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Digital camera simulation


Abstract and Figures

We describe a simulation of the complete image processing pipeline of a digital camera, beginning with a radiometric description of the scene captured by the camera and ending with a radiometric description of the image rendered on a display. We show that there is a good correspondence between measured and simulated sensor performance. Through the use of simulation, we can quantify the effects of individual digital camera components on system performance and image quality. This computational approach can be helpful for both camera design and image quality assessment.
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Joyce Farrell, Manu Parmar, Peter Catrysse and Brian Wandell
Customers judge the image quality of a digital camera by viewing the final rendered output.
Achieving a high quality output depends on the multiple system components, including the
optical system, imaging sensor, image processor and display device. Consequently, analyzing
components singly, without reference to the characteristics of the other components, provides
only a limited view of the system performance. In multi-device systems, a controlled simulation
environment can provide the engineer with useful guidance that improves the understanding of
the system and guides design considerations for individual parts and algorithms.
In this chapter, we describe a system approach to modeling and simulating the image processing
pipeline of a digital camera, beginning with a radiometric description of the scene captured by
the camera and ending with a radiometric description of the final image as it appears on an LCD
display. The chapter is organized into five sections.
The section on the Scene describes how to create a radiometric description of the scene. We
represent a scene as a multidimensional array describing the spectral radiance
(photons/sec/nm/sr/m2) at each pixel in the sampled scene. We use linear models of surfaces and
lights to simplify and compress the representation of spectral radiance emitted from a scene. We
also describe several methods for acquiring high dynamic multispectral images of natural scenes
to generate radiometric image data (surfaces and illuminants).
The section on Optics describes how scene radiance data are converted into an irradiance image
at the sensor surface. The conversion from radiance to irradiance is determined by the properties
of the optics, which gather the diverging rays from a point in the scene and focus them onto the
image sensor [1] .
The section on the Sensor describes how the irradiance image is converted into electrons. This
transformation includes a model of the optical and electrical properties of the sensor and pixel.
The image sensor model includes a great many design parameters, only some of which will be
discussed here. Among the various factors accounted for are the spatial sampling of the optical
image by the image sensor, the size and position of pixels and their fill-factor. The wavelength
selectivity of color filters, intervening filters (such as an infrared filter), and the photodetector
spectral quantum efficiency are also included in the simulation. After converting photons into
electrons, we add sources of noise and quantize the data to generate digital sensor values.
The section on the Processor describes how digital values generated by the two-dimensional
sensor array are converted into a three-dimensional (RGB) image that can be rendered on a
specified display. This is accomplished by interpolating missing RGB sensor values
(demosaicking) and transforming sensor RGB values into an internal color space for encoding
and display (color-balancing and display rendering). In this chapter, we describe the general
principles of auto-exposure, demosaicing and color-balancing. Other chapters describe
algorithms for these processes in more detail.
The section on Displays describes how to generate a radiometric description of the final image
as it appears on an LCD display. The spectral radiance for any image rendered on a typical
display can be predicted by three functions – the display gamma, the spatial point spread
functions for the red, green and blue pixel components and the spectral power distributions of the
display color primaries [2]. Modeling the display makes it possible to use the radiance field as
the input to objective image quality metrics. Calculating the spatial-spectral radiance is useful
because, unlike the digital image values, the radiance field is the stimulus that actually reaches
the eye.
In the final section of this chapter, we use an integrated suite of Matlab® software tools to
explore the full imaging pathway, including scene, optics, and sensor of a calibrated digital
camera. We illustrate how it is possible to use simulation tools to characterize imaging sensors
and explore novel designs.
Many of the ideas we describe in this chapter are an extension of work that we and our
colleagues have described in previous publications. For example, Chen et al [3] developed a
grayscale digital camera simulator to evaluate the effects of pixel size on dynamic range, signal-
to-noise ratio and camera modulation transfer functions. Vora et al [4] and Longere and Brainard
[5] developed a color digital camera simulator that combined hyperspectral scene data with a
linear model of imaging sensors. They demonstrated the viability of their simulator by
comparing simulated and measured sensor performance. And Longere and Brainard [5] used
their digital camera simulator to evaluate the effect that sensor spectral sensitivity, noise and
color filter array patterns have upon image quality. We recommend these previous publications
as an introduction to the ideas developed more completely in this chapter.
Digital camera simulation requires a physically accurate description of the light incident on the
imaging sensor. We represent a scene as a multidimensional array describing the spectral
radiance (photons/sec/nm/sr/m2) at each pixel in the sampled scene. The spectral radiance image
data are assumed to arise from a single image plane at a specified distance from the optics.
There are several different sources of scene data. For example, there are synthetic scenes, such
as the Macbeth ColorChecker, spatial frequency sweep patterns, intensity ramps and uniform
fields. When used in combination with image quality metrics, these synthetic target scenes are
useful for evaluating specific features of the system, such as color accuracy, spatial resolution,
intensity quantization, and noise.
Another important source of scene data is calibrated representations of natural scenes. In the
section below, we describe several methods for acquiring high dynamic multispectral images of
natural scenes which can be used to generate radiometric image data.
Finally, scene radiance image data can be estimated from RGB data. In the section on Displays,
we show that it is possible to use linear models to predict the spectral radiance of a displayed
We describe several methods for creating high dynamic range spectral (HDRS) radiance scene
data using calibrated cameras and a variety of different filters and light sources.
We extend the dynamic range of a single image exposure by combining unprocessed or so-called
“raw” camera images across multiple exposures [2, 4-7]. If we exclude specular highlights, the
dynamic range of the vast majority of natural scenes is on the order of 5000:1[8]. Hence, a
capture system based on a 12-bit ADC that uses a few exposure durations to bracket the range
can capture the full dynamic range of most natural scenes [7-10].
We can also extend the spectral information in the image data by combining raw camera images
from more than three color channels. Additional color channels can be created by interposing
different color filters between successive images. The color filters can either be placed between
the light and the scene (creating different colored lights) [11, 12] or between the scene and the
camera (adjusting the spectral sensitivity of the camera) [13-15]. In both cases we can use linear
models of surfaces and illuminants, briefly described below and more thoroughly in other
publications[16-23] to estimate the spectral radiance.
Data gathered over the last fifteen years suggests that for scenes containing only one or two light
sources, the dimensionality of the scene spectral radiance functions will be captured accurately
(within 5% rmse) using as few as five or six basis functions for a particular illumination
condition [23-26]. This number is much smaller than the 61 values necessary to represent
spectral radiance measurements from 400 to 700nm, in 5 nm steps, but it is greater than the three
values captured with a traditional RGB digital camera.
As noted earlier, there are several ways to increase the number of spectral samples. In the
example described here, we use a three-color camera in combination with three filters. One of
the captures is made with no color filter, and two are made with color filters placed between the
camera and the scene. This procedure yields nine different spectral samples. While these
samples are not independent, they do provide enough independent information to obtain a
reasonable spectral approximation to many scenes.
The spectral radiance of a surface measures the rate of photons emitted (or scattered) from a
surface as a function of area, angular direction, and wavelength. In most cases, such as for a
simple reflective surface, the spectral radiance (in units of photons/sec/nm/m2) can be
represented by a vector, s. If we have measurements from 400 to 700nm, in 5 nm steps, then the
vector s will have 61 entries. This representation requires a lot of memory for images of any
significant spatial size. For example, a 512 x 512 image with 16-bits per wavelength plane is
stored in about 30 MB. A 1Kx1K image is stored in about 120 MB of data. It is possible,
however, to use low-dimensional linear models to summarize efficiently the scene spectral
radiance data. This data compression takes advantage of the fact that spectral radiance functions
in most natural images are regular functions of wavelength.
The spectral representation of the light reflected from each point a scene can be approximated by
a linear combination of a small set of spectral basis functions, Bi, where wi are the weights
chosen to minimize the error between the spectral radiance and its linear model approximation,
and N is the dimensionality of the spectral representation.1
The low-dimensional linear models of spectral radiance reduce the amount of data that must be
stored when the number of basis functions, N, is less than 61. The accuracy of the low-
dimensional spectral representations are limited by 1) how well the spectral radiance data
conform to a low dimensional linear model, 2) the dimensionality of the image capture system
which may or may not have enough sensors to capture the scene information, and 3) the quality
of the algorithms for inferring the spectral radiance functions from the sensor data.
To illustrate high dynamic range multispectral imaging, we used a Nikon D100 digital SLR
camera with two carefully chosen filters to record high dynamic range multispectral
measurements of natural scenes. Because of its low noise and dark current, the Nikon D100 is
close to a true 12-bit camera. We used the Nikon D100’s exposure auto-bracketing function to
take three images in rapid sequence at exposure settings each separated by a factor of four. The
shortest exposure duration was based on photometric measurements of the maximum luminance.
For each scene, we captured three exposure auto-bracketing pictures each with and without the
two color filters in the imaging path. This resulted in a total of 9 images (3 exposure settings x 3
color filter conditions) for each scene. To reduce image registration problems, the aperture
setting (f-stop) was fixed for all 9 images. The camera was positioned on a tripod for stability
across the capture. The images were spatially registered when necessary, though very little
camera motion was present. Stable scenes, without significant image motion, were selected.
