Sin ttulo de diapositiva

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Universitat Politècnica de Catalunya Departament
de Llenguatges i Sistemes Informàtics Programa de
Doctorat en Software
An Information-Theory Framework for the study of
the Complexity of Visibility and Radiosity in a
Scene
Miquel Feixas i Feixas Director Mateu Sbert i
Casasayas
Departament dInformàtica i Matemàtica
Aplicada Universitat de Girona
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Contents
1. Introduction 2. Previous Work 3. Scene
Visibility Entropy 4. Scene Visibility
Complexity 5. Scene Radiosity Entropy and
Complexity 6. Refinement Criteria 7. Conclusions
and Future Work
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Objective
1. Introduction

• In this thesis, information-theory tools are
  applied to visibility and radiosity in order to
• quantify the complexity of a scene
• obtain new refinement criteria
• The three fundamental pillars of this thesis are

Information Theory (IT)
Complexity
Radiosity
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First question
1. Introduction

• How can we apply IT to the study of a scene?

When a photon is emitted from a light source
and then strikes an object, that photon has
effected the transfer of some information ...
(Glassner, 1995)

• Information is considered as a purely
  probabilistic concept

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Radiosity
1. Introduction

• Radiosity method only considers diffuse surfaces
• 1. discretisation of the surfaces into patches
• 2. form factor computation
• 3. solution of the system of linear equations
• 4. visualization of the solution

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Radiosity two main problems
1. Introduction

• Scene meshing has to accurately represent
  illumination variations
• But it has also to avoid unnecessary
  subdivisions of the surfaces
• that would increase the number of form factors to
  be computed ?
• computational cost

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Complexity
1. Introduction

• A very active research area in many different
  areas
• Various interpretations of the term
• But, what is complexity?
• A complex object is an arrangement of parts, so
  intricate as to be hard to understand or deal
  with (Webster, 1986)

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Scene complexity and an accurate solution
1. Introduction
The difficulty in obtaining an accurate solution
mainly depends on the degree of dependence
between all the surfaces of the scene
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Information theory
1. Introduction

• IT deals with the transmission, storage and
  processing of information
• It is used in many different fields
• physics, computer science, economics, neurology,
  learning, etc.
• medical image processing, computer vision and
  robot motion
• Information ? Shannon entropy uncertainty,
  diversity
• Information transfer ? mutual information
  dependence, correlation

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Contents
1. Introduction 2. Previous Work 3. Scene
Visibility Entropy 4. Scene Visibility
Complexity 5. Scene Radiosity Entropy and
Complexity 6. Refinement criteria 7. Conclusions
and Future Work
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Radiosity method
2. Previous work

• The radiosity method solves the problem of
  illumination in an environment of diffuse surfaces

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Discrete radiosity equation
2. Previous work

• Discrete radiosity equation

Form factor Fij
fraction of energy i ? j
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Form factor computation
2. Previous work

• Analytical solutions
• Between two spherical patches

• Monte Carlo computation
• Uniform area sampling
• Uniformly distributed lines

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Random walk
2. Previous work

• Random walk in a scene ? Markov chain
• Markov chain stochastic process
• defined over a set of states 1,2, ..., n
• described by a transition probability matrix

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4
2
1
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Random walk in a scene
2. Previous work

• Discrete Markov chain the states form a
  countable set
• states n ? patches np
• Pij ? Fij
• wi ? ai Ai /AT

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4
F42
F13
2
1

• Continuous Markov chain the states form an
  uncountable set
• states ? dAx
• transition probabilities ? F(x,y)
• stationary distribution ? w(x) 1 /AT

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Refinement criteria for HR
2. Previous work

• In hierarchical radiosity, the mesh is generated
  adaptively
• Oracles based on
• Transported power
• Kernel-smoothness

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Entropy
2. Previous work
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Discrete channel
2. Previous work
pij pi pj i
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Important inequalities
2. Previous work

• Jensens inequality if f (x) is a convex
  function
• Log-sum inequality
• Data processing inequality if X ? Y ? Z is a
  Markov chain, then

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Continuous channel
2. Previous work

• Continuous entropy
• Continuous mutual information

• Ic(X,Y) is the least upper bound for I(X,Y)
• refinement can never decrease I(X,Y)

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What is complexity?
2. Previous work

• The difficulty in constructing an object, in
  describing a system, in reaching a goal, in
  performing a task, and so on (W.Li, 91)
• A theory of complexity can be seen as a theory
  of modelling
• object ? model (condensed information)
• A system is not complex by some abstract
  criterion but because it is intrinsically hard to
  model (Badii and Politi, 1997)

• To define complexity of an object we must
• divide it into parts which may be further split
  into subelements (hierarchical model)
• establish the interactions at different levels
  of resolution

• As we can model the object from different
  perspectives, there cannot be a unique indicator
  of complexity

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Complexity measures
2. Previous work

• Many different ways to quantify complexity from
  different fields (automata, information theory,
  computer science, physics, biology, neuroscience,
  )

• How hard is it to describe? entropy,
  algorithmic, ...
• How hard is it to create? computational,
  logical depth, ...
• What is the degree of organization?
• difficulty of describing organizational
  structure effective complexity
• amount of information shared between the parts
  of a system mutual information

