Linearisation of RGB Camera Responses for Quantitative

Image Analysis of Visible and UV Photography: A

Comparison of Two Techniques
Jair E. Garcia1,2*, Adrian G. Dyer2, Andrew D. Greentree1, Gale Spring1, Philip A. Wilksch1
1 School of Applied Sciences, RMIT University, Melbourne, Victoria, Australia, 2 School of Media and Communication, RMIT University, Melbourne, Victoria, Australia

Abstract
Linear camera responses are required for recovering the total amount of incident irradiance, quantitative image analysis,
spectral reconstruction from camera responses and characterisation of spectral sensitivity curves. Two commercially-
available digital cameras equipped with Bayer filter arrays and sensitive to visible and near-UV radiation were characterised
using biexponential and Bézier curves. Both methods successfully fitted the entire characteristic curve of the tested devices,
allowing for an accurate recovery of linear camera responses, particularly those corresponding to the middle of the
exposure range. Nevertheless the two methods differ in the nature of the required input parameters and the uncertainty
associated with the recovered linear camera responses obtained at the extreme ends of the exposure range. Here we
demonstrate the use of both methods for retrieving information about scene irradiance, describing and quantifying the
uncertainty involved in the estimation of linear camera responses.

Citation: Garcia JE, Dyer AG, Greentree AD, Spring G, Wilksch PA (2013) Linearisation of RGB Camera Responses for Quantitative Image Analysis of Visible and UV
Photography: A Comparison of Two Techniques. PLoS ONE 8(11): e79534. doi:10.1371/journal.pone.0079534
Editor: Christof Markus Aegerter, University of Zurich, Switzerland
Received August 1, 2013; Accepted September 30, 2013; Published November 18, 2013
Copyright: ß 2013 Garcia et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: Jair E. Garcia was partially supported by Colfuturo 200818772 (Colombia). A.G.D. was supported by Australian Research Council DP0878968/
DP0987989. A.D.G. acknowledges the Australian Research Council for financial support (Project No DP0880466). The funders had no role in study design, data
collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors declare that one of the authors, Adrian G. Dyer, is a PLOS ONE Editorial Board member. This does not alter their adherence to
all the PLOS ONE policies on sharing data and materials.
* E-mail: jirgarci@gmail.com

Introduction this relationship. Departures from linearity in the camera response

may be built into the camera’s hardware and software to satisfy
With recent advances in optical and digital technology, the several purposes, such as the historical practice of gamma
consumer-level digital camera has become a convenient and cost- correction, aesthetic and perceptual considerations relating to
effective instrument for acquiring images for quantitative analysis image display, and for increasing the dynamic range of the sensor
[1,2]. One major issue with using consumer-level cameras is [2,18]. Furthermore, the techniques employed by the camera
obtaining a linear response, which is a prerequisite for tasks such manufacturers are usually proprietary, and response curves are not
as deriving spectral sensitivity curves [3], spectral reconstruction generally available.
[4–9] and colorimetric evaluation [10]. Furthermore, quantitative Linear responses from consumer-level cameras can be recov-
analysis on images representing the linear sensor response has ered by fitting a function to a plot of camera response versus
applications in various biological studies including: characterisa- incident radiance, the Opto-Electronic Conversion Function curve
tion of animal colour patterns [11], and the evolution of signaller- (OECF), and subsequently inverting the fitting function via
receiver interactions through the analysis of the spectral compo- analytical or graphical methods, or look-up tables (LUTs) [19].
nent of images representing naturally-occurring visual signals [12]. Polynomial, power and exponential functions have been previ-
In particular, measurements with digital cameras can be of high ously suggested as fitting functions [20,21]. Nevertheless the
value for qualifying non-visible regions of the spectrum like the implementation of these functions does not guarantee an accurate
ultraviolet (UV) [13]. There are also new and emerging fit of the entire OECF curve for all camera models. For example,
applications of using digital images for quantifying subject matter. for cameras with extended dynamic or spectral ranges, the OECF
For example digital imaging can be useful for measuring the curve may present two distinct regions: linear and saturation
occurring turbidity of fluids for quantifying bacteria counts [14], separated by an ‘inflexion’ point corresponding to the amount of
measuring spectral information from different inorganic salts [15] energy required for activating the electron drainage mechanism
or in forensic applications for accurately documenting bite marks [22]. Consequently, there is no a priori reason to expect a particular
on skin through the use of the various penetration levels in
camera sensor to obey any specific analytical function for its
different wavebands of radiation [16]. Although digital cameras
OECF curve. For this reason, it is necessary to carry out
designed for technical purposes usually maintain the linear
measurements to find a function that is able to accurately fit the
relationship between the incident radiance and the camera
entire OECF curve if high quality quantifiable data is to be
response typical of most CCD and CMOS sensors [17],
recovered.
consumer-level digital camera models do not necessarily maintain