The Nikon sensor contains a Bayer mosaic with two green, one red and one blue photodetector
type in a 1012x1517 grid. We down-sampled the Nikon sensor images to obtain rgb images with
a spatial resolution of 253x380. (We use higher resolution digital cameras and various sampling
rates to obtain rgb images with higher resolution.)
To estimate the intensity recorded by each pixel, we used the last sample before saturation
(LSBS) algorithm [6]. This sample has the highest signal-to-noise ratio. From this estimate, we
created a single high dynamic range image with 9 color channels.
We use the 9-color high dynamic range images captured for each scene to estimate a spectral
radiance image for that scene. Spectral power distributions of many natural images can be
modeled as a linear combination of a relatively small (~5) set of spectral basis functions [24-26].
To improve the estimation of the spectral radiance of the image, we acquired a measurement of
the spectral power distribution of the scene illumination. Using this measurement, we could
simulate the light scattered from a standard test chart (the Macbeth Color Checker) and, using the
singular value decomposition, derive three scene basis functions for that illumination. We added
two spectral basis functions to these, a DC and a linear ramp function (see Figures 1, 2 and 3,
below). These functions are not orthogonal, but they are independent. Through empirical
evaluations, described below, we found that these basis functions provide a high quality
representation of the image data. Each HDRS image, then, includes a set of spectral bases, Bi,
along with the basis coefficients (weights), wi,. The basis coefficients, wi, are calculated from the
nine color channels recorded by the multicapture imaging system.
We estimate the basis coefficients in the following way. We represent the wavelength
information at 61 samples, from 400 to 700 nm in 5 nm steps. Let r be a 9x1 vector representing
the output of nine color channels for a single pixel a sampled scene. Let T be a 9 x 61 matrix
representing the spectral responsivities of the nine color channels in the image capture device.
Let B be a 61 x 5 matrix representing scene spectral basis functions. Finally, let w be a 5x1
vector representing the weights on B. Then,
Solve for w using the pseudoinverse of TB,
= (1.3)
In this example, we used 5 spectral basis functions in the linear model approximations for the
spectral radiance measured from the 24 color patches in the Macbeth ColorChecker under
tungsten and fluorescent illumination (Figure 1). Two spectral basis functions were a constant
(DC) and ramp signal. The remaining 3 spectral basis functions correspond to the 3 most
significant eigenvectors derived from the singular value decomposition (SVD) of the spectral
radiance measured from the 24 surfaces illuminated by tungsten or fluorescent lights.
Figure 1. Spectral basis functions for diffusely reflecting surfaces when surfaces are illuminated by tungsten(A)
and fluorescent (B) illumination.
Figures 2 and 3 compare the estimated and measured spectral radiance of 24 color patches in the
Macbeth Color Checker illuminated by tungsten (Figure 2) and fluorescent (Figure 3). The rmse
between the estimated and measured data is less than 3.5 %, though in some parts of the
spectrum, particularly above 650 nm, the Nikon D100 has poor wavelength sensitivity and the
errors are larger. In subsequent work, we used a modified Nikon D200 digital camera that can
capture energy in both the visible and near infrared regions of the spectrum [27].
Figure 2. Measured (solid line) and estimated spectral radiance of the twenty-four surfaces of a Macbeth
ColorChecker illuminated by a tungsten light. The estimation was made using the five linear model basis functions
shown in Figure 1a and data from the Nikon D100, as described in the text.
Figure 3. Measured (solid line) and estimated scene radiance of the twenty-four surfaces of a Macbeth
ColorChecker illuminated by a fluorescent light. The estimation was made using the five linear model basis
functions shown in Figure 1b and data from the Nikon D100, as described in the text.
In the example described above, we created additional color channels by placing color filters
directly on the camera, effectively adjusting the spectral sensitivity of the camera. It is also
possible to place the color filters directly in front of a light, effectively changing the spectral
composition of the scene illuminant.
Many researchers have used methods based on optical filters to acquire multispectral images of
artwork [9-12], scenes with IR energy [27] and natural scenes [28]. Since a number of
acquisitions are required with different optical filters, this method is best suited for imaging
static scenes. Long exposure times preclude the use of this method for acquiring scenes that
include people and other dynamic objects. The time required to change filters may be reduced by
the use of filters mounted on a color wheel, but this renders the device physically cumbersome
and expensive.
An alternate approach is to use multiple acquisitions of a scene illuminated by lights with
different spectral power distributions. Since lighting systems may be designed to switch at high
rates, this approach can be used to acquire multispectral scenes with people and other animate
objects. We have created a system that uses multiple LEDs for capturing multispectral
scenes[29]. The spectral data may be within the visible or extend into the infrared[27].
The spectral radiance image can be stored in a compact wavelength representation as a data file
that contains a representation of the spectral basis functions and coefficients. When the scene
illumination is known, it can be stored in the file as well. Hence, a relatively small (four, five or
six) chromatic samples – along with the modest overhead of including the basis functions -
represent the full spectral radiance information. It is further possible to compress the spatial
images of the coefficients, of course.
Figure 4 shows an image derived from a multispectral representation of a female face, and the
spectral reflectance data for a selected region of her face. In order to illustrate the multispectral
image, we summed the energy in long-, middle- and short-visible wavebands and assigned these
to the R, G and B primaries, respectively. The graph in Figure 4 compares the measured and
estimated spectral reflectance of a region selected from the subject’s forehead.
Figure 4. The spectral image of a female face is rendered by summing the energy in long-, middle- and short-visible
wavebands and assigning these to the R, G and B primaries, respectively. The graph compares the measured and
estimated spectral reflectance of a region selected from her forehead.
In the current implementation, image data are modeled as arising from a plane. Extending the
image representation to include depth information would allow the simulation to account for
several additional factors, such as the optical depth of field. Also, the image radiance is assumed
to radiate equally in all directions, while in a true natural scenes the scattering depends on the
geometry of the light surface, surface normals, and material specularities. Consequently, while
the current spectral radiance data represent one particular view, they are not easily generalized to
other camera viewpoints. To lift these limitations, it is possible to capture multispectral images
of objects in a scene from multiple vantage points. For example, one can use multiple cameras
in different locations [30] or rotate the objects in the scene [31] into different positions. It would
then be possible to render different instances of the scene, captured from different viewpoints.
In the scene radiance, photons are emitted in multiple directions by light sources or scattered in
multiple directions. The lens gathers up these divergent rays so that they converge to an
irradiance image at the sensor surface. We call the irradiance image at the sensor surface, just
prior to capture, the optical irradiance. To compute the optical irradiance image, we must
account for a number of factors. First, we account for the lens f-number and magnification.
Second, we account for lens shading (relative illumination), the fall-off in intensity with lens
field height. Third, we blur the optical irradiance image. The blurring can be performed with one
of three models: a wavelength-dependent shift-invariant diffraction-limited model, a wavelength-
dependent general shift-invariant model (arbitrary point spread), and a general ray trace
calculation which further incorporates geometric distortions and a wavelength-dependent blur
that is not shift-invariant. The reader can find a thorough description of imaging optics in
Chapter 2 of this book [1].
The camera equation [32] defines a simple model for converting the scene radiance function,
L, to the optical irradiance field at the sensor, I. The camera equation is
(,,) (,,)
4( /#)
image scene mm
Ixy L
The term f/# is the effective f-number of the lens (focal length divided by effective aperture); m
is the lens magnification; and ()T
is the transmissivity of the lens. The camera equation holds
with good precision for the center of the image (i.e., on the optical axis). For all the other
locations in the optical image, we need to apply an off-axis correction.
The fall-off in illumination from the principal axis is called the relative illumination or lens-
shading, ( , , )
. There is a simple formula to describe the shading in the case of a thin lens
without vignetting or (geometric) distortion. In that case, the fall-off is proportional to 4
is the off-axis angle
(, , ) cos d
Rxy S
The term S is the image field height (distance from on-axis) and d is the distance from the lens to
the image plane. The (x,y) coordinates specify the position with respect to the center of the
image axis. This is formula is often called the Cosine-fourth Law. In real lenses, or lens-
collections, the actual off-axis correction may differ from this function. It is often used, however,
as a good guess for the irradiance decline as we measure off-axis.
The irradiance image, (, , )
cannot be a perfect replica of the scene radiance,
(, , )
. Imperfections in the lens material or shape, as well as fundamental physical
limitations (diffraction), limit the precision of the reproduction. The imperfections caused by
these factors can be modeled by several types of blurring calculations of increasing complexity.
When a wave encounters a small obstacle, the wave’s path is changed. Similarly, as waves pass
through a small aperture their direction of travel is influenced by the aperture. This phenomenon
is known as diffraction. The effect of the aperture on the passage of the light waves imposes a
fundamental limit on the possibility of forming an accurate image of the scene using
conventional optics.
A diffraction-limited system can be modeled as having a wavelength-dependent, shift-invariant
point spread function [33, 34]. Diffraction-limited modeling uses a wave-optics approach to
compute the blurring caused by a perfect lens with a finite aperture. The point spread function of
a diffraction-limited lens is quite simple, depending only on the f/# of the lens and the
wavelength of the light. It is particularly simple to express the formula in terms of the Fourier
Transform of the point spread function, which is also called the optical transfer function (OTF).
