100.1840.173

The following parameters have been used: Standard deviation error - 0.25 meter An expected MSE of the coordinates - 0.05 meter Maximum generations (iterations) - < 100

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Simulation results cont.

Solution improvement throughout the generations

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Simulation results cont.

Param- etersminmaxSTD

X0.0390.0490.004

Y0.0370.0450.002

Solution stability examination results (meters)

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Introduction

The problem at hand

The proposed algorithm

Implementation of GAs – cadastral analogy

Results

Summary

Future work

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Summary

The proposed method presents a new approach for achieving homogeneous coordinates by using evolutionary algorithms - GAs

GAs imitate the natural process of evolving solutions

Applying the GAs to synthetic data yields satisfactory results

Repeated simulation executions showed similar results

The GAs method is more accurate and provides better results than those of the traditional LS approach

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Introduction

The problem at hand

The proposed algorithm

Implementation of GAs – cadastral analogy

Results

Summary

Future work

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Future work

The simplicity of the algorithm enables considering additional cadastral and geometric conditions without altering its fundamental mechanism

Objectives: more detailed analysis of single blocks expansion of the dimensions of the problem implementation of the algorithm on "real" data

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Thank you