Making choices in multi-dimensional parameter spaces PhD thesis - - PowerPoint PPT Presentation

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Making choices in multi-dimensional parameter spaces PhD thesis - - PowerPoint PPT Presentation

Aug 23, 2022 542 likes •1.98k views

Making choices in multi-dimensional parameter spaces PhD thesis defence Steven Bergner Model adjustment at different levels User-driven experimentation: Use cases for paraglide Criteria optimization: Lighting design Theoretical

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SLIDE 1

Making choices in multi-dimensional parameter spaces

Steven Bergner

PhD thesis defence

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SLIDE 2

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Model adjustment at different levels

2

  • User-driven experimentation: Use cases for paraglide
  • Criteria optimization: Lighting design
  • Theoretical analysis: Sampling in volume rendering
  • Discretizing a region: Lattices with rotational dilation
  • Summary and conclusion
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SLIDE 3

Data acquisition and visualization

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SLIDE 4

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Turning code into data

  • Computer simulation code
  • Function abstraction

Variables: input, output, and

algorithm specific

Deterministic code

4

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Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Biological aggregations

5 (c) Sareh Nabi Abdolyousefi

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Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Biological aggregations

5

Input Output

(c) Sareh Nabi Abdolyousefi

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SLIDE 7

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Biological aggregations

5

Input Output 1D+time model 14 parameters

  • attraction, repulsion, and

alignment coefficients

  • turning rates

internal:

  • space-time resolution

influences cost patterns:

steady state bifurcation and stability:

(c) Sareh Nabi Abdolyousefi

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SLIDE 8

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Biological aggregations

5

Input Output 1D+time model 14 parameters

  • attraction, repulsion, and

alignment coefficients

  • turning rates

internal:

  • space-time resolution

influences cost patterns:

steady state bifurcation and stability:

time space

2 u∗ qal Q∗ | Q∗∗ |

A 2

( u∗

1,u∗ 5)

( u∗

5,u∗ 1)

( u∗

3,u∗ 3)

( u∗

2,u∗ 4)

( u∗

4,u∗ 2)

( a) qh |

(c) Sareh Nabi Abdolyousefi

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SLIDE 9

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

More cases

  • Parameter space segmentation
  • Bio-medical imaging algorithm
  • Fuel cell design
  • Scene lighting configuration
  • Raycasting step size parameter

6

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SLIDE 10

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Paraglide design

7

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SLIDE 11

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Paraglide design

7

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SLIDE 12

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Paraglide design

  • Setup compute node

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SLIDE 13

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Paraglide design

  • Setup compute node
  • Choose variables

7

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SLIDE 14

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Paraglide design

  • Setup compute node
  • Choose variables
  • Choose region

7

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SLIDE 15

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Paraglide design

  • Setup compute node
  • Choose variables
  • Choose region
  • Sample and compute

7

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Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Paraglide design

  • Setup compute node
  • Choose variables
  • Choose region
  • Sample and compute
  • Compute features

7

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SLIDE 17

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Paraglide design

  • Setup compute node
  • Choose variables
  • Choose region
  • Sample and compute
  • Compute features
  • View, predict, diagnose

7

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Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Sampling the region of interest

8

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SLIDE 19

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Sampling the region of interest

  • Tensor product of value levels for each

dimension

Nested for-loops Cost is exponential in #dims

8

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SLIDE 20

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Sampling the region of interest

  • Tensor product of value levels for each

dimension

Nested for-loops Cost is exponential in #dims

  • Separate range specification from

sample generation

8

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SLIDE 21

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Sampling the region of interest

  • Tensor product of value levels for each

dimension

Nested for-loops Cost is exponential in #dims

  • Separate range specification from

sample generation

8

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SLIDE 22

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Paraglide summary

  • Longitudinal study showed use of parameter

space partitioning

  • Requirements informed follow-up projects
  • Alternative user interaction

Dimensionally reduced slider embedding Mixing board

  • Video demo

9

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SLIDE 23

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Model adjustment at different levels