The formula for the diffraction-limited OTF is
2cos( ) ( 1 ), 1
0, 1
ρρ ρρ
Where ρ = f(A/ (λ d)) (normalized frequency)
f = frequency in cycles/meter
A = aperture diameter in meters
λ = wavelength
d = distance between the lens aperture and detector
The diffraction-limited point spread function is a specific instance of a shift-invariant linear
model. It is natural to generalize this model by allowing the point spread function to vary. For
example, rather than having a point spread defined by physical bounds the user may define a
wavelength-dependent point spread function. There are various reasons to model such blur in a
typical camera path; for example, many cameras include a blurring glass on the sensor surface
that includes protection from infrared rays and reduces aliasing from detector sampling.
In optics, the term isoplanatic is used to define conditions when a shift-invariant model is
appropriate. Specifically, an isoplanatic patch in an optical system is a region in which the
aberrations are constant; experimentally, a patch is isoplanatic if translation of a point in the
object plane causes no change in the irradiance distribution of the PSF except its location in the
image plane. This is precisely the idea behind a shift-invariant linear system.
The lens transformation from a shift-invariant system can be computed much more efficiently
than a shift-variant (anisoplanatic) system. The computational efficiency arises because the
computation can take advantage of the Fast Fourier Transform to calculate the spatial
distribution of the irradiance image.
Specifically, we can convert the image formation and point spread function into the spatial
frequency domain. The shift-invariant convolution in the space domain is a point-wise product
in the spatial frequency domain. Hence, we have
(, , ) (, , ) (, , )
image ideal
FT I x y FT PSF x y FT I x y
=⋅ (1.5)
(,,) ( , ,) (,,)
image x y ideal
FT I x y OTF f f FT I x y
=⋅ (1.6)
(, , ) (, , )
image ideal
=⋅ (1.7)
where {}
FT is the Fourier transform operator and the optical transfer function, OTF, is the
Fourier transform of the PSF. Because no photons are lost or added, the area under the PSF is
one, or equivalently, (0,0, ) 1
=. In this shift-invariant model, we assume that the point
spread is shift-invariant for each wavelength, but the point spread function may differ across
wavelengths. Such differences are common because of factors such as longitudinal chromatic
The ray-trace method model replaces the single, shift-invariant, point spread function with a
series of wavelength-dependent point spread functions that vary as a function of field height and
angle. These functions can be specified by the user, or more likely they can be calculated using
lens design software. Ray trace programs also specify the geometric distortion from the lens.
This takes the form of a displacement field that varies as a function of input image position
(d(x,y)). The displacement field is applied first, and then the result is blurred by the local point
spread function, wavelength-by-wavelength.
Image sensors transform the optical irradiance image into a two-dimensional array of voltage
samples, one sample from each pixel. Each sample is associated with a position in the image
space. Most commonly, the pixel positions are arranged to form a regular, two-dimensional
sampling array. This array matches the spatial sampling grids of common output devices,
including displays and printers.
In most digital image sensors the transduction of photons to electrons is linear: specifically, the
photodetector response (either CCD or CMOS) increases linearly with the number incident
photons. Depending on the material properties of the silicon substrate, such as its thickness, the
photodetector wavelength sensitivity will vary. But even so, the response is linear in that the
detector sums the responses across wavelengths. Hence, ignoring device imperfections and
noise, the mean response of the photodetector to an irradiance image ( ( , )
photons/sec/nm/m2) is determined by the sensor spectral quantum efficiency ( ( )S
, e-/photon),
aperture function across space ( )
Ax, and exposure time (T, sec). For the ith photodetector, the
number of electrons will be summed across the aperture and wavelength range
() ()(,)
eTS AxI xddx
=∫∫ (1.8)
A complete sensor simulation must account for the device imperfections and noise sources.
Hence, the full simulation is more complex than the linear expression in Equation 1.7. Here, we
outline the factors and computational steps that must be incorporated in a simulation.
Computing the signal current density image. One approach to simulation is to first convert the
sampled irradiance image, whose values represent the irradiance in units of quanta per sec per
square meter, to a signal current density image (amps/m2). This image represents what we would
measure if the surface of the sensor were one continuous photodetector.
The irradiance image already includes the effects of the imaging optics. To compute the signal
current density we must further specify the effect of several additional optical factors within the
sensor and pixel. For example, most consumer cameras include an infrared filter that covers the
entire sensor. This filter is present because the human eye is not sensitive to infrared
wavelengths, while the detector is. For consumer-imaging the sensor is designed to capture the
spectral components of the image that the eye sees – and to exclude those parts that the eye fails
to see. The infrared filter helps to accomplish this goal, and thus it covers all of the pixels.
We must also account for the color filters placed in front of the individual pixels. While the
photodetectors in the array are typically the same, each is provided with a color filter that permits
certain wavelengths of light to pass and rejects others.
The geometric structure of a pixel, which is something like a tunnel, also has a significant impact
on the signal current density image. The position and width of the opening to the tunnel
determine the pixel aperture. Ordinarily the photodetector is at the bottom of the tunnel in the
silicon substrate. In modern CMOS imagers, usually built using multiple metal layers, the pixel
depth can be as large as the pixel aperture. Imagine a photon that must enter through the
aperture and arrive safely at the photodetector at the bottom. If the pixel is at the edge of the
sensor array, the photon’s direction as it travels from the center of the imaging lens must be
significantly altered. This re-direction is accomplished by a microlens, positioned near the
aperture. The position of each microlens with respect to the pixel center varies across the array
because the optimal placement of the microlens depends on the pixel position with respect to the
imaging lens.
As the photon travels from the aperture to the photodetector, the photon must pass through a
series of materials. Each of these has its own refractive index and thus can scatter the light or
change its direction. The optical efficiency of each pixel depends on these materials.
Space-time integration. After accounting for the photodetector spectral quantum efficiency, the
various filters, the microlens array, and pixel vignetting, we can compute the expected current
per unit area at the sensor. This signal current density image is represented at the same spatial
sampling density as the optical irradiance image.
The next logical step is to account for the size, position and exposure duration of each of the
photodetectors by integrating the current across space and time. In this stage, we must
coordinate the spatial representation of the optical image sample points and the pixel positions.
Once these two images are represented in the same spatial coordinate frame, we can integrate the
signal across the spatial dimensions of each pixel. We also integrate across the exposure duration
to calculate the electrons accumulated at each pixel.
Incorporating sensor noise. At this stage of the process, we have a spatial array of pixel
electrons. The values are noise-free. In the third step we account for various sources of noise,
including the photon shot noise, electrical noise at the pixel, and inhomogeneities across the
Photon shot noise refers to the random (Poisson) fluctuation in the number of electrons captured
within the pixel even in response to a nominally identical light stimulus This noise is an
inescapable property of all imaging systems. Poisson noise is characterized by a single rate
parameter that is equal both the mean level and the variance of the distribution.
There are a variety of electrical imperfections in the pixels and the sensor. Dark voltage refers to
the accumulation of charge (electrons) even in the absence of light. Dark voltage is often referred
to as thermally generated noise because the noise increases with ambient temperature. The
process of reading the electrons accumulated within the pixel is noisy, and this is called read
noise. Resetting the pixel by emptying its electrons is an imperfect process, and this noise is
called reset noise. Finally, the detector only captures a fraction of the incident photons in part
because of the material properties and in part because the photodetector only occupies a portion
of the surface at the bottom of the pixel. The spectral quantum efficiency is a wavelength-
dependent function that describes the fraction of the photons that produce an electron. The fill-
factor is the percentage of the pixel that is occupied by the photodetector.
Finally, there is the inevitable variance in the electrical linear response function of the pixels.
One variation is in the slope of the response to increasing light intensity; this differs across the
array and is called photodetector response nonuniformity (PRNU). Second, the offset of the
linear function differs across the array, and this variance in the offset is called dark signal
nonuniformity (DSNU). PRNU and DSNU are a type of fixed pattern noise – another source of
FPN is due to variation in column amplifiers
Over the years, circuit design has improved greatly and very low noise levels can be achieved.
Also, various simple acquisition algorithms can reduce sensor noise. An important example is
correlated double sampling. In this method the read process includes two measurements – a
reference measurement and a data measurement. The reference measurement includes certain
types of noise (reset noise, FPN). By subtracting the two measurements, one can eliminate or
reduce these noises. Correlated double sampling does not remove and may even increase, other
types of noise (e.g., shot noise or PRNU variations) [35, 36]
Analog to digital conversion. In the fourth step, we convert the current to a voltage at each pixel.
In this process we use the conversion gain and we also account for upper limit imposed by the
voltage swing. The maximum deviation from the baseline voltage is called the voltage swing.
The maximum number of electrons that can be stored in a pixel is called the well-capacity. The
relationship between the number electrons and the voltage is called conversion gain (volts/e-).
In many cases, the output voltage is further scaled by an analog gain factor; this too can be
specified in the simulation. Finally, the voltages are quantized into digital values.
