Assessment of measurement uncertainties for imaging luminance measurements

Abstract

This paper presents an approach to assess the measurement uncertainty of luminance measurements obtained by imaging luminance measurement devices (ILMDs). A common strategy for users doing measurements in photometry is to characterise and correct for residual errors, that in this domain are often significant. To apply this strategy to measurements with ILMDs leads to issues because the user only has limited knowledge on the device internals and the effort for such characterisations would exceed the capabilities of most users. To illustrate this, the complexity of the devices’ internal operation is shown, and tasks of the manufacturer to build and adjust an ILMD based on a camera system are outlined. From this we concluded, that for most users, the best approach is to assume that all significant effects have been corrected for by the manufacturer, and only the remaining uncertainties need to be estimated. An example is given for this estimation and for the handling of correlations between measurements.

1. Introduction

Imaging luminance measurement devices (ILMDs) are used in many fields that rely on the evaluation of luminance distributions of emitted or reflected light, for example, glare evaluation, assessment of lighting installations, characterisation of displays and many more. They are complex devices based on microelectronic pixel matrix sensors as a key-enabling technology combined with an imaging optic. ILMDs are based on the technology of digital cameras, that are omnipresent these days and deliver remarkable image quality. Their quality criteria, however, target visual perception of the images by humans via structural interpretation of relative signals in the image and are often related to artwork. In contrast to this, ILMDs focus on quantitative luminance measurements.

Results of independent measurements are metrologically comparable when they are metrologically traceable to the same measurement unit. Among other things, this traceability consists of assigning uncertainty statements to the measurement results. Developing guidelines on the determination or estimation of measurement uncertainties for luminance distributions measured with an ILMD was an objective of the normative joint research project EMPIR ‘19NRM02 RevStdLED’, ¹ which resulted in a good practice guide. ² The target audience were testing laboratories and other users that use ILMDs to characterise light emitting products or scenes and are required to provide traceable measurement results.

This paper will present and explain the general approach that results from the project’s research, discuss the principle of the required characterisations of an ILMD and show how to get to statements on measurement uncertainty.

2. Operating principle and sources of errors

For each pixel of the image, the hardware of an ILMD implements an imaging of a scene element onto the pixel, resulting in a pixel signal that corresponds to the average luminance at that scene element. The subsequent control software implements the inversion of the imaging process to derive a measured quantity value from the pixel signal that best matches the corresponding original scene luminance. On the complex optical and electronic signal path, many effects can distort the intermediate signals. To illustrate its complexity, this section will give a short summary of effects that may occur for a static scene, without claiming completeness.

The first part of the signal path is the optical path from the imaged area in the focus plane to the sensor pixel to which the light gets directed by the objective lens. The effective solid angle, and therefore the captured luminous flux, is limited by the aperture of the lens and is different between the sensor pixels. The stability and repeatability of the aperture influence the luminous flux when changing configurations (e.g. switching to a different aperture and back).

Typically, a planar $V(\lambda )$ -filter is placed inside the optical path, that is usually, between the lens and sensor, whose spectral transmission matches, together with the spectral responsivity of the sensor pixel itself, the aspired overall relative spectral responsivity $s_{\mathrm{rel}}\left(\lambda \right)$ of the ILMD to the $V(\lambda )$ -function, the spectral luminous efficiency for photopic vision. Because this matching filter in practice does not realise a perfect fit to the desired relative spectral responsivity, a spectral mismatch with respect to $V(\lambda )$ will remain. The filter’s characteristic typically has a significant temperature dependence, and the effective optical path length through the filter will be different for different pixels, which leads to a systematic difference of the spectral mismatch between the sensor pixels. A similar effect can be expected when additional neutral density filters are present in the optical path.

Modern pixel sensors often include an array of microlenses; a microlens in front of each pixel increases its responsivity by focussing the incident light into the photosensitive volume. The level of this increase depends on the pixel design, the shape and alignment of the microlens, and the direction of the incident light.

The correspondence of an ideal imaging between an area element of the scene in the ILMD’s focus plane and the pixel area gets disturbed by different effects like optical aberrations, reflections from optical boundaries and diffraction at the ILMD aperture that can be subsumed as stray light. Most of these effects relate to light originating from inside the measurement field, but there is also internal stray light from areas outside of the measurement field whose luminance does not get imaged directly onto the pixels. Instead, the light gets directed to outside of the sensor area or onto inner structures of the lens housing, and then gets further reflected or diffusely scattered multiple times until it reaches the pixel sensor. Even for light from the measurement field reflections can occur on the surfaces of the individual lenses of an objective, that lead to stray light, for example, in the form of ‘ghost’-images. For lenses having a variable focus, all these undesired signal contributions will depend on the focus setting.