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 Linearisation of UV-Visible RGB Camera Responses

Here we compare the use of (parametric) cubic Bézier curves Materials and Methods
and biexponential functions for characterising two camera models:
(i) a Canon D40 camera sensitive to visible radiation and (ii) a Definitions
Nikon D70s camera modified for recording near-ultraviolet In an ideal system, the camera response at each pixel site of a
radiation. Although both methodologies allow the recovery of CCD or CMOS sensor is defined by the total number of
linear camera responses, they differ in the model assumptions, the photoelectrons generated by input radiance and the combined
interpretation of the recovered camera responses and the size of effect of the analogue to digital conversion, signal amplifiers and
the uncertainty bounds associated with the recovered responses. software balancing in the system. The response per pixel is [18]:
We compare performance using both methods and provide some
recommendations for selecting the appropriate method depending r gpe
on the intended use of the recovered linear responses. ~G( ), ð1Þ
rmax gmax

Figure 1. Cubic Bézier curves (dashed lines) and biexponential functions (solid lines) fitting the camera responses (circle markers)
making up the OECF curves for the red (a), green (b) and blue (c) colour channels of a Canon 40D digital camera and the red colour
channel of a Nikon D70s camera modified for ultraviolet recording (d). Exposure values corresponding to the total incident irradiance were
calculated from Equation (2). Values were normalised by dividing the total amount of irradiance required for each camera response (g) by the amount
of energy required to attain a camera response equal to the selected maximum pixel response rmax (gmax ). See text for details.
doi:10.1371/journal.pone.0079534.g001

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 Linearisation of UV-Visible RGB Camera Responses

where G is the OECF, which expresses the digital output r of a the replacement of the standard hot mirror filter by a Baader U
pixel as a function of gpe , the number of generated photoelectrons. filter (Company Seven, USA), cutting off radiation at wavelengths
The function is normalised such that the output reaches its longer than 398 nm, and adjusting the focusing point. The Canon
maximum value of rmax when gpe ~gmax . In the simplest case, camera was equipped with a 100 mm Electro-Focus (EF) lens
gmax is the maximum number of photoelectrons that can be stored (Canon Inc., Japan) fitted with a skylight filter (Hoya, Philippines).
in the electron well of the photoelement and rmax is the maximum The modified Nikon D70s camera was equipped with a Micro
output determined by the bit-depth of the converter. However, we Nikkor 105 mm quartz lens (Nikon Corporation, Japan) to ensure
found it to be an advantage to define these two constants to be a free transmission of near-ultraviolet radiation [24,25].
smaller than each of these two limits, about 250 intensity levels and
their corresponding exposure values as detailed in the Results Reconstruction of the OECF Curves
section, to avoid anomalous behaviour close to saturation of the We reconstructed OECF curves corresponding to the different
electron well i.e. clipping [2]. In any case we define G such that colour channels of each test camera by plotting the camera
r = rmax when gpe = gmax ; i.e., G(1)~1. response (r), in pixel intensity values, against signals of varying
The number of photoelectrons generated at each pixel site intensity calculated from Equation (2), following a protocol similar
depends on the scene radiance, the characteristics of the lens, the to the one specified by the ISO 14524:2009 standard [26]. Most
selected exposure parameters, the transmissive properties of the photographic lenses have a uniform spectral transmittance within
optics and the spectral sensitivity of the material making up the the 400{710 nm spectral interval [25]; therefore, TOp (l) in
sensor [18]: Equation (2) was treated as a constant for the calculations. The
same property characterises quartz optics in the 300{400 nm
ðl spectral interval [24,25] so the same procedure was implemented
b p : Lq (l)AD : for the calculations corresponding to the UV-sensitive channel of
gpe ~ Rq (l):TOp (l)tdl, ð2Þ
la 4 f 2 (1zM) the Nikon camera. The irradiation source was a xenon arc lamp
type VX150-1f-2b-L (Siemens, Germany) continuously emitting
where Lq (l) is the spectral radiance incident on the camera lens, radiation between 300{800 nm.
AD the effective detector area, f is the lens f -number, M is the The radiance of each signal was measured with an NIST
optical magnification, Rq (l) is the spectral sensitivity, TOp (l) is the traceable ILT-900 spectroradiometer (International Light Tech-
combined spectral transmittance of the lens and any hardware nologies, USA) equipped with a narrow acceptance-angle collector
filters (colour filters, polariser, hot mirror filter, etc.) and t is the (International Light Technologies, USA). Each radiance reading
integration time, set by the shutter speed [18]. The wavelength was the average of five different scans between 250 and 950 nm at
integration is carried out over the range for which Rq (l) is non- 1 nm intervals. Raw spectral radiance data were expressed as a
zero. photon flux (mmol:m{2 :s{1 :nm{1 :sr{1 ). Converted data were
subsequently binned at 5 nm intervals. Data corresponding to the
Camera Systems 395{710 nm interval were used for the characterisation of the
OECF curves were reconstructed for the three colour channels Canon camera, whilst 300{400 nm spectral data were used for
of a Canon 40D (Canon Inc, Japan) and the ‘red’ colour channel characterising the Nikon camera.
of a Nikon D70s (Nikon Corporation, Japan) modified for reflected Signals required to reconstruct the OECF curves of the Canon
ultraviolet image recording. By selecting these two cameras we camera were obtained by employing a set of four neutral density
ensured that the proposed methodology is applicable to different filters (Newport, USA) with nominal values of optical density (OD)
consumer-level cameras equipped with Bayer filter arrays inde- of 0.1, 0.2, 0.5 and 1.0. Additional densities of OD 0.3, 0.7, 1.2
pendently from their spectral sensitivity range. The ‘red’ channel and 1.5 were obtained by combining the filters. Filters were
of the Nikon D70s camera was selected as this shows the highest mounted on a holder located at 0.12 m from the xenon arc lamp.
sensitivity to near-ultraviolet radiation [23]. Camera modification The lamp output was projected through a baffle onto a glass
for ultraviolet recording was carried out by a professional camera diffuser screen (Edmund Optics, USA) placed on a filter holder
technician (Camera Clinic, Melbourne, Australia) and included positioned 0.46 m from the xenon lamp.