Next, we review the general principles of auto-exposure, demosaicking and color-balancing.
Other chapters cover these and other topics (e.g. noise removal and sharpening) in more detail.
The most important aspect of the exposure duration is to guarantee that the acquired image falls
in a good region of the sensor’s sensitivity range. In many devices, the selected exposure value
is the main processing step for adjusting the overall image intensity that the consumer will see.
In these types of applications, the exposure duration must not only guarantee that the image falls
within the sensor range, but it is also part of the image rendering pipeline that should guarantee a
pleasing image. This is particularly true for imaging products in which the acquired data are
rendered almost immediately, say on a mobile phone. Hence, auto-exposure algorithms are a key
element of the image-rendering pipeline that influence image quality.
There is a very small academic literature on auto-exposure algorithms[37-39], although there is a
large patent literature [40-42]. Many of the first digital cameras used a separate metering system
to set exposure duration, rather than using data acquired from the sensor chip. More recently, the
exposure-metering function has been integrated onto the main imaging sensor (usually called
through-the-lens, or TTL, metering).
Methods for auto-exposure in digital cameras are based on methods first developed for film
cameras. Experts and photographers in film photography use a measure called exposure value
(EV) to specify the relationship between the f-number, F, and exposure duration, T :
log 2log ( ) log ( )
⎝⎠ (1.9)
The exposure value becomes smaller as the exposure duration increases, and it becomes larger as
the f-number grows.
Most auto exposure algorithms work in the following way. First, take a picture with a pre-
determined exposure value (EVpre). Then, calculate a single statistic from the brightness image,
Bpre, such as the center-weighted mean in the average of the red and green channels. Assume
that we know what the ideal brightness statistic, Bopt, should be. This value is typically selected
empirically. The optimum exposure value, EVopt , is then defined as
log ( ) log ( )
opt pre pre opt
EV EV B B=+ (1.10)
Auto-exposure algorithms differ in how they derive the single number Bpre from the picture. For
example, Bpre might correspond to the mean brightness across the whole image or a center-
weighted mean that puts more weight on the center part than the surrounding area. And, of
course, there are many variants of these methods. One could average multiple areas in an image,
calculate the median instead of the mean brightness, and analyze the green channel only. Notice
that the methods we described assume that only exposure duration varies. This is a valid
assumption for digital cameras that have a fixed aperture.
Typically, each pixel in the sensor image is red, green OR blue. To display an image, each pixel
must have a red, green AND blue value. We create the display image from the sensor pixel
mosaic by interpolating the missing values. This interpolation process is called “demosaicking”.
(a) (b) (c) (d)
Figure 5. An illustration of color filter array (CFA) sampling. Each pixel captures information about only one color
band. (a) A cropped image from a Mackay ray chart, (b-d) The red, green, and blue CFA samples, respectively, from
a Bayer CFA.
Demosaicking has attracted interest from a wide range of people in signal processing and applied
mathematics. The reader can find an accessible survey in a recent article [43]. Demosaicking
algorithms draw on a diverse array of signal processing techniques, e.g., inverse problems [44],
neural networks [45], Wavelets [46], Bayesian statistics[47, 48], convex optimization [49] etc.
Here we present a brief overview of several approaches to demosaicking. We classify these
algorithms into a few categories that loosely follow the progression of algorithm development in
the field.
(a) Within channel interpolation
The first group of algorithms interpolates the color planes separately. That is, only the data
from, say, the red pixels are used to interpolate the red values at the missing locations. The only
prior information that can be used in this approach is the smoothness of each color channel. This
is a major disadvantage since some very useful information about inter-channel correlation,
which is critical to good demosaicing performance, is neglected. Methods such as bilinear
interpolation, channel-wise spline interpolation, etc. fall in this category.
Figure 6 shows results of interpolating each channel separately using two within channel
interpolation methods. Figure 6b shows a nearest neighbor interpolation and Figure 6c shows a
bilinear interpolation. In this case the reconstruction produces visible color fringing that is not
present in the original image. These methods are simple but as demosaicking algorithms
developed the limitations of within channel interpolation quickly became apparent.
(a) (b) (c)
Figure 6. Comparison of within channel interpolation methods. (a) Original image, and channel-wise demosaicked
images using (b) the nearest neighbor method, and (c) bilinear interpolation.
(b) Combined channel interpolation
A second generation of algorithms uses data from all color planes to interpolate each missing
value. There is a wide range of principles that have been offered to guide this interpolation.
One key observation, common to these algorithms, is that the response at a pixel is correlated
with other nearby pixels even when they are in different color planes. The reason for this
correlation can be understood from first principles. The value of a pixel response is the sum over
wavelength of the local irradiance weighted by the pixel’s spectral quantum efficiency (Section
‘Sensor’, Eq. 1.7). The irradiance signal is typically a smoothly varying function of wavelength.
Further, in most cameras, the spectral bandwidths of color filters overlap. Consequently, the
neighboring pixel values, even those in different color channels, are themselves correlated.
Even more correlated than the mean pixel levels are the local differences in pixel levels. This
spatio-chromatic correlation is easily revealed by simple experiments with pictures. We
separated an image into its R, G, and B bands. Within each band, we calculated the second
difference in the horizontal and vertical directions. These local differences measure the
horizontal and vertical high frequency components. The scatter plots in Figure 7bc show that the
local differences in the R-channel are highly correlated with the local differences in the G-
channel. This correlation is present for both the horizontal and vertical directions. In our
experience, the correlation coefficients are generally higher than 0.95 [46].
(a) (b) (c)
Figure 7. Scatter diagrams of color channels for image 9 from the dataset in Kodak’s PhotoCD PCD0092.
Axes colors correspond to channel colors. (a) RGB color image (b) horizontal high frequency component
of the red channel vs. the similar component of the green channel, and (c) vertical high frequency
component of the red channel vs. the similar component of the green channel.
The correlation of the mean and of the local differences has been used to motivate many
demosaicing algorithms [50-54]. Practical implementations of demosaicking that exploit high
inter-channel correlations rely on the higher rate of sampling in the green CFA channel in the
ubiquitous Bayer array. In general terms, most algorithms derive the best estimate of high
frequency image information using the green channel. They then use the inter-channel
correlation to improve the interpolation in the red and blue channels.
A second key observation common to demosaicking algorithms concerns identifying locations in
an image where the correlations break down. These failures typically happen at object edges.
Edges can be approximately identified by finding locations where there are larger differences
between adjacent pixel levels. Many algorithms proceed by identifying these locations and
interpolating along the edge, but preventing interpolation across the edge. Such algorithms are
often called adaptive because the interpolation method adapts to the local image properties. We
illustrate three adaptive algorithms in Figure 8.
(a) (b) (c)
Figure 8. The Mackay image in Fig 1a demosaicked using (a) adaptive Laplacian (b) projections on convex sets
[49] , and (c) adaptive homogeneity [50]
(c) Combined color and space interpolation
The field of demosaicking continues to produce new and innovative ideas. An interesting idea
suggested by Alleysson and colleagues [55] interprets the CFA-based acquisition in the spatial
frequency domain.
Consider a frequency representation of the CFA without regard to the color of individual pixels.
In the transform domain, different spatial frequency terms capture information about different
color combinations. Consider the most trivial case of a one Bayer super-pixel, (G, R; B, G). The
difference between the left column (G + R) and the right column (B + G) is referred to as the 1
cylce vertical term. This difference is B – R because the two G terms cancel. Hence, the
amplitude of the CFA transform coefficient at 1 cycle per super-pixel represents a measurement
difference between the B and R samples.
There are additional complexities when we consider the full array. Alleysson and colleagues
explain how to extract color information from the Fourier transform of the full color filter array
of a Bayer pattern. Dubois [56] proposed an adaptive extension of this framework. Hirakawa
[57] further extended the framework to design novel CFA patterns optimized for this
computational approach. The results of demosaicking the Mackay ray image in Fig. 1a using
Dubois’s method [56] based on these ideas is shown in Fig. 9.
Figure 9. The Mackay image in Fig 1a demosaicked using Dubois’s method [56].
The choice of demosaicking algorithm can have a significant impact on the final image quality. It
is not possible, however, to assess demosaicking in the absence of knowledge about other
imaging components. For example, in a very low noise system simple adaptive algorithms may
perform well. In a high noise system, it may be better to use non-adaptive interpolation methods.
Further, it is necessary to consider the interaction between demosaicking and other image
processing algorithms, such as denoising and color balancing. Demosaicking changes the image
statistics by introducing structure into the noise. This can adversely affect the performance of
later denoising and sharpening algorithms. Considering the capabilities of the sensor, and the
color imaging pipeline in its entirety is important when evaluating demosaicking methods.
If one simply copies the sensor pixel values into the display values, the resulting image will not
generally be a good color representation of the original scene. A color transform, also called a
color correction, is used to convert the sensor data to display values that create a perceptual
match between the original and displayed images. Figure 10 compares corrected and uncorrected
color images.
