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Visual Perception and the
Statistical Properties of
Natural Scenes
Wilson S. Geisler
Center for Perceptual Systems and Department of Psychology, University of Texas
at Austin, Austin, Texas 78712-1062; email: [email protected]

Annu. Rev. Psychol. 2008. 59:10.1–10.26

The Annual Review of Psychology is online at
http://psych.annualreviews.org

This article’s doi:
10.1146/annurev.psych.58.110405.085632

Copyright c© 2008 by Annual Reviews.
All rights reserved

0066-4308/08/0203-0000$20.00

Key Words
natural scene statistics, spatial vision, motion perception, color
vision, ideal observer theory

Abstract
The environments in which we live and the tasks we must perform
to survive and reproduce have shaped the design of our percep-
tual systems through evolution and experience. Therefore, direct
measurement of the statistical regularities in natural environments
(scenes) has great potential value for advancing our understanding
of visual perception. This review begins with a general discussion of
the natural scene statistics approach, of the different kinds of statis-
tics that can be measured, and of some existing measurement tech-
niques. This is followed by a summary of the natural scene statistics
measured over the past 20 years. Finally, there is a summary of the
hypotheses, models, and experiments that have emerged from the
analysis of natural scene statistics.

10.1

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Contents
RATIONALE FOR MEASURING

NATURAL SCENE
STATISTICS . . . . . . . . . . . . . . . . . . . . 10.2

Roots of the Natural Scene
Statistics Approach . . . . . . . . . . . . . . . 10.3

Within-Domain Statistics . . . . . . . . . . . . 10.4
Across-Domain Statistics . . . . . . . . . . . . 10.4
MEASURING NATURAL SCENE

STATISTICS . . . . . . . . . . . . . . . . . . . . 10.6
NATURAL SCENE STATISICS . . . . 10.7

Luminance and Contrast . . . . . . . . . . 10.8
Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8
Spatial Structure . . . . . . . . . . . . . . . . . 10.9
Range . . . . . . . . . . . . . . . . . . . . . . . . . . .10.12
Spatiotemporal Structure . . . . . . . . .10.13
Eye Movements and Foveation . . . .10.14

EXPLOITING NATURAL SCENE
STATISTICS . . . . . . . . . . . . . . . . . . . .10.15
Coding and Representation of the

Visual Image . . . . . . . . . . . . . . . . . .10.16
Grouping and Segregation . . . . . . . .10.18
Identification . . . . . . . . . . . . . . . . . . . . .10.19
Estimation . . . . . . . . . . . . . . . . . . . . . . .10.20

CONCLUSION . . . . . . . . . . . . . . . . . . . .10.21

RATIONALE FOR MEASURING
NATURAL SCENE STATISTICS
The process of natural selection guarantees a
strong connection between the design of an
organism’s perceptual systems and the prop-
erties of the physical environment in which
the organism lives. In humans, this connec-
tion is implemented through a mixture of fixed
(hardwired) adaptations that are present at
birth and facultative (plastic) adaptations that
alter or adjust the perceptual systems during
the lifespan.

The link between perceptual systems and
environment is most obvious in the design
of sensory organs. The physical properties of
electromagnetic waves, acoustic waves, and
airborne molecules and their relation to the
properties of objects and materials are clearly

a driving force behind the evolution of eyes,
ears, and noses. Not surprisingly, central per-
ceptual mechanisms that process the outputs
of sensory organs also tend to be closely re-
lated to specific physical properties of the
environment.

The design of a perceptual system is also
constrained by the particular tasks the organ-
ism evolved to perform in to survive
and reproduce. For example, mammals that
suffer high rates of predation have a strong
need to detect predators and hence tend to
have laterally placed eyes that maximize field
of view, whereas mammals that are predators
have a strong need to capture moving prey
and hence tend to have frontally placed eyes
that maximize binocular overlap (Walls 1942).
Furthermore, there are purely biological con-
straints on the design of perceptual systems,
including the biological materials available to
construct the sensory organs and competition
for space with other organs and systems within
the body.

