Wien Approximation Research Articles

Analysis of statistical uncertainties of non-contact temperature and emissivity measurements in the optical and near infrared ranges

The aim of this report is the analysis of the uncertainties of non-contact measurement of temperature and emissivity determined by the least sum of squares (LSS) method. Analytical expressions are obtained for the uncertainties of the temperature T and the emissivity ε by three methods: (1) linear regression using the Wien approximation, (2) non-linear regression with covariance matrix calculation using the Planck’s equation, and (3) hybrid method that combines linear and non-linear regression methods. It was found that the better the condition for the Wien’s approximation is fulfilled, the worse the accuracy of the emissivity measurement compared to the accuracy of the temperature measurement. The new optimal strategy is proposed to determine the T and the emissivity ε for non-contact measurements. It consists of three steps: (1) the temperature of the radiating body is found by the Wien approximation, TW; (2) the obtained value TW is used as an initial approximation for one-dimensional “hybrid” minimization. Minimization is carried out by one of the numerical methods of one-dimensional minimization. As a result, the values of ε0 and T0 are found at with the sum of the squared residuals, S0, reaches a minimum; (3) Measurement uncertainties are estimated using the covariance matrix method.

COMPUTER MODELS OF INCANDENSCENT LAMPS

The article is devoted to the research of existing and the development of new computer models of the hemispherical and normal spectral coefficient of emission of optical incandescent lamps. Optical lamps (filament electric lamp, FEL) remain a prevalent source of radiation in various optoelectronic systems today. They provide radiation in the ultraviolet, visible, near-infrared (IR-A, NIR), and mid-infrared (IR-B, SWIR) spectral ranges and have typical color temperatures of 2300 K, 2850 K, 3200 K.An analysis of two groups of existing FEL models was carried out.The models of the first group are oriented primarily for existing calibration and certification installations of optoelectronic systems.The models of the second group are more universal and comprehensive.They reproduce the element-by-element structure of the modeled optoelectronic system and use the physical and constructive parameters of significant elements in the modeled system. Planck's formula or Wien's approximation is the basis for all types of temperature source models. Existing models are designed for the visible range and the range of 0.34 μm – 2.6 μm for temperatures of 1600 K – 2800 K, which does not take into account all the capabilities of modern FELs.The aim of the work is to investigate known models and polynomial models for use in calculations of modern optical lamps with temperatures ranging from 2300 K to 3200 K across a wide range of wavelengths, including ultraviolet, IRA, and IR-B.The accuracy of the Larabi and Pon models has been analyzed and their parameters have been modified. The obtained dependencies ensure an increase in accuracy by 1.4 - 4 times.The accuracy and cost-effectiveness of polynomial models of the normal spectral coefficient of radiation in the ultraviolet, visible, IR-A and IR-B spectrum ranges were analyzed. Model coefficients have been calculated based on the criterion of minimizing the root mean square deviation. It has been demonstrated that, according to the criterion of average errors in the range of 0.23 μm - 2.7 μm, all dependencies have errors of less than 5 % throughout the entire range and provide a minimum of 5 correct digits.According to the criterion of maximum errors with a relative error of 5%, linear expressions cannot be applied, and an error of less than 1 % is provided only by cubic polynomials. A combined polynomial model was formed, which provides a methodological error of approximation of less than 1 % in all spectral sub-ranges, and in the visible range of 0.3 μm - 0.94 μm - less than 0.5 %.The time costs of the models are determined and the combined models are formed for use in processing systems with limited computing capabilities.

Approach to multispectral thermometry with Planck formula and hybrid metaheuristic optimization algorithm.