10

  • User-driven experimentation: Use cases for paraglide
  • Criteria optimization: Lighting design
  • Theoretical analysis: Sampling in volume rendering
  • Discretizing a region: Lattices with rotational dilation
  • Summary and conclusion
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SLIDE 24

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Model adjustment at different levels

10

  • User-driven experimentation: Use cases for paraglide
  • Criteria optimization: Lighting design
  • Theoretical analysis: Sampling in volume rendering
  • Discretizing a region: Lattices with rotational dilation
  • Summary and conclusion
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SLIDE 25

Roadmap

From light to colour Efficient light model Designing spectra for lights and materials Evaluation Applications

11

Bergner, Drew, Möller - Generating Light and Reflectance Spectra - ACM Trans. on Graphics 2009

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SLIDE 26

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!

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From Light to Colour

12

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SLIDE 27

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From Light to Colour

12

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SLIDE 28

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From Light to Colour

12

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SLIDE 29

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From Light to Colour

12

B G R

=

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SLIDE 30

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From Light to Colour

12

B G R

= component-wise product in RGB

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SLIDE 31

Use for Visualization

13

Light 1 Light 2

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SLIDE 32

Use for Visualization

Metamers Different Spectra give same RGB

13

Light 1 Light 2

{

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SLIDE 33

Use for Visualization

Metamers Different Spectra give same RGB Constant Colours Metamers under changing light

13

Light 1 Light 2

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SLIDE 34

Use for Visualization

Metamers Different Spectra give same RGB Constant Colours Metamers under changing light Metameric Blacks Spectra give RGB triple = 0

13

Light 1 Light 2

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SLIDE 35

Use for Visualization

Metamers Different Spectra give same RGB Constant Colours Metamers under changing light Metameric Blacks Spectra give RGB triple = 0 Effective choice of light & material palette needed!

13

Light 1 Light 2

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SLIDE 36

Roadmap

From light to colour Efficient light model Designing spectra for lights and materials Evaluation Applications

14

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SLIDE 37

Roadmap

From light to colour Efficient light model Designing spectra for lights and materials Evaluation Applications

15

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SLIDE 38

Illumination Dependent Colour Picker

16

? ? ? ? ? ? ?

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SLIDE 39

Illumination Dependent Colour Picker

16

? ? ? ? ? ? ?

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SLIDE 40

Illumination Dependent Colour Picker

16

? ? ? ? ? ? ?

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SLIDE 41

Illumination Dependent Colour Picker

16

? ? ? ? ? ? ?

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SLIDE 42

Illumination Dependent Colour Picker

16

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SLIDE 43

Quality Criteria

Colour

– Fit the desired colour or metamer

Smoothness

– Regularize solution and reduce extrema

Minimal error in linear model

– Minimal colour difference when illumination bounce is

computed in linear subspace

Positivity

– Produce physically plausible spectra

17

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SLIDE 44

Quality Criteria

18

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SLIDE 45

Quality Criteria

Instead of equation system for spectrum Solve normal equation

18

M x = y

  • x

argmin xM x − y

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SLIDE 46

Quality Criteria

Instead of equation system for spectrum Solve normal equation

– Colour:

18

M x = y

  • x

argmin x

 mred mgreen mblue   diag( S) x −   cr cg cb  

  • argmin

xM x − y

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SLIDE 47

Quality Criteria

Instead of equation system for spectrum Solve normal equation

– Colour: – Smoothness:

18

M x = y

  • x

argmin x

 mred mgreen mblue   diag( S) x −   cr cg cb  

  • argmin

xM x − y

argmin x

    −1 2 −1 · · · −1 2 −1 · · · ... · · · −1 2 −1      x −      . . .     

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SLIDE 48

Quality Criteria

Instead of equation system for spectrum Solve normal equation

– Colour: – Smoothness:

Weight the criteria and combine as stacked matrix

– Global minimum error solution via pseudo-inverse of – Positivity through quadratic programming

18

M x = y

M

  • x

argmin x

 mred mgreen mblue   diag( S) x −   cr cg cb  

  • argmin

xM x − y

argmin x

    −1 2 −1 · · · −1 2 −1 · · · ... · · · −1 2 −1      x −      . . .     

