Temporal Scale Spaces

Daniel FagerströmCVAP/NADA/KTHdanielf@nada.kth.se

Measurement of Temporal Signals

• Real time measurement• Temporal causality• Structure at different resolution• Well defined derivatives

Earlier Approaches

• Koenderink (88), Florack (97), ter Haar Romeny et. al. (01) – Logaritmic mapping of the past time-line, then ordinary scale space

• Lindeberg & Fagerström (96) – Noncreation of local maxima with increasing scale

• Salden (99) – Diffusion on the temporal time-line with absorbing or reflecting boundary condition

Current Approach

• Using axioms from Pauwels et. al. 95, replacing reflectional symmetry with temporal causality

Space vs. Time

SpaceAll directions simultaneously available

TimeThe past can be memorized, the future can not

Memory

Overview

• Motivation of axioms• Form of temporal scale space kernels• Time recursive realization• Numerical scheme• Comparison with earlier work

Principles

• Non committed representation• Embodiment of symmetries in the

surroundings• Extended point

Measurement

Posf : xf

Sensorf : f

Meaurement

0|, 1 CL

•Choose well behaved sensor functions

Endf

fff ,

•Linearity

Examples

dtttgf

•Regular distribution

0

•Dirac ”function”

fffff

•Derivation

Causality

•Causality

0|1 tttt

•Measurement time

: t

Embodiment of Geometry

• The structure of the sensor system should correspond to regularities in the environment

• Translation covariance: the measurement process is the same at each moment

• Scaling covariance: all time spans are treated in the same way

Covariance

af

axfD

xafxfSaxfxfT

a

a

a

11 :Dilatation

:Scaling :nTranslatio

•Transformation groups

aa

aa

DffSTffT

•Corresponding action on a sensor

Orbits

0','' tt TDT

Convolution

'

'

f

dsstsfdstssfTf t

•Definition

ff

•Associativity

Cascade Property (Semi-Group)

Extended Point

•Unit area1

•Positivity 00 ff

ss 0

lim

Temporal Scale Space Kernels

• Continuity• Positivity• Unit area• Causality• Dilatation covariance• Convolution semi-groupPauwels et al used the same axioms, with (4) replaced by reflexion

symmetry, to characterize spatial scale-spaces.

Characterization

Laplace transform

ˆˆˆ

Semi-group and causality

:,ˆ ges sg

Continuity

(Cauchy’s functional equation)

Characterization (Cont) sges

ˆ

Dilatation covariance

sesˆ

Positivity

(Bernstein)

10

Temporal Scale Space Kernels0.3 0.5 0.8

set -1, L

Extremal Stable Density Functions

•Limit densities for sum of stochastic variables (Levy)

•Infinite first and second moments

•Increasing popularity in fysics, finance …

Explicit Form

0,

,1

23

2

,21

,0

!1

24exp4

k

kk

t

tkkt

t

tt

kt

tt

tt

Markov Property?

• The memory is the only access to the past• Convolution directly with the input signal is

unrealistic• Need for an evaluation equation!

t

L(0,)=0

L(t,)=f(t)

LALt tf

tLftL

,,,,

Fractional Derivatives

fxfsfxf xx ,

1, L

•Linear

•Equal to ordinary derivatives for integer order

xxx

•Generalized Leibniz property

gfgf kx

kx

kx

0 k

Evaluation Equations

,

1

-1,

expLxpe

L

t

s

set

tftLL

LL t

,lim0,0

0

,•Signaling equation for

tftLL

LLt

,lim0,0

0

22,

•Fractional Brownian motion

•Diffusion for

L

LLL

t

tt

2,

,,

2

tftLL

LLt

,lim0,0

0

1,

•Scale space as state

•Diffusion for

Uniquenes of the Signaling Equation

tftLL

LLt

,lim0,0

0

1,

•Locality: is a differential operator if 1/ is an integer

•Positivity: have a positive Greens function if 1/

•Locality and positivity is only satisfied for

L1

,

L1

,

Signaling Equation

tftL

LLLt

,lim0,0

0

2

•The temporal scale space is the only memory of earlier input

•The memory diffuses over time

Numerical Scheme

t

3/2-1/2

•Implicit scheme

•Second order stability in time and scale

•4(add)+4(multiply)+2(divide) per mesh point

Example

Example

Comparison with Previous Work

• Florack (97) requires the measurement kernel to be a positive Schwartz test function, this rules out all stable density functions except Gaussians

• Ter Haar Romeny et. al. (01) requires the measurement kernel to have finite first and second moment, only fulfilled for Gaussians, extremal (causal) stable density functions even has infinite first moment

Comparison with Earlier Work

• Koenderink (88) and Lindeberg & Fagerström (96) uses stronger requirements (than semi-group) on causality in the resolution domain

• Salden (99) does not require translational covariance on the kernel only on the generator

Comparison with Earlier Work

• Lindeberg & Fagerström (96) and the current approach are the only ones that has a (known) time recursive formulation

• Lindeberg & Fagerström (96) requires either time or scale to be discrete and lack scale covariance

• The kernels in the current approach are less well localized than in the other approaches

Conclusion

• Causal scale invariant convolution semi group

• Time recursive realization• Signaling equation• Efficient numerical scheme