Accurate temperature measurement has significant implications for product quality, industrial process control, and scientific research. As a non-contact temperature measurement method with broad application prospects, multispectral thermometry still poses significant challenges in data processing. Currently, most multispectral thermometry methods use the Wien approximation equation to construct the objective function. However, the use of the Wien approximation equation is conditional and generally applicable only to low temperatures or short wavelengths. In this paper, what we believe is a new data processing model of multispectral thermometry is established based on the Planck formula; Additionally, a feasible region constraint method is proposed to constrain the emissivity range; By utilizing a hybrid metaheuristic optimization algorithm based on differential evolution (DE) and multi-population genetic (MPG) algorithms, the simulation results of six different models and experimental results of silicon carbide demonstrate that the proposed algorithm achieves an average relative error in temperature measurement within 0.42% and a random relative error within 0.79%. The average computation time for each temperature inversion is approximately 0.26 seconds. The accuracy and efficiency of the algorithm ensure that it can be applied to real-time temperature measurement in industrial field.

Approximating Planckian Black-body Lights using Wien's Approximation

Planck’s law and Wien’s approximation of this law are both widely used to calculate the spectral radiation of a black-body based on its color temperature. The Wien approximation is a slightly simpler equation and has the advantage that the logarithm of a Wien light can be written as a linear sum of two basis vectors plus an offset (a fact that is exploited in some computer vision algorithms). In this paper, we show that the Wien formulation can, in general, be used to approximate Planckian lights assuming there is a mapping function taking Planckian to corrected Wien temperature. Significantly, we show that a correction function <i>f</i>() exists and for the range of color temperatures of interest the Wien spectrum calculated for <i>f</i>(T) has a very similar shape to the actual spectrum of a Planckian light with temperature T. We find that defining <i>f</i>() as a polynomial-type function models to a good extent the relationship between the color temperatures of Planckians and their closest Wien-Planckians lights both in terms of the angular error between their two respective spectral functions and their projections to u’v’ coordinates.

Locus filters.

In this paper, directly following from Gage [J. Opt. Soc. Am.23, 46(1993)10.1364/JOSA.23.000046], we study the design of a particular theoretical filter for photography, that we call the locus filter. It is built in such a way that a Wien-Planckian light (of any temperature) is spectrally mapped to another Wien-Planckian light. We provide a physical basis for designing such a filter based on the Wien approximation of Planck's law, and we prove that there exists a unique set of filters that have the desired property. While locus filtered Wien-Planckian lights are on the locus, the amount they shift depends both on the locus filter used and on the color temperature of the light. In experiments, we analyze the nature of temperature change when applying different locus filters and we show that real lights shift more or less as if they were Planckians in terms of the changes in their correlated color temperatures. We also study the quality of the filtered light in terms of distance from the Planckian locus and color rendering index.

Imaging pyrometry for most color cameras using a triple pass filter.

A simple combination of the Planck blackbody emission law, optical filters, and digital image processing is demonstrated to enable most commercial color cameras (still and video) to be used as an imaging pyrometer for flames and explosions. The hardware and data processing described take advantage of the color filter array (CFA) that is deposited on the surface of the light sensor array present in most digital color cameras. In this work, a triple-pass optical filter incorporated into the camera lens allows light in three 10-nm wide bandpass regions to reach the CFA/light sensor array. These bandpass regions are centered over the maxima in the blue, green, and red transmission regions of the CFA, minimizing the spectral overlap of these regions normally present. A computer algorithm is used to retrieve the blue, green, and red image matrices from camera memory and correct for remaining spectral overlap. A second algorithm calibrates the corrected intensities to a gray body emitter of known temperature, producing a color intensity correction factor for the camera/filter system. The Wien approximation to the Planck blackbody emission law is used to construct temperature images from the three color (blue, green, red) matrices. A short pass filter set eliminates light of wavelengths longer than 750nm, providing reasonable accuracy (±10%) for temperatures between 1200 and 6000K. The effectiveness of this system is demonstrated by measuring the temperature of several systems for which the temperature is known.

Quantitative experiments in a distance lab: studying blackbody radiation with a smartphone

In this paper we present an experiment aimed at investigating the thermal radiation law. The spectrum of a filament light bulb is acquired at different temperatures using a low-cost grating spectrometer based on a smartphone camera, while the measurement of filament temperature is obtained indirectly from the temperature dependence of the filament resistance. Two different methods for measuring the light intensity are tested: one is based on the use of the ambient light sensor of a smartphone, the other on a home-made grating spectrometer coupled with the smartphone camera. We observe a good agreement of the experimental data with the theoretical predictions of the Planck distribution in the limit of the Wien approximation restricted to the wavelength range characterizing the response of the smartphone light sensor and camera.