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SLIDE 49

Roadmap

From light to colour Efficient light model Designing spectra for lights and materials Evaluation Applications

19

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SLIDE 50

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Materials and lighting

20

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SLIDE 51

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Materials and lighting

20

Given: Output Goal: Input

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SLIDE 52

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Materials and lighting

20

Given: Output Goal: Input

  • 3x5 combination colours

with 3 components each

  • 7x31 component SPDs

lights reflectances ? ? ? ? ? ? ?

!"" #"" $"" %"" " "&# )*+,-./01)$#
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Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Materials and lighting

20

Given: Output Goal: Input

  • 3x5 combination colours

with 3 components each

  • 7x31 component SPDs

lights reflectances ? ? ? ? ? ? ?

!"" #"" $"" %"" " "&# )*+,-./01)$# !"" #"" $"" %"" " "&# ' !"" #"" $"" %"" " "&' !"" #"" $"" %"" " "&# !"" #"" $"" %"" " "&( !"" #"" $"" %"" " "&# !"" #"" $"" %"" " "&# )*+,-./01)$# !"" #"" $"" %"" " "&# ' 23,-04-./0' !"" #"" $"" %"" " "&# ' 23,-04-./0(
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SLIDE 54

Image based re-lighting

21

a)

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SLIDE 55

Image based re-lighting

21

a) b)

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SLIDE 56

Image based re-lighting

21

a) b) c)

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SLIDE 57

Applications in Graphics and Visualization

22

Additional texture details appear under changing illumination

400 500 600 700 0.2 0.4 0.6 0.8 400 500 600 700 0.5 1 refl 1 refl 2 400 500 600 700 0.2 0.4 sodium hp 400 500 600 700 0.1 0.2 D65 daylight

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SLIDE 58

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Model adjustment at different levels

23

  • User-driven experimentation: Use cases for paraglide
  • Criteria optimization: Lighting design
  • Theoretical analysis: Sampling in volume rendering
  • Filling a region: Lattices with rotational dilation
  • Summary and conclusion
slide-59
SLIDE 59

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Model adjustment at different levels

23

  • User-driven experimentation: Use cases for paraglide
  • Criteria optimization: Lighting design
  • Theoretical analysis: Sampling in volume rendering
  • Filling a region: Lattices with rotational dilation
  • Summary and conclusion
slide-60
SLIDE 60

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 24

Volume Rendering

  • Map data value f to optical properties

using a transfer function

  • Then shading+compositing

f

Opacity

g(f(x)) g f

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SLIDE 61

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 24

Volume Rendering

  • Map data value f to optical properties

using a transfer function

  • Then shading+compositing

f

Opacity

g(f(x)) g f

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SLIDE 62

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 24

Volume Rendering

  • Map data value f to optical properties

using a transfer function

  • Then shading+compositing

f

Opacity

g(f(x)) g f

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SLIDE 63

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 24

Volume Rendering

  • Map data value f to optical properties

using a transfer function

  • Then shading+compositing

f

Opacity

g(f(x)) g f

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SLIDE 64

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 24

Volume Rendering

  • Map data value f to optical properties

using a transfer function

  • Then shading+compositing

f

Opacity

g(f(x)) g f

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SLIDE 65

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 25

Example of g(f(x))

Intuition Analysis Application

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SLIDE 66

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 25

Example of g(f(x))

Original function f(x)

Intuition Analysis Application

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SLIDE 67

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 25

Example of g(f(x))

Original function f(x) Transfer function g(y)

Intuition Analysis Application

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SLIDE 68

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 25

Example of g(f(x))

Original function f(x) Transfer function g(y) g(f(x)) sampled by

Intuition Analysis Application

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SLIDE 69

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 25

Example of g(f(x))

Original function f(x) Transfer function g(y) g(f(x)) sampled by g(f(x)) sampled by

