Identification of Finite Lattice Dynamical Systems

Transcription

1 Identification of Finite Lattice Dynamical Systems Junior Barrera BIOINFO-USP University of São Paulo, Brazil

2 Outline Introduction Lattice operator representation Lattice operator design Lattice dynamical systems (LDS) A simulator for chain dynamical systems LDS identification System identification examples Conclusion

3 Introduction

4 Dynamical Systems initial state x 2 = y 2 t t u u Final state Final state x 1 = y 1

5 Sequential Switching Circuits

6 Mathematical Morphology studies operators between complete lattices, what includes switching functions lattice operators are decomposed in terms of simple morphological operators: erosion, dilation, anti-erosion, anti-dilation Any lattice operator can be decomposed in a canonical morphological representation

7 Lattice Dynamical Systems We present the notion of Lattice Dynamical System (LDS) Give a representation for LDSs, based on canonical morphological representations Formalize the problem of statistical identification of LDSs,

8 Operator Representation

9 ψ : Fun[W, L] L

10 Intervals Let a, b Fun[W,L], a b iff a(x) b(x), x W W = 2 a b Interval [a,b] = {u Fun[W,L]: a u b}

11 Binary Sup-generating Sup-generating operator: ( u) = 1 u [ a, ] λ a, b b [a,b] λ a,b

12 Kernel Kernel of ψ at y: K(ψ)(y) = {u Fun[W,L]: y ψ(u)} K(ψ)(-2) K(ψ)(-1) K(ψ)(0) K(ψ)(1) K(ψ)(2)

13 Basis Basis of ψ at y: B(ψ) is the set of maximal intervals contained in K(ψ) K(ψ)(-2) K(ψ)(-1) K(ψ)(0) K(ψ)(1) K(ψ)(2) B(ψ)(-2) B(ψ)(-1) B(ψ)(0) B(ψ)(1) B(ψ)(2)

14 Canonical Representation ( u) = U{ y M : { ( ) :[, ] ( )( )} 1}, u a b B y = ψ U λ a ψ b K(ψ)(-2) K(ψ)(-1) K(ψ)(0) K(ψ)(1) K(ψ)(2) ψ(-1,-1) = 1

15 Operator Design

16 Optimization problem Design goal is to find a function ψ opt : Fun[W, L] L with minimum error. Error (expected loss) of a function : Er(ψ) = E[l(ψ(X),Y)] Loss function X is a random function Y is a random variable l : L L R +

17 Estimation problem The distribution P(X,Y) is unknown P(X,Y), Er(ψ) and ψ opt should be estimated from realizations of X and Y. For m > m( ε, δ ) examples Pr( Er(ψ) - Er(ψ opt ) < ε) > 1 - δ ε, δ (0,1)

18 The constrained estimation problem Error ψ opt Ψ Sample size

19 Stack filter Spatially translation invariant Spatially locally defined Increasing Commutes with threshold

20 Noise elimination training images

21 test image iteration 1

22 test image iteration 5

23 test image iteration 1

24 test image iteration 5

25 Apertures Spatially translation invariant Spatially locally defined Range translation invariant Range locally defined

26 Deblurring training images

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28 Resolution Enhancement

29 (0,0) (0,1) Ψ 2 Ψ 0 Ψ 1 Ψ 3

30 Original Aperture: 3x3x21x51 Linear Bilinear

31 Zoom Original Aperture: 3x3x21x51 Linear Bilinear

32 Independent Constraints Constraints Restriction of the operators space K(ψ opt ) Q P(P(W)) Independent Constraint Let be A,B P(W) with A B: h ψ (x)=1 x A & h ψ (x)=0 x B, ψ :K(ψ) Q P(P(W)) K(ψ opt ) {1} B Q A {0,1} Q B P(W) {0} P(P(W)) A

33 Independent Constraints Proposition: if Q is an independent restriction then exist a par of operators (α,β) such that, for any ψ Ψ W K(ψ) Q α ψ β where K(α) = A and K(β) = B All independent constraint is characterized by two operators α and β The pair (α,β) is called Envelope h α {1} h β {1} A {0} A {1} B {0} B {0} P(W) P(W)