A new wavelength selection criterion for two-color pyrometer interfered with participating media

Two-color pyrometry is well established and widely applied to surface temperature measurements. However, its accuracy degrades significantly when the transmission path between the object surface and the pyrometer is full of high concentration participating media, which attenuates the surface radiation signal to the pyrometer. To improve measurement accuracy, a theoretical optimization wavelength selection criterion for two-color pyrometry working with the interference of a participating medium is established on the basis of the Plank’s law and Wien's approximation. The closer to zero the criterion value is, the smaller the measuring error is. Therefore, the optimum choice of operating wavelengths combination for two-color pyrometer is obtained through inter-comparison of the criterion values. Evaluation of the practical applicability of the new criterion is performed by employing an analytical radiation thermometry model considering radiative attenuation and augmentation in participating media. Theoretical estimations confirm that the new criterion has good applicability and scientific quality considering two typical media ordinarily encountered in engineering applications: water vapor and water mist. This selection criterion not only provides solid reference information for the proper selection of wavelengths combination, but also helps one to better understand the fundamentals of measurement errors for two-color pyrometer interfered with participating media. The analyses in the paper provide the necessary theoretical supports for the design and application of a two-color pyrometer.

Using spectral distribution of radiance temperature and relative emissivity for determination of the true temperature of opaque materials

A method for determining the thermodynamic (true) temperature of opaque materials by the registered spectrum of thermal radiation under the conditions when we do not know emissivity of a free-radiating body is presented. A special function, which is a product of relative emissivity of tungsten by the radiation wavelength, was used as the input data. The accuracy of results is analyzed. It is shown that when using relative emissivity, the proposed algorithm can be used both within the range of applicability of the Wien approximation and the Planck formula.

Multi-spectral image processing for the measurement of a spatial temperature distribution on the surface of a laser-heated microscopic object

In this paper, we demonstrate that combining a laser heating (LH) system with a tandem acousto-optical tunable filter (TAOTF) allows us to measure the temperature distribution (TD) across a laser-heated microscopic specimen. Spectral image processing is based on one-dimensional (1D) non-linear least squares fitting of the Planck radiation function. It is applied for determining the temperature T at each point (x, y) of the specimen surface. It is shown that spectral image processing using the 1D non-linear least squares fitting allows measurement of the TD of the laser-heated microscopic specimen with higher precision and stability than those of the conventional linear least-squares fitting of the Wien approximation of Planck’s law.

A temperature and emissivity separation algorithm based on maximum entropy estimation of alpha spectrum's scaling and translation

In the thermal infrared (TIR) waveband, solving the target emissivity spectrum and temperature leads to an ill-posed problem in which the number of unknown parameters is larger than that of available measurements. Generally, the approaches developed for solving this kind of problems are called, by a joint name, the TES (temperature and emissivity separation) algorithm. As is shown in the name, the TES algorithm is dedicated to separating the target temperature and emissivity in the calculating procedure. In this paper, a novel method called the new MaxEnt (maximum entropy) TES algorithm is proposed, which is considered as a promotion of the MaxEnt TES algorithm proposed by Barducci. The maximum entropy estimation is utilized as the basic framework in the two preceding algorithms, so that the two algorithms both could make temperature and emissivity separation, independent of experiential information derived by some special data bases. As a result, the two algorithms could be applied to solve the temperature and emissivity spectrum of the targets which are absolutely unknown to us. However, what makes the two algorithms different is that the alpha spectrum derived by the ADE (alpha derived emissivity) method is considered as priori information to be added in the new MaxEnt TES algorithm. Based on the Wien approximation, the ADE method is dedicated to the calculation of the alpha spectrum which has a similar distribution to the true emissivity spectrum. Based on the preceding promotion, the new MaxEnt TES algorithm keeps a simpler mathematical formalism. Without any doubt, the new MaxEnt TES algorithm provides a faster computation for large volumes of data (i.e. hyperspectral images of the Earth). Some numerical simulations have been performed; the data and results show that, the maximum RMSE of emissivity estimation is 0.017, the maximum absolute error of temperature estimation is 0.62 K. Added with Gaussian white noise in which the signal to noise ratio is measured to be 11, the relative RMSE of emissivity estimation is 2.67%, the relative error of temperature estimation is 1.26%. Conclusion shows that the new MaxEnt TES algorithm may achieve high accuracy and fast calculating speed, and also get nice robustness against noise.