Our often-veridical perceptions of the
world give the impression of a deterministic
connection between perception and environ-
ment; however, this is largely an illusion. Most
perceptual capabilities depend upon combin-
ing many very different sources of stimulus
information, each of which is only proba-
bilistically predictive in the task the organism
is trying to perform. For example, our esti-
mates of physical object size and shape are of-
ten based upon a combination of information
sources, including lighting/shading, texture,
occlusion, motion, and binocular disparity.
Each of these sources is only probabilistically
related to object shape and size, but together
they provide us with very robust perceptions
and perceptual performance. Furthermore, all
visual measurements are noisy due to the in-
herent randomness of light absorption and
chemical events within the photoreceptors.
Consequently, the appropriate way to char-
acterize natural stimuli is in statistical terms.

The primary aim of this review is to
demonstrate the great potential value of an-
alyzing the statistical properties of natural

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scenes,1 especially within the context of de-
veloping statistical models of perception. An-
other aim is to demonstrate that the Bayesian
framework for statistical inference is partic-
ularly appropriate for characterizing natural
scene statistics and evaluating their connec-
tion to performance in specific tasks. In prin-
ciple, measuring natural scene statistics allows
one to identify sources of stimulus informa-
tion available for performing different per-
ceptual tasks and to determine the typical
ranges and reliabilities of those sources of in-
formation. Analyzing natural scene statistics
within the Bayesian framework allows one
to determine how a rational visual system
should exploit those sources of information.
This approach can be valuable for generating
hypotheses about visual mechanisms, for de-
signing appropriate experiments to test those
hypotheses, and for gaining insight into why
specific design features of the visual system
have evolved or have been learned.

Roots of the Natural Scene
Statistics Approach
The natural scene statistics approach origi-
nates in physics. Historically, physics has been
concerned with topics of direct relevance to
understanding the design of visual systems in-
cluding the properties of light, the laws of
image formation, the reflectance, scattering,
and transmittance properties of natural ma-
terials, and the laws of motion and gravity.
Against this backdrop, biologists began asking
how visual systems are adapted to the physical
environment and to the tasks that the organ-
ism performs. Most early work on the ecology
and evolutionary biology of vision was con-
cerned with the optics and the photoreceptors
of eyes (e.g., Cronly-Dillon & Gregory 1991,
Lythgoe 1979, Walls 1942). This early work
emphasized the relationship between design,

1 Natural scenes refer to real environments, as opposed to
laboratory stimuli, and may include human-made objects.
Most of the studies described here concern measurements
of outdoor environments without human-made objects.

function, and the properties of the environ-
ment, but because of the issues being investi-
gated, gave little consideration to the statisti-
cal properties of natural stimuli.

Interest in the statistical properties of nat-
ural visual stimuli began with the discovery
in physics of the inherent Poisson random-
ness of light (quantal fluctuations). Human
and animal studies by early sensory scien-
tists subsequently showed that under some
circumstances behavioral and neural perfor-
mance is limited by a combination of quantal
fluctuations and internal sources of noise (e.g.,
Barlow 1957, Barlow & Levick 1969, Hecht
et al. 1942). This work, along with parallel
work in audition, led to the development of
signal detection theory and Bayesian ideal ob-
server theory (e.g., see Green & Swets 1966),
which provides an appropriate formal frame-
work for proposing and testing hypotheses
about the relationship between perceptual
performance and the statistical properties of
stimuli and neural responses. However, early
work on the statistical properties of visual
stimuli and neural responses focused on sim-
ple detection and discrimination tasks, paying
little attention to sources of stimulus varia-
tion other than quantal fluctuations and pixel
noise.

Some early perception scientists (e.g.,
Gibson 1966, 1979) did appreciate the im-
portance of the complex properties of natu-
ral stimuli for solving perceptual tasks, but
they paid little attention to statistical varia-
tions of those properties in natural scenes. An
exception was Brunswik (1956), who realized
that the relationship between distal and prox-
imal stimuli is inherently statistical; in fact,
he demonstrated, by analyzing natural images,
that perceptual biases such as the Gestalt rule
of proximity have a statistical basis in natural
scenes (Brunswik & Kamiya 1953).