The light flux that reaches the pixel and gets absorbed results in a charge generation rate inside the semiconductor. A huge fraction of these charges gets collected and accumulated in the pixel’s photodiode between its reset and readout phase. The charge collection is affected by the photodiode's drift voltage (that relates to the momentary fill level) to a degree that can vary with the wavelength of the incident photons. This can lead to an effect that resembles a spectral-dependent non-linearity of the pixel.

During the integration time, when the charges are accumulated, some charges undergo a recombination by the diffusion (forward) current through the pixel diode. For very long integration times, this may lead to a significant non-linearity of the pixel. The pixel also collects unwanted thermally generated charges, representing a reverse leakage current, during the integration time. The accuracy of the realisation of specific integration/exposure times directly influences the result.

Subsequent to the integration time interval, the accumulated charges are read out and converted into a voltage which is then converted into a digital value. For CCD sensors, the number of collected charges may be influenced by losses or additionally generated charges (smear effect) in the transfer registers. All these steps of readout, amplification and AD-conversion add some systematic non-linearity to the system and add noise to the signal.

After all these steps, the firmware or control software of the ILMD applies corrections for some effects to the pixel signal and inverts this imaging process to provide a luminance image. This is done using an adjustment factor that is determined by the manufacturer measuring a calibrated luminance source. Therefore, the uncertainty of this standard itself and this measurement of the standard jointly give an uncertainty contribution for all measurement results obtained with the ILMD used as provided by the manufacturer.

In some cases, post-processing is applied to the luminance image after all the steps mentioned above, such as composite images (averaging or high dynamic range), ³ smoothing across pixels, etc., which, if directly implemented in the ILMD or its control software, can be considered as a special operation mode. This adds a layer of complexity that is not discussed in the following sections but can be dealt with similarly. In the following sections, it is assumed that no further post-processing, by means of corrections or composites by the user, is applied, and only the manufacturer-configured adjustment is active.

3. Initial adjustment by the manufacturer

3.1 Overall approach

The manufacturer of an ILMD has to assess which of the internal effects are significant for the chosen hardware and for the intended measurement tasks. They have to design and integrate a model of evaluation that implements the transformation of the sensor signal into a luminance signal, under consideration of internal configuration settings and data supplied by the operator of the ILMD. Part of this model are proprietary corrections for most significant insufficiencies of the hardware by means of significant systematic errors (signal distortions) and therefore reduce corresponding measurement errors in the ILMD luminance signal.

This modelling for correcting the internal effects is usually more phenomenological than a detailed physical model, while the level of detail and the types of correction depend on the external needs, which are typically determined by a cost–benefit analysis. Indeed, only limited information is available from the datasheets, especially for the sensor and its analogue front end, even for manufacturers of ILMDs. The provided information is sufficient to implement the access to the functions of the sensor, but no details are known about its internal operation, which prevents detailed modelling.

For each ILMD, the model needs to be parametrised by an individual adjustment procedure. The characterisation for this adjustment is a complex task that requires a suitable setup and access to internal components, and takes a significant amount of effort. Beside the complexity of the model, the actual effort required depends on the number of different configurations that need to be characterised.

However, it has to be noted that often neither the proprietary correction nor an associated uncertainty are reported, so this can therefore only be considered a proprietary adjustment rather than a metrologically traceable correction.

3.2 Internal correction of systematic effects

In Section 2, different sources for errors are listed. If the aim is to model these effects by functions that can be used to correct the signal and to determine the function or its parameters, as well as the dependencies on the individual configuration by means of dynamic quantities, then all quantities on which these effects depend, must be known and be accessible. This can be a limiting factor for the level of detail and complexity of the implemented corrections.

Some sources of errors depend on known internal quantities (i.e. device parameters and configuration) or quantities that can be estimated during measurement. These are candidates to be corrected for. Other sources of error depend on environmental conditions or the scene to be measured, which are not known during the measurement. Here, an intrinsic correction is difficult or impossible.

Some corrections may be possible in general but require significant computational effort. The temporal stability of the device’s properties is also a limiting factor for the level of detail. The manufacturer has to, therefore, select an internal model and apply corrections that balance the effort during characterisation and application, and the long-term benefit that can be obtained. The weighting of these boundary conditions may depend on the targeted measurement task. An ILMD that is to be used in a very specialised setup, for example, a near-field goniophotometer, can be optimised to this measurement task, but a general-purpose ILMD, which has to support a wider range of applications (which necessitate the use of different lenses, variable focus, different/more filters, range of relative spectral distributions, etc.), requires other design criteria.

