Dissertation / Doctoral Thesis

DISSERTATION / DOCTORAL THESIS Titel der Dissertation /Title of the Doctoral Thesis „Slicing multi-dimensional spaces“ verfasst von / submitted by Thomas Torsney-Weir angestrebter akademischer Grad / in partial fulfilment of the requirements for the degree of Doktor der Technischen Wissenschaften (Dr. techn.) Wien, 2018 / Vienna, 2018 Studienkennzahl lt. Studienblatt / A 786 880 degree programme code as it appears on the student record sheet: Dissertationsgebiet lt. Studienblatt / Informatik field of study as it appears on the student record sheet: Betreut von / Supervisor: Univ.-Prof. Dr. Torsten Möller, PhD To whomever finds this ii Acknowledgements I would like to acknowledge the invaluable support of my supervisor, Torsten Möller, as well as Michael Sedlmair. Their assistance and experience have been invaluable these last five years. In addition, I want to thank all past and present members of the Visualization and Data Analysis reserach group at the University of Vienna. In particular, I want to thank Johanna Schlereth, Michael Phillips, Alireza Ghane, Raphael Sahann, Christoph Kralj, Bernhard Fröler, Mohsen Kamalzadeh, Hamid Younesy, Elena Rudkowsky, Patrick Wolf, Manfred Klaffenböck, Chrtistoph Langer, Peter Ruch, Jennifer Prengel, Monika Gregor, Michaela Parzer, and Anne Marie Faisst. The discussions I’ve had over these past few years have been many and varied. I want to thank all of them for their suggestions and assistance in performing my research. And last but certainly not least, I want to thank my family for their love, support, and encouragement throughout my life. Without their support I would not have left a lucrative but boring career in finance in order to become a happy student again. Furthermore, I would not have made it through the most difficult times of my PhD without their support. iii Contents List of Figures viii List of Tables x List of Algorithms xi 1 Motivation 1 1.1 Multi-dimensional spaces . 3 1.1.1 Manifolds . 4 1.1.2 Shapes . 5 1.2 Visualizing multi-dimensional continuous spaces . 6 1.2.1 Encoding multi-dimensional data . 7 1.2.2 Methods . 8 1.3 Slices . 9 1.4 Upcoming . 11 2 1D slices 13 2.1 Motivation . 14 2.2 Related Work . 16 2.2.1 Discretization . 16 2.2.2 Local methods . 17 2.2.3 Global methods . 18 2.3 Sliceplorer . 18 2.3.1 Design requirements . 19 iv Contents 2.3.2 Intuition . 19 2.3.3 Focus point projection . 20 2.3.4 Linked selection . 21 2.3.5 Clustering . 21 2.4 Task-based evaluation . 22 2.4.1 Study design . 24 2.4.2 Results . 25 2.4.3 Summary . 28 2.5 Usage scenarios . 28 2.5.1 2D sinc function . 29 2.5.2 Neural networks . 30 2.5.3 Optimization algorithm . 32 2.6 Discussion . 34 2.7 Limitations and future work . 35 2.8 Conclusion . 36 3 2D slices 39 3.1 Motivation . 40 3.2 Related work . 42 3.2.1 Multi-objective optimization . 42 3.2.2 Multi-dimensional objects . 43 3.3 Algorithm . 45 3.3.1 Conceptual overview . 45 3.3.2 Algorithm details . 47 3.4 Interface . 50 3.4.1 Global view . 50 3.4.2 Local view . 51 3.5 Case studies . 51 3.5.1 Polytopes . 52 3.5.2 Positive and Bernstein polynomials . 53 v Contents 3.5.3 Pareto fronts . 57 3.6 Algorithm performance . 59 3.7 Conclusion and future work . 61 4 Fast slicing 63 4.1 Motivation . 63 4.2 Related work . 66 4.2.1 Multi-D visualization . 66 4.2.2 Multi-D interpolation . 68 4.3 Problem description . 69 4.3.1 1D analysis example . 69 4.3.2 Applications in multi-D . 72 4.3.3 Pipeline description . 74 4.4 Requirements . 75 4.4.1 Scene geometry . 76 4.4.2 HyperSlice effect on scene geometry . 77 4.4.3 Algorithm . 78 4.5 Derivation of scene geometry . 80 4.5.1 Filtering . 80 4.5.2 Rendering . 81 4.5.3 Expected total time . 81 4.6 Fitting . 83 4.6.1 Sampling . 84 4.6.2 Final model . 86 4.7 Timing results . 86 4.7.1 Data fitting . 86 4.7.2 Accuracy . 88 4.8 Application scenarios . 90 4.8.1 Constrain sampling . 91 4.8.2 Subsample points . 91 vi Contents 4.9 Limitations and future work . 93 5 Conclusion 95 5.1 Future . 96 5.2 Implications . 97 Bibliography 99 A Full sliceplorer views 113 B The expected number of points in a parameter space 115 B.1 Expected number of fragments . 115 B.2 Expected 3D quad size . 118 B.3 Expected quad size for the d> 3 case . 119 B.3.1 Identities . 121 B.3.2 Component derivation . 123 B.3.3 Final derivation . 125 C Derivation of Q(r, d) 127 C.1 Polar coordinatesb in n dimensions . 129 C.2 Derivation . 129 Abstract 135 Zusammenfassung 137 vii List of Figures 2.1 The evolution of Sliceplorer. 13 2.2 500 projected slices of the 5th dimension of the 5D Zakharov [7] function. 22 2.3 The four techniques we used to compare with 1D slices. 23 2.4 Different views of the 2D sinc function. 29 2.5 Different views of the predictions of four different machine learning re- gression models . 31 2.6 1D slice and HyperSlice views of optimization algorithm traces . 34 3.1 The interface for browsing slices created by the Hypersliceplorer algorithm. 39 3.2 An overview of the Hypersliceplorer algorithm . 45 3.3 Views of regular polytopes . 53 3.4 3-, 4-, and 5-dimensional hypercubes. 54 3.5 3-, 4-, and 5-dimensional hyperspheres. 54 3.6 Using Hypersliceplorer to examine differences in spaces . 56 3.7 Differences in the space of general positive polynomials and Bernstein polynomials with positive Bernstein coefficients. 57 3.8 Hypersliceplorer views of spherical Pareto fronts in 3D and 5D. 57 3.9 Visualization of the 3-objective and 5-objective DLTZ1 problems in Hy- persliceplorer. 58 3.10 Slicing operations per simplex . 60 4.1 The function f(x)= sin(x)+ cos(2x) uniformly sampled with 10 points . 70 4.2 Example of different interpretation methods . 71 viii List of Figures 4.3 Gaussian process interpolation example . 72 4.4 Tuner’s HyperSlice view . 75 4.5 Example of 2D slices of 2D and 3D spheres . 78 4.6 Scatterplots of the time to render using the HyperSlice method. 87 4.7 Actual and relative error rates of prediction . 89 4.8 Using rendering time prediction to constrain sampling . 92 4.9 Interactive rendering times for various dimensions . 93 B.1 Kernel/slice interaction . ..

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