There are several factors that must be taken into account in designing the color transform. First,
imaging sensors do not generally have the same spectral response as human viewers. Thus, the
device and human eye weight the spectral image differently. Second, the properties of the display
medium must be taken into account; for example, the transformation must be adjusted for
different display primaries. Third, the camera often acquires the scene under ambient lighting
conditions that differ from those in which humans view the display. The human visual system
adjusts its sensitivities (adapts) in response to ambient lighting conditions. The differences in
adaptation can be very large, and adaptation often causes the same physical stimulus to appear
different under the two conditions. As a first-order approximation, adaptation between ambient
viewing conditions generally preserves surface colors. Hence, the consequence of visual
adaptation is commonly referred to as “color constancy”[58]. The color transform must account
both for the physical properties of the sensor and display, as well as for the physiological
adaptation between different ambient viewing conditions.
The complexity of the problem has spawned a large, complex and interesting literature that inter-
mixes device and physiological modeling. The scope of that literature is too large for this
review. Here we describe a simple but common approach for designing a color transform. To
explain this approach, we first describe how to set the display primary intensities to create a
radiance whose appearance matches the light reflected from a surface illuminated by a known
illuminant, say 6500 Kelvin daylight. We refer to this process as display rendering. Then we
describe how to calculate the expected sensor rgb values to this surface under a known light. We
refer to this process as image capture. Finally, we describe a computational approach for
converting the captured image data to display values. We refer to this process as color
Figure 10. The top left image (a) is the demosaicked (but unbalanced) sensor image of a Macbeth ColorChecker
illuminated with daylight (D65). The top right image (b) is based on a sensor image of the Macbeth ColorChecker
illuminated with a tungsten source. The bottom images have been corrected using two different 3x3 matrix
transforms, one optimized for daylight (c) and the other for tungsten (d).
Given a surface reflectance, ( )S
, and an illuminant with spectral energy, ( )E
, the light
scattered toward the viewer will be ( ) ( )SE
. The impact of this light on the human eye is
described by three numbers, the tri-stimulus coordinates, (X,Y,Z). The X value is computed
using the simple formula:
()() ()
= (1.11)
The integral is taken over the visible wavelengths, usually over the range from 370 to 770 nm.
To compute the Y and Z terms, one simply replaces ()x
with similar standardized
functions (), ()
z. These functions are an international standard found in many references [59].
To produce the same visual effect on a human subject as the surface and illuminant combination,
we adjust the display rgb values to emit a light with the same XYZ values. The relationship is
easiest to see in matrix form. Suppose that we place sampled values (assume N samples) of the
CIE tristimulus functions, (),(),()xyz
in the columns of an Nx3 matrix, V. We place the
color signal ()()SE
in a vector, c. Then XYZ tristimulus values in the column vector, t, are
computed from the matrix product t
tVc=. Suppose there is a display with three primary lights
with peak spectral power distributions ( )
. We place these three sampled wavelength
functions into the column of a matrix P. We combine the three relative intensities of the primary
lights, i
winto a column vector w. To match the surface on the display, we require the color
signal and the display to produce the same tristimulus coordinates
tVcVPw== (1.12)
When an observer views the display and the original color signal in the same context, they will
produce the same tristimulus values and thus match one another. We can solve for the intensities
of the three primaries using the equation
= (1.13)
There are two practical notes. First, the primary intensities (weights) for most displays are
related to the digital values by a nonlinear function called the display gamma. This must be
accounted for during a calibration process [60, 61] . Second, in some instances the solution for
the weights will be outside of the range (0,1). In that case, the color signal is beyond the gamut
for that display.
The vector of camera rgb values,u, produced by exposure to c can be calculated using a similar
set of equations. Suppose that the spectral responsivity of the red channel is ()r
. Then the red
pixel level will be proportional to
= (1.14)
Shifting to matrix notation, if we place the sampled wavelength functions of the three pixel
spectral responsivities into the columns of the Nx3 detector matrix, D, the camera rgb values in
the vector will be
The goal of the color transform is to specify a means for converting the camera rgb values uinto
display primary intensities w. The simplest case is when the camera and display viewing
contexts are the same. In that case, a color signal c produces a camera response uand is
matched by display primary intensities w. One solution is to create a pair of matrices, Uand W,
whose columns are corresponding pairs of camera responses and display primaries. We then find
a color transform matrix
that minimizes the least squares error of the equation WMU=, or
minimizes the error in a perceptual space (e.g., CIELAB). Nonlinear functions or various more
complex transforms based on look-up tables can also be used.
When the ambient lights in the display context and the scene differ, the situation is more
complex. In that case, a tristimulus match between the original scene and the display will not be
a satisfactory perceptual match. Consequently, we must find a new way to identify the display
rgb values that represent the appearance of the original. It is during this calculation that the
principles of human color constancy play a role.
As an example, assume that we know the spectral reflectance of several surfaces in the original
scene. In that case, we can render the scene on the display as if it were illuminated by a standard
light for display viewing, such as a daylight spectral power distribution (e.g., D65). This
provides the target display values, W. The rationale for this choice is that color constancy
implies that perceived color is predicted by surface reflectance, not color signal. A simple way
to implement this strategy is this: 1) Render a specific color target on the display under the
standard illuminant, 2) acquire sensor images of that target under a series of different ambient
illuminants, and 3) find a transformation between each of these sensor images and the display.
The transform will vary as we change the ambient illumination for the camera image. For
example, when the surfaces are illuminated by tungsten, we calculate a transform, T
, that
maps the linear camera rgb values into the linear display rgb values that generate the perception
that the surfaces were illuminated by daylight. When these surfaces are illuminated by a
fluorescent light, we calculate a different transform F
. In general, the transform depends on
the spectral power of the scene illuminant, the standard display viewing illuminant, the camera,
the surfaces used in the target, and the criterion that we use for the minimization – it might be a
least-squares minimization or a minimization with respect to another error metric (e.g.,
At this stage, we have described how to specify a transformation from the sensor data to the
display data for a range of illuminants. There is one final step required to complete the color
balancing algorithm. Given a camera image, we must estimate the illuminant so we know which
transform to apply. Figure 11 shows what happens when a camera image is corrected using a
color transform that is inappropriate for the scene illumination. Many illuminant estimation
algorithms are described in the literature [62-70]. Some of these algorithms are simple and
based on image statistics such as the mean rgb value or the ratio of the red and blue sensors.
Others involve more elaborate Bayesian computations [65] or Retinex-style algorithms [71, 72].
Figure 11. Color images that have been corrected using a 3x3 transform that is optimized for daylight (D65). The
left image (a) is based on a camera image of the Macbeth ColorChecker under the D65 daylight illumination and the
right image (b) is based on a camera image of the Macbeth ColorChecker illuminated by tungsten.
People judge the quality of a digital imaging system by viewing the final output rendered on a
display or on a printer. Hence, we cannot evaluate the quality of a digital imaging system
without simulating (or measuring) the intended display. In this section, we describe how to
calculate the spatial-spectral radiance of a displayed image. This is the visual stimulus that
actually reaches the eye.
A display can be satisfactorily characterized if emissions from the three color channels are
independent [60]. Specifically, the emission from one channel should not be influenced by the
levels in the other channels. When this condition is met, the spatial-spectral radiance for any
image rendered on the display can be predicted by three types of measurements – the display
gamma, the spatial point spread functions for the red, green and blue pixel components and the
spectral power distributions of the display color primaries [2]. The display gamma converts
digital values into a measure of the linear intensity. The point spread functions describe the
spatial spread of light for each component within a color pixel, which we call the sub-pixels.
The spectral power distributions describe the spectral radiance emitted from each sub- pixel. .
Figure 12. Display gamma for a calibrated LCD monitor. The gamma is comparable for other LCD, CRT and
Plasma displays measured in our laboratory.
Figure 13. Magnified images of a white pixel in 6 different displays. Top images (a, b and c) are pixels in different
LCD displays. The bottom images show pixels in d) a CRT with an aperture grille, f) a CRT with an hexagonal
pixel array, and g) a plasma display.
Figure 14 Spectral power distributions of red, green and blue color primaries in an LCD (top), CRT(middle) and
Plasma display (bottom).
We model the sub-pixels as (a) space-wavelength separable and (b) independent. Separable
means that the spectral power distribution emitted by a component is the same across the entire
pixel. Formally, the spatial-spectral distribution of light emitted from the ith sub-pixel can be
defined by the product of two functions:
(, , ) (, ) ()
pxy sxyw
The function ( )
describes the spectral power distribution of the ith sub-pixel. The function
(, )
sxyis the spatial spread of the light from that sub-pixel. This additivity assumption means
that the light emitted from the ith sub-pixel does not depend on the intensity of the other sub-
pixels. In this case, the radiance emitted by a pixel is the sum of the radiance emitted by the
(, , ) (, , )
xy p xy
We introduce notation to describe the static nonlinearity between the digital value, v = (R,G,B),
and the emitted light, referred to as the display gamma. We describe the static nonlinearity for
the ith sub-component as ()
gv. The ith gamma function converts the digital controller values, v =
(R,G,B), into the intensity of each sub-pixel. Taking all of these assumptions together, we
expect the spatial-chromatic image from a pixel, given a digital input, (R,G,B), to be
(, , ) () (, , )
() (, ) ( )
ii i
pxy g vp xy
These equations apply to the light emitted from a single pixel. We create the full display image
by repeating this process across the array of display pixels. In so doing, we assume that the light
emitted from a pixel is independent of the values at adjacent pixels. We refer to the spatial-
spectral independence between pixels as display-independence [60].
