Time/Space noise and thermal processing of temperature signal - - PowerPoint PPT Presentation

Time/Space noise and « thermal » processing of temperature signal

J.C. Batsale, C. Pradere Lecture 6

1

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

Introduction

• The processing of space and time temperature fields is more and

more necessary (not only in heat transfer but also in all domains related to continuous media such as solid or fluid mechanics…)

• Simple devices are now currently available in order to quickly

measure, store and process thermal information ( Infrared thermography , optical or mechanical scans, …)

• “how to process” and “how to estimate” thermophysical

properties from a great amount of thermal data, such as temperature fields?

– Difficulties occurring with such instruments (noise and signal perturbation, systematic errors…) – Difficulties related to the manipulation of a great amount of data and the suitable processing of such data

2

The processing of space and time temperature fields is more and more necessary

• Velocity , strain or temperature fields, are any more

experimentally accessible for Continuum Mechanics…

• In a near future, several different fileds will be simultaneously

recorded and processed.

3

PIV velocity fields Shearography: strain fields Thermography: temperature fields

For thermal analysis, simple devices are now currently available!

• Cameras
• Scanners and in the future …tomography

With a lot of active heating possibilities (Laser, flash lamps, acoustic or electromagnetic sources….)

FLIR-CEDIP Orion, Titanium… Irisys FLIR-Indigo-A10

x y

Insulating foam Black painted glass wafer: 2 inch diameter 170 µm of thickness

z

Visible mirror Transmitting IR Laser diode IR Camera IR microscope Laser beam

4

– Part1-Difficulties occurring with the instruments (noise and signal perturbation, systematic errors, filters, resolution…) – Part2-Difficulties related to the manipulation of a great amount of data and the suitable processing of such data, by considering a heat transfer model.

“How to process” and “how to estimate” thermophysical properties from a great amount of thermal data, such as temperature fields?

5

1-1 Noise characterization 1-1-1Monosensor stationnary signal-simple observation

10 20 30 40 50 60 70 80 90 100

• 3
• 2
• 1

1 2 3 10 20 30 40 50 60 70 80 90 100

• 5
• 4
• 3
• 2
• 1

1 2 3 4 10 20 30 40 50 60 70 80 90 100

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 2.5 3

Perfectly gaussian noise Correlated signal (parasitic periodic noise superposed to the signal) Digitized noise All of them have the same rough statistical characteristics (zero mean value, standard deviation), oher ways to study such signal?

• 4
• 3
• 2
• 1

1 2 3 4 5 500 1000 1500 2000 2500 3000 3500

histogram

6

Hypothesis :

• zero mean and additive errors

− − − −β β β β constant and unknown before the estimation and Xij known without error

• constant variance (σ

σ σ σ known) and uncorrelated errors

Studies with Linear least squares theorem

^ ^

β β β βoptimum minimize the sum squares function S between theory and experiment

T = X β β β β ^ ^ β β β β = (Xt X)-1 Xt T cov(eβ

β β β) = (Xt X)-1 σ 2

Estimator Estimation error

7

Real value measurement

T (°C)

Time (s)

Example : Stationary Signal -Estimation of the mean value

β                 =                 1 : : 1 1 : :

2 1 N

T T T N T

N i i

∑

=

=

1

ˆ ˆ β

N / σ σ β =

The processing of a great amount N of noisy and stationary data improves the accuracy of the estimation.

8

Example :Estimation of several parameters from the previous signal Case where f and g are orthogonal

( )

β

∧ ∧

= ⋅       ⋅

1

f T f f

t t

( )

β

∧ ∧

= ⋅       ⋅

2

g T g g

t t

( )

( ) ( ) 

       =

− − 1 1 2

ˆ cov g g f f B

t t

σ

( )

                ≈

− − − 2 2 1 max 2

ˆ cov g f t N σ B

( ) ( )

2 2

ˆ cov cond g f ≈ B

• Ti regularly spaced, N must be chosen as great as possible!
• The conditioning number is non-dependant on N !

9

T1 T2 : : TN                 = f1 g1 f2 g2 : : : : fN gN                 β1 β2      

Signal Power Spectral density fft(T).*conj(fft(T)) N=100

• Even if the signal is noisy, it is advantageous to process a great amount of data
• The function: sin(wt) is « orthogonal » to the function: f(t)=1

N=800

Example: T=B1+B2*sin(wt+phi)+NOISE

10

1-1-2 Sensor array- stationnary observation

5 10 15 20 25 30 5 10 15 20 25 30

• 5

5 5 10 15 20 25 30 5 10 15 20 25 30

• 2

2 10 20 30 5 10 15 20 25 30

• 5

5 10 20 30 5 10 15 20 25 30

• 10
• 5

5

Digitized noise Random Space correlation Non-uniformity distortion When a signal is multidimensional, instead of projecting on an orthogonal basis, it is possible to set out a singular value decomposition (SVD).