As a consequence, even if internal corrections are applied, the devices will still show residual systematic deviations that lead to measurement uncertainties depending explicitly on the extent and the quality of the internal corrections.

4. Issues in estimating the measurement uncertainty

A correction of all significant systematic effects is required to assess the uncertainty in measurement (Section 3.2.4 of the ‘Guide to the expression of uncertainty in measurement’ (GUM)), ⁴ and only stochastic components are considered to contribute to the measurement uncertainty (further details are in CIE 198:2011). ⁵ To this end, the user of an ILMD may first start to identify effects that influence the result and functions that are suitable to describe these effects, and then determine their parameters, for example by a series of measurements. Information on the effects to be considered may come from literature such as the normative joint research project EMPIR ‘19NRM02 RevStdLED’ ¹ or technical documents such as CIE 244:2021. ⁶ These functions need to cover the whole parameter space that follows from the measurement conditions at which the ILMD will be used. Later, at their application to measurements by a model of evaluation, these functions give correction values relating to each effect.

This approach seems straightforward: (1) invest effort for characterisation of an ILMD; (2) later use these functions to eliminate systematic errors; (3) then use the residuals from the determination of the functions as information on the remaining stochastic errors; and (4) sum up to the measurement uncertainty. However, its implementation is not generally viable when the aim is to reduce systematic errors, due to the following issues.

The user has to work with limited knowledge about the internal properties and configuration of the ILMD. Manufacturers usually give only very limited information about them, particularly, not about the proprietary internal corrections/adjustments. The operator is supposed to use the device ‘as is’ as a black box delivering luminance images. This means that the user’s assumptions about properties and behaviour contradict the intended use as a black box. Especially, directly transferring correction methods used with devices with lower complexity, for example, photometers, to devices with larger complexity, like ILMDs, will be problematic.

The correction functions determined are only valid for the internal configuration active during the characterisation for their parametrisation. A change of the internal configuration might drastically change the ILMD’s properties. For example, if the device can alter internal gain settings automatically, the corrections determined for non-linearity effects will not be correct for configurations with other gain settings.

Another issue is that it is very challenging to stimulate the system so that the change of the acquired signal (luminance value) can be attributed to only one specific influential effect. However, this selective stimulation is required to determine the associated characteristic without considering other cross-sensitive effects. If each influential effect cannot be handled independently, the parameter space to be sampled gets exceedingly large. A change of the scene/stimulus will usually affect multiple effects simultaneously. The significance of this issue increases because the manufacturer has already corrected some effects. This means that, to the user, only residual errors of these effects are visible. If the residual systematic measurement errors are determined, the uncertainty of this correction and the remaining stochastic components need to be quantified. This characterisation might be possible for some devices and effects, but the chances are high that another systematic error will supersede the so-far determined one.

For example, a single-point calibration with a known standard could be done to determine an adjustment correction factor $c_{\mathrm{adj}}=L_{\mathrm{ref}}/L_{\mathrm{meas}}$ that is intended to correct future measurements to $L_{\mathrm{corr}}=c_{\mathrm{adj}}·L_{\mathrm{meas}}$ . However, this measurement can be distorted by effects other than a change in the average luminous responsivity, such as:

• Non-linearity correction. It is not generally known if a non-linearity correction is implemented inside the ILMD. The adjustment factor $k_{\mathrm{adj}}$ that converts the pixel's (partially corrected) count rate y to a luminance signal ( $L=y·k_{\mathrm{adj}}$ ) is determined by the manufacturer at a specific integration time and count signal, intrinsically normalising the non-linearity to this operation point. Determining the adjustment correction factor at a different operation point (count signal) and applying it for correction will normalise the non-linearity characteristics to this new operating point by shifting it for all other operating points. Therefore, one error gets replaced by another.

• Size-of-source effect. During determination of the adjustment factor $k_{\mathrm{adj}}$ by the manufacturer, a calibrated luminance source is measured. This has a specific size in the image and the measurement region has a specific fraction of this size. If a different configuration is used for the determination of the adjustment correction factor, for example, by utilising a luminance source of a different size, the influence of the size-of-source effect will be different to the manufacturer’s adjustment. The resulting error will be transferred to future measurements.

The stability of the device properties also influences the stability of the corrections. A prerequisite for achieving state-of-the-art small uncertainties is to use an ILMD configuration (resolution, measurement field, luminous responsivity) that the manufacturer recommends for the measurement application. Furthermore, it is essential that the parameter set used for internal corrections in the software (i.e. lens, focus, neutral density filter) corresponds to the selected hardware configuration.

5. Proposed approach to estimate measurement uncertainty

According to the International Vocabulary of Metrology (VIM), ⁷ ‘measurement uncertainty’ is a ‘non-negative parameter characterising the dispersion of the quantity values being attributed to a measurand, based on the information used’. In our view, the user of an ILMD has to work with very limited information about the device, as it is not the task of the user to reverse-engineer the internals of an ILMD.

Based on the issues with the determination of correction functions, we generally propose for users of ILMDs to refrain from trying to determine and correct remaining errors of the device by an additional subsequent correction. This suggestion of a reduced approach is motivated by the issue that an invested effort for another subsequent correction will probably not lead to a reduction of the systematic error but only a distortion/shift of it.

Instead of trying to carry out further corrections, it should be assumed that all significant systematic effects are already reasonably characterised and corrected by the manufacturer. The remaining systematic errors are taken as non-significant by definition and therefore do not need to be corrected by the user but instead can be considered through uncertainty contributions associated with the already applied internal corrections.

Another argument to support this as a valid and reasonable assumption is that it is exactly the task of the manufacturer to implement a development and manufacturing process for the device that reduces all significant internal effects to a level where they are sufficiently small for the targeted applications. In case of ILMDs, this motivates for buying an ILMD instead of just a consumer camera or industrial camera. What constitutes significant effects and necessary corrections is a matter of discretion by the manufacturer. If a user concludes, for whatever reasons, that not all significant internal errors are corrected sufficiently or not at all, then the user has not chosen an adequate device for the intended measurement task. Then a device that is optimised by the manufacturer for the intended measurement task might be preferable instead of a general-purpose one.

The fact that the manufacturers do not themselves determine functions for a second or even third layer of corrections for a device gives strong evidence that just applying another layer of corrections by the user will probably not be successful, especially if knowledge on its internals and an already included level of corrections is missing.

The approach of not applying further corrections is in concordance with the conclusions of Klauenberg et al., ⁸ who highlight the metrological issues that arise from not correcting for systematic effects, because there, known significant systematic effects are discussed. For any given application it will therefore often be sufficient to estimate measurement uncertainty contributions for an expected variability range of the measurement conditions. For similar measurements of this kind, where the measurement conditions stay within the considered variability range, the estimated measurement uncertainty contributions might be reused, rather than needing to be determined for every measurement.

The proprietary adjustment procedure for an ILMD is usually not described and depends on the purchased options or configuration. However, relying on device specifications, which are affected by potentially significant internal corrections (rather than just the hardware design), is considered too limited to support the correct estimation of uncertainty contributions with respect to a specific application. In addition, the reliability of such device specifications as provided by the manufacturer is difficult to judge as they often lack assigned uncertainty or tolerance statements. Therefore, verifications by means of specific characterisations performed by the user themself are needed to reliably estimate uncertainty contributions and thereby establish traceability.

For a single luminance measurement, the measurand $L$ is given by the model of evaluation as shown in Equation (1):

$$L=Y·c_{\mathrm{a}}·c_{\mathrm{b}}·\dots$$

(1)

where $Y$ is the device’s luminance signal reading, averaged over an evaluation region, and $c_{\mathrm{a}},c_{\mathrm{b}}$ , … are correction factors for errors a, b, …; with all $c_i\equiv 1$ (no additional correction applied to the luminance reading) but with assigned uncertainty contributions $u_{\mathrm{a}},u_{\mathrm{b}},\dots$ (Note that additive corrections can be transformed into multiplicative form: $c_{\mathrm{mult}}=1+\frac{c_{\mathrm{add}}}{Y}$ . Not applying correction then means $c_{\mathrm{add}}=0$ and $c_{\mathrm{mult}}=1$ )

The effort of the user should focus on the identification and estimation of the remaining uncertainty contributions $u_i$ resulting from different effects. To do this, for the possibly relevant effects, the maximum relative errors ${ϵ}_{\max ,i}$ that can be attributed to these effects need to be determined. With this, a uniform distribution $\mathcal{U}(1,{ϵ}_{\max ,i})$ can be defined to describe the distribution of the residual errors for each effect $i$ expected in the future. For an ILMD, the determination of each ${ϵ}_{\max ,i}$ is done by certain relative measurements with an appropriate change of the measured scene or device configuration to trigger mostly just one of the effects. Here, the issue still exists that it is not easy to stimulate only one effect in an isolated way. This will also influence the determination of ${ϵ}_{\max ,i},$ but this cross-sensitivity is considered not to be significant for just estimating the uncertainty contributions rather than a correction.