34 Noise Edge Detection Ground Image Noise Addition Noisy Image Restoration Filtered Image Edge Detection Edge Detected Direct Edge Detection Edge Detected

35 Restoration a) Machine design of the restoration ψ pac designed by examples b) Human-machine design of the restoration ψ con = (ψ pac β) α α = δ B B ε B B δ B ε B and β = ε B B δ B B ε B δ B α and β are alternating sequential filters with P[ α(s) I β(s) ] 1 B is the 3x3 square Machine design of the restoration Human-Machine design of the restoration 0.28 % 0.13 % 0,3 0,25 0,2 0,15 0,1 0,05 Machine design of the restoration Human design of the restoration 0 Error (%)

36 Noise Edge Detection Edge Detection a) Machine design over noisy images ζ pac designed by examples from noisy images b) Human design after restoration ζ = I d - ε B B is the 3x3 square c) Machine design after restoration ζ pac designed by examples from restored images Machine design over noisy images Human design after restoration Machine design after restoration 0.65 % 0.27 % 0.24 % 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 Error (%) Machine design over noisy images Human design after restoration Machine design after restoration

37 Noise Edge Detection Machine design over noisy images Error = 0.65% Human design after restoration Error = 0.27% Machine design after restoration Error = 0.24%

38 Multiresolution Constraint x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 8 variables z 1 z 2 z 3 z 4 4 variables w 1 w 2 2 variables variables: 2 2 =4 4 variables: 2 4 =16 8 variables: 2 8 =256

39 Multiresolution Constraint D 1 = P(W 1 ) D 0 = P(W 0 ) W 0 z i = p i (x i1,..., x i9 ), z = p(x), p=(p 1,.. p 9 ) Let φ:d 1 {0,1}, it defines the operator Ψ φ on D o by Ψ φ (x) = φ(p(x)) W 1 The operador Ψ φ is constrained by resolution to D 1 Equivalence classes defined by p(x) = p(y) D 0

40 Multiresolution Constraint W 1 p D 2 = P(W 2 ), D 1 = P(W 1 ), D 0 = P(W 0 ) W x D 0, z D 1, v D 2, 0 z i = p i (x i,1,..., x i,9 ), z = p(x), p=(p 1,.. p 81 ) v i = w i (x i,1,..., x i,81 ), v = w(x), w=(w 1,.. w 9 ) W 2 w The equivalence classes defined by p may be different by the ones defined by w.

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44 Multiresolution Noise image noise image + noise

45 3x3 window Pyramid Restauration Persisting noise

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47 Dynamical Systems

48 Finite Lattice Dinamical System

49 Representation The component functions have canonical morphological representations

50 System: input-output

51 Filter For example, processing of motion images.

52 Input-free systems

53 Causal systems

54 Time translation invariant systems

55 Causal, time translation invariant

56 A simulator for chain dynamical systems

57 Simulator Architecture

58 Functions Representation

59 System Description

60 Cell cycle simulation Division Steps Ingredients Control External Signal Inhibition p53 Receptors Excitation Excitation or Inhibition

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66 A model for system identification

67 Model vector of functions S( Φ, Ψ) S opt ( Φ, Ψ) each component function has a basis S

68 System Error

69 Stationary Conditions

70 Component Error

71 Additive Loss Function

72 Independence Condition Under additive loss function optimize the system error is equivalent to optimize the system components error The problem of system identification is reduced to a family of problems of lattice operator design.

73 Identification of Dynamical Systems

74 A Boolean System

75 System simulation

76 System identification: system error

77 System identification: transition error

78 Ideal Est.: 100 training examples Est.: 500 training examples Est.: 1500 training examples

79 Motion Segmentation

80 Mask Predictor result

81 Filtering Color Composition

82 Watershed Color Composition

83 Motion Segmentation

84 Motion Segmentation

85 Conclusion

86 Presented the notion of Lattice Dynamical System Proposed a model for LDS identification Under additive condition, system identification reduces to a family of problems of lattice operator design Some examples were presented This perspective unifies theories such as switching theory, discrete automatic control and reinforcement learning.