From Reality to Generalization Working with Abstractions Research Seminar Mohammad Reza Malek...
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![Page 1: From Reality to Generalization Working with Abstractions Research Seminar Mohammad Reza Malek Institute for Geoinformation, Tech. Univ. Vienna malek@geoinfo.tuwien.ac.at.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec15503460f94bccba3/html5/thumbnails/1.jpg)
From Reality to Generalization Working with Abstractions
Research Seminar
Mohammad Reza Malek
Institute for Geoinformation, Tech. Univ. Vienna
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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)
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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
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Specific Problem Specific SolutionSpecific Method
General Problem
Abstraction/Generalization
General SolutionGeneral Method
Specification/Instantiation
Introduction (Methodology)
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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
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Structure
• Introduction
• Related work
• Functional analysis
• Functional analysis as a toolbox in GIS
• Some remarks with examples
•Summarize
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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
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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.
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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
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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:
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(*)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
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Functional Analysis as a toolbox
Analog-to-digital conversion
Func. desc. Value desc. Xc Xd
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Functional Analysis as a toolbox
Key concept:
Function spaces Analog situation
Dual spaces Digital situation
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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 )()(
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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
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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
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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
…
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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
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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
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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.
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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
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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)
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