From Reality to Generalization Working with Abstractions

1. From Reality to Generalization Working with Abstractions Research Seminar Mohammad Reza Malek Institute for Geoinformation, Tech. Univ. Vienna malek@geoinfo.tuwien.ac.at

2. Introduction (Definition) 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.

3. 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

4. General Problem General Solution General Method Specification/Instantiation Abstraction/Generalization Specific Method Introduction (Methodology) Specific Problem Specific Solution

5. 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

6. Structure • Introduction • Related work • Functional analysis • Functional analysis as a toolbox in GIS • Some remarks with examples • Summarize

7. 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

8. X functinal: Scalar Field Vector Space L:XnR  A Functional Analysis Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. • Dual Sapce is created (spanned) by functionalas themselves.

9. Functional Analysis (continue) -dirac functional at a specified point returns the value of the function at that point. x f=f(x)  Example: A raster map (digital image) can be considered as : Nearly all kind of measurements such as temp., dist., angle can be interpreted as a  functional on a Hilbert space. L:HER

10. ? Xn Lm A- A Px Pl X’ L’ X= (At.Pl.A) -1.At.Pl.l At Functional Analysis (example)  Parametric Model Adjustment: (*)x=(Px)-1.At.Pl.(*)l Px= (At.Pl.A)

11. ? Wn Lm B- B Pw Pl W’ L’ l= Pl-1.Bt.(B.Pl -1.Bt)-1.w Bt Functional Analysis (example)  Observation condition equation: (*)l=(Pl)-1.Bt.Pw.(*)w Pw= (B. Pl-1. Bt)-1

12. Func. desc. Value desc. Xc Xd Functional Analysis as a toolbox Analog-to-digital conversion

13. Functional Analysis as a toolbox Key concept: Function spaces Analog situation Dual spaces Digital situation

14. Functional Analysis as a toolbox (spectral description) Digital process means using spectral descriptions Base function Eigenvector Example: (Linear Filter) An important theorem in functional analysis

15. 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, …)

16. Functional Analysis as a toolbox (Generalized spatial interpolation) L2 L3 L0=? L1 L4 L5 L f=l ; O(L)=n×1  Given n linear, independent and bounded functional (not necessary  functional): - Estimate the vale of a functional (Local Interpolation) -Estimate the function(Global Interpolation)

17. 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 …

18. 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

19. Notes in Abstraction/Generalization (isomorphism) Network design orders Structure design  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.

20. 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.

21. Notes in Abstraction/Generalization (limitation) Euclidean space, D=[-1,1] with Known: = = d = l L f f 1 1 1 1 = - x 1 2 Required:  Be aware of the limitation of the selected tool • Example: A method maybe too general to apply.

22. 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)