Table 1. Coefficients of biexponential functions fitting the OECF curves for two camera models.

Equation parameter Canon 40D Nikon D70s

Channel Channel

Coefficient statistics ‘red’ ‘green’ ‘blue’ ‘red’

b m 103634 85.5621 75.8619 248616

(pixel intensity level) 95% CI
c m 36406960 420061000 524061460 1540620
(mol21) 95% CI
d m 157633 172622 179620 12613
(pixel intensity level) 95% CI
g m 9816159 961699 10656110 14700627300
(mol21) 95% CI

Mean coefficients (m) and 95% confidence intervals (CI) of biexponential curves fitting the three colour channels of a Canon 40D camera and the red channel of a Nikon
D70s modified for reflected ultraviolet recording, rlim = 255 in all cases.
doi:10.1371/journal.pone.0079534.t001

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 Linearisation of UV-Visible RGB Camera Responses

Table 2. Coordinates and 95% confidence intervals for the four control points defining each Bézier curve fitting the OECF curves
for two camera models.

Channel Parameter P0 P1 P2 P3

‘Red’ g m 1.0961022 1.3161022 2.7461022 1.8761022

gmax 95% CI +1.9161024 +5.1961024 +1.0761023 +1.3061023
r m 2.2061021 2.7361021 8.6661021 9.6261021
rmax 95% CI +3.8561023 +6.1061023 +3.5561023 +1.0961023
‘Green’ g m 1.2161022 1.1861022 4.6861022 1.9661021
gmax 95% CI +2.4061024 +5.1561024 +9.3561024 +5.6161024
r m 2.2961021 3.0261021 8.3461021 9.4061021
rmax 95% CI +4.0361023 +6.0061023 +3.5161023 +5.4461024
‘Blue’ g m 1.2561022 1.4561022 5.4061022 2.3361021
gmax 95% CI +2.4961024 +8.4661024 +1.8661023 +1.6761023
r m 2.1861021 3.1161021 8.2861021 9.4261021
rmax 95% CI +3.6561023 +5.6161023 +3.2961023 +1.0561023
‘Red’-UV g m 2.0261022 3.7061022 7.0461022 2.3761021
gmax 95% CI +4.6161024 +1.5161023 +3.4961023 +3.2061023
r m 2.4061021 3.2161021 8.6561021 9.5861021
rmax 95% CI +4.6661023 +7.3461023 +3.8361023 +2.5261023

Mean coordinates (m) and 95% confidence intervals (CI) for the four control points defining cubic Bézier curves fitting the OECF curves reconstructed for the colour
channels of a Canon 40D camera and the ‘red’-UV channel of a modified Nikon D70s. Coordinates of the first and last control points correspond to the normalised
minimum and maximum camera responses included in the OECF and the normalised exposure required to obtain them. Exposure values (g) are expressed in m mol units
and camera responses (r) in normalised pixel intensity levels.
doi:10.1371/journal.pone.0079534.t002

