Identification of Finite Lattice Dynamical Systems Junior Barrera BIOINFO-USP University of São Paulo, Brazil
Outline Introduction Lattice operator representation Lattice operator design Lattice dynamical systems (LDS) A simulator for chain dynamical systems LDS identification System identification examples Conclusion
Introduction
Dynamical Systems initial state x 2 = y 2 t t u u Final state Final state x 1 = y 1
Sequential Switching Circuits
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
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,
Operator Representation
ψ : Fun[W, L] L 2 1 0-1 -2 0 1 2 2 2 0 1 2 2 2-1 1 2 2 2-1 1 1 1 1-2 -1-1 -1-1 -2-1 0 1 2
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}
Binary Sup-generating Sup-generating operator: ( u) = 1 u [ a, ] λ a, b b [a,b] 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 λ a,b
Kernel Kernel of ψ at y: K(ψ)(y) = {u Fun[W,L]: y ψ(u)} 2 1 0-1 -2 0 1 2 2 2 0 1 2 2 2-1 1 2 2 2-1 1 1 1 1-2 -1-1 -1-1 -2-1 0 1 2 K(ψ)(-2) K(ψ)(-1) K(ψ)(0) K(ψ)(1) K(ψ)(2)
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)
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
Operator Design
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 +
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)
The constrained estimation problem Error ψ opt Ψ Sample size
Stack filter Spatially translation invariant Spatially locally defined Increasing Commutes with threshold
Noise elimination training images
test image iteration 1
test image iteration 5
test image iteration 1
test image iteration 5
Apertures Spatially translation invariant Spatially locally defined Range translation invariant Range locally defined
Deblurring training images
Resolution Enhancement
(0,0) (0,1) Ψ 2 Ψ 0 Ψ 1 Ψ 3
Original Aperture: 3x3x21x51 Linear Bilinear
Zoom Original Aperture: 3x3x21x51 Linear Bilinear
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
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)
Noise Edge Detection Ground Image Noise Addition Noisy Image Restoration Filtered Image Edge Detection Edge Detected Direct Edge Detection Edge Detected
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 (%)
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
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%
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 0 1 0 1 00 01 10 11 0 0 0 1 1 0 1 1 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 2 variables: 2 2 =4 4 variables: 2 4 =16 8 variables: 2 8 =256
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
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.
Multiresolution Noise image noise image + noise
3x3 window Pyramid Restauration Persisting noise
Dynamical Systems
Finite Lattice Dinamical System
Representation The component functions have canonical morphological representations
System: input-output
Filter For example, processing of motion images.
Input-free systems
Causal systems
Time translation invariant systems
Causal, time translation invariant
A simulator for chain dynamical systems
Simulator Architecture
Functions Representation
System Description
Cell cycle simulation Division Steps Ingredients Control External Signal Inhibition p53 Receptors Excitation Excitation or Inhibition
A model for system identification
Model vector of functions S( Φ, Ψ) S opt ( Φ, Ψ) each component function has a basis S
System Error
Stationary Conditions
Component Error
Additive Loss Function
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.
Identification of Dynamical Systems
A Boolean System
System simulation
System identification: system error
System identification: transition error
Ideal Est.: 100 training examples Est.: 500 training examples Est.: 1500 training examples
Motion Segmentation
Mask Predictor result
Filtering Color Composition
Watershed Color Composition
Motion Segmentation
Motion Segmentation
Conclusion
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.