High Precision Multicolorimetric Pyrometer With a Novel Photoelectric MOSFET

Colorimetric thermometry is a useful method for high temperature measurement. In this paper, a high precision multicolorimetric pyrometer is introduced. A multicolorimetric calculation without the gray body assumption is deduced. A novel photoelectric MOSFET with direct ac signal output is employed as the sensor, which works near the visible waveband and brings smaller uncertainty under the Wien approximation than the infrared radiation detectors. A dual-processor embedded signal process circuit is constructed, which ensures the pyrometer's portability as well as supports the accuracy of colorimetric calculation. Through the tests, the precision of the pyrometer is better than 30/00 in the temperature range 1323-1923 K.

1/f noise in the UV solar spectral irradiance

The investigation of the intrinsic properties of the solar spectral irradiance as a function of the ultraviolet (UV) wavelength is attempted by exploiting rare observations performed at the Villard St. Pancrace station of the Lille University of Sciences and Technology ranging from 278 to 400 nm with a step of 0.05 nm every half an hour from nearly sunrise to sunset. To achieve this goal, the modern method of the detrended fluctuation analysis was applied on the solar spectral irradiance values versus wavelength. This analysis revealed that the solar incident flux at the top of the atmosphere and the solar spectral irradiance at the ground during two overcast sky days fluctuate with the UV wavelength exhibiting persistent long-range power-law behavior. More interestingly, the exponent of the power-law relationship between the fluctuations of the solar spectral irradiance versus UV wavelength at both the top of the atmosphere and the ground is consistently close to unity (of 1/f-type) throughout the day. This 1/f behavior has been detected in many complex dynamical systems, but despite much effort to derive a theory for its widespread occurrence in nature, it remains unexplained so far. According to the above-mentioned findings we speculate that the 1/f property of the incident solar UV flux at the top of the atmosphere could probably drive both the 1/f behavior depicted in the atmospheric components and the solar UV irradiance at the Earth's surface. The latter could influence the UV-sensitive biological ecosystems, giving rise to a 1/f-type variability in the biosphere, which has already been proven by recent observational data. We finally propose that Wien approximation could be multiplied by a 1/f function of wavelength (e.g., of the type of the fractional Brownian motion) in order to reproduce the aforementioned 1/f feature of the solar UV flux.

基于维恩近似修正的热红外温度和发射率反演算法

基于维恩近似修正的热红外温度和发射率反演算法

COMBUSTION OF COAL CHARS IN OXYGEN-ENRICHED ATMOSPHERES

This work pertains to the high-temperature combustion of pulverized coal chars under oxygen-enriched atmospheres. Single char particles were burned in a drop-tube furnace, electrically-heated to 1300–1500 K, in 21%, 50% and 100% O2, in a balance of N2. Their luminous combustion histories were observed with two-color ratio pyrometry. A solution of the Planckian ratio-pyrometry equation for temperature was implemented, extending on Wien's approximation. The temperature and time histories for 45–53 µm bituminous chars experienced wide particle-to-particle disparity, and varied depending on oxygen mole fraction and furnace temperature. Average char surface temperatures increased from 1600–1800 K in air, to 2100–2300 K in 50% O2, to 2300–2400 K in 100% O2, at gas temperatures of 1300–1500 K, respectively. Combustion durations decreased from 25–45 ms in air, to 8–17 ms in 50% O2, to 6–13 in 100% O2. Thus, average particle temperatures increased by up to 45%, whereas burnout times decreased by up to 87% as combustion was progressively enriched in O2 until 100% was attained. The apparent and intrinsic reactivity of the chars burning at 1500 K gas temperature were found to increase by factors of to 8 and 35, respectively, as the oxygen mole fraction increased by a factor of five, from 21% to 100%.