In theory, the user could use a single correction factor in the model of evaluation of Equation (1) and scan the whole parameter space while measuring a calibrated luminance source to find the overall maximum error ${ϵ}_{\max }$ . In practice, however, this is not feasible: due to the huge parameter space, many measurements would be required to sample the residual error sufficiently densely to detect a local maximum. Splitting the error effects in mostly independent parts can massively reduce this effort. To trigger a specific effect with a measurement still requires assumptions on the inner operation principle, that is, blocks of equivalent circuits/models with the necessary level of details, including possible corrections implemented by the manufacturer. From this, the user can derive assumptions regarding which measurement conditions typically produce the largest errors. These assumptions help to reduce the number of measurements required to properly characterise residual errors.

To check if these assumptions are valid, the user must also measure a few other configurations that are randomly distributed. With respect to available experience and reports, a test or cross-check on a general-purpose parameter field might be sufficient to evaluate the uncertainty contributions relevant in a specific application, that is, limited to a reduced parameter range. If the maximum error is not found at the expected configurations, then the density of measurement configurations needs to be increased. The case in which the maximum error is not measured is accepted as residual risk of this approach.

A good source of information on relevant sources of error, their parameter dependencies and measurement methods for their characterisation is the Technical Report CIE 244:2021 ‘Characterization of Imaging Luminance Measurement Devices (ILMDs)’. ⁶ The quality indices introduced there are designed to characterise different error sources and quantify the device’s performance regarding these error sources. They are designed to compare devices. The quality indices are explicitly not usable to correct measurement results or to estimate the measurement uncertainty for a specific measurement because their definitions are in general related to very specific standardised measurement situations and ILMD configurations that will in general not correspond to the measurement application for which uncertainties are to be estimated. If some quality indices are provided for a device, they are not necessarily determined in the most critical configuration nor the configuration used for a measurement.

6. Example for determination of error boundaries: Shading error and its focus dependence

From the effects briefly presented in Section 2, it can be deduced that the responsivity to luminance varies between the pixels of the ILMD and is non-uniform across the image. One part of this variation of responsivity, also called ‘shading’, is caused by the changing transmissivity of the optical path through the objective lens (lenses and aperture) onto the pixel matrix sensor. This effect leads to a decline of the pixel signal with increasing viewing angle and therefore with larger distance to the optical axis (image centre). Another contribution to the shading comes from the changing path length through the optical filters and other local variations or structures, for example, the pixel structure including the shape and alignment of its microlens.

To initially adjust an ILMD, the relation of the sensor signal (after some corrections) to the scene’s luminance needs to be determined. For this, one would ideally use a calibrated and uniform luminance source that fills the whole measurement field of the ILMD. This combination is in general not available nor affordable. Therefore, most often the adjustment is divided between the determination of the responsivity of the pixels relative to each other (i.e. the non-uniformity) and an absolute link to the SI unit for a small (usually central) pixel region. The relative responsivity can, for example, be determined by imaging into an integrating sphere.

Figure 1 shows horizontal profiles through the image centre of the responsivity relative to the image centre (shading) of the raw pixel signal and its dependence on the lens’ focus setting. In the vertical direction, this effect has a similar characteristic, but rotational symmetry to the image centre or optical axis cannot be assumed.

Figure 1

Focus dependence of the relative shading characteristic (non-uniformity of the responsivity relative to the image centre) across the image; examples can vary strongly between configurations and devices

The correction of this effect is often implemented by the manufacturer using one dataset that is most representative of this set of shading characteristics. From this, it follows that some residual shading error can be expected for other focus settings than the corrected one. Figure 2 shows an example of the measured residual relative responsivity (shading) errors of the luminance signal for horizontal and vertical profiles through the image centre at three different focus settings. It can be seen that the implemented shading correction will leave some residual error for focus settings that differ from the one used for adjusting the correction. The critical points to test for this effect are the image corners relative to the image centre.