A different approach was required for reconstructing the OECF IEC61966-2.1 colour space (Nikon camera). Raw image process-
curves for the modified Nikon D70s camera. Because of the low ing was performed employing the Camera Raw Plug-in v.6.7 for
near-UV irradiation transmittance of the neutral density filters and Photoshop CS5 (Adobe Incorporated, USA). Processed images
diffuser screen, the irradiation produced by the xenon arc lamp were subsequently encoded into uncompressed 8-bit TIFF files.
was projected onto five diffuse achromatic targets, each one Camera responses were calculated from the average pixel intensity
reflecting different amounts of incident irradiation, to obtain in a 50 times 50 pixel sample area located at the centre of each
signals of varying intensity. The achromatic targets were image. Sampling was performed on the TIFF files employing the
constructed by mixing barium sulphate with different proportions ImageJ processing software version 1.42q (National Institutes of
of activated charcoal following published protocols [27] yielding Health, USA) [28].
reflectance values of approximately 86, 60, 51, 15 and 2% for
incident near-UV irradiation, thus covering a wide range of Biexponential, Cubic Bézier Curve Fitting and
camera responses up to the saturation point. Spectral radiance Linearisation
readings were obtained after placing each calibration target
Biexponential and cubic Bézier curves were fitted to the OECF
0.25 m away from the xenon arc lamp and irradiating the targets
curves reconstructed for the two tested cameras. A biexponential
at normal incidence. The narrow-angle acceptance collector of the
function was selected as it provides a good model for the apparent
spectroradiometer was placed at 0.07 m from each one of the
dual-region instrument response function of many consumer-level
targets and oriented 45o from the target normal.
digital cameras, namely the observed high sensitivity to low light
Camera responses for each signal were obtained by taking a
levels and the saturation response at high light levels, as suggested
series of images of either the diffuser screen or the achromatic
by the use of non-linear expressions including several exponentials
reflective target, from the same direction as the spectroradiometer
to model the gain function of these cameras [22]. The observed
measurements. Ten f-apertures were selected for testing the Canon
compression of camera response at high radiance levels is used to
40D camera including complete, half and third stops from f-
extend dynamic range [29]. The Bézier functions produce flexible
aperture 8 to 22. For the modified Nikon camera seven f-apertures
curves for fitting different data distributions [30], are intuitive, and
were selected representing complete stops from f-aperture 32 to
easily inverted with LUTs as as shown in the Results section;
4.0 and including f-aperture 4.5. Shutter speed (integration time)
however, these functions do not have such a close physical
was fixed in both cameras at 0.017 seconds for the Canon camera
connection with the voltage response from the camera sensor.
and 2 seconds for the Nikon camera. ISO 200 was selected in both
devices. White balance programs were set at 5100 K for the A cubic Bézier curve is defined by the position of four control
Canon camera and the pre-set ‘flash’ program (approximately points (P0 , . . . ,P3 ) and it is constructed by evaluating an
5400 K) for the Nikon camera. A dark image, with the lens cap independent parameter t in a ½0,1 interval. If Equation (1) is
on, was recorded at the beginning of each image-recording run to rewritten as y~G(x), then the Bézier curve is described
account for dark noise. The dark image was subsequently parametrically by Equation 3.
subtracted from each camera response image at each pixel
location over the entire image. Images were recorded in the native gpe
x~ ~(1{t)3 P0x z3(1{t)2 tP1x z3(1{t)t2 P2x zt3 P3x ,
RAW file format for each camera and encoded either into the gmax
Adobe 1998 colour space (Canon camera), or the sRGB

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 Linearisation of UV-Visible RGB Camera Responses

constructed by inverting the x and y-axes of the Bézier curve,

and calculating point coordinates along the curve as t took on 256
uniformly spaced values between 0 and 1.
The form of biexponential fitting function that was used is
shown in Equation (4). The parameter rlim is the notional limiting
output approached as gpe becomes very large, but the function is
only applied for gpe ƒgmax . Coefficients b and d are in pixel
response units, and may take any positive value up to rmax , whilst
coefficients c and g are in inverse photoelectron-number units
(mol{1 ) and may take any positive value.

gpe 1 gpe gpe

G( )~ ½r {b: exp ({c ){d : exp ({g ),
gmax rmax lim gmax gmax ð4Þ
0ƒgpe ƒgmax :

To comply with the normalisation of G, there are two

conditions:

