Lambertian reflection is a common assumption in eliminating substitution errors for an integrating sphere [
15,
16]. In Stage I, we also employed this assumption in modeling the lighting deviation caused by the imaged samples. We used a white board, which is hereafter referred to as the white standard or simply the standard, to correct the spatial non-uniformity. We further employed a small white patch to model the change of lighting intensity caused by the imaged sample.
Figure 2 shows the layout of the reference white patch and a sample. The size of the white patch was much smaller than that of the sample.
3.1. Using the White Standard to Correct Spatial Non-Uniformity
We denote the radiant flux entering the sphere in the wavelength range of
by
, where
is a quite small interval. Due to the existence of the diffuser and baffles, the flux can be assumed to strike the sphere wall uniformly. For the Lambertian sphere wall whose spectral reflectance is
, the flux in its first reflection is:
Among the flux reflected by the sphere wall, the percentages of the part striking the sphere wall and the part incident on the sample holder are denoted by and , respectively. Due to the existence of the entrance port and detector port, we have .
For calibration, the white standard was placed on the sample holder to acquire the spatial distribution of the lighting. The standard had the same size as the sample and was much larger than the white patch illustrated in
Figure 2. The reflectance of the standard,
, is also referred to as average reflectance. The flux in the first reflection of the standard under the integrating sphere lighting is computed as:
A part of the flux reflected by the standard strikes the sphere wall, at a percentage of
. The flux in the second reflection of the sphere wall can then be computed as:
where
.
The flux in the second reflection of the standard is:
Then, the flux in the sphere wall’s third reflection is computed as:
The flux in the reflection of the sphere wall, , and that of the standard, , can be computed in a similar manner.
We first consider the imaging model of the white standard. For a pixel position
in the standard, we denote its spectral reflectance by
. A part of the flux reflected by the standard at
reaches the image sensor, at a percentage of
. Note that
is spatially varying, due to the spatial non-uniformity as mentioned in
Section 2. The intensity of the pixel
in the captured image, i.e., the camera response, is then computed as:
where
is the factor between the detected flux and camera response,
N is the number of corresponding pixels of the standard, and
is only determined by equipment specifications. Note that we subtract the dark current from the camera response in practical measurements. Since
and
, we have
. Equation (
6) can be further computed as:
Then, we consider the imaging model of a Lambertian sample whose average reflectance is
. Assume that the radiant flux entering the sphere becomes
due to the possible fluctuation of the lamp irradiance. The camera response at
becomes:
where
is the sample reflectance at
, and
is computed as:
The background variation, which corresponds to the different reflectances
of samples, leads to the change of
. Besides, the illuminant fluctuation introduces a time-variant
. The lighting deviation in both cases influences the camera response
of the sample, according to Equation (
8).
The camera response of the sample
can be normalized with respect to the response of the white standard at
,
where:
Note that varies with the lighting deviation just mentioned, due to the change of and at the time of measurement.
Thanks to the uniformity of the white standard, we actually have
. By using the white standard, we eliminate the spatially-varying term
from camera response
and thus correct the spatial non-uniformity in Equation (
10). However, the normalized camera response
still suffers from the lighting deviation, due to the existence of term
.
3.2. Using the White Patch to Correct Lighting Deviation
We correct the lighting deviation using a reference white patch as illustrated in
Figure 2. Thanks to the relatively large field of view of the imaging system, the camera responses of both the sample and reference white patch can be measured simultaneously. Let
be the spectral reflectance of the white patch at pixel
, its camera response when acquiring the image of the standard is:
and its camera response when acquiring the image of the sample is:
Our aim is to characterize lighting deviation when imaging various samples; thus, we normalize
with respect to
, yielding:
which has exactly the same form of
. We introduce a white-patch ratio defined as:
Then, the normalized camera response of the sample in Equation (
10) can be further normalized, yielding:
When compared with Equation (
10), it is observed that in Equation (
16),
is completely eliminated. We note that the white patch ratio actually characterizes the lighting change in the integrating sphere and can be regarded as another form of introducing a reference beam. Hence, lighting deviation can be corrected by employing a white patch for reference.
Based on the above theoretical analysis, we present the procedure of lighting correction using the white standard and white patch as follows. We first acquired the image of the white standard, obtaining the camera response
of the white standard at position
and the camera response
of the reference white patch at position
. Then, we acquired the image of a sample, getting the camera responses of the sample and white patch, which are respectively denoted as
and
. Based on the white patch ratio
, the camera response after the correction is computed as: