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From Reality to Generalization Working with Abstractions
Research Seminar
Mohammad Reza Malek
Institute for Geoinformation, Tech. Univ. Vienna
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)