G(0)~0 [rlim ~bzd,

G(1)~1 [rlim ~rmax zb: exp ({cgmax )zd : exp ({ggmax ):ð5Þ

Biexponential fitting procedures were performed using the trust-

region effective algorithm available in the optimization toolbox for
Matlab release 2009b (The Mathworks, USA). The biexponential
function is not, in general, invertible, however numerical inversion
of the function can be efficiently performed, and we used custom-
written code based on the fzero routine in Matlab release 2009b. A
Wilcoxon signed rank test was employed to compare the results
obtained from the two methods using routines available in IBM
SPSS Statistics V.20.0 (IBSS Corporation, USA).
When reconstructing images representing linear camera
responses, the camera output must be normalised first by dividing
each pixel intensity value by the selected rmax value. The
linearised response is then obtained with the use of the inverted
G function. Finally photoelectron numbers can be found from the
Figure 2. Observed camera responses for the red channel of a linearised results by multiplication by gmax .
Canon 40D digital camera (red 6 markers) and fitting results
including values below the minimum pixel response value rmin . Reconstruction of Confidence Bounds for the Recovered
(A) Biexponential fit (black circle markers), (B) 19 Bézier segments (black
squares). Linear Camera Responses
doi:10.1371/journal.pone.0079534.g002 For the biexponential method, confidence bounds for the linear
responses recovered for the 256 possible camera response levels
from each colour channel were reconstructed by implementing
simulation methods. A total of 1000 linear camera responses were
r
y~ ~(1{t)3 P0y z3(1{t)2 tP1y z3(1{t)t2 P2y zt3 P3y , recovered for each r value in a 0{255 interval after inverting
rmax ð3Þ Equation 4, using coefficients drawn in a pseudorandom manner
t[½0,1 from a Gaussian (normal) distribution following a Monte Carlo
simulation method [32]. The standard deviation of the distribution
where Pnx and Pny are the coordinates of control point Pn for was calculated from the upper and lower limits of the 95%
n~0,1,2,3. confidence interval for each one of the different coefficients.
The coordinates of the first and last control points P0 and P3 Confidence bounds of the control points defining the Bézier
correspond to the normalised minimum and maximum exposure curve were obtained from a set of 1000 control points
values and their corresponding camera responses; the other two corresponding to the same number of Bézier curves fitting OECF
control points are found by minimisation in a least-squares sense. curves constructed from sub-sets of 32 points each. Sub-sets were
Cubic Bézier curves were fitted implementing the Cubic Bézier least constructed by randomly selecting camera responses and their
square fitting algorithm [31] written for Matlab. corresponding exposure values from a total of 96 data points
When implementing cubic Bézier curves, linear camera measured for reconstructing the OECF curves. Subsequently,
responses were recovered by employing LUTs. These were confidence bounds for the linear responses were constructed

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 Linearisation of UV-Visible RGB Camera Responses

Figure 3. Recovered linear camera responses and confidence bounds for the (A–B) red, (C–D) green and (E–F) blue channels of a
Canon 40D digital camera and; (G–H) the red channel of a Nikon D70s camera modified for ultraviolet recording, using cubic Bézier
curves (left column) and biexponential functions{ (right column). Linear camera responses were obtained by inverting the biexponential
fitting function (Equation 4) (squares) and implementing a look up table derived after evaluating a cubic Bézier curve (Equation 3)(circles). Confidence
bounds represent the standard deviation in all cases. { Standard deviation of the biexponential function |5 for display purposes.
doi:10.1371/journal.pone.0079534.g003

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 Linearisation of UV-Visible RGB Camera Responses

Figure 4. Standard deviation of linear camera responses (cross markers) as a function of increasing values of g/gmax recovered
implementing a biexponential function (dotted line left column) and cubic Bézier curves (solid and dashed lines right column) for
the (A–B) red, (C–D) green and (E–F) blue channels of a Canon 40D digital camera and; (G–H) the red channel of a Nikon D70s
camera modified for ultraviolet recording. Standard deviations for each g/gmax recovered by the biexponential function were obtained after
simulating 1,000 normally-distributed random coefficients within the 95% confidence intervals for each of the four parameters in Table 1. Standard
deviation for each g/gmax recovered by the cubic Bézier curve were obtained from the LUTs constructed after simulating 1,000 normally-distributed
pseudorandom coefficients within the 95% confidence intervals for the eight parameters in Table 2. Solid line in panels B, D, F and H corresponds to
the standard deviation of the normalised camera responses (r), whilst the dashed line represents the standard deviation of the recovered normalised
exposure value (g).
doi:10.1371/journal.pone.0079534.g004

following the same procedure implemented for the biexponential OECF curve: camera responses and irradiation input, with the
method. latter defined by the selected exposure parameters as expressed by
Equation 2.
Results Pixel intensity values, representing the camera output, were
normalised by dividing each camera response by the maximum
OECF curves were reconstructed for the three different colour intensity level attainable in the selected colour-bit depth scale. This
channels of the Canon camera and the red channel of the modified value, rmax , equals 255 intensity levels for the 8-bit colour
Nikon device. All the reconstructed OECF curves present a similar encoding scheme selected for characterising the two cameras.
form that are entirely fitted by implementing either biexponential Normalisation of the input exposure was done by dividing the
functions or cubic Bézier curves (Figure 1); nevertheless, the use of exposure value g corresponding to each camera response included
Bézier curves requires an additional normalisation step prior to in the OECF by the exposure required to obtain rmax for each
fitting as these curves are solely defined in a [0, 1] interval [30]. characterised colour channel.
Normalisation was carried out on the two variables defining the