Farrell et al [2] describe four conditions that are necessary for display-independence. First, the
relative spectral power distribution of the display color primaries should be invariant as digital
value increases (spectral homogeneity). Second, the spectral power distribution of any
combination of pixel components can be predicted by the sum of the spectral power distribution
of the individual pixel components measured separately (spectral additivity). Third, the relative
spatial spread of each pixel component should be unchanged as digital values increase (spatial
homogeneity). And fourth, the spatial distribution of light emitted by any combination of pixels
is predicted by the sum of the spatial light distribution of the individual pixels (spatial additivity).
Many LCD displays meet all four conditions. For CRT displays, however, spatial additivity
holds only for vertically adjacent pixels and fails for horizontally adjacent pixels[2, 73]. The
failure of spatial additivity for CRT displays can be explained by sample and hold circuitry.
CRT manufacturers use sample and hold circuitry to compensate for the slew rate limitations of
the electron beam as it moves horizontally across the screen.
It is possible to create a more complex model for CRTs. For example, the failures of additivity
can be used as an empirical basis for modeling the distortion introduced by the sample and hold
circuitry. Our measurements and display simulations show that this added complexity is
necessary to adequately characterize these CRT displays and probably many others. But LCD
displays are rapidly replacing CRT displays, both in the home and in the office. They are also an
integral part of the imaging pipeline for digital cameras and camera phones. It is fortunate that
most of the LCD devices we have tested satisfy the simple model properties. This makes it easier
to calibrate, control and model these displays and thus predict their effect in the imaging
We have created software models for the scene, optics, sensor, processor and display in an
integrated suite of Matlab® software tools - the Image Systems Evaluation Toolbox, or
ISET[74]. The ISET simulation begins with scene data; these are transformed by the imaging
optics into the optical image, an irradiance distribution at the image sensor array; the irradiance
is transformed into an image sensor array response; finally, the image sensor array data are
processed to generate a display image
In this section we use ISET to model the scene, optics and sensor of a 5 megapixel CMOS digital
camera that we calibrated in our laboratory.
The sensor manufacturer provided many of the simulation parameters. Other parameters, such as
read noise, dark voltage, DSNU and PRNU were estimated from measurements in our laboratory
(see Appendix A). We modeled the camera lens using a diffraction-limited model2 with a lens f-
number of 4 and a focal length of 3 mm. We simulated the effects of an optical diffuser that
filters out signals above the Nyquist frequency limit of the imaging sensor. This was
accomplished using a Gaussian filter with full-width half maximum equal to the pixel width.
Table 1 lists the sensor parameters along with the reference source for the data.
Table 1: Sensor parameters for ISET simulations
Sensor Parameter Parameter Value Reference Source
Pixel width (µm) 2.2 manufacturer
Pixel height (µm) 2.2 manufacturer
CFA pattern gbrg manufacturer
Spectral quantum efficiencies Measured (see Appendix B)
Dark voltage (V) 4.68 mV/sec Measured (see Appendix A)
Read noise (mv) 0.89 Measured (see Appendix A)
DSNU (mv) 0.83 mV. Measured (see Appendix A)
PRNU (%) 0.736 Measured (see Appendix A)
Fill factor 45% manufacturer
Well capacity (electrons) 9000 manufacturer
Voltage swing (V) 1.8 manufacturer
Conversion Gain (µV/e) 2.0000e-004 manufacturer
Analog gain 1.0 software setting
Exposure duration (s) 100 measured
Scene luminance (cd/m2) 61 measured
Lens f-number 4 lens setting
Lens focal length (mm) 3 lens calibration software[75]
The sensor spectral quantum efficiencies for the red, green and blue pixels were calculated by
combining the effects of the lens transmittance, color filter arrays and photodiode quantum
efficiency into one spectral sensitivity function for each red, green and blue pixels, respectively
(Figure 15). Appendix B describes the laboratory measurements and calculations used to
estimate these spectral curves.
Figure 15. Spectral quantum efficiencies of red, green and blue channels for a calibrated imaging sensor.
To evaluate the quality of the simulation, we first measured the radiance image of a Macbeth
ColorChecker using the methods described in the Scene section. We used these measured
radiance data as the scene input in ISET simulations. Second, we acquired an image of this scene
using the real camera. We compared the simulations with the real acquisition in several ways.
Figure 16 compares the predicted and simulated sensor images of the Macbeth ColorChecker
after it is demosaicked using bilinear interpolation. These processed images (not color-balanced)
illustrate the similarity between the measured and simulated sensor images. For a quantitative
comparison, we calculated the mean and standard deviation of the pixels values for the 24 color
patches in the sensor images of the Macbeth ColorChecker. A scatter plot of the signal-to-noise
(mean divided by the standard deviation) in the measured and simulated images also shows good
agreement for each of the twenty four patches in the Macbeth Color Checker (Figures 17 and
Figure 16. Measured (a) and simulated (b) sensor images. The images have been demosaicked but not color-
Figure 17. Mean pixel values averaged for each of the twenty four patches in the Macbeth Color Checker in the
measured sensor images plotted against mean pixels values for the same patches in the simulated sensor images.
Figure 18. Standard deviation in pixel values for each of the twenty four patches in the Macbeth Color Checker in
the measured sensor images plotted against standard deviation in pixels values for the same patches in the simulated
sensor images.
An important simulation objective is to understand how accurately a sensor can render a color
scene. The color accuracy of the measured and simulated imaging sensors are characterized in
Figures 19 and 20, respectively. In this example we render a sensor image of a Macbeth
ColorChecker. To perform the rendering we specify the linear sRGB [76] display values of an
image that match a Macbeth ColorChecker under D65 illumination. These are the desired sRGB
values. Then, we find the 3x3 matrix that best transforms (in a least squares sense) the sensor
RGB values into the desired sRGB values. We compare the transformed and desired sRGB
values for the measured (Fig 19) and simulated sensor images (Fig 20). Histograms of the
CIELAB color difference error (E) are also shown. Again, the simulation predicts the color
accuracy of the sensor quite well.
Figure 19. Color accuracy of measured imaging sensor. The graph on the left plots the desired sRGB values against
the color-balance sRGB values derived from the measured sensor images. The histograms show the distribution of
ΔE color errors for the 24 color patches in the Macbeth ColorChecker.
Figure 20. Color accuracy of simulated imaging sensor. The graph on the left plots the desired sRGB values against
the color-balanced sRGB values derived from the simulated sensor images. The histograms show the distribution of
ΔE color errors for the 24 color patches in the Macbeth ColorChecker.
Figures 16 to 20 show that there is a good correspondence between measured and simulated
sensor performance. The mean and variance in pixel values are nearly the same for simulated
and measured sensor images (see Figures 17 and 18). And both simulated and measured sensor
images have comparable color accuracy (see Figures 19 and 20).
The simulation parameters were derived from a few fundamental measurements that characterize
sensor spectral sensitivity and electrical properties including dark current, read noise, dark signal
non-uniformity and photoreceptor non-uniformity. While we estimate these parameters from a
modest set of calibration measurements (see Appendix A and B), these sensor parameters are
often provided by the sensor manufacturer in the form of a product data sheet.
Once we have confidence in the simulations, it is further possible to estimate how the sensor will
perform when acquiring scenes that are difficult to recreate or calibrate (high dynamic range, low
light levels, and so forth). It is also possible to evaluate the effects that optical and sensor
components have upon perceived image quality. As a simple example, Figure 21 shows the
effects of reducing scene luminance; Figure 22 illustrates the effect that pixel size has on image
quality. At any fixed exposure duration, the number of photons captured by the imaging sensor
decreases with pixel size (Figure 22) and the results will be equivalent to reducing scene
luminance, as illustrated in Figure 21.
We have used ISET simulations to investigate the tradeoff between pixel size and light
sensitivity[77], the effects of camera motion [78], different sources of sensor noise [79], novel
color filter arrays [80], and many other system parameters. Continuing validation of the
simulation technology and methods for estimating the sensor characteristics should lead to high
efficiencies for sensor evaluation. Ultimately, as we gain increasing confidence in the
simulation, it can be used to design novel sensor designs as part of the manufacturing process.
Figure 21. Processed (demosaicked and color-balanced) images based on ISET simulations. The left image shows
the results when the mean scene luminance is 5 cd/m2 and the exposure duration is 20 msec. The right image shows
the results when the mean scene luminance is 100 cd/m2 and exposure duration is 20 msec. The pixel size is 2.2
micron for both simulations. Other sensor parameters are listed in Table 1.
Figure 22. Processed (demosaicked and color-balanced) images based on ISET simulations of a sensor with 2.2
micron pixels with 45% fill factor (left) and a sensor with 4.4 micron pixels with 90% fill factor (right). The mean
scene luminance is 5 cd/m2 and the exposure duration is 20 msec. in both simulations. All other sensor parameters
are listed in Table 1.