11

SVD decomposition of the previous images

5 10 15 20 25 30 5 10 15 20 25 30

• 5

5 5 10 15 20 25 30 5 10 15 20 25 30

• 2

2 10 20 30 5 10 15 20 25 30

• 10
• 5

5 5 10 15 20 25 30 1 2 3 4 5 6 7 8 9 10 sigular values 5 10 15 20 25 30 35 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 singular values 5 10 15 20 25 30 5 10 15 20 25 30 35 singular values 5 10 15 20 25 30

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 U1(x) and V1(y)

Random Space correlation Non-uniformity distortion The Singular values are giving an idea about the « complexity » of the images U1(x) and V1(y) Σ values Uniform singular values distribution (non-compressible signal) High order zero singular values (compressible signal)

12

ˆ T = UnxnΣnxnVnxm

T

1-2 Systematic errors 1-2-1 Monosensor (thermocouple, resistor, pyrometer…)

• In the best case, the sensor is measuring the « temperature of the sensor »! (see

Bourouga and Bardon, 2000), several illustrations:

– Perturbation of the Isothermal lines and Position error – Inertia of the sensor

• B. Bourouga, V. Goizet, J.-P. Bardon, Les aspects théoriques régissant

l’instrumentation d’un capteur thermique pariétal à faible inertie, International Journal of Thermal Sciences 39 (1) (January 2000) 96– 109.

Isothermal lignes Well positioned sensor Badly positioned sensor Real temperature evolution Observed signal (noisy but also delayed)

13

1-2-1 Systematic error: inertia of a thermocouple

∫

−        − =

t

d t U cL h cL K t Y ) ( exp ) ( τ τ τ ρ ρ

∑

= −

=

i j j j i i

t U H Y

1

• 

                               =                

− m m m m

U . . U U H H H H . . . H H H H H H t Y . . Y Y

2 1 1 2 1 1 2 3 1 2 1 2 1

• U(t): real temperature

behaviour Y(t): observed behaviour

The observable signal Y(t) is a correlated signal! Even if N observations are made, only less than N informations are really available.

h: exchange coefficient ρcL: heat capacity

14

1-2-2 Systematic errors with thermographic sensor arrays

Calibration emission and reflexion, with infrared thermography

• Radiative balance between the sensor and the environment (proper emission, reflexion

and influence of the environment) (see [3], [4]).

• Luminance function of the temperature of the surface (calibration with Planck’s law)

Non uniformity correction (NUC) A distribution of gain and offset for each pixel must be regularly re-estimated (Non Uniformity Correction). Bad or dead Pixels Generally, these pixels are recognized initially by the device provider and corrected by a signal averaged from the neighbouring pixels (Bad Pixel Replacement). Time recording, dead time step Thermal stability of the instrument The freezing of the detector array and the thermal regulation is not always stable (1 to 5 mK). Space resolution

15

Time recording, dead time step

The integration time of FPA cameras with quantum detectors is generally about 100µs. The time for electronic recording and storage is greater (about 40 ms at 25 Hz). If an accurate triggering is possible, the heterodyne methods can be implemented.

16

Principle of the stroboscopic effect or heterodyne technics

cam exc

f k f . =

(k integer) (k and N integers)

( )

N 1 k f f

exc acq

+ =

It is possible to reconstruct very fast transient periodic phenomena with a 25 Hz camera!

17

Space resolution

• Slit response function

Tslit Tplate Tslit x T x SRF − − = ) ( ) (

∫

− =

L

d x U p x Y ) ( ) ( ) ( χ χ χ

                                ∆ =                

m m

U U U p p p p p p p p p p p p x Y Y Y . . . . . .. .. . .

2 1 1 2 2 1 2 3 2 1 2 3 2 1 2 1

• 60
• 40
• 20

20 40 60

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 x=width of the slit SRF(x) real signal Measured signal symetrical part

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0,5 1 1,5 2 2,5 3 3,5 4

Pix el SR F

E dge C enter

The pixels of an infrared camera are generally correlated (N pixels, but less than N informations)!