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 Linearisation of UV-Visible RGB Camera Responses

Table 3. Statistical comparison of the linear camera respons- Linear camera responses recovered by implementing the two
es obtained with two characterisation methods. methods are presented in Figure 3. The uncertainty associated
with the recovery of the linear camera responses varies with the
exposure, reaching its maximum value at rmax for the two
SSE (mmol) Wilcoxon signed rank test methods. Such a behaviour is not surprising, as large changes in
Channel
exposure only produce slight changes in camera responses near
Cubic Significance
rmax as expected from the asymptotic behaviour of the OECF
Biexponential Bézier Statistic (2-tailed)
curve (Figure 1); however, an important difference between the
Red Canon 4.2161024 2.0261023 4580 0.751 two methods is the number of dimensions associated with the
24
Green Canon 4.35610 1.7561023 4450 0.761 uncertainty of the recovered linear camera responses. Whilst the
Blue Canon 1.1261023 2.6161023 4720 0.764 uncertainty of the linear camera responses recovered by imple-
Red Nikon 1.0961023 3.4061024 326 0.825
menting a biexponential function is only associated with the
recovered exposure value, i.e. variation in the y-axis (Figure 3,
Sum of squared errors for the values predicted by the functions fitting the OECF right column), the uncertainty of the recovered linear camera
curves (second and third column) and results of the statistical comparison responses by using Bézier involves both the g=gmax and r=rmax
between the camera responses predicted by the two methods.
parameters (Figure 3, left column), as these are required to define
doi:10.1371/journal.pone.0079534.t003
each Pn control point of the Bézier curve (Equation 3).
The magnitude of the uncertainty associated with the recovered
linear camera responses is not uniform, but varies with the
The maximum exposure values (gmax ) obtained for the four different values of g irrespective of the employed linearisation
characterised colour channels were: 0.0122 mmol, 0.0125 mmol, method (Figure 4). However, differences do exist in the total
0.0124 mmol and 0.0081 mmol, corresponding to the Canon magnitude of the standard deviation obtained by implementing
camera red, green, blue channels and the modified Nikon D70s each method and in the precise g values where it is higher. In the
UV-sensitive red channel respectively. gmax values were obtained case of the biexponential function, the magnitude of the standard
from a biexponential function fitted to the OECF curves expressed deviation increases in a relatively linear manner after reaching
in the original (not normalised) scale; however, these values can about 10% of gmax and up to the saturation region where it rapidly
also be directly obtained from the OECF curve either by visual increases until reaching gmax (Figure 4, left column). This
inspection or by linear interpolation of the OECF data points, behaviour is also observed for the Bézier curves with an additional
provided that there are enough points at the upper end of the increase in the uncertainty of the recovered g values at low
curve up to the rmax value. Note that a biexponential function can irradiance levels arising from the high standard deviation
be fitted to the OECF curve expressed either in the original or a associated with the r=rmax parameter (Figure 4, right column).
normalised scale. The sum of squared errors (SSE) between the measured
Regardless of the method selected to fit a given OECF curve, irradiance input (exposure) and the linear camera responses
linear camera responses, i.e. the intensity of the irradiance signal at recovered by the two fitting functions is presented in the second
a given pixel location corresponding to a given r value, can be and third columns of Table 3. Even though the implementation of
recovered by inverting the equation of the selected fitting function. biexponential functions always resulted in predicted camera
The parameters defining the two fitting functions, namely the response values which are closer to the exposure calculated from
coefficients of the biexponential function and the coordinates of the measured irradiance, particularly at the extreme ends of the
the control points for the Bézier curve, are presented in Tables 1 exposure range, a comparison of the median differences between
and 2 along with their 95% confidence intervals. Equations 1 and the camera response values predicted by the two methods for the
2 present the general form of the two fitting functions. Whilst the entire exposure interval did not prove significantly different
biexponential function coefficients and associated 95% confidence (Table 3 fourth and fifth column). However, significant differences
intervals (Table 1) were obtained directly from the output of the between the two methods do exist in the computational time
biexponential fitting procedure, implementation of simulation required for applying the two methods. Calculation of the
techniques were necessary for obtaining the coordinates of the confidence bounds for 256 linear responses, as required to
control point defining the Bézier and their 95% confidence reconstruct the LUT employed for linearising images, took a
intervals (Table 2) as detailed in the Methods section. median of 131 seconds for the biexponential function compared to
Another important difference between the two fitting functions a median of 4.10 seconds required for the implementation of the
is the minimum camera responses included in the OECF: the rmin Bézier approach.
value. The precise value for rmin was found to be a factor
influencingg the number of Bézier segments required to accurately Discussion
fit the OECF curve (Figure 2, panel B). Although complex curves
With the growing use of digital imaging for quantifying the tonal
can be accurately fitted using several Bézier segments rather than a
and spectral characteristics of radiations reflected from various
single Bézier curve [30], for the purpose of camera characterisa-
object matter [1,2,11,12,14–16], it is important to have accurate
tion, it is desirable to fit the entire OECF curve using a single methods for specifying the relationships between input irradiance
segment in such way that the LUT required for recovering the signal and camera output for quantitative analyses. In spite of
linear camera values can be constructed applying an equation- being sensitive to different regions of the spectrum, the OECF
based interpolation (Equation 3) from a single Bézier segment. curves of the two tested cameras present a notable similarity in
rmin values were set at 31 and 37 pixel intensity values for the their general form (Figure 1). This result indicates a close likeness
Canon and Nikon camera respectively, corresponding to the first between the gain functions applied to the sensor response of the
control point (P0 ) on the Bézier curve. On the other hand, the two cameras. The use of non-linear gain functions which
biexponential function accurately fitted the entire OECF curve, asymptotically approach to rmax is characteristic of different
eliminating the need for a rmin value (Figure 2, panel A). consumer-level digital cameras as a strategy for increasing their

