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 Solution Specific Method General Problem Abstraction/Generalization General Solution General 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.  A X Vector Space Scalar Field functinal: L:X n  R  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:H  E  R  Example: A raster map (digital image) can be considered as :

X  n L  m A L’X’ AtAt PlPl PxPx (*) x =(P x ) -1.A t.P l.(*) l P x = (A t.P l.A) X= (A t.P l.A) -1.A t.P l.l A-A- ? Functional Analysis (example)  Parametric Model Adjustment:

(*) l =(P l ) -1.B t.P w.(*) w P w = (B. P l -1. B t ) -1 l= P l -1.B t.(B.P l -1.B t ) -1.w B-B- ? Functional Analysis (example)  Observation condition equation: W  n L  m B PlPl PwPw L’W’ BtBt

Functional Analysis as a toolbox  Analog-to-digital conversion Func. desc.Value desc. X c XdXd

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 Base functionEigenvector  Example: (Linear Filter)  An important theorem in functional analysis

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, …)

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 L5 L4 L3 L2 L0=? L f=l ; O(L)=n×1

Functional Analysis as a toolbox (summary) subjectTool 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. 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: Where i, j, k are hyper imagery numbers.  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] with 1ffLl 2 1 x 11 1   Known: Required:

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)