18

Conclusions of part 1

• The experimental space/time temperature signal is coming from an experiment

(with unknowns and non perfect characteristics)

• Tree categories of errors can be globally considered:

– the random noise : with zero mean value is an unwanted perturbating noise but able to be processed with simples asumptions (related to the uniform covariance matrix). – the systematic errors: (NUC, time derive, parasitic effects, sensor positions ...) which must be fought, detected or bypassed by the experimenter. – the space and time convolutions and correlations of the signal acting on the real time and space resolution limit.

• The signal is then not-only noisy but also filtered, and truncated in space and in time.
• A great amount of data does not significate that all the possible data are available.
• Nevertheless a multidimensional signal gives more processing possibilities than a

monodimensional one.

• It will be assumed for the next steps that the systematic errors are mastered!

19

• 2. Thermal » processing of a T(x,y,t) field

In a lot of cases, the heat transfer models will consist in derivating the space and time temperature field.

20

2.1 Strategies for the estimation of the time and space derivative of the signal

) cos( ) exp( ) sin( / 2 / ) , (

2 1

x t a b L L b t x T

n n N n n n

α α α α − + =

∑

=

L n

n

/ π α =

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

• 0.2

0.2 0.4 0.6 0.8 1 1.2 x T(x) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

• 140
• 120
• 100
• 80
• 60
• 40
• 20

20 40 x Derivative of T(x)

Temperature field from the previous analytical expression at a given time

at time t=0.5s; a=10-5 m2 s-1; b=L/2; L=0.1m; (continuous line: real signal, ‘o’: discrete noisy signal);

Finite differences space-derivative

21

2.1.1 Finite differences

) ( lim = → x x x

i

ε

Unfortunately, when the space step ∆x is tending to zero, the approximation error is effectively tending to zero, but the random error is tending to infinity! Random variable : “measurement noise”: Approximation error:

) ( i x ε

22

ˆ T 'i = ˆ T

i+1 − ˆ

T

i

∆x

ˆ T

i = Ti + eTi

eT

i

ˆ T 'i = T(xi+1) − T(xi ) ∆x + ε(xi+1) + eTi+1 − eTi ∆x

2.1.2 Polynomial fitting

∑

=

=

N n n nx

x T ) ( β

T n ]

.... , , [

3 2 1

β β β β = B

( )

T X X X B

t 1 t

ˆ

−

=

              = ... ... ... 1 . 1 1

2 2 2 2 2 1 1 m m

x x x x x x X

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

• 0.2

0.2 0.4 0.6 0.8 1 1.2 x T(x) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

• 200
• 150
• 100
• 50

50 100 150 x Derivative of T(x)

Polynomial fitting Derivation of the polynomial expression The projection of the signal on a reduced polynomial basis is giving quite good results (excepted with the boundaries) when the rank of the polynom is adapted.

23

2.1.3 Fourier cosinus basis

) cos( ) ( x x T

n n M n

α β

∑

=

=

L n

n

/ π α =

T M ]

.... , , [

3 2 1

β β β β = B

X X t

matrix is orthogonal,

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

• 0.2

0.2 0.4 0.6 0.8 1 1.2 x T(x) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

• 140
• 120
• 100
• 80
• 60
• 40
• 20

20 x derivative of T(x)

Fourier fitting Derivation of the serie The derivation of the “Fourier estimated expression” is giving good results when the rank

• f the serie is adapted.

24

2.1.4 Filtering with a convolution kernel

∫

− =

L

d x T p x T ) ( ) ( ) ( ~ χ χ χ

1 ) ( =

∫

L

d p χ χ

∫ ∫

− = − =

L L

d dx x dp T d dx x dT p x dx T d ) ( ) ( ) ( ) ( ) ( ~ χ χ χ χ χ χ The discrete approximation of the derivative is then conveniently considered by a convolution with a « derived » kernel.