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 Linearisation of UV-Visible RGB Camera Responses

dynamic range [22,29]; a commonly desired feature for commer- accurately linearised. Camera responses below rmin follow a
cial photography, but a limitation for quantitative image analysis distribution different from the remaining OECF curve [2,36], and
[2]. Therefore the present method is potentially applicable to other including them may prevent attaining an adequate fit with the
camera models presenting a similar gain function, including those selected programming code. The precise value of rmin varies from
cameras capable of producing images from reflected near- one camera to another and must be found empirically, which is
ultraviolet radiation [1,23]. again a limitation compared with the biexponential approach
Even though the two proposed characterisation and linearisa- (Figure 2, panel A). Although it is possible to fit the entire OECF
tion methods accurately recover the linear camera response curve, including the low response region, implementing several
(Figure 3, Table 3), they differ in the magnitude of the uncertainty Bézier segments rather than a single Bézier curve (Figure 2, panel B),
associated with the recovered radiometric information. Irrespec- this approach has the limitation of producing LUT tables whose
tive of the selected linearisation method, a graphical depiction of values do not uniformly cover the entire OECF curve, but are
the gain function, i.e. the OECF curve (Figure 1), in conjunction clustered along different regions of varying length along the curve
with a plot of the standard deviation as a function of exposure level corresponding to the different segments (Figure 2, panel B). This
(Figure 4), provides a guideline for establishing the maximum arrangement of the LUT values makes it necessary to resort to
camera response included in a given image and its corresponding interpolation techniques to recover linear values corresponding to
exposure value. By establishing these two criteria it is possible to r values located on non-sampled regions of the OECF, thus
define precise exposure parameters, f-number and shutter speeds, introducing an additional step in the computation and increasing
for attaining a standardised exposure, which in turn allows for an the uncertainty bounds of the recovered linear response. Contrary
objective comparison among images recorded with the same to the use of Bézier fitting techniques, the implementation of a
camera. biexponential function does not require the use of a rmin value as it
Selecting r values corresponding to g values located before the accurately fits the entire OECF curve including extremely low
region of increasing standard deviation has the advantage of camera responses (Figure 2, panel A). This characteristic of the
ensuring the recovery of linear camera responses with the lowest biexponential function is particularly convenient when recon-
possible uncertainty for a given camera system/colour channel structing spectral sensitivity curves, as it removes the necessity to
combination; however, other factors such as the intensity of the modify the exposure parameters to increase the camera’s response
signals produced by study object itself should also be considered at wavelengths where the sensitivity is very low.
when selecting the gmax value. Even though the two methods differ in the number of
One of the most common applications of linear camera parameters that need to be estimated to fit a curve, in the present
responses is for reconstructing spectral sensitivity curves [3], application, only four parameters need to be estimated by either
defined as the ratio of linear camera response to incident energy at method. Camera characterisation by means of a biexponential
different wavelengths across a given spectral interval [33]. Camera function requires estimating four parameters, corresponding to the
characterisation by means of a biexponential function and the two coefficients included on each of the biexponential terms here
subsequent recovery of linear camera responses and their represented by the letters b, c, d and g (Table 1 and Equation 4).
associated standard deviation after inverting the fitting function Even though in principle camera characterisation by cubic Bézier
(Equation 4) is particularly useful in this case, as the linearised curves requires finding a total of eight parameters represented by
responses are expressed in the same units as the energy input the g=gmax and r=rmax values for each of the four control points
(Table 2). Furthermore, the number of camera responses required defining the curve in Equation (3), two of these points, P0 and P3 ,
for this application allows for a precise recovery of the linear are predefined by setting rmin and by the highest r included in the
camera responses whilst keeping the computational time at OECF, so again there are only four free parameters.
reasonable levels. On the other hand, the use of Bézier curves From our results it can be concluded that both biexponential
for this purpose not only requires an extra step represented by the functions and cubic Bézier curves overcome the limitations of
multiplication of the recovered linear response by a separately- power and exponential functions to completely characterise the
measured value of gmax , but has the shortcoming of the wide OECF curve of cameras equipped with a Bayer filter array.
uncertainty bounds associated with extremely low and high Although either of the two methods can be used for accurately
exposure values (Figure 3). recovering total irradiance at a given pixel location, the selection
When the objective is to quantitatively analyse images of a particular method should be based on: (i) the final objective of
representing complex scenes including large areas widely varying using linear camera responses, and (ii) the potential implications of
in irradiance levels (brightness), or when the entire photographic differences in the magnitude of the uncertainty associated with the
frame has to be analysed, a researcher faces different require- recovered linear camera responses.
ments. In these, and other biology-related studies involving When the objective is to reconstruct spectral sensitivity curves,
imaging such as characterisation of animal colour patterns, camera characterisation by means of biexponential functions is the
camouflage studies, modelling non-human visual spaces and best approach. These functions accurately model the entire OECF
animal-plant interactions [1,2,11,12,34,35], the efficiency of the curve including the extremely low camera response and saturation
Bézier technique may overcome the wider uncertainty levels region thus making unnecessary the use of ad hoc parameters,
associated with this methodology (Figure 3); in particular, when namely the rmin value. Moreover camera characterisation by this
the digital images to be linearised consist of several megapixels. method allows for a precise estimation of the normalisation
Yet in this case a biexponential linearisation function can be parameters required for the implementation of Bézier fitting
efficiently implemented if a LUT is constructed for linearising the techniques.
images rather than directly inverting the function for the camera On the other hand, cubic Bézier curves have the advantage of
response at each pixel location as was done here. permitting the recovery of linear camera responses and their
In contrast to the biexponential fitting function, the cubic Bézier associated uncertainty bounds in a computationally-efficient
curve requires establishing a minimum pixel response value (rmin ). This manner through the implementation of a formula-based interpo-
value corresponds to the first control point of the fitting curve lation. When implementing this method, the required look-up-
(Figure 1) and represents the lowest camera response that can be tables are constructed by simply inverting the axes, making