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Contents
1. Introduction 2. Previous Work 3. Scene
Visibility Entropy 4. Scene Visibility
Complexity 5. Scene Radiosity Entropy and
Complexity 6. Refinement criteria 7. Conclusions
and Future Work
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Scene discrete channel
3. Scene visibility entropy

• We model the scene visibility as an information
  channel

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Discrete visibility entropy
3. Scene visibility entropy
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Randomness vs correlation
3. Scene visibility entropy

• How much uncertainty is there about the next
  patch?

randomness, unpredictability

• Information transfer in a scene

correlation, dependence
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Randomness vs correlation results
3. Scene visibility entropy
A
B
C
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Results
3. Scene visibility entropy
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Entropy and error
3. Scene visibility entropy

• Scene entropy and variance of the form factor
  estimators

For a given error, we need to cast more lines for
a scene with more entropy
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Contents
1. Introduction 2. Previous Work 3. Scene
Visibility Entropy 4. Scene Visibility
Complexity 5. Scene Radiosity Entropy and
Complexity 6. Refinement criteria 7. Conclusions
and Future Work
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Complexity of a scene
4. Scene visibility complexity
How difficult is it to compute the visibility and
radiosity of a scene with sufficient accuracy?
Why analyze scene complexity? scene
classification and optimal discretisation
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Continuous visibility mutual information
4. Scene visibility complexity
By discretising (modelling) a scene, a distortion
or error is introduced

• From discrete to continuous
• ? ? ?
• Fij ? F(x,y)
• ai Ai / AT ? 1 / AT

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Monte Carlo computation
4. Scene visibility complexity
x
?x
Lines cast K
Total area AT
Line segments N
?y
y
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Results
4. Scene visibility complexity
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Complexity and discretisation
4. Scene visibility complexity
Two basic results 1. If any patch is
subdivided, IS increases or remains the same 2.
ISc is the least upper bound to IS
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Discretisation accuracy
4. Scene visibility complexity
discretisation error
information transfer loss
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Discretisation accuracy
4. Scene visibility complexity
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Discretisation accuracy
4. Scene visibility complexity
Two fundamental proposals
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Contents
1. Introduction 2. Previous Work 3. Scene
Visibility Entropy 4. Scene Visibility
Complexity 5. Scene Radiosity Entropy and
Complexity 6. Refinement criteria 7. Conclusions
and Future Work
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From visibility to radiosity
5. Scene radiosity entropy and complexity

• Analogy null variance probability transition
  matrix

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Results
5. Scene radiosity entropy and complexity
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Continuous radiosity mutual information
5. Scene radiosity entropy and complexity

• Scene radiosity complexity

• Monte Carlo computation with constant values
  over all patches

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Patch refinement
5. Scene radiosity entropy and complexity
Increase in mutual information between two
patches i and j when subdividing a patch i into m
subpatches
Same treatment for visibility, radiosity and
importance
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Contents
1. Introduction 2. Previous Work 3. Scene
Visibility Entropy 4. Scene Visibility
Complexity 5. Scene Radiosity Entropy and
Complexity 6. Refinement criteria 7. Conclusions
and Future Work
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Mutual information maximization
6. Refinement criteria

• Objective to maximize the discrete mutual
  information
• Feasibility of IT tools for scene discretisation

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Mutual information maximization
6. Refinement criteria
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Mutual information matrix
6. Refinement criteria
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Discretisation error between two patches
6. Refinement criteria
Discretisation error loss of information
transfer
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Mutual-information-based oracle
6. Refinement criteria
patch-to-patch discretisation error
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Mutual-information-based oracle
6. Refinement criteria
Oracle
Discretisation error benefit to be gained by
refining
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Results
6. Refinement criteria

• Advantages
• it preserves illumination details
• it avoids overrefinement in smoothly lit areas
• it is more robust than classic smoothness-based
  oracles

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Results
6. Refinement criteria
Kernel-smoothnes-based
MI-based
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Results
6. Refinement criteria
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Contents
1. Introduction 2. Previous Work 3. Scene
Visibility Entropy 4. Scene Visibility
Complexity 5. Scene Radiosity Entropy and
Complexity 6. Refinement criteria 7. Conclusions
and Future Work
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Summary
1. Introduction
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Future work
7. Conclusions and future work

• Some concepts presented in this thesis can be
  extended to the interior points of a scene and to
  the points in the environment
• entropy and mutual information fields
• mutual information density

Adaptive supersampling
View-point selection

• The concept of entropy can be applied to
  viewpoint selection

• Non diffuse environments, mesh simplification,
  etc.

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Gràcies per la vostra atenció! Thanks!
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Universitat Politècnica de Catalunya Departament
de Llenguatges i Sistemes Informàtics Programa de
Doctorat en Software
An Information-Theory Framework for the study of
the Complexity of Visibility and Radiosity in a
Scene
Miquel Feixas i Feixas Director Mateu Sbert i
Casasayas
Departament dInformàtica i Matemàtica
Aplicada Universitat de Girona