Figure 2

Residual relative shading error in the horizontal and vertical directions for different focus settings, with respect to the value at the image centre with a device-internal shading correction corresponding to an intermediate focus scale value of 15

In summary, the maximum relative errors can be estimated for each ILMD–lens combination by measuring the average luminance of a small spot source which is imaged successively at the centre, and near the corners, of the measurement field, with the measurements being repeated for different focus distances. Figure 3 shows these combined five locations of the small circular luminance source and the evaluation regions. To ensure that the source luminance is the same for all measurements, the changes of the location of the source on the sensor should be carried out by rotating the ILMD around its projection centre rather than by lateral or vertical movement. By doing this, the measurement direction with respect to the source is kept nearly constant, as opposed to the change of direction resulting from a translation of the ILMD. Especially for small focus distances and a large measurement field angle, the use of a nodal point adapter can be useful to realise this rotation.

Figure 3

Evaluation regions (dashed circles) of homogenous light source (white spot) achieved by five rotations around the ILMD projection centre

These measurements give one value for the centre $Y_{\mathrm{centre}}$ and multiple values $Y_i$ for the corners. Additional locations to measure $Y_i$ are possible (especially at the diagonals, see also definition of quality index $f_{22}$ in CIE244). ⁶ This series of single-point measurements needs to be repeated for different focus settings and corresponding source distances $k$ . The focus distances should cover the smallest and largest possible distances that are limited by the objective lens (where the largest errors are expected) or should cover the intended measurement distances and a few distances and locations in between to ensure that the assumptions on assessing the critical measurement configurations hold. For each distance, the correct focus setting of the objective lens also needs to be set in the ILMD control software. During the variation of the distances, the imaged size of the source needs to be held in a similar range in terms of pixels to ensure a sufficient local resolution. This can be achieved by using different sources or placing apertures in front of the source during the measurements at shorter distances.

The maximum error is then given by Equation (2):

$${ϵ}_{\max ,\mathrm{shading}}=\max \left({\left|\frac{Y_{\mathrm{centre}}-Y_i}{Y_{\mathrm{centre}}}\right|}_k\right)$$

(2)

For this shading example, the central measurement value $Y_{\mathrm{centre}}$ acts as a reference point where the lowest or indeed no residual error can be expected. For some effects, the choice of the reference point is ambiguous because the corresponding choice during the adjustment by the manufacturer is not known. This choice can also be implicit by a specific measurement condition during the initial adjustment. The measurements will give a set of values between $Y_{\min }$ and $Y_{\max }$ . Then possible choices for the reference point can be the maximum (or minimum), the average over all measured points $\overline{Y}$ , or the centre between minimum and maximum, which can easily change the value of ${ϵ}_{\max }$ by a factor of 2. This degree of freedom is left to the user.

It is evident that this characterisation requires some effort, but it is strongly reduced compared to a detailed determination of residual error functions. For most effects, a characterisation procedure can be designed that can be performed by the operator of the ILMD and does not require very special and expensive equipment.

However, an exception to this is the estimation of the spectral mismatch error. Its calculation is equivalent to that of the spectral mismatch correction factor of photometers ⁹ and requires knowledge of the spectral responsivity of the ILMD and the relative spectral distribution of the measured source. The determination of the spectral responsivity of ILMDs is beyond the capabilities of most users. Instead, the user has to rely on information provided by the manufacturer that may be based on its characterisation for the individual device or on typical data of what to expect, resulting from the design of the $V(\lambda )$ -filter. For white solid state lighting products (phosphor-type and RGB-type white LED products), an estimation of the uncertainty accounting for the spectral mismatch might be made based on a specification of the general V(λ) mismatch index $f_1^{\prime }$ , as described in Section C.3.5 of CIE S025:2015; ¹⁰ however the investigation presented there only covers illuminance photometers, and it is unclear whether these characteristics are also representative for ILMDs. The relative spectral distribution of the source can be measured or obtained from datasheets. Even if this information is given without uncertainty statements, it can be used for a first-order estimation of the spectral mismatch error.

The spectral mismatch error may in some cases be the largest effect, but it is still a residual error because the main correction is achieved by the $V(\lambda )$ -filter. Although the spectral mismatch error is calculated similarly to the spectral mismatch correction factor for photometers, it should not be used for correction because the spectral responsivity data are only given for one operating condition (e.g. sensor region, temperature, focus setting, individual device vs. class of devices).