Intuition Analysis Application

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SLIDE 70

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

·

26

Composition in Frequency Domain

Intuition Analysis Application

y

y

·

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SLIDE 71

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

·

26

Composition in Frequency Domain

Intuition Analysis Application

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SLIDE 72

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

H(k) = 1 2π

  • R
  • R

G(l)eil·f(x)dle−ik·xdx

·

26

Composition in Frequency Domain

Intuition Analysis Application

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SLIDE 73

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

H(k) = 1 2π

  • R
  • R

G(l)eil·f(x)dle−ik·xdx

·

26

Composition in Frequency Domain

Intuition Analysis Application

slide-74
SLIDE 74

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

H(k) = 1 2π

  • R
  • R

G(l)eil·f(x)dle−ik·xdx

·

26

Composition in Frequency Domain

Intuition Analysis Application

H(k) = 1 2π

  • R

G(l)

  • R

eil·f(x)e−ik·xdxdl

slide-75
SLIDE 75

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

H(k) = 1 2π

  • R
  • R

G(l)eil·f(x)dle−ik·xdx

·

26

Composition in Frequency Domain

Intuition Analysis Application

H(k) = 1 2π

  • R

G(l)

  • R

eil·f(x)e−ik·xdxdl

slide-76
SLIDE 76

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

H(k) = 1 2π

  • R
  • R

G(l)eil·f(x)dle−ik·xdx

·

26

Composition in Frequency Domain

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π

  • R

G(l)

  • R

eil·f(x)e−ik·xdxdl

slide-77
SLIDE 77

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 27

Visualizing P(k,l)

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π < G(·), P(k, ·) >

slide-78
SLIDE 78

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 27

Visualizing P(k,l)

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π < G(·), P(k, ·) >

slide-79
SLIDE 79

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 27

Visualizing P(k,l)

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π < G(·), P(k, ·) >

slide-80
SLIDE 80

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 27

Visualizing P(k,l)

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π < G(·), P(k, ·) >

slide-81
SLIDE 81

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 27

Visualizing P(k,l)

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π < G(·), P(k, ·) >

slide-82
SLIDE 82

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 27

Visualizing P(k,l)

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π < G(·), P(k, ·) >

slide-83
SLIDE 83

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 27

Visualizing P(k,l)

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π < G(·), P(k, ·) >

slide-84
SLIDE 84

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 27

Visualizing P(k,l)

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π < G(·), P(k, ·) >

slide-85
SLIDE 85

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 27

Visualizing P(k,l)

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π < G(·), P(k, ·) >

slide-86
SLIDE 86

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 27

Visualizing P(k,l)

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π < G(·), P(k, ·) >

slide-87
SLIDE 87

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 27

Visualizing P(k,l)

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π < G(·), P(k, ·) >

slide-88
SLIDE 88

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 27

Visualizing P(k,l)

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

H(k) = 1 2π < G(·), P(k, ·) >

slide-89
SLIDE 89

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 28

Visualizing P(k,l)

Intuition Analysis Application

slide-90
SLIDE 90

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 28

Visualizing P(k,l)

Intuition Analysis Application

slide-91
SLIDE 91

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 28

Visualizing P(k,l)

Intuition Analysis Application

slide-92
SLIDE 92

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 28

Visualizing P(k,l)

Intuition Analysis Application

  • Slopes of lines in P(k,l) are related to 1/f‘(x)
slide-93
SLIDE 93

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 28

Visualizing P(k,l)

Intuition Analysis Application

  • Slopes of lines in P(k,l) are related to 1/f‘(x)
  • Extremal slopes bounding the wedge are 1/max(f’)
slide-94
SLIDE 94