In this chapter we describe how to model and simulate the complete image processing pipeline of
a digital camera, beginning with a radiometric description of the scene captured by the camera
and ending with a radiometric description of the final image rendered on an LCD display. The
ability to simulate sensor performance provides several advantages compared to a simulation
entirely on the experimental bench.
First, creating and calibrating images to test components is essential; yet this is a major
bottleneck in the laboratory. The availability of digitally accurate radiometric scenes for use in a
simulator enables the user to predict sensor performance for a wide range of scenes that are
difficult to create in the laboratory.
Second, a simulator makes it possible to evaluate the effects that different optical and sensor
components have upon image quality, as well as different post-processing algorithms[80]. For
example, the simulator could be used to verify the properties of a particular sensor in different
imaging conditions or when coupled with different types of lenses.
Third, a simulator offers a common standard between laboratories. In principle, different
laboratories can communicate about a sensor by simply sending the data files that characterize
the sensor to one another. In summary, accurate simulations can determine which design
changes will meet customers’ expectations of image quality, without incurring the cost of
building a new module.
We gratefully acknowledge support from Hewlett Packard, Logitech International, Olympus
Corporation, Microsoft Corporation, Nikon Corporation and the Samsung Advanced Institute of
Technology. We thank Norihiro Aoki from Nikon for providing us with many wonderful Nikon
cameras and lenses and Bernd Girod for providing us with lab facilities. And we thank our many
colleagues who have contributed to this work, including David Brainard, Tien Chen, Jeff
DiCarlo, Keith Fife, Abbas El Gamal, Kunal Ghosh, Francisco Imai, Max Klein, Ricardo Motta,
Amnon Silverstein, Poorvi Vora, Feng Xiao and Xuemei Zhang.
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We describe how to estimate several types of sensor noise from unprocessed (“raw”) sensor data.
Dark voltage or dark current is thermally-generated electron noise in the absence of light. To
characterize dark voltage, capture many images in the dark at a set of different exposure
durations. The rate of increase in pixel digital values (DV) over time is proportional to the dark
voltage. The dark voltage (volts/sec) can be derived from the DV/time data using the voltage
swing and number of quantization levels.
Read noise is the variance in digital values from repeated reads of the same pixel. To measure
read noise, capture many images in the dark with the same very short exposure duration. The
exposure duration should be as short as possible in order to avoid contributions from dark
voltage. Read noise is the standard deviation in the multiple measurements obtained in the dark
with the same (short) exposure duration. It has units of volts.
Dark signal non-uniformity (DSNU) is the variability across pixels in dark voltage. DSNU can
be estimated by averaging multiple measurements in the dark with constant exposure duration
and then calculating the standard deviation of the mean pixel value across the array of pixels.
Averaging many dark images reduces the contribution of read noise. DSNU has units of volts.
Photoreceptor non-uniformity (PRNU) is the standard deviation in sensitivity across pixels.
PRNU can be estimated by analyzing raw sensor images of a uniform light field captured at a
series of exposure durations. Do not include sensor images that are dominated by noise at short
durations or saturated at long exposure durations. Then, for each pixel, calculate the increase in
mean digital value as exposure duration increases. The slope differs between color channels
because each has its own light sensitivity. The standard deviation of the slope, measured as a
proportion of the mean slope for that channel, is the same across the colored pixels. This
standard deviation is the PRNU and is dimensionless.
The spectral sensitivity of a color channel depends on the spectral quantum efficiency of the
photodetector and the spectral transmittance of optical elements and filters in the imaging path.
For instrument design, one would like to measure the spectral transmission of all of these.
Measuring the components, however, requires access prior to assembly or disassembling the
device; often neither of these options is practical. Here, we describe how to measure the
combined effect of these components which is called the spectral efficiency of the channel.
To measure the channel spectral efficiency we record the channel response to a series of
narrowband lights. The wavelength range of these lights should span 400 and 700 nm, matching
the range of human vision and the typical range of consumer cameras. One method of creating
such lights is to use a monochromator which separates broadband light into monochromatic
wavelength components. Another way to create narrowband lights is to use a set of LEDs with
different peak wavelengths that sample the visible range [81]
Figure 23 illustrates a procedure for measuring the spectral efficiency of the rgb color channels
in a digital camera. We illuminate a surface with a Lambertian reflectance, such as a flat piece
of magnesium-oxide chalk, with narrowband light from a monochromator. Using a
spectrophotometer, we measure the spectral radiance of each of the narrowband lights (Fig 23a).
The spectral radiance measured for each of the narrowband lights is combined into a matrix, L.
The matrix is N x M, where N is the number of wavelength samples M is the number of spectral
Second, place the camera at the same location used for the spectrophotometer measurement (Fig
23b); capture an image of each of the narrowband lights. For each wavelength band, use the
camera exposure duration that yields the highest SNR. Normalize the linear camera values (i.e.,
correct for differences in exposure duration by dividing the rgb values to obtain a total response
per unit time). To avoid the effects of chromatic aberration, calculate the mean r, g and b values
for the central region of the normalized camera image. Place the corrected rgb values for each of
the M lights in the columns of a 3xM matrix, C. We estimate the channel spectral sensitivity by
using a robust method to solve for S in the linear equation: C = S’L.
We recommend measuring the lens spectral transmissivity independently. Manufacturers use
unique lens coatings that can strongly influence the channel spectral efficiency. Hence, the
channel spectral efficiency can depend strongly on the imaging lens.
Figure 23.A) Illuminate a Lambertian surface with narrowband lights spanning 400 – 800 nm using a
monochromator. Measure the spectral radiance of each of the narrowband lights using a spectrophotometer. B) Place
the camera at the same location used for the spectrophotometer measurement and capture an image of each of the
narrowband lights.
... The commercial, automotive simulation tools, such as Microsoft AirSim [33], Vector DYNA4 [34], Hexagon VTD [35], dSPACE ASM [36], IPG CarMaker [37], and Siemens PreScan [38], besides the vehicle and traffic model, usually include a simplified simulation of specific sensor types using game engines, such as Unreal [39] or Unity [40]. The less complex rasterization and ray casting techniques are commonly used in these render engines, simulating digital cameras [41]- [44], and raw LiDAR point clouds [45]- [48]. There are a lot of alternative approaches for the simulation of raw radar data as well [49], [50], such as time-domain electromagnetic models as in [51]. ...
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The gradually evolving automated driving and ADAS functions require more enhanced environment perception. The key to reliable environmental perception is large amounts of data that are hard to collect. Several simulators provide realistic, raw sensor data based on physical sensor models. However, besides their high price, they also require very high computation capacity. Furthermore, most sensor suppliers provide high-level data, such as object detections, that is complicated to reproduce from simulated raw sensor data. This paper proposes a method that directly simulates the detections or object tracks provided by smart sensors. The model involves several uncertainties of the sensors, such as missed-, false detections, and measurement noise. In contrast to the conventional sensor models, this method tackles with state-dependent clutter model and considers the field of view in the detections model. The parameters of the proposed model are identified for an automotive smart radar and camera based on pre-evaluated real-world measurements. The resulting model provides synthetic object-level data with higher fidelity than the conventional probabilistic models, differing less than 2% from the precision and recall metrics of the actual sensors.
... It is also possible to include the factor of spherical aberration, coma aberration, and astigmatism to illustrate the effect of these parameters on the image quality (J. Farrell, 2012). Motion factor is one of the important influences that affect image quality. ...
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In this work, optical system with different aperture shapes (circular, square, elliptical and triangle aperture) has been used for efficiency evaluation when the system involved moving factor in ideal case (aberration free). The optical system evaluate far moving object, therefore the image forming at image plane due to point spread function (image formula of incoherently illuminated point object). A mathematical treatment has been used to getting results by Gaussian numerical calculations method. The results show priority of circular aperture when optical system that submits of moving factor.
... The Image Systems Engineering Toolbox (ISET) for Cameras is one of the most comprehensive toolkits for assessing the quality of images, including handling consistent radiometric units for physical accuracy [176], [177]. ISET has a wide user base (500+ people) working across 24 different countries. ...
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Huge advancements have been made over the years in terms of modern image-sensing hardware and visual computing algorithms (e.g., computer vision, image processing, and computational photography). However, to this day, there still exists a current gap between the hardware and software design in an imaging system, which silos one research domain from another. Bridging this gap is the key to unlocking new visual computing capabilities for end applications in commercial photography, industrial inspection, and robotics. In this survey, we explore existing works in the literature that can be leveraged to replace conventional hardware components in an imaging system with software for enhanced reconfigurability. As a result, the user can program the image sensor in a way best suited to the end application. We refer to this as software-defined imaging (SDI), where image sensor behavior can be altered by the system software depending on the user’s needs. The scope of our survey covers imaging systems for single-image capture, multi-image, and burst photography, as well as video. We review works related to the sensor primitives, image signal processor (ISP) pipeline, computer architecture, and operating system elements of the SDI stack. Finally, we outline the infrastructure and resources for SDI systems, and we also discuss possible future research directions for the field.