Recent years have seen rapid growth in the
statistical analyses of natural images (for a re-
view, see Simoncelli & Olshausen 2001) as
well as in the analysis and modeling of com-
plex perceptual tasks within the framework of
Bayesian ideal observer theory (for reviews,

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ANRV331-PS59-10 ARI 8 August 2007 17:33

see Geisler & Diehl 2003, Kersten et al. 2004,
Knill & Richards 1996). A central tenet of
this review is that combining measurements
of natural scene statistics with Bayesian ideal
observer analysis provides an important new
approach in the study of sensory and percep-
tual systems.

Within-Domain Statistics
Natural scene statistics have been measured at
various stages (domains) along the path from
physical environment to behavioral response.
The simpler and more common kinds of
measurements are what I call within-domain
statistics (first column in Figure 1A, see color
insert). The purpose of within-domain statis-
tics is to characterize the probability distri-
bution of properties within a specific domain
such as the (distal) environment or (proximal)
retinal image. The more complex and less
common kinds of measurements are what I
call across-domain statistics (other columns in
Figure 1A). Their purpose is to characterize
the joint probability distribution of properties
across specific domains. Across-domain statis-
tics are essential for analyzing natural scene
statistics within the Bayesian ideal observer
framework.

In the case of within-domain statistics for
the environment, a vector of physical scene
properties ! is selected and then those prop-
erties are measured in a representative set of
scenes in to estimate the probability dis-
tribution of the properties p (!). For example,
! might be the reflectance function at a point
on a surface in a natural scene; that is, a vec-
tor giving the fraction of light reflected from
the surface for a number of different wave-
lengths, ! = [!(“1), L, !(“n )]. Making these
physical measurements for a large number of
surface points in natural scenes would make
it possible to estimate the probability distri-
bution of natural surface reflectance. Simi-
larly, in the case of within-domain statistics
for images, a vector of retinal image proper-
ties s is selected and their distribution mea-
sured. For example, s might be a vector rep-

resenting the wavelength spectrum at a reti-
nal image location, s = [I (“1), L, I (“n )] (see
plots in Figure 1B ). For the domain of neu-
ral response, a set of response properties for
a population of neurons is selected and their
distribution measured for a representative set
of natural stimuli. In this case, z might be a
vector of the spike counts of each neuron,
z = [count1, L, countn ]. Finally, for the domain
of behavior, a vector of properties for some
class of behavior is selected and their distri-
bution measured for a representative set of
natural stimuli. For example, r might be the
eye fixation locations in a natural image dur-
ing free viewing, r = [fixation1, L, fixationn ].

Measurements of within-domain statistics
are highly relevant for understanding neural
coding and representation. A plausible hy-
pothesis is that the retina and other stages
of the early visual pathway have evolved (or
learned) to efficiently code and transmit as
much information about retinal images as pos-
sible, given the statistics of natural images and
biological constraints such as the total num-
ber of neurons and the dynamic range of neu-
ral responses. Variants of this efficient cod-
ing hypothesis have been widely proposed and
evaluated (Atick & Redlich 1992; Attneave
1954; Barlow 1961, 2001; Field 1994; Laugh-
lin 1981; van Hateren 1992). For example, the
efficient coding hypothesis predicts many of
the response characteristics of neurons in the
retina directly from the joint probability dis-
tributions of the intensities at two pixel loca-
tions in natural images, p (s) = p (I1, I2), mea-
sured for various separations of the pixels in
space and time. The measurement of within-
domain statistics is central to the enterprise of
testing the efficient coding hypothesis: To de-
termine what would be an efficient code, it is
essential to know the probability distribution
of the image properties to be encoded.

Across-Domain Statistics
Within-domain statistics say nothing about
the relationship between the domains listed
in Figure 1A, such as the relationship

10.4 Geisler

!

!

!

!

ANRV331-PS59-10 ARI 8 August 2007 17:33

between properties of the environment and
the images formed on the retina. Natural
visual tasks generally involve making infer-
ences about specific physical properties of the
environment from the images captured by
the eyes. These tasks include classifying ma-
terials, discriminating object boundary con-
tours from shadow contours, estimating ob-
ject shape, identifying objects, estimating the
distance or motion of an object, or estimating
one’s own motion direction and speed. The
relevant statistics for understanding how the
visual system performs such tasks are the joint
distributions of physical scene properties and
image properties, p (!, s), or equivalently,
p (!) and the conditional distribution p (s|!)
for each value of !. [Note that p (!, s) =
p (s|!) p (!).] Using Bayes’ rule, these across-
domain statistics specify the posterior proba-
bilities of different states of the world (phys-
ical environment properties) given particular
observed retinal image properties p (!|s). It is
the characteristics of the posterior probabil-
ity distributions that visual systems evolve or
learn to exploit in performing specific natural
visual tasks.

Suppose an organism’s task is to identify
(for a given species of tree) whether a ran-
domly selected location (small patch) in the
retinal image corresponds to the surface of a
fruit or the surface of a leaf, based solely on
the information available in the wavelength
spectrum at that randomly selected location
(see Figure 1B ). In this case, there are just
two relevant states of the environment (distal
stimuli): ! = fruit and ! = leaf , and due to
variations in lighting and reflectance, a large
number of possible wavelength spectra (prox-
imal stimuli) for each distal stimulus. In prin-
ciple, p (!) could be measured by randomly
selecting lines of sight from a large number of
example scenes and counting the proportion
of times a fruit surface is the first surface en-
countered along that line of sight. Similarly,
p (s|!) could be measured by recording the
wavelength spectra for each of the randomly
sampled lines of sight, sorting them accord-
ing to whether they are from fruit or leaf, and

then analyzing them separately to estimate the
two conditional probability distributions. The
statistical regularities represented by p (s|!)
and p (!) could be exploited by the visual sys-
tem for identifying fruits and leaves from the
wavelength spectra that reach the eye. On the
other hand, knowing only the within-domain
statistics, p (s) and p (!), is not useful for iden-
tifying fruits and leaves because the statis-
tics do not specify the relationship between
the image properties (wavelength spectra) and
the physical objects (fruits and leaves) of rel-
evance in the task. This example illustrates
the use of across-domain statistics for charac-
terizing the connection between environmen-
tal properties and image properties; compara-
ble examples can be generated for the other
kinds of across-domain statistics (see table in
Figure 1A).

Bayesian ideal observer theory provides
an appropriate formal framework for under-
standing how across-domain statistics might
be exploited by the visual system to perform
specific tasks (Geisler & Diehl 2003). The
Bayesian approach in perception research has
been discussed at length elsewhere and is only
briefly summarized here, as a prelude to some
of the studies described below. An “ideal ob-
server” is a theoretical device that performs a
task in an optimal fashion given the available
information (and possibly other constraints).
Deriving an ideal observer can be very use-
ful because (a) the derivation usually leads
to a thorough understanding of the compu-
tational requirements of the perceptual task,
(b) the ideal observer provides the appropriate
benchmark for comparison with behavioral
performance, and (c) ideal observers often re-
duce to, or can be approximated by, relatively
simple decision rules or procedures that can
serve as initial hypotheses for the actual pro-
cessing carried out in a perceptual system.

The logic behind deriving an ideal ob-
server is straightforward. Consider an ideal
observer that wishes to perform a specific task
in its current environment and has access to
some vector of properties in the retinal image.
Upon receiving a particular stimulus vector

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S, an ideal observer should make the response
that maximizes the utility (gain/loss) averaged
over all possible states of the environment,

ropt (S) = arg max
r

!
”

!

# (r, !) p (S|!) p (!)
#

(1)
where # (r, !) is the utility of making re-
sponse r when the true state of the environ-
ment is !, and the function arg maxreturns
the response that maximizes the sum in the
brackets. In other words, once the relevant
across-domain statistics have been estimated
and the utility function for the task has been
specified, then Equation (1) can be used to
determine (via simulation or calculation) the
optimal performance in the task.

Think back to the hypothetical task de-
scribed above: identifying whether a small
patch of retinal image corresponds to a fruit
or a leaf. There are two possible responses,
r = fruit and r = leaf . To maximize accuracy,
an appropriate utility function is # (r, !) = 1,
if r = !, and # (r, !) = !1, if r “= ! (i.e.,
equal but opposite weights for corrects and
errors). Substituting this utility function and
the measured probability distributions into
Equation (1) gives the (parameter-free) op-
timum decision rule. The performance accu-
racy of this decision rule can be determined by
applying the rule to random samples (!, S)
from the across-domain probability distri-
bution, p (!, s), or alternatively, by directly
calculating the probability that ropt (S) = !.
The optimal decision rule (or an approxi-
mation to it) could serve as a principled hy-
pothesis about the perceptual mechanisms in
the organism that discriminate fruits from
leaves.

