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. 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 )

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

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. 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.

9. 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 :

10. 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:

11. (*) 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

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

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 functionEigenvector  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)  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

17. 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 …

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)  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

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)  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:

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)