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 29

Method of stationary phase

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

slide-95
SLIDE 95

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 29

Method of stationary phase

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

slide-96
SLIDE 96

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 29

Method of stationary phase

Intuition Analysis Application

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

slide-97
SLIDE 97

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 29

Method of stationary phase

Intuition Analysis Application

  • Taylor expansion around

points of stationary phase

  • Exponential drop-off at

maximum

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

l · max |f ′| = k

slide-98
SLIDE 98

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 29

Method of stationary phase

Intuition Analysis Application

  • Taylor expansion around

points of stationary phase

  • Exponential drop-off at

maximum

P(k, l) =

  • R

ei(l·f(x)−k·x)dx

l · max |f ′| = k

slide-99
SLIDE 99

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 30

Adaptive Raycasting

Same number of samples

Intuition Analysis Application

slide-100
SLIDE 100

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 31

Adaptive Raycasting SNR

Ground-truth: computed at a fixed sampling distance

  • f 0.06125

Intuition Analysis Application

slide-101
SLIDE 101

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence 32

Summary

  • Proper sampling of combined signal g(f(x)):
  • Solved a fundamental problem of rendering
  • Composition is a general data processing
  • peration

Intuition Analysis Applications

slide-102
SLIDE 102

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Model adjustment at different levels

33

  • User-driven experimentation: Use cases for paraglide
  • Criteria optimization: Lighting design
  • Theoretical analysis: Sampling in volume rendering
  • Filling a region: Lattices with rotational dilation
  • Summary and conclusion
slide-103
SLIDE 103

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Model adjustment at different levels

33

  • User-driven experimentation: Use cases for paraglide
  • Criteria optimization: Lighting design
  • Theoretical analysis: Sampling in volume rendering
  • Filling a region: Lattices with rotational dilation
  • Summary and conclusion
slide-104
SLIDE 104

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Point lattices

  • Definition via basis

34

R

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

slide-105
SLIDE 105

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Point lattices

  • Definition via basis {Rk : k ∈ Zn}

34

R

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " ! !! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

slide-106
SLIDE 106

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

R =

  • 1

1

  • 35

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&$*%&#+&!(%

slide-107
SLIDE 107

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

R =

  • 1

1

  • dyadic subsampling

35

det K = 2n = 4 K =

  • 2

2

  • = 2I

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&$*%&#+&!(%

slide-108
SLIDE 108

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

R =

  • 1

1

  • dyadic subsampling

35

det K = 2n = 4 K =

  • 2

2

  • = 2I

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&$*%&#+&!(%

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)#&%*%&#+&!(%

slide-109
SLIDE 109

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

R =

  • 1

1

  • dyadic subsampling

35

det K = 2n = 4 K =

  • 2

2

  • = 2I

Reduction factor is exponential in

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&$*%&#+&!(%

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)#&%*%&#+&!(%

n

slide-110
SLIDE 110

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

R =

  • 1

1

  • dyadic subsampling

35

det K = 2n = 4 K =

  • 2

2

  • = 2I

Reduction factor is exponential in

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&$*%&#+&!(%

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)#&%*%&#+&!(%

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)#&%*%&#+&!(%

n

slide-111
SLIDE 111

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)#&%*%&#+&!(%

R =

  • 1

1

  • 36

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

slide-112
SLIDE 112

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)#&%*%&#+&!(%

R =

  • 1

1

  • K =
  • 1

−1 1 1

  • quincunx subsampling

det K = 2

36

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

slide-113
SLIDE 113

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)#&%*%&#+&!(%

R =

  • 1

1

  • K =
  • 1

−1 1 1

  • quincunx subsampling

det K = 2

36

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

slide-114
SLIDE 114

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)#&%*%&#+&!(%

R =

  • 1

1

  • K =
  • 1

−1 1 1

  • quincunx subsampling

det K = 2

36

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

This low rate dilation does not exist for integer lattices with

[Van De Ville, Blu, Unser, SPL 05]

n > 2

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

slide-115
SLIDE 115

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)#&%*%&#+&!(%

R =

  • 1

1

  • K =
  • 1

−1 1 1

  • quincunx subsampling

det K = 2

36

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

This low rate dilation does not exist for integer lattices with

[Van De Ville, Blu, Unser, SPL 05]

However, possible for irrational !