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

• 0.2

0.2 0.4 0.6 0.8 1 1.2 x T(x) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

• 100
• 50

50 100 x derivative of T(x)

Filtering of the signal Derivation by a convolution with a “derivated kernel” Filtering of the signal Derivation by “derivated kernel”

25

1.2.5 Singular value decomposition of the whole space and time signal

20 40 60 80 100 20 40 60 80 100

• 0.5

0.5 1 1.5 t x T(x,t)

10 20 30 40 50 60 70 80 90 100

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 x Uk(x) U1(x) U2(x) U3(x) 10 20 30 40 50 60 70 80 90 100

• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 t Vk(t) V1(t) V2(t) V3(t)

10 20 30 40 50 60 70 80 90 100

• 0.2

0.2 0.4 0.6 0.8 1 1.2 x T(x,0.02) 10 20 30 40 50 60 70 80 90 100

• 250
• 200
• 150
• 100
• 50

50 x Space derivative ot T(x,0.02)

Consider not only the Space but also the time signal! 3 dominant Singular values Reconstructed signal with a reduced rank of SVD decomposition Space derivative of the compressed signal Uk(x) Vk(t)

26

ˆ T = UnxnΣnxnVnxm

T

2.2 Estimation of a transverse diffusivity field from flash experiments (comparison of classical Non Destructive Evaluation methods):

Experimental situation and temperature evolution

IR camera « Flash » lamp Composite sample Ox Oy Oz e l L IR camera IR camera « Flash » lamp Composite sample Ox Oy Oz Ox Oy Oz e l L

Flash uniforme y=0

( )

, , T x y e t =

27

Transverse flash method and IR thermography

28

• 1 T(t) measurement
• measurement with contact
• very accurate measurement
• >10000 T(t) measurements
• measurements without contact
• very noisy measurements

Metrology Imaging Laboratory method

Sample

Thermocouple

1 T(t) measurement

0.1 T (°C) 10 20 30 t (s) 40

Industrial method

Sample

IRcamera

>10 000 pixels T(t) measurement

2.5 T (°C) 10 20 30 t (s) 40

29

Can we discern two very noisy thermograms ?

• 0.5

0.5 1 1.5 20 40

Temperature level Time (s)

2.2.1 Estimation with physical asymptotic expansions:

        ∂ ∂ ∆ + =

) , , (

) , ( ) , , ( ) , ( ) , , , (

λ

λ λ β

t z

f y x t z f y x q t z y x T

With q(x,y) : spatial distribution of energy, and ∆λ (x,y): spatial thermal conductivity variation

( )

( )

2 1 2

/ ² ² exp 2 1 , L at f cL Q L t a n cL Q t z T

n

ρ π ρ =               ⋅ ⋅ ⋅ ⋅ − ⋅ + ⋅ = =

∑

∞ =

1D température response: Asymptotic expansion by considering a small thermal conductivity variation Other possibility : consider the conductivity sensitivity function as a logarithmic time derivative

( )

) / ( 2

2

/ ) , (

cL t i i

i

t f t cL Q cL t f cL Q t T

ρ λ

∂ ∂ λ λ ρ ρ λ ρ ∆ + ≈

Other possibilities with other thermophysical properties :

( ) ( )

        + ∆ − ≈

) / ( 2 2

2

2 / / ) , (

cL t i i i

i

t f t cL t f L L cL Q cL t f cL Q t T

ρ λ

∂ ∂ ρ λ ρ ρ λ ρ

( ) ( )

        + ∆ − ≈

) / ( 2 2

2

/ / ) , (

L c t i i i

i

t f t L c t f c c L c Q L c t f L c Q t T

ρ λ

∂ ∂ ρ λ ρ ρ ρ ρ λ ρ

30

Linear Estimation method, if X is well known

) ( ) ( ) , (

2 1

2 1 i i i

t X t X t T

β β

β β + ≈

T = T 0,t1

( )

.. T 0,tN

( )

[ ]

t

t N N

t X t X t X t X       = ) ( ... ) ( ) ( ... ) (

2 2 1 1

1 1 β β β β

X

( )

T X X X

t 1 t

ˆ ˆ ˆ

2 1 −

=         β β

Is suitable only if X is perfectly known. If f is only a reference curve obtained from experiment, the time logarithmic derivative will noisy and the estimation bad! Shepard proposed to decompose the signal with a polynomial fitting, such as:

Ln(T(x,y,z=0,t))=β0(x,y)+β1(x,y)Ln(t)+β2(x,y)Ln2(t)+…

31

2.2.3 The SVD decomposition

=

( ) ( )

, , ,

réarrangement

T x y t T X t →

• 1. Arrangement of the information cube in a

space time matrix

• 2. SVD decomposition of the resulting matrix

( ) ( ) ( )

1

,

P T k k k k

T X t U X V t λ

=

= ⋅ ⋅

∑

• 3. Arrangement of the spatial U vectors in spatial matrices

( ) ( )

,

réarrangement k k

U X U x y →

32

SVD and signal compression

Flash Milieu hétérogène Caméra IR séquence IR

y x

t e m p s (SVD)

y x

« M

• d

e s » From a sequence (about 100 or 1000 images), the SVD will give sometime 3

• r 4 images related to the structure of the sample.