Exemplary procedures for the characterisation of different uncertainty contributions can be found in a good practice guide available on the website of project EMPIR ‘19NRM02 RevStdLED’. ²

7. Measurement uncertainty and correlations

The determined maximum relative errors ${ϵ}_{\max ,i}$ can then be used to derive the overall standard uncertainty. For each effect $i$ , the resulting relative standard deviation for a uniform distribution in the interval $[1-{ϵ}_{\max ,i},1+{ϵ}_{\max ,i}]$ can be determined according to clause 4.3.7 of the GUM ⁴ using Equation (3):

$${\sigma }_{\mathrm{rel},i}=u_{\mathrm{rel},i}=\frac{2{ϵ}_{\max ,i}}{\sqrt{12}}=\frac{{ϵ}_{\max ,i}}{\sqrt{3}}$$

(3)

For an effect that scales the transfer function of the pixels by a single factor (as in the measurement model above, Equation (1)), the uncertainty contribution is best expressed as a relative contribution $u_{\mathrm{rel}}$ . This can be transferred to the absolute uncertainty contribution by multiplying it with the output quantity $Y$ (ILMD luminance signal) related to the pixel (or the evaluation region). For additive effects, this is not helpful because the sensitivity of the output quantity to the uncertainty of the input quantity, and therefore the value of the absolute uncertainty contribution $u_{\mathrm{abs}}$ , does not just scale with the pixel signal itself. It can also depend on the signal of other pixels or on the luminance of regions completely out of the measurement field. Such additive effects have to be estimated during the measurements for the specific device configuration and scene, for example, the straylight floor (offset) caused by surrounding luminance.

In this approach, all uncertainty contributions of a single absolute luminance measurement $L$ are treated as uncorrelated for the sake of simplicity. Then, the overall standard uncertainty for a single luminance measurement according to Equation (1) is given by Equation (4):

$$u_{\mathrm{abs}}(L)=\sqrt{Y^2(u_{\mathrm{rel},1}^2+u_{\mathrm{rel},2}^2+\dots )+u_{\mathrm{abs},1}^2+\dots }$$

(4)

When derived quantities must be calculated from multiple luminance measurements, for example, a luminance ratio or a difference of luminance values, the model of evaluation is given by the equation of this derived quantity, for example, Equation (5) for the luminance ratio:

$$R_L=\frac{L_1}{L_2}=\frac{Y_1·c_{\mathrm{a},1}·c_{\mathrm{b},1}·\dots }{Y_2·c_{\mathrm{a},2}·c_{\mathrm{b},2}·\dots }$$

(5)

or Equation (6) for the luminance difference:

$$\begin{array}{c}\multicolumn{1}{c}{D_{\mathrm{L}}=L_1-L_2=(Y_1·c_{\mathrm{a},1}·c_{\mathrm{b},1}·\dots )}\\ \multicolumn{1}{c}{-(Y_2·c_{\mathrm{a},2}·c_{\mathrm{b},2}·\dots )}\end{array}$$

(6)

Equations (5) and (6) serve just as simple examples of how to transform the model of evaluation into a form that users can apply to uncertainty propagation. The model of evaluation can, of course, be more complex, but the general structure will be the same: according to Equation (1), the luminances $L_i$ will be replaced by the product of the constant signal reading $Y_i$ and the correction factors $c_{*,i}$ that have the expectation value of 1 but carry the uncertainty. The correction factors are the variables for which the partial derivatives have to be calculated for analytical error propagation ⁴ or random samples have to be drawn from the given distribution (here uniform distribution) for the Monte-Carlo method. ¹¹ This principle is not limited to these two examples. It can be applied to all models of evaluation that contain luminance measurements.

Equation (5) might be wrongly identified as the substitution method. But just because a ratio is calculated, this is not necessarily equivalent to a substitution method, where most uncertainty contributions cancel out. The substitution method is a very special case where both measurements are performed with the same device under nearly the same measurement conditions. It requires that both measurement objects are very similar but does not require an absolute calibrated ILMD. The other extreme case is when the luminances $L_1$ and $L_2$ are measured with different devices for different sources. Here, all uncertainty contributions are treated as uncorrelated between the measurements. For the intermediate cases where luminance measurements are carried out with the same device, the errors related to a specific configuration can occur in each of them and statements on full correlations between some of the uncertainty contributions associated with these measurements can be made, to be used in the calculation of the combined uncertainty. In this case, if the critical measurement conditions for the individual measurements are the same, then the corresponding uncertainty contributions $u_{*,1|2\dots }$ are fully correlated. For example, the uncertainty of the initial adjustment factor $u_{\mathrm{adj}}$ is fully correlated between all measurements with the same device. If the measurement regions for successive measurements are identical (or at a very similar region of the image) and the focus setting is identical, then the uncertainty contribution caused by the shading error is likewise fully correlated. For measurement regions near the image centre, this is also true for different focus settings. Measurement regions at different locations in the image have to be treated as uncorrelated regarding the shading error.

