Universitat Politècnica de Catalunya Departament de Llenguatges i Sistemes Informàtics

1. 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 d’Informàtica i Matemàtica Aplicada Universitat de Girona

2. 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

3. 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 refinementcriteria • The three fundamental pillars of this thesis are Complexity InformationTheory (IT) Radiosity

4. dependenceorcorrelation • A scene contains information • which is exchanged between its different parts 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) • Informationis considered as a purely probabilistic concept

5. Accurate computation of the global illumination in a scene, intensities of light over all its surfaces • Realistic images precise treatment of lighting effects  physical simulation of the light propagation Main problems Radiosity 1. Introduction • Radiositymethod only considers diffusesurfaces • 1. discretisation of the surfaces into patches • 2.formfactorcomputation • 3. solution of the system of linear equations • 4. visualization of the solution

6. We look at these problems from an IT approach 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

7. Scene • Global illumination simulates theinterreflectionof lightbetween all the surfaces of a scene • This simulation shows a complex behaviour : • orderanddisorder • unpredictability • interactionsbetween subsystemswhich change independenceon how the system issubdivided 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)

8. We introduce acomplexity measurewhich quantifies the degree ofstructureor dependencein a scene This measure is also used to obtain newrefinement criteria 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

9. Both measures capture different aspects of scene complexity : entropy randomness or uncertainty mutual information structure or correlation 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

10. 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

11. Continuous radiosity equation Radiosity method 2. Previous work • The radiosity method solves the problem of illumination in an environment of diffuse surfaces

12. Form factor properties • Reciprocity • Energy conservation Discrete radiosity equation 2. Previous work • Discrete radiosity equation Form factor Fij • fraction of energy ij

13. Local lines Global lines Form factor computation 2. Previous work • Analytical solutions • Between two spherical patches • Monte Carlo computation • Uniform area sampling • Uniformly distributed lines

14. P42 P13 • In each step, an imaginary particle makes a transitionijwith probability Pij • The probabilities of finding the particle in each state iconverge to a stationary distribution w = ( w1, w2, ..., wn ) 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 3 4 2 1

15. 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 3 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

16. Refinement criteria for HR 2. Previous work • In hierarchical radiosity, the mesh is generated adaptively • Oracles based on • Transported power • Kernel-smoothness

17. Discrete random variableX • X: {x1, x2, …, xn}, pi=Pr { X = xi } • Shannon entropyof X : uncertainty, ignorance Entropy is also related to how difficult it is to guess the values of a random variable • Continuous random variable p(x)= density probability of X p(x)= density probability of X 1.0 1.0 0.0 1.0 0.0 1.0 high entropy low entropy Entropy 2. Previous work

18. Conditional entropy: uncertainty pj|i Information channel X Y {pi} {qj} H(X) H(Y) H(X|Y) I(X,Y) H(Y|X) H(X,Y) Mutual information: shared information, information transfer Discrete channel 2. Previous work pij= pi pj|i

19. Important inequalities 2. Previous work • Jensen’s inequality: if f (x) is a convex function • Log-sum inequality • Data processing inequality : if X  Y  Z is a Markov chain, then

20. Continuous channel 2. Previous work • Continuous entropy • Continuous mutual information • Ic(X,Y)is the least upper bound forI(X,Y) • refinement can never decrease I(X,Y)

21. 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

22. Statistical complexity: degree of structure, correlation, pattern … 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

23. 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

24. pj|i Fi j X X Y Y {pi} { ai} {qj} { aj} Scene: discrete channel 3. Scene visibility entropy • We model the scene visibility as an information channel

25. Positional entropy Scene entropy ? ? ? ? Scene mutual information Discrete visibility entropy 3. Scene visibility entropy

26. Maximum correlation:maximum predictability, privileged visibility directions • Maximum randomness: • no privileged visibility directions, minimum correlation narrow spaces independence 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

27. Lines HS IS HP A 106 6.370 5.171 11.541 A 107 6.761 4.779 11.541 B 106 5.072 6.469 11.541 B 107 5.271 6.270 11.541 C 106 4.674 6.867 11.541 C 107 4.849 6.692 11.541 Randomness vs correlation: results 3. Scene visibility entropy A B C

28. randomness increases with the number of patches correlation increases with the number of objects Results 3. Scene visibility entropy

29. Entropy is closely related to computational error 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

30. 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

31. This difficulty depends on: • degree of dependence between all the surfaces • how the interactions change in dependence when the system is subdivided complexity is more than the number of patches • degree of unpredictability entropy mutual information scene complexity 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 analyzescene complexity?: scene classification and optimal discretisation

32. information loss continuous mutual information expresses with maximum accuracy the scenecomplexity 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

33. contribution of each segment Monte Carlo computation 4. Scene visibility complexity x x Lines cast =K Total area =AT Line segments =N y y

34. ISc=3.837 ISc sphere 0 ISc=4.102 icosahedron 0.543 dodecahedron 0.825 ISc=5.044 octahedron 1.258 cube 1.609 tetrahedron 2.623 Cornell box 3.274 Results 4. Scene visibility complexity

35. These results can be extended to radiosity and importance Complexity and discretisation 4. Scene visibility complexity Two basic results : 1. If any patch is subdivided, ISincreases or remains the same 2. IScis the least upper bound to IS

36. Discretisation accuracy 4. Scene visibility complexity discretisation error information transfer loss

37. Discretisation accuracy 4. Scene visibility complexity

38. 1 Between different discretisations of the same scene the most precise one will be the one that has the highestIS 2 IScexpresses the difficulty of the discretisation The higher the ISc, the more difficult it is to obtain an accurate discretisation is greater in the more complex scenes Discretisation accuracy 4. Scene visibility complexity Two fundamental proposals

39. 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

40. Discrete entropy and mutual information From visibility to radiosity 5. Scene radiosity entropy and complexity • Analogy: null variance probability transition matrix

41. Results 5. Scene radiosity entropy and complexity

42. From discrete to continuous • Discretisation error Continuous radiosity mutual information 5. Scene radiosity entropy and complexity • Scene radiosity complexity • Monte Carlo computation with constant values over all patches

43. Patch refinement 5. Scene radiosity entropy and complexity Increase in mutual information between two patchesiandjwhen subdividing a patch iintomsubpatches Same treatment for visibility, radiosity and importance

44. 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

45. j Partially occluded pair of patches: maximum increase in mutual infomation corresponds to the discontinuity meshing i Two square patches with common edge Mutual information maximization 6. Refinement criteria • Objective: to maximize the discrete mutual information • Feasibility of IT tools for scene discretisation

46. Three square patches with common edges Empty cube Mutual information maximization 6. Refinement criteria

47. information transfer between patches i and j information transfer from patch i Mutual information matrix 6. Refinement criteria

48. visibility log-sum inequality Monte Carlo integration Discretisation error between two patches 6. Refinement criteria Discretisation error: loss of information transfer

49. Information theory principles applied to visibility Refinement increases discrete mutual information Mutual information maximization radiosity equation kernel-smoothness based oracle Radiosity Mutual-information-based oracle 6. Refinement criteria patch-to-patch discretisation error

50. Discretisation error is weighted by Mutual-information-based oracle 6. Refinement criteria Oracle Discretisation error: benefit to be gained by refining