33

SVD and variable separation

34

temporal covariance matrix only depending on

M X

Only depending on

U X

Space covariance matrix only depending on

N t

Only depending on

V t

( )

,

SVD T

T X t U V = Σ

( )

( )

2 ,

, ,

Diagonalisation T i j i j t

M T X t T X t dt U U     = ⋅ ⋅ ⇒ Σ    

∫

( )

( )

2 ,

, ,

Diagonalisation T i j i j X

N T X t T X t dX V V     = ⋅ ⋅ ⇒ Σ    

∫

Comparison between SVD and asymptotic expansions

35

( ) ( ) { } { } ( ) ( )

( )

{ } ( )

max max

,

x moy x moy x

T T x T x t T t x t T τ τ τ   ∂ ≈ + − ⋅   ∂    

Asymptotic expansion:

( ) { } ( )

2 x moy

T V t t τ ∂ ≈ ∂

( ) ( )

( )

( )

( )

2 moy moy

U x x x τ τ τ τ ≈ − − U1 (x) and V1(t) are very near from the time and space average signal. U2(x) and V2(t) are the space and time deviation from the space and time average signal.

Practical example: NDE and tensile test on a composite medium

36

Relative evolution of versus the stress = transvers diffusivity variations.

( )

2

U x

Remarks about the previous methods

• From the great amount of data, the most

previous methods consisted in trying to compress the data, by a projection on a suitable basis (very often non related to the physics)…

• Is it possible to look directly to the

phenomena of physical interest?

37

2.3 Estimation of in-plane diffusivity field-Time-space correlation and elimination of the non useful data

Randomly flying spot

38

39

Experiment

IR Camera Material Laser diode Motorized displacements in X and Y directions

40

« Nodal » method in the case of a source point

• Only a few pixels are available on an image at a given time
• The in-plane diffusion is approximated with a finite difference scheme:

If the system is in pure relaxation it gives The diffusivity is to be estimated only if the correlation coefficient is near from 1:

k j i k j i k j i j i

T T Fo

, , , ,

δ = Φ + ∆

( )

k j i k k k k k j i

T T T T T T

, 1 j i, 1 j i, j , 1 i j , 1 i ,

4 − + + + = ∆

− + − +

k j i k j i k j i

T T T

, 1 , ,

− =

+

δ

with ;

2 , ,

x t a Fo

j i j i

∆ ∆ =

;

k j i k j i j i

T T Fo

, , ,

δ = ∆

∑ ∑ ∑

∆ ∆ =

t t t t

F k j i F k j i F k j i k j i F j i

T T T T

2 , 2 , , , ,

δ δ ρ

Correlation indicator method

41

ΔT δT ΔT δT

∑ ∑ ∑ ∑

∆ = ∆ =

t t t t t t

F k j i F k j i F j i F k j i F k j i k j i F j i

T T T T T Fo

2 , 2 , , 2 , , , ,

1 δ ρ δ δ

• 1.8
• 1.6
• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2

• 0.2

0.2 0.4 0.6 0.8 1 1.2 Laplacian(T) derivative(T) Heat source on Pure diffusion No diffusion or bad pixel

• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2 1.4 Laplacian(T) derivative(T) Heat source on Pure diffusion No diffusion or bad pixel

Perfect data Noisy data

ρi, j

Ft =

∆Ti, j

k δTi, j k Ft

∑

∆Ti, j

k 2 Ft

∑

δTi, j

k 2 Ft

∑

→ 1

Foi, j∆Ti, j

k = δTi, j k

Example with an in-plane source point

42

Temperature field Evolution of a central pixel Correlation field Correlation at a central pixel Diffusivity field Diffusivity estimation at a central pixel

Results for a heterogeneous plate

43

Nombre de Pixels en x Nombre de Pixels en x

Foi, j∆Ti, j

k = δTi, j k

ρi, j

Ft =

∆Ti, j

k δTi, j k Ft

∑

∆Ti, j

k 2 Ft

∑

δTi, j

k 2 Ft

∑

→ 1

General Conclusion

• Space/ time signal=great amount of noisy and

« non-perfect » data.

• Several strategies:

– Analysis of the different kinds of noise and bias of the signal. – Compression (projection, filtering, averaging…) and estimation with a model by the implementation of a « suitable » basis. – Direct use of the physical model

(example: Correlation analysis)

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