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Computational Aesthetics in Graphics, Visualization and Imaging (2005)
L. Neumann, M. Sbert, B. Gooch, W. Purgathofer (Editors)
Benford’s Law for Natural and Synthetic Images
E. Acebo, and M. Sbert
Institut d’Informàtica i Aplicacions, Universitat de Girona, Spain
Abstract
Benford’s Law (also known as the First Digit Law) is well known in statistics of natural phenomena. It states
that, when dealing with quantities obtained from Nature, the frequency of appearance of each digit in the first
significant place is logarithmic. This law has been observed over a broad range of statistical phenomena. In this
paper, we will explore its application to image analysis. We will show how light intensities in natural images,
under certain constraints, obey this law closely. We will also show how light intensities in synthetic images follow
this law whenever they are generated using physically realistic methods, and fail otherwise. Finally, we will study
how transformations on the images affect the adjustment to the Law and how the fitting to the law is related to the
fitting of the distribution of the raw intensities of the image to a power law.
Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computing Methodologies]: Computer Graph-
icsPicture/Image Generation; I.4.8 [Computing Methodologies]: Image Processing and Computer VisionScene
Analysis
1. Introduction
Image statistics is a growing field in image analysis
[ERT01], [SO01], [AW05], with many potential applica-
tions. We contribute to this field by studying the fitting of
images (both natural and synthetics) to Benford’s Law. This
law (also known as the First Digit Law) is well known in
statistics of natural phenomena. It states that, when dealing
with quantities obtained from Nature, the frequency of ap-
pearance of each digit in the first significant place is loga-
rithmic. This law has been observed over a broad range of
physical phenomena, from city populations to river lenghts
and flows. It has been applied to fraud detection in tax pay-
ing, to the design of computers, to the analysis of roundoff
errors and as a statistical test for naturalness. We will show
in this paper how natural images, under certain constraints,
follow closely this law. We also show how synthetic (com-
puter generated) images follow this law whenever they are
generated using physically realistic methods, and fail other-
wise. On the other hand, a transformation on the image, such
as filtering, will affect the adjustment to the law. We finally
discuss how the fitting to Benford’s Law is related to the fit-
ting of raw intensities to a power law.
The rest of the paper is organized as follows. In section
2 we review the Benford’s law, in section 3 we study its ap-
plicability to images, both synthetic and natural, in section 4
we present an explanation for the obtained results and finally
in section 5 we present our conclusions and future research.
2. Benford’s Law
2.1. The uneven distribution of digits in Nature
Suppose that we have a table with the populations of all the
villages and towns in the world. Count how many of these
numbers begin with the digit 1, how many with the digit 2
and so on. Contrary to what we can intuitively expect, the
relative frequency of the populations starting with each of
the nine digits is not the same. Actually, we will find that
about 30% of populations begin with the digit 1, and only
about 5% with the digit 9. That phenomenon is not partic-
ular of populations, a vast amount of "natural" quantities,
ranging from river lengths and flows to molecular weights
of chemical compounds and stock prices, exhibit the same
uneven distribution of first digits. It is not new. It was ob-
served by Simon Newcomb in 1881 and then rediscovered
by Frank Benford in 1938.
Benford’s Law (also known as logarithmic significant
digit distribution), in its most general form (base 10, D
i
means the ith significant digit) tells that for all positive in-
tegers k, all d
1
∈ {1,2,..., 9} and all d
j
∈ {0,1,2, ... ,9},
j = 2, .. .,k, probability P is given by
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The Eurographics Association 2005.
E. Acebo, M. Sbert, & / Benford’s Law for images
Figure 1: First digit distribution according to Benford’s
Law.
P(D
1
= d
1
,. .., D
k
= d
k
) = log
10
[1 + (
k
∑
i=1
d
i
· 10
k−i
)
−1
] (1)
or equivalently
P(mantissa ≤
t
10
) = log
10
t (2)
for t ∈ [1 . . .10).
Hence the first significant digit law:
P(first significant digit = d) = log
10
(1 +
1
d
) (3)
We can see in the graph in Fig.
1 the values of the prob-
abilities of each of the nine possible first digits. A perhaps
surprising corollary to the general law is that significant dig-
its are not independent, that is, the probability for the nth
significant digit to take a certain value depends on the values
of the n − 1 leading significant digits [Hil96].
Some interesting properties of the logarithmic distribution
are the following:
Let X
1
,X
2
be random variables with logarithmic signifi-
cant digit distribution, then
• K · X
1
for K > 0
• 1/X
1
• X
1
+ X
2
• X
1
· X
2
• X
K
1
for K > 0
also have logarithmic significant digit distribution.
Also, let X
1
,X
2
,. .., X
n
be random variables, then, under
certain (rather general) conditions
∏
i
X
i
(4)
tends to have logarithmic significant digit distribution when
n tends towards infinity.
Benford’s law has been so far applied to the fields of com-
puter design:
• Data compression [
Sch88]
• Floating point computing optimization [FT86], [Knu69],
[BB85]
and to test for "naturalness":
• Model validation [Var72], [NW95]
• Tax fraud detection [Nig96], [NM97]
It has also been applied in gambling [Che81].
2.2. Explanations for Benford’s Law
While Benford’s law unquestionably applies to many situa-
tions in the real world, a satisfactory explanation has been
given only recently through the work of Hill [Hil96]. An
extensive account of the early efforts to explain the phe-
nomenon can be found in [Rai76]. Perhaps the first impor-
tant step in the right direction was the realization that Ben-
ford’s law is applied to data that are not dimensionless (so
the numerical values of the data depend on the units). If there
is a universal probability distribution over such numbers,
then it must be invariant under scale changes. If somehow
Benford’s Law has to be of universal applicability, it has to
hold independently of the units used in the measurements.
If you have a set of random variables following Benford’s
Law and you multiply them by some constant, the resulting
values will also obey Benford’s Law (after all, "God is not
known to favor either the metric system or the English sys-
tem" [Rai76]). It turns out that Benford’s Law is the only
scale-invariant probability distribution for significant digits
(see [Hil96] for a deep analysis of this point). So, if natural
quantities have to follow a given probability law for signif-
icant digits, it has to be Benford’s Law. The question about
why those quantities would have to follow such a fixed law
remains open, however.
The most convincing explanation of the phenomenon, by
now, comes from Hill [Hil96]. The main result of his work
is a sort of central-limit-like theorem for significant digits
which says, putting it in the author’s own plain words, "if
probability distributions are selected at random, and random
samples are then taken from each of these distributions in
any way so that the overall process is scale (or base) neu-
tral, then the significant digit frequencies of the combined
sample will converge to the logarithmic distribution". Two
key concepts in the above statement are those of scale and
base neutrality. Let’s focus on the former. According to Hill,
a sequence of random variables X
1
,X
2
,. .. has scale-neutral
mantissa frequency if
n
−1
| #{i ≤ n : X
i
∈ S} −#{i ≤ n : X
i
∈ sS} |→ 0 a.s. (5)
for all s > 0 and all S ∈ M, where M is the sub-sigma alge-
bra of the Borels where the probability measure P is defined
(see [Hil96] for details):
S ∈ M ⇐⇒ S =
∞
[
n=−∞
B · 10
n
(6)
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The Eurographics Association 2005.
E. Acebo, M. Sbert, & / Benford’s Law for images
for some Borel B ⊆ [1, 10).
We will return to it later in the discussion section, where
we will make use of the theorem to justify the fitting of
a broad class of images to Benford’s Law attending to the
shape of their histograms.
3. Benford’s Law and images
Our aim in this section is to study to what extent and in what
aspects images, both syntethic and real, obey Benford’s Law.
This should allow us to design new methods for image anal-
ysis and classification based in this previous study. Appli-
cation of Bendford’s Law to image analysis is quite innova-
tive. To our knowledge, there is only one other attempt in
this way [Jol01]. In that paper, however, the author quickly
discards the idea of pixel intensity values of general images
to follow Benford’s Law and focuses on the gradient of these
values, instead. We will study the matter from another per-
spective. Obviously not all the images obey Benford’s Law,
but in this paper we will show how a great variety of natural
and artificial images do, and will characterize them in terms
of the shape of their histograms.
3.1. Benford’s Law and synthetic physically realistic
images
Radiosity and ray-tracing provide us with physically realistic
images [DBB03]. In Radiosity three radiosity values (RGB)
are computed for each patch (or polygon of the scene mesh)
in the scene. In ray tracing three intensity values are com-
puted for each pixel in the screen. These values closely fol-
low the logarithmic first digit distribution in several scenes
that we have tested. This can be seen in the set of images
in Fig.2 and Fig.3. Scenes in Fig.2 contain 49124 (top) and
22718 (bottom) patches, respectively. The quality of the fit-
ting is measured with the χ
2
divergence:
9
∑
i=1
( f
i
− log
10
(1 +
1
i
))
2
log
10
(1 +
1
i
)
(7)
where f
i
is the relative frequency of i as first digit in the set
of values (radiosities or intensities).
3.2. Benford’s Law and synthetic non-physically
realistic images
When non-physically realistic rendering methods are used
in addition with radiosity or ray tracing methods (ambient
occlusion or obscurances [IKSZ03], extended ambient term
[CNS00], textures) the resulting values diverge from Ben-
ford’s Law. Thus, Benford’s Law could be used as a test for
physical realism in synthetic image rendering. In Fig.4 we
see the discrepancy between Benford’s Law and two texture
images, obtained by PovRay [Pov].