... Using the ISETCam software, [14][15][16][17] we convert the sensor spectral irradiance into electrons captured at each pixel, which are then converted into digital values. ISETCam simulates multiple sensor properties including photon noise, electrical noise, color filter arrays, and sensor geometry. ...
... Using the ISETCam software [14,15,16,17], we convert the sensor spectral irradiance into electrons captured at each pixel, which are then converted into digital values. ISETCam simulates multiple sensor properties including photon noise, electrical noise, color filter arrays, and sensor geometry. ...
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The design and evaluation of complex systems can benefit from a software simulation - sometimes called a digital twin. The simulation can be used to characterize system performance or to test its performance under conditions that are difficult to measure (e.g., nighttime for automotive perception systems). We describe the image system simulation software tools that we use to evaluate the performance of image systems for object (automobile) detection. We describe experiments with 13 different cameras with a variety of optics and pixel sizes. To measure the impact of camera spatial resolution, we designed a collection of driving scenes that had cars at many different distances. We quantified system performance by measuring average precision and we report a trend relating system resolution and object detection performance. We also quantified the large performance degradation under nighttime conditions, compared to daytime, for all cameras and a COCO pre-trained network.
Surveillance systems are image acquisition systems that use commercial security cameras fixed at specific locations in stores, houses, or warehouses to enhance safety and protect staff, assets, and buildings. Although these systems are good security options, a temporal secure area or nearby mobile needs an alternative personal monitoring solution that can be constructed by using the smartphone multimedia resources to access a PC's machine vision software, providing a tool for surveillance. In this chapter, an application is developed using only a smartphone camera and a pattern matching algorithm to detect presence of people in a secure area; when presence occurs, a signal is sent to start a recording and/or activate an alarm to warn the users. A virtual instrument algorithm application was able to detect objects in a room, recognizing who are residents and who are guests. The functionality of this project with encouraging results obtained provides a flexible security instrument.
We present optimized modulation and coding for the recently introduced dual modulated QR (DMQR) codes that extend traditional QR codes to carry additional secondary data in the orientation of elliptical dots that replace black modules in the barcode images. By dynamically adjusting the dot size, we realize gains in embedding strength for both the intensity modulation and the orientation modulation that carry the primary and secondary data, respectively. Furthermore, we develop a model for the coding channel for the secondary data that enables soft-decoding via 5G NR (new radio) codes already supported by mobile devices. The performance gains for the proposed optimized designs are characterized via theoretical analysis, simulations, and actual experiments using smartphone devices. The theoretical analysis and simulations inform our design choices for the modulation and coding, and the experiments characterize the overall improvement in performance for the optimized design over the prior unoptimized designs. Importantly, the optimized designs significantly increase usability of DMQR codes with commonly used QR code beautification that cannibalizes a portion of the barcode image area for the insertion of a logo or image. In experiments with a capture distance of 15 inches, the optimized designs increase the decoding success rates between 10% and 32% for the secondary data while also providing gains for primary data decoding at larger capture distances. When used with beautification in typical settings, the secondary message is decoded with a high success rate for the proposed optimized designs, whereas it invariably fails for the prior unoptimized designs.
Image systems simulation software can accelerate innovation by reducing many of the time-consuming and expensive steps in designing, building and evaluating image systems. To realize this potential, it is necessary to build trust in physics-based end-to-end image systems simulations. Towards this goal, we describe and experimentally validate an end-to-end physics-based image systems simulation of a digital camera. The simulation models the spectral radiance of 3D-scenes, formation of the spectral irradiance by multi-element optics, and conversion of the irradiance to digital values by the image sensor. We quantify the accuracy of the simulation by comparing real and simulated images of a precisely constructed, three-dimensional high dynamic range test scene.
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The booming mobile imaging market is aggressively pursuing CMOS imagers with higher spatial resolution and smaller form factors. Given a fixed die size, reducing pixel size seems like a straightforward way to increase spatial resolution. The reduced sensitivity of smaller pixels, however, must be compensated by longer exposure durations. Longer exposure durations result in camera motion blur when the camera is handheld, effectively reducing spatial resolution. We empirically measured how the exposure duration, camera mass and subject's skill level affect camera motion. We then used these measurements to calculate the effect that camera motion has upon the system MTF for CMOS imagers with different pixel sizes at different scene illuminance levels. We described the scene illuminance conditions for which imaging sensors with larger pixels produce sharper images than imagers with smaller pixel sizes.
Spectral distributions of 622 samples of daylight (skylight, and sunlight plus skylight) have been subjected to characteristic vector analysis, as composite data and in three subgroups (99 distributions measured by Budde; 249, by Condit; and 274, by Henderson and Hodgkiss). The chromaticity coordinates (x,y) computed from these distributions have been compared with direct visual determinations of chromaticity coordinates of daylight by Nayatani and Wyszecki, and by Chamberlin, Lawrence, and Belbin. It was found that the chromaticities indicated by the spectral distributions and by direct visual colorimetry cluster about the curve: y = 2.870x-3.000x2-0.275. This curve of typical daylight chromaticities falls slightly on the green side of the Planckian locus. From the mean and the first two characteristic vectors of the composite data, spectral distribution curves have been reconstituted by choice of scalar multiples of the vectors such that the chromaticity points fall on the curve of typical daylight chromaticities at places corresponding to correlated color temperatures of 4800°, 5500°, 6500°, 7500°, and 10000° K. The representative character of these reconstituted spectral-distribution curves has been established by comparison with the measured curves from each subgroup yielding the closest approximation to the same chromaticities. The agreement so found suggests that this family of curves is more representative of the various phases of daylight between correlated color temperatures 4800° and 10000°K than any previously derived distributions.
Single-chip digital cameras use a color filter array and subsequent interpolation strategy to produce full-color images. WHile the design of the interpolation algorithm can be grounded in traditional sampling theory, the fact that the sampled data is distributed among three different color planes adds a level of complexity. Previous ways of treating this problem were based on computationally intensive approaches, such as iteration. Such methods, while effective, cannot be implemented in todays crop of digital cameras due to the limited computing resources of the cameras and the accompanying host computers. These previous methods are usually derived from general numerical methods that do not make many assumptions about the nature of the data. Significant computational economies, without serious losses in image quality, can be achieved if it is recognized that the data is image data and some appropriate image model is assumed. To this end, the design of practical, high- quality color filter array interpolation algorithms based on a simple image model is discussed.
The charge-coupled device dominates an ever-increasing variety of scientific imaging and spectroscopy applications. Recent experience indicates, however, that the full potential of CCD performance lies well beyond that realized in devices currently available. Test data suggest that major improvements are feasible in spectral response, charge collection, charge transfer, and readout noise. These properties, their measurement in existing CCDs, and their potential for future improvement are discussed in this paper.
We describe a method for simulating the output of an image sensor to a broad array of test targets. The method uses a modest set of sensor calibration measurements to define the sensor parameters; these parameters are used by an integrated suite of Matlab software routines that simulate the sensor and create output images. We compare the simulations of specific targets to measured data for several different imaging sensors with very different imaging properties. The simulation captures the essential features of the images created by these different sensors. Finally, we show that by specifying the sensor properties the simulations can predict sensor performance to natural scenes that are difficult to measure with a laboratory apparatus, such as natural scenes with high dynamic range or low light levels.
When the size of a CMOS imaging sensor array is fixed, the only way to increase sampling density and spatial resolution is to reduce pixel size. But reducing pixel size reduces the light sensitivity. Hence, under these constraints, there is a tradeoff between spatial resolution and light sensitivity. Because this tradeoff involves the interaction of many different system components, we used a full system simulation to characterize performance. This paper describes system simulations that predict the output of imaging sensors with the same dye size but different pixel sizes and presents metrics that quantify the spatial resolution and light sensitivity for these different imaging sensors.
In this paper an end-to-end hyperspectral imaging system model is described which has the ability to predict the performance of hyperspectral imaging sensors in the visible through to the short-wave infrared regime for sub-pixel targets. The model represents all aspects of the system including the target signature and background, the atmosphere, the optical and electronic properties of the imaging spectrometer, as well as details of the processing algorithms employed. It provides an efficient means of Monte-Carlo modelling for sensitivity analysis of model parameters over a wide range. It is also capable of representing certain types of non-Gaussian hyperspectral clutter arising from heterogeneous backgrounds. The capabilities of the model are demonstrated in this paper by considering Uninhabited Airborne Vehicle scenarios and comparing both multispectral and hyperspectral sensors. Both anomaly detection and spectral matched-filter algorithms are characterised in terms of Receiver Operating Characteristic curves. Finally, some results from a preliminary validation exercise are presented.
Understanding signal and noise quantities in any practical computational imaging system is critical. Knowledge of the imaging environment, optical parameters, and detector sensitivity determine the signal quantities but often noise quantities are assumed to be independent of the signal and either uniform or Gaussian additive. These simplistic noise models do not accurately model actual detectors. Accurate noise models are needed in order to design optimal systems. We describe a noise model for a modern APS CMOS detector and a number of noise sources that we will be measuring. A method for characterizing the noise sources given a set of dark images and a set of flat field images is outlined. The noise characterization data is then used to simulate dark images and flat field images. The simulated data is a very good match to the real data thus validating the model and characterization procedure.