MEASURING NATURAL SCENE
STATISTICS
A variety of devices and techniques has
been used to measure natural scene proper-
ties. Spectrophotometric devices measure the
wavelength distribution (radiance as a func-
tion of wavelength) of the light that reaches

their sensors. They can be used to measure re-
flectance spectra of materials, irradiance spec-
tra of light sources (illuminants), as well as ra-
diance spectra that reach the eye. Spectropho-
tometers collect light over only one small
patch at a time, making them impractical for
collecting data from a large number of loca-
tions in a scene. Hyperspectral cameras can
measure radiance spectra at each camera pixel
location, but require relatively long exposure
time, and thus are practical only for conditions
where effects of object and shadow motion are
minimized (e.g., long distances or indoor en-
vironments). The most common method of
measuring natural scene properties has been
to analyze images captured by digital still cam-
eras and digital movie cameras. Digital cam-
eras usually provide either 8-bit grayscale or
24-bit color (8 bits per color) images, although
some high-end cameras provide 36-bit color
(12 bits per color) images, which is desirable
for some kinds of measurements. A weakness
of standard digital cameras is that they can-
not provide detailed spectral (chromatic) in-
formation, although with proper calibration
it is possible to obtain images that give, for
each pixel location, the approximate lumi-
nance and/or the approximate amount of radi-
ant power absorbed in each of the three classes
of cone photoreceptor, the long (L), middle
(M), and short (S) wavelength cones. Many
studies have analyzed uncalibrated camera im-
ages, which is justifiable if the scene statistics
of interest (e.g., contour geometry) are little
affected by monotonic transformations of the
camera’s color responses.

Another useful class of device is the range
finder, which measures distance to each point
in a scene by measuring return time for an
emitted pulse of infrared laser light. These
devices are accurate for large distances (a
few meters to a kilometer or more). A re-
lated class of device is the three-dimensional
scanner, which uses triangulation rather than
time-of-flight, and is useful for making pre-
cise range measurements at near distances
(e.g., measuring the shape of a face). A weak-
ness of both devices is that the scans take

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substantial time (typically seconds), and thus
motion in the scene can produce significant
distortions.

The above devices are the most common
for measuring natural scene statistics, and
they can be used in a fairly straightforward
manner to measure within-domain statistics
for the image or environment. Measuring
across-domain statistics is more difficult be-
cause both image and environment properties
must be measured for the same scene. One
approach is to combine environment mea-
surements from one instrument with image
measurements from another instrument. For
example, monocular across-domain statistics
for depth can be measured by combining a
calibrated camera image with a distance map
obtained with a range finder.

An expedient approach for measuring
across-domain statistics involves hand seg-
mentation. The central assumption of this ap-
proach is that under some circumstances hu-
mans are able to make veridical assignments
of image pixels to physical sources in the en-
vironment. When this assumption holds, the
pixel assignments are useful measurements of
environmental properties.

For example, consider the close-up im-
age of foliage in Figure 2A (see color in-
sert). When observers are asked to segment
individual leaves and branches that are within
or touching the orange dashed circle, the re-
sult in Figure 2B is obtained. The colored
leaves and branches show the segmented ob-
jects, the red/brown shaded leaf shows one in-
dividual segmented object. These segmenta-
tions are performed with high confidence and
repeatability, and hence generally provide an
accurate measurement of the physical source
(the specific leaf or branch) that gave rise to
a given pixel. Many across-domain statistics
can be measured with a large set of such seg-
mentation data. To take one simple example,
it is straightforward to measure the poste-
rior probability of the same or different object
[! = same or ! = different], given the dis-
tance between a pair of image pixels and their
luminance values [s = (d12, l1, l2)].