3.3. Benford’s Law and real images
We can’t work with real images, only with pictures of real
images. Digital pictures represent an imperfect measurement
of natural phenomena. We face potential data corruption
from:
• Noise
• Over/underexposure
• Discretization
• Interpolation
• Gamma correction
• Retouching
In order to overcome these problems as much as possible we
will work with raw images (no interpolation, no gamma cor-
rection, no retouching, 16 bits per channel). Not all "real"
images obey Benford’s Law, see for instance Fig.5. How-
ever, they tend to do it quite often and quite well. (See Figure
6). In Fig.7 we show a photograph of a painting fitting also
Benford’s Law. In Fig.8 we show the effect on the fitting of
a flash light, and in Fig.9 we show the effect of applying dif-
ferent filters to the images. Fig.9 top-left corresponds to the
original raw, unfiltered image. Fig.9 top-right is the result of
applying to the previous image the PhotoShop contrast and
brightness autobalancing, in Fig.9 bottom-left we applied the
PhotoShop high-pass filter, and finally in Fig.9 bottom-right
we applied the PhotoShop equalizing filter.
4. Discussion
A broad class of synthetic and real images tend to obey Ben-
ford’s Law. Those that better fit this law seem to have multi-
ple heterogeneous interacting objects (from the point of view
of lighting), see Fig.2,3& 6(bottom). Images reflecting only
a small part of a scene or a detail do not fit so well or do not
fit at all, see Fig.5. The logs image in Fig.6(middle) is obvi-
ously an exception to this rule. We can, however, try to give a
more objective justification to the phenomenon. Recall from
section 2 Hill’s theorem, which states that if probability dis-
tributions are selected at random, and random samples are
then taken from each of these distributions in any way so that
the overall process is scale (or base) neutral, then the signifi-
cant digit frequencies of the combined sample will converge
to the logarithmic distribution. We can deduce from this that
if we take samples from a single distribution and they re-
sult to be distributed in a scale neutral way in the sense of
(5) then we can expect the samples to follow the logarithmic
distribution. In other words, samples taken from a scale neu-
tral probability distribution will follow Benford’s Law. One
such probability distribution is p(x) = 1/x. In effect we have
Z
b
a
1
x
dx =
Z
kb
ka
1
x
dx (8)
for a, b,k ∈ <, 0 < a ≤ b, k > 0, and this implies, in lack of
a rigorous proof, scale neutrality in the sense of (
5).
Now consider the histograms of the images in the paper.
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The Eurographics Association 2005.
E. Acebo, M. Sbert, & / Benford’s Law for images
Figure 2: Left: radiosity images. Middle: graphs demonstrate the fitting of left image with Benford’s law. For the top image
the fitting corresponds to χ
2
=0.00703, and for the bottom one χ
2
=0.01549. Right: Histograms. Images courtesy of Francesc
Castro and Roel Martinez.
They plot frequencies against intensity values of pixels (or
patches in the case of radiosity), so we can see them as prob-
ability distributions of pixel (patch) intensities. By the above
discussion, histograms with shape similar to the 1/x func-
tion will correspond to images whose pixels values are more
scale-neutrally distributed. We expect those images to fol-
low closely Benford’s Law. And that is the case, indeed, as
we can see in Figs.2,3, 6, 7& 8.
We can draw some additional conclusions. On the one
hand, Benford’s Law tends not to hold well in images ob-
tained by means of non-physically realistic rendering meth-
ods. On the other hand, in real images, the fit with the law
is very sensitive to the application of filters and retouching
algorithms, see Fig.9.
5. Conclusions and future work
A broad class of synthetic and real images tend to obey
Benford’s Law. The images that fit the law better seem to
be those with multiple heterogeneous interacting (from the
point of view of lighting) objects. Benford’s Law tends not
to hold in images obtained by means of non physically real-
istic rendering methods. In real images, the fit with the law
is very sensitive to the application of filters and retouching
algorithms.
We envisage the following applications:
• Hardware/Software design: digital camera sensors,
graphic cards, graphic algorithms
• Data compression. (Not much, about half a bit per floating
point number in FP16)
• Test for "naturalness" in synthetic images
Further analysis of when and why images follow Benford’s
Law is needed. We intend to further study specialized image
types, including astronomical and medical images, as well
as other types of data (do natural sounds follow Benford’s
Law?).
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c
The Eurographics Association 2005.
E. Acebo, M. Sbert, & / Benford’s Law for images
Figure 3: Left: ray-tracing images. Middle: graphs demonstrate the fitting of left image with Benford’s law. For the top image
the fitting corresponds to χ
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Figure 4: Left: textured images. Middle: graphs demonstrate the unfitting of left image with Benford’s law. For the top image
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c
The Eurographics Association 2005.
E. Acebo, M. Sbert, & / Benford’s Law for images
Figure 5: Left: real images. Middle: graphs demonstrate the unfitting of left image with Benford’s law. For the top image the
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The Eurographics Association 2005.
E. Acebo, M. Sbert, & / Benford’s Law for images
Figure 7: Left: photograph of a painting. Middle: graph demonstrate the fitting of left image with Benford’s law χ
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= 0.02998.
Right: Histogram.
Figure 8: Top/bottom rows correspond to a photograph (left images) taken without/with flash light, respectively. Middle: graphs
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2
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The Eurographics Association 2005.
E. Acebo, M. Sbert, & / Benford’s Law for images
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