n > 2 R

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

slide-116
SLIDE 116

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

R =

  • 1

1

  • K =
  • 1

−1 1 1

  • quincunx subsampling

det K = 2

37

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

slide-117
SLIDE 117

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

R =

  • 1

1

  • K =
  • 1

−1 1 1

  • quincunx subsampling

det K = 2

37

fractional subsampling RKs for s = 0..2

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

slide-118
SLIDE 118

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

R =

  • 1

1

  • K =
  • 1

−1 1 1

  • quincunx subsampling

det K = 2

37

fractional subsampling acts like a scaled rotation with RKs for s = 0..2

lattice, that is, it can with QT Q = α2I

QR

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

!! !" !# !$ % $ # " ! !! !" !# !$ % $ # " !

&&'&(&)$&%*%&$+ &,&(&)$&!$*$&$+&!(!-

slide-119
SLIDE 119

Construction

slide-120
SLIDE 120

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Similarity of Q and K

QR = RK

lattice, that is, it can with QT Q = α2I

with

39

slide-121
SLIDE 121

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Similarity of Q and K

QR = RK

R−1QR = K

lattice, that is, it can with QT Q = α2I

with

39

slide-122
SLIDE 122

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Similarity of Q and K

  • and have same characteristic

polynomial

QR = RK

R−1QR = K

lattice, that is, it can with QT Q = α2I

with

nomial d(λ) = det(K − λI) = det(Q − λI) ( ) = 0

between K and Q

39

slide-123
SLIDE 123

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Similarity of Q and K

  • and have same characteristic

polynomial

QR = RK

R−1QR = K

lattice, that is, it can with QT Q = α2I

with

nomial d(λ) = det(K − λI) = det(Q − λI) ( ) = 0

between K and Q

) = n

k=0 ckλk

case n = even with

) ∈ Z[λ]

39

slide-124
SLIDE 124

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Similarity of Q and K

  • and have same characteristic

polynomial

QR = RK

R−1QR = K

lattice, that is, it can with QT Q = α2I

with

nomial d(λ) = det(K − λI) = det(Q − λI) ( ) = 0

between K and Q

) = n

k=0 ckλk

case n = even with

) ∈ Z[λ]

and thus agree in eigenvalues and determinant.

39

slide-125
SLIDE 125

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

Diagonalizing rotation Q

  • cos θ

− sin θ sin θ cos θ

  • = 1

2

  • 1

1 j −j ejθ e−jθ 1 j 1 −j

  • J2 = J−1

2 ∆J2.

40

slide-126
SLIDE 126

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

Diagonalizing rotation Q

=        ejθ1 e−jθ1 ejθ2 e−jθ2 ...       

∆ =      1 ejθ1 e−jθ1 ...     

∆ =

  • cos θ

− sin θ sin θ cos θ

  • = 1

2

  • 1

1 j −j ejθ e−jθ 1 j 1 −j

  • J2 = J−1

2 ∆J2.

Different eigenvalue structure for even and

  • dd dimensionality

With analogue block-wise construction of Jn

40

slide-127
SLIDE 127

Steven Bergner et al. - Sampling Lattices with Similarity Scaling Relationships - SampTA 2009

Diagonalizing rotation Q

  • cos θ

− sin θ sin θ cos θ

  • = 1

2

  • 1

1 j −j ejθ e−jθ 1 j 1 −j

  • J2 = J−1

2 ∆J2.

Different eigenvalue structure for even and

  • dd dimensionality

d(λ) = λn + Cλ

n 2 + αn

that C 2 < 4αn

d(λ) = λn − αn

n n

with

  • even:
  • odd:

restricts characteristic polynomial:

40

slide-128
SLIDE 128

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Finding suitable K

41

slide-129
SLIDE 129

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Finding suitable K

  • Fulfill conditions implied by

41

QR = RK

slide-130
SLIDE 130

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Finding suitable K

  • Fulfill conditions implied by
  • Exhaustive search over range of K ∈ Zn×n

41

QR = RK

slide-131
SLIDE 131

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Finding suitable K

  • Fulfill conditions implied by
  • Exhaustive search over range of
  • Companion matrix

