From Reality to Generalization Working with Abstractions

Research Seminar

Mohammad Reza Malek

Institute for Geoinformation, Tech. Univ. Vienna

malek@geoinfo.tuwien.ac.at

There is no science and no knowledge without abstraction.

Abstraction is an emphasis on the idea, qualities and properties rather than particulars.

Generalization is a broadening of application to encompass a larger domain of objects.

Introduction (Definition)

Introduction (Motivation)

Advantages:

- To open new windows

- To ease solving problems:

* in abstraction by hiding irrelevant details

* in generalization by replacing multiple entities which perform similar

functions

In GIS:

- A framework for open systems

* Standards

* Software programming

Specific Problem Specific SolutionSpecific Method

General Problem

Abstraction/Generalization

General SolutionGeneral Method

Specification/Instantiation

Introduction (Methodology)

Introduction (Aim)

The main aim of the current presentation is:

To give some important and practical remarks about abstraction and generalization based on mathematical toolboxes

Structure

• Introduction

• Related work

• Functional analysis

• Functional analysis as a toolbox in GIS

• Some remarks with examples

•Summarize

Related Work

… How people do get abstract concepts? (Epistemology)

Any work in the spatial theory

Frank’s approach:

- GIS is pieces of a puzzle

- Describe your model by an algebra

- Algebras can be combined

Functional Analysis

Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions.

AX

Vector Space Scalar Field

functinal:

L:XnR

Dual Sapce is created (spanned) by functionalas themselves.

Functional Analysis (continue)

-dirac functional at a specified point returns the value of the function at that point.

Nearly all kind of measurements such as temp., dist., angle can be interpreted as a functional on a Hilbert space.

x f=f(x)

L:HER

Example: A raster map (digital image) can be considered as :

),(.),( , kl

klkk

yyxxfyxf

Xn Lm

A

L’X’ At

PlPx

(*)x=(Px)-1.At.Pl.(*)l

Px= (At.Pl.A)

X= (At.Pl.A) -1.At.Pl.l

A-?

Functional Analysis (example)

Parametric Model Adjustment:

(*)l=(Pl)-1.Bt.Pw.(*)w

Pw= (B. Pl-1. Bt)-1

l= Pl-1.Bt.(B.Pl

-1.Bt)-1.w

B-?

Functional Analysis (example)

Observation condition equation:

Wn Lm

B

PlPw

L’W’ Bt

Functional Analysis as a toolbox

Analog-to-digital conversion

Func. desc. Value desc. Xc Xd

Functional Analysis as a toolbox

Key concept:

Function spaces Analog situation

Dual spaces Digital situation

Functional Analysis as a toolbox (spectral description)

Digital process means using spectral descriptions

fftfxdfftfxyxf ),(),(),()(),(_

Base function Eigenvector

Example: (Linear Filter)

k

kik

k

niknikn

kknkn

ef

eefy

xfy

)(

;)()(

;

kkk xAX

XAX

;

An important theorem in functional analysis

kk xfXAf )()(

Functional Analysis as a toolbox (numerical solvability)

Is there a solution for the specific problem?

Does this procedure converge?

Fixed point theorem (Banach theorem, Schauder theorem, …)

)(

:

00 xTx

T

Functional Analysis as a toolbox (Generalized spatial interpolation)

Given n linear, independent and bounded functional (not necessary functional): - Estimate the vale of a functional (Local Interpolation) - Estimate the function (Global Interpolation)

L1

L5L4

L3

L2

L0=?

)( 11 1pffl p

)( 22 2pff

dx

dl p

)( 32

2

3 3pff

dx

dl p

b

adxxfl )(4

00lfLp

L f=l ; O(L)=n×1

lGbLfGbfLl TT 1100

ˆˆ

Functional Analysis as a toolbox (summary)

subject Tool in functional

Digitizing

Digital description

Process

A distance minimization

Convergence

New problem

Finding optimal solution

Distance

Multi type interpolation

…

Functional

Eigenvalue

Operator

Approximation

Fixed point theorem

Linearization

Orthogonal projection theorem

Meter

Generalized interpolation

…

Notes in Abstraction/Generalization (similarity)

Look to similarities - A reasonable start point - It maybe necessary but not sufficient

Example: Similarities between a geodetic network and a cable framework

Notes in Abstraction/Generalization (isomorphism)

Look for isomorphism - Note to fundamental properties

Example: The weight matrix in the least squares adjustment procedure and the stiffness matrix in the framework structure analysis by finite element method.

ASAT .. VPV T ..

Network design orders

Structure design

Notes in Abstraction/Generalization (change)

Change the selected tools with another suitable and consist tool

Example: Using 4-dimensional Hamilton algebra in place of traditional matrix rotational methods: - The gimbal lock problem in navigation and virtual reality - A quaternion is defined as follow:

3210 ... qkqjqiqq

Where i, j, k are hyper imagery numbers.

1 kkjjii

The newer does not mean the better.

Notes in Abstraction/Generalization (limitation)

Be aware of the limitation of the selected tool

Example: A method maybe too general to apply.

Euclidean space, D=[-1,1] with2

11 x

2

10 x

dx

dL x002

1

103 (.)dxL001 xL

1ffLl2

1x

111

Known:

Required:

lGbl̂ 1T0

1

1 1

1

10

031

021

011

3

2

1

)2

3

2

1(

2

32

3

2

1

,

,

,

dxxx

x

xx

LL

LL

LL

b

b

b

bT

121

1

10

03

02

01

0 31

2

12

32

3

2

1

ˆ

ˆ

ˆ

ˆ lx

x

xx

l

l

l

l

7

87

67

1

0l

Summary

Abstraction/generalization is an important part of preparing an open system.

Functional analysis is introduced.

The following notes play an important role in abstraction:

- similarities

- fundamental common concepts or properties

- to be dare to change the selected tool

- familiarity with limitation of the selected tool

We need a type of experts who work as a bridge between pure science and engineering (after Grafarend: operational expert)