However, partial correlations cannot be derived by comparing measurement configurations. This would require detailed determination of the systematic residual errors, which might each be further split up into fully correlated and uncorrelated contributions with respect to the manufacturer’s underlying corrections and adjustments. On the other hand, using the knowledge of which critical measurement conditions are identical between different evaluation regions in one or multiple luminance images, a correlation matrix can be created to hold the correlation information in a standardised way, as we describe in the following example.

We assume for this example that a measurement of the average luminance of two sources of the same type are taken in one image, with one evaluation region in the centre and one near the corner. For this situation, the calibration uncertainty $u_{\mathrm{adj}}$ (associated to the global adjustment factor $k_{\mathrm{adj}}$ ), a residual shading uncertainty $u_{\mathrm{shading}}$ and a residual spectral non-linearity $u_{\mathrm{nl}}$ were identified as significant uncertainty contributions, which are labelled here as $u_{\mathrm{a}}$ , $u_{\mathrm{b}}$ and $u_{\mathrm{c}}$ for the sake of simplicity. The suffix 1 or 2 is used to differentiate between the light sources to be measured. Furthermore, because both measurements are obtained with the same device, $u_{\mathrm{a},1}$ and $u_{\mathrm{a},2}$ are fully correlated. Because both measurements are obtained at different locations in the image, there is no full correlation for the shading errors, and we consider $u_{\mathrm{b},1}$ and $u_{\mathrm{b},2}$ as fully uncorrelated to each other. Both sources are the same type and therefore have the same spectral distribution and a similar luminance. Therefore, the sensor will give similar count signals for both regions (where differences due to different light paths through the filter package are neglected here). This means that $u_{\mathrm{c},1}$ and $u_{\mathrm{c},2}$ are treated as fully correlated. This information can be rendered in a correlation matrix $P$ by setting the corresponding non-diagonal elements to 1, as shown in Equation (7) (note that elements outside the brackets are row/column captions and not matrix content).

$$\begin{array}{l}P=u_{a,1}{u}_{b,1}{u}_{c,1}{u}_{a,2}{u}_{b,2}{u}_{c,2}\\ \begin{array}{c}{u}_{a,1}\\ u_{b,1}\\ u_{c,1}\\ u_{a,2}\\ u_{b,2}\\ u_{c,2}\end{array}\left[\begin{array}{cccccc}1& 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 1\\ 1& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 1\end{array}\right]\end{array}$$

(7)

With the diagonal matrix of the input uncertainties given by Equation (8):

$$\begin{array}{l}D=\mathrm{diag}(u_{a,1},u_{b,1},u_{c,1},u_{a,2},u_{b,2},u_{c,2})\\ =\left[\begin{array}{cccccc}{u}_{a,1}& 0& 0& 0& 0& 0\\ 0& u_{b,1}& 0& 0& 0& 0\\ 0& 0& u_{c,1}& 0& 0& 0\\ 0& 0& 0& u_{a,2}& 0& 0\\ 0& 0& 0& 0& u_{b,2}& 0\\ 0& 0& 0& 0& 0& u_{c,2}\end{array}\right]\end{array}$$

(8)

the covariance matrix $\mathbf{\Sigma }$ is then given by Equation (9):

$$\mathbf{\Sigma }\mathbf{=}\mathit{D}\mathbf{P}\mathbf{D}$$

(9)

This matrix then can be used for uncertainty propagation according to the GUM^4,11,12 with consideration of correlation.

8. Summary

To establish traceability for ILMD-based measurements, uncertainty statements have to be assigned. The common approach, which is to characterise systematic effects and then correct for them, leads to issues caused by the inner complexity of ILMDs and the limited information about device internals, that is, proprietary adjustments implemented by the manufacturer. These issues have been described, and we conclude that the best approach for most users will be to assume that all significant errors are already corrected in an ILMD by internal corrections and only residual errors remain. These errors have to be estimated by searching for the maximum relative errors, for each different effect within the parameter range for the targeted application. These measured maximum errors set the boundaries for a uniform distribution used to estimate future uncertainty contributions in measurements in this specific application/configuration. The handling of correlations between multiple luminance measurements that are evaluated in more complex measurement models have been discussed. The chosen approach balances the need to state uncertainties for measurements and the required effort for the characterisation to be performed by the user.