Comparison of fire temperature retrieved from SWIR and TIR hyperspectral data

This paper addresses the problem of active-fire temperature retrieval from hyperspectral remotely sensed data. A simple theoretical model able to retrieve the temperature of burning objects employing spectral data collected in the short-wave infrared interval is described and analysed. This algorithm assesses hot-spot temperature by processing the finite difference of at-sensor spectral radiance between nearly local spectral channels, and it is not able to retrieve the temperature of warm or cold targets at standard environment conditions (e.g. T∼300 K). Spectral radiance finite difference is modelled by a power series stopped to the first order, the expression of which is further simplified by Wien's approximation. The model assumes that spectral emissivity change between the two selected wavelengths is negligible and it would be used with hyperspectral data by averaging the estimates obtained at several channel couples. Performance of the new algorithm is compared with that of the “Grey-Body-Emissivity method”, which computes the pixel temperature from thermal infrared spectral radiance data. The paper shows results obtained applying the above discussed methods to hyperspectral images gathered by the airborne MIVIS spectrometer over a natural fire broken out in July 1999 over the Alps, Northern Italy.

Illuminant invariance from a single reflected light

AbstractThis article presents an algorithm for computing an illuminant‐invariant quantity inherent in a single pixel of an imaged object color. The invariance pertains for two different assumptions about the illuminant spectrum: the photoreceptor sensitivities and the reflectance spectrum of the object. For one regime the illuminant spectrum is exponential, the photoreceptor sensitivities are equal‐width Gaussians, and the reflectance is also Gaussian. For the other regime the illuminant is a Wien approximation to a blackbody radiator, the photoreceptor sensitivities are narrow band in wavelength, and the reflectance is unconstrained. The existence of two regimes for the invariant testifies to its robustness. Computing this invariant over all pixels in an image will assist object‐color recognition (machine‐vision color constancy) without resorting to the usual assumption that illuminant variation over a scene is gradual compared to reflectance variation over that scene. © 2002 John Wiley & Sons, Inc. Col Res Appl, 27, 45–48, 2002

Color invariant for daylight changes: relaxing the constraints on illuminants.

We extend previous work that addressed the problem of color changes on reflective surfaces resulting from changes in the daylight spectrum. In that work, we constrained the illuminants to a family represented by the Wien approximation to Planck's formula in order to derive a function of the three camera outputs that is invariant to daylight changes. In this work, we show that the constraint on the form of the illuminants can be relaxed and that a much more general form is permissible. We use principal components analysis on the logarithm of the illumination to represent the CIE standard in the more general form and show that the result closely represents the standard. We recalculate the exponent used in the invariant for our camera from the extended theory and obtain a result that duplicates the one found by empirical means used in our previous work.

Emissivity of condensed substances from their thermal radiation spectra

A technique for measuring the temperature and spectral emissivity of condensed substances is described. With this technique, measurements can be made in any spectral range, rather than only at short wavelengths, where the Wien approximation for the Planck formula holds. In particular, in the spectral range around the peak-emission wavelength, the highest signal-to-noise ratio can typically be attained, which raises the accuracy in temperature and emissivity determination. The proposed approach to processing the experimental emissivity curves offers the possibility of analyzing the effects of the major factors (signal-to-noise ratio, spectral range of measurements, type of the model function representing the spectral emissivity, and others) on the accuracy in temperature measurements. The potentialities of the technique are demonstrated by measuring the temperatures and emissivities of W, Re, and Ta strip lamps in the spectral range 0.9 to 2.1 Μm.

A new approach for temperature and emissivity separation

A Taylor expansion can be used to linearize the Planck function that governs the emission from a material at a given temperature. Such a linearization is more accurate than using Wien's approximation. This approach can be used to improve the accuracy of those logarithms that currently utilize Wien's approximation, such as the alpha residual and alpha emissivity techniques.