Hand segmentation methods are useful for
measuring across-domain statistics only to the
extent that the segmentations are veridical
(i.e., represent physical “ground truth”), and
there are cases (e.g., distant images of foliage)
where some image regions are ambiguous and
difficult to segment.2 In cases where hand seg-
mentation methods fail, accurate ground truth
measurements require more direct physical
measurement. Another strategy for measuring
across-domain statistics combines computer-
graphics simulation with direct measurements
of scene statistics.

NATURAL SCENE STATISICS
It is difficult to know ahead of time which
specific statistics will prove most informa-
tive for understanding vision. At this time,
progress is being made by selecting statis-
tics based on intuition, historical precedence,
and mathematical tractability. It is impor-
tant to note that the number of samples re-
quired for estimating a probability distribu-
tion grows exponentially with the number of
properties/dimensions (“the curse of dimen-
sionality”), and hence most studies measure
natural scene statistics for only one or a few
properties at a time. This is a significant lim-
itation because perceptual mechanisms ex-
ploit complex regularities in natural scenes
that may only be fully characterized by mea-
suring joint distributions over a substantial
number of dimensions. Nonetheless, the pub-
lished work has demonstrated that much can
be learned from low-dimensional measure-
ments and that there are useful methods for
learning about the structure of probability dis-
tributions in high-dimensional spaces. This
section presents a somewhat eclectic summary
of some of the natural scene statistics that have
been measured.

2 Hand segmentation has also been used to measure how
humans segment scenes into regions without specific in-
structions to be exhaustive or identify physical sources;
in this case, the aim is not to precisely measure physical
ground truth but rather to obtain a useful data set for train-
ing image-processing algorithms (e.g., Martin et al. 2004).

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PCA: principal
components analysis

Luminance and Contrast
Luminance and contrast, fundamental stim-
ulus dimensions encoded by visual systems,
vary both within a given scene and across
scenes. Most studies have involved measuring
the statistics of luminance and contrast within
images of natural scenes (e.g., Brady & Field
2000, Frazor & Geisler 2006, Laughlin 1981,
Ruderman 1994, Tadmor & Tolhurst 2000).
The distribution of local luminance within a
given image is typically obtained by first divid-
ing the luminance at each pixel by the average
for the whole image.3 Combining these dis-
tributions across images and then scaling to
the average luminance of the images gives the
distribution of luminance in the typical natu-
ral image. As shown in Figure 3A (see color
insert), this distribution is approximately sym-
metric on a logarithmic axis and hence posi-
tively skewed on a linear scale (Brady & Field
2000, Laughlin 1981, Ruderman et al. 1998).
In other words, relative to the mean lumi-
nance, there are many more dark pixels than
light pixels.

The distribution of local contrast within
images has been measured using various defi-
nitions of contrast. Figure 3B shows the dis-
tribution of local root-mean-squared contrast
(the standard deviation of luminance divided
by the mean luminance in a small neighbor-
hood) in the typical natural image. Another
more specialized measure is an equivalent
Michelson contrast—the Michelson contrast
of a sine wave grating (sine wave amplitude di-
vided by mean) that would produce the same
contrast response as the local image patch,
where the contrast response is from a filter
designed to mimic a typical receptive field at
some level of the visual system (Brady & Field
2000, Tadmor & Tolhurst 2000). These latter
distributions tend to be similar in shape to
the one in Figure 3B, but (as expected given

3 One could regard the ratio of pixel luminance to global
luminance as a form of Weber contrast, but here the
term “contrast” is reserved for measurements of lumi-
nance variation relative to the average luminance in a small
neighborhood.

the selectivity of the filter) are shifted toward
lower contrasts.

There are large variations of local lumi-
nance and local contrast in natural images,
and these variations tend to be statistically in-
dependent. The average joint distribution of
luminance and contrast has a slight negative
correlation (r = !0.2) primarily due to the
fact that sky regions tend to be both bright
and low in contrast (Figure 3C ). Low cor-
relations between luminance and contrast are
also observed within the constituents of natu-
ral images. For example, the joint distribution
of luminance and contrast in purely foliage
regions (Figure 3D) has a slight positive cor-
relation (r = 0.15). As discussed below, the
large variations in local luminance and con-
trast and their low correlation have important
implications for neural coding.

Color
Interest in natural scene statistics was stim-
ulated by the discoveries that the …