K =         −c0 1 −c1 1 . . . ... ... −cn−2 1 −cn−1        

K ∈ Zn×n

41

QR = RK

slide-132
SLIDE 132

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Finding suitable K

  • Fulfill conditions implied by
  • Exhaustive search over range of
  • Companion matrix
  • More with unimodular similarity transforms

K =         −c0 1 −c1 1 . . . ... ... −cn−2 1 −cn−1        

K ∈ Zn×n

41

QR = RK

KT = T−1KT with det T = 1 and T ∈ Zn×n

slide-133
SLIDE 133

Results

slide-134
SLIDE 134

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

2D

2 4 1 2 3 4 5 1: R = [0.71 0;0.71 1.4] K = [2 2;1 0] θ=45 2 4 1 2 3 4 5 2: R = [0 0.58;1.7 0.65] K = [2 1;4 1] θ=69.3 2 4 1 2 3 4 5 3: R = [0 0.84;1.2 0] K = [0 1;2 0] θ=90

|det K| = 2

Dilation factor

43

slide-135
SLIDE 135

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

2D

Dilation factor |det K| = 3

5 1 2 3 4 5 1: R = [0 0.93;1.1 0.54] K = [1 2;2 1] θ=90 5 1 2 3 4 5 2: R = [0 0.84;1.2 0] K = [1 1;2 1] θ=54.74 5 1 2 3 4 5 3: R = [0 0.74;1.3 0.22] K = [1 1;3 0] θ=73.22 5 1 2 3 4 5 4: R = [0.66 0;1.1 1.5] K = [3 4;3 3] θ=90

44

slide-136
SLIDE 136

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Rotational grid summary

  • First time low-rate admissible dilation

matrices are available for n>2

  • Additional degrees of freedom in the design

enable further optimization

  • Current results allow optimized

constructions up to n=9

45

slide-137
SLIDE 137

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Model adjustment at different levels

46

  • User-driven experimentation: Use cases for paraglide
  • Criteria optimization: Lighting design
  • Theoretical analysis: Sampling in volume rendering
  • Filling a region: Lattices with rotational dilation
  • Summary and conclusion
slide-138
SLIDE 138

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Model adjustment at different levels

46

  • User-driven experimentation: Use cases for paraglide
  • Criteria optimization: Lighting design
  • Theoretical analysis: Sampling in volume rendering
  • Filling a region: Lattices with rotational dilation
  • Summary and conclusion
slide-139
SLIDE 139

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Data taxonomy

  • primary: field measurements
  • secondary: synthetic data or human input
  • tertiary: rules provided by theoretical study
  • r statistical inference

47

slide-140
SLIDE 140

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Model adjustment at different levels

48

  • User-driven experimentation: Use cases for paraglide
  • Criteria optimization: Lighting design
  • Theoretical analysis: Sampling in volume rendering
  • Filling a region: Lattices with rotational dilation
slide-141
SLIDE 141

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Model adjustment at different levels

48

  • User-driven experimentation: Use cases for paraglide
  • Criteria optimization: Lighting design
  • Theoretical analysis: Sampling in volume rendering
  • Filling a region: Lattices with rotational dilation

User input Theoretical input

slide-142
SLIDE 142

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

Acknowledgements

  • Torsten Möller and Derek Bingham
  • GrUVi-Lab members
  • Collaborators: SFU -CS, -Maths, and -Stats

BIG @ EPFL

  • Funding: SFU, Precarn, and NSERC

49

slide-143
SLIDE 143

Steven Bergner - Making choices in multi-dimensional parameter spaces - PhD thesis defence

!! !" !# $ $%& #

Thank you! Questions?

50

! " # #

!! !" # " ! !! !$ !" !% # % " $ !

400 500 600 700 0.2 0.4 0.6 0.8 400 500 600 700 0.5 1 refl 1 refl 2 400 500 600 700 0.2 0.4 sodium hp 700 400 500 600 700 0.1 0.2 D65 daylight