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

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011 Time/Space noise and « thermal » processing of temperature signal J.C. Batsale, C. Pradere L ecture 6 1

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. PIV velocity Shearography: strain Thermography: 3 fields fields temperature fields

For thermal analysis, simple devices are now currently available! • Cameras Irisys FLIR-Indigo-A10 FLIR-CEDIP Orion, Titanium… With a lot of active heating possibilities (Laser, flash lamps, acoustic or electromagnetic sources….) • Scanners and in the future …tomography IR Camera IR microscope Visible mirror Laser diode Laser beam Transmitting IR Black painted glass wafer: 2 inch diameter y 4 170 µm of thickness x Insulating foam z

“How to process” and “how to estimate” thermophysical properties from a great amount of thermal data, such as temperature fields? – 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. 5

1-1 Noise characterization 1-1-1Monosensor stationnary signal-simple observation 3 3 4 2.5 3 2 2 2 1.5 1 1 1 0 0.5 0 -1 0 -1 -2 -0.5 -1 -3 -2 -1.5 -4 -2 -3 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 -5 0 10 20 30 40 50 60 70 80 90 100 Perfectly gaussian noise Correlated signal Digitized noise (parasitic periodic noise superposed to the signal) 3500 3000 2500 2000 histogram 1500 1000 500 0 -4 -3 -2 -1 0 1 2 3 4 5 All of them have the same rough statistical characteristics (zero mean 6 value, standard deviation), oher ways to study such signal?

Studies with Linear least squares theorem T = X β β β β Hypothesis : -zero mean and additive errors − − − −β β β β constant and unknown before the estimation and X ij known without error -constant variance ( σ σ known) and uncorrelated errors σ σ ^ ^ β β optimum minimize the sum squares function S between theory and experiment β β ^ ^ β = ( X t X ) -1 X t T β β β Estimator β ) = ( X t X ) -1 σ 2 Estimation error cov( e β β β 7

Example : Stationary Signal -Estimation of the mean value T (°C) Real value measurement Time (s)     T 1 1     N ∑ T 1   ˆ   T 2 i   =   β σ β = σ β = = : : ˆ / i 1 N     N :   :           T 1 N 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     ∧ ∧     t ⋅ f T g T t ⋅     ∧ ∧     β = β = T 1 f 1 g 1 ( ) ( )     t ⋅ f f 1 t g ⋅ g 2 T 2 f 2 g 2       β 1     =   : : : β 2       ( ) : : :       ( ) − 1 t f f 0       = σ B ˆ     2 T N f N g N cov ( )   − 1 t g g   0 ( ( ) ) 2 −   − ( ) 1   2 f f 0 N   ≈ B ˆ   ≈ σ B ˆ 2 cond cov cov     − 2 2   t 0 g   g max - T i regularly spaced, N must be chosen as great as possible! -The conditioning number is non-dependant on N ! 9

Example: T=B1+B2*sin(wt+phi)+NOISE Power Spectral density Signal fft(T).*conj(fft(T)) N=100 N=800 -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 10

1-1-2 Sensor array- stationnary observation 5 2 0 0 -5 30 -2 30 25 25 20 20 15 15 10 10 30 25 5 20 5 15 30 25 10 20 15 5 0 0 10 5 0 0 Space correlation Random 5 5 0 0 -5 -10 -5 30 30 25 25 30 30 20 20 15 15 20 20 10 10 10 10 5 5 0 0 0 0 Digitized noise 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 The Singular values are giving an idea about the « complexity » of the images T = U nxn Σ nxn V nxm ˆ T 10 5 9 0 8 -5 7 30 6 sigular values 25 Uniform singular values distribution 5 20 4 (non-compressible signal) 15 3 10 2 1 5 30 25 20 0 15 0 10 0 5 10 15 20 25 30 5 0 Random 5 4.5 2 4 0 3.5 High order zero singular values -2 singular values 3 30 2.5 (compressible signal) 25 2 20 1.5 15 1 10 30 0.5 25 20 5 15 0 10 0 5 10 15 20 25 30 35 5 0 0 Space correlation 0.2 35 0.1 30 0 25 5 U1(x) and V1(y) singular values -0.1 0 20 -5 -0.2 15 -10 -0.3 10 30 25 30 -0.4 20 5 15 20 10 0 -0.5 0 5 10 15 20 25 30 0 5 10 15 20 25 30 10 5 0 12 0 Σ values Non-uniformity distortion U1(x) and V1(y)

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 Isothermal lignes Well positioned Badly positioned sensor sensor – Inertia of the sensor Observed signal Real temperature evolution (noisy but also delayed) 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 . 13

1-2-1 Systematic error: inertia of a thermocouple t  −  K h ∫ =  τ  − τ τ h: exchange coefficient Y ( t ) exp U ( t ) d   ρ cL: heat capacity ρ ρ   cL cL 0 i ∑ U(t): real temperature = � Y H U t − i i j j behaviour = j 1 Y(t): observed behaviour       Y H 0 0 U 1 1 1       Y H H 0 U       2 2 1 2       = � . t H H H . 3 2 1             . . . . 0 .             Y H H H H U − m m m 1 2 1 m The observable signal Y(t) is a correlated signal! Even if N observations are made, only less than N informations are really available. 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). Thermal stability of the instrument The freezing of the detector array and the thermal regulation is not always stable (1 to 5 mK). Time recording, dead time step 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 = ( k integer ) f k . f exc cam ( ) = + f f k 1 N ( k and N integers) acq exc It is possible to reconstruct very fast transient periodic phenomena with a 25 Hz camera! 17