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 Linearisation of UV-Visible RGB Camera Responses

unnecessary the implementation of numerical approximation responses. However the main differences between the two methods
algorithms such as those required for inverting the biexponential consist on the amount of uncertainty associated with the recovered
fitting function. Nevertheless when implementing this method it is irradiance and the means by which the two functions are inverted.
still important to consider the wider uncertainty bounds compared Recovering of irradiance values by implementing biexponential
to those obtained by implementing the biexponential approach. functions results in consistently reduced uncertainty bounds, but
Finally, by selecting adequate rmin and rmax values it is possible the inversion of such a function requires resorting to optimisation
to establish precise and standardised minimum and maximum techniques requiring longer computational times. On the other
exposure parameters thus permitting the objective comparison and hand, recovery of irradiance values employing Bézier curves
quantitative analysis of the reconstructed images. These images requires shorter computational times, is more intuitive and easily
accurately reconstruct two-dimensional information from real, achieved with linear interpolation through the use of LUTs. The
complex scenes, which should have high value for biological application of these methodologies makes it possible to accurately
imaging and other quantitative image analysis applications. recover total irradiance information from complex scenes,
enabling investigations such as the study of animal vision in
Conclusions natural settings.
Our results introduce two different methodologies for recover-
ing irradiance information, at each pixel location, within a digital Author Contributions
image recorded with RGB cameras sensitive to visible and UV Conceived and designed the experiments: JEG ADG GS PAW. Performed
irradiation. Both methods achieve this by fitting a mathematical the experiments: JEG. Analyzed the data: JEG ADG PAW. Contributed
function to the OECF curve (gain function) of the camera and reagents/materials/analysis tools: GS PAW. Wrote the paper: JEG AGD
subsequently inverting it to solve for exposure from camera ADG GS PAW.

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PLOS ONE | www.plosone.org 10 November 2013 | Volume 8 | Issue 11 | e79534