Journal of Visual Communication and Image Representation 11, 266–277 (2000)
doi:10.1006/jvci.1999.0443, available online at http://www.idealibrary.com on
The Hausdorff Dimension and Scale-Space
Normalization of Natural Images
Kim Steenstrup Pedersen and Mads Nielsen
DIKU, Universitetsparken 1, 2100 Copenhagen E, Denmark
Received October 29, 1999; accepted November 14, 1999
Fractal Brownian motions have been introduced as a statistical descriptor of natural images. We analyze the Gaussian scale-space scaling of derivatives of fractal
images. On the basis of this analysis we propose a method for estimation of the
fractal dimension of images and scale-space normalization used in conjunction with
automatic scale selection assuming either constant energy over scale or self-similar
energy scaling. °C 2000 Academic Press
Key Words: fractal dimension; natural images; self-similarity; Gaussian scalespace; image derivatives; scale-selection; feature detection.
1. INTRODUCTION
In the literature [1–8] one finds several investigations into the fractal nature of natural
images and in this paper we will look at some scale-space properties of images of natural
scenes. Here we use the term natural image to denote any image of a real world scene which
may be assumed to have a fractal intensity surface (or volume). A function is said to be
fractal if it has a so-called fractal dimension that differs from the topological dimension in
a fractional manner. For that reason one uses the fractal dimension to characterize fractal
functions. A fractal function is self-similar, which means that if one looks at the function
as a random function then its distribution is scale independent.
The fractal dimension of an image intuitively describes the roughness of the image
intensity graph and the fractal dimension of the intensity surface of 2D images must a priori
lie between 2 and 3. There has for some time been a general consent that 2D images of
natural scenes have a fractal dimension (Hausdorff dimension1 ) D H = 2.5, which is the
same dimension as the classical 2D Brownian motion. But recent studies by Bialek et al.
[6] have shown that 2D images of natural scenes2 do not necessarily have to come from a
Gaussian process and that the fractal dimension can vary freely in the interval between 2
and 3.
1 In this paper we will use the Hausdorff dimension as the definition of the fractal dimension. See [9] for a
mathematical definition of the Hausdorff dimension.
2 The studies by Bialek et al. are based on a series of images in a forest.
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HAUSDORFF DIMENSION AND SCALE-SPACE NORMALIZATION
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Fractal Brownian motions (fBm) can be used as a model for images of natural scenes.
By using this model we have the freedom to model images of any fractal dimension. The
classical Brownian motion is a special case of the fBm. The fBms are in general continuous,
but not differentiable. In the limit D H → 2, the 2D fBms generically become smooth (C ∞ ),
whereas in the limit D H → 3, the 2D fBms become spatially uncorrelated.
The estimation of the fractal dimension of regions of interest in images has different
interesting prospects. It has been proposed [7, 8] that the fractal dimension of X-ray images
of trabecular bone can give an indication of the microstructure of the bone and thereby
also the biomechanical strength of the bone. This can be a helpful tool for the research of
osteoporosis and other bone diseases. Other uses of the fractal dimension could be as a
quality measure of surfaces produced in different kinds of industries, e. g., metal plates and
wood.
Linear scale-space [10–12] is a mathematical formalization of the concept of scale (or
aperture) in physical measurements. By Gaussian convolution, images at higher inner scales
than the measurement scale can be simulated, enabling us to create an artificial scale-space
of an image. By using this type of scale-space we bypass the problem of differentiability
of digital images, because differentiation of the image in scale-space may be obtained by
differentiation of the Gaussian prior to convolution.
By the use of a nonlinear combination of image derivatives, called measures of feature
strength, it is possible to detect features in images [13]. In order to get dimensionless
derivatives Florack et al. [14] has proposed normalization of image derivatives where the
derivatives are multiplied by the scale σ , (∂/∂ x)norm = σ ∂/∂ x. Lindeberg [15–17] operates
with scale-normalized derivatives in order to detect the most significant scale for the features.
He uses a normalization which is defined through a scaling exponent γ . In application
to feature detection, this normalization exponent γ depends on the feature in question.
Lindeberg determines this parameter on the basis of analysis of feature models. In this
analysis the parameter varies in the interval [ 12 ; 1]. Our intuition3 is that this parameter must
reflect the local complexity of the image and may be modeled through the fractal dimension
of the local image. In this paper we conjecture a simple relation between the topological
dimension of a feature and the fractal dimension of the local image for determining the
scale-normalization.
We will in this paper assume that the fBms constitute a model of images of natural scenes.
Using this model we establish a method of scale-space normalization of derivatives, changing the analytical expression of Lindeberg’s γ -normalization. This expression includes the
fractal dimension of the image in a neighborhood of the feature we want to detect. Furthermore, we can use this normalization method for estimation of the fractal dimension of
images.
2. FRACTAL BROWNIAN MOTIONS AND NATURAL IMAGES
The 1D fBm was first defined by Mandelbrot and van Ness [18] in an integral form, which
was later restated in terms of self-similarity of a distribution function. In the latter form it is
x ) : R N 7→ R
straightforward to state the fBm defined over an N -D space [5]. A function f H (E
3
Developed during discussions with Lindeberg.
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PEDERSEN AND NIELSEN
is called an N -D fBm if for all positions xE ∈ R N and all displacements λE ∈ R N
µ
P
E − f H (E
f H (E
x + λ)
x)
<y
E H
kλk
¶
= F(y),
where F(y) is a cumulative distribution function and P(·) the probability. The scaling
exponent H ∈ ]0; 1[ determines the fractal dimension of the fBm. This definition implies
x ) is a 1D fBm of the same
that any 1D straight line in the domain of the function f H (E
scaling exponent H .
One can find experimental data to support the assertion that images of natural scenes
may satisfy the definition of the fBm [1, 3, 4, 6], but F(y) is not in general a cumulated
Gaussian distribution [6] as commonly presumed [1, 3].
x ) is given by
The power spectrum of the N -D fBm f H (E
| f˜ H (ω)|
E 2 ∝ |ω|
E −α ,
(1)
where f˜ H (ω)
E is the Fourier transform of the fBm and α = 2H + 1, which is independent of the dimensionality N [5, 18, 19]. Voss and Pentland [5, 19] note the relation
x ) and
D H = N + (1 − H ) between the Hausdorff dimension4 D H of an N -D fBm f H (E
the scaling exponent H . The estimation of α in (1) is, together with the relation between
D H and H , a well-known method for estimation of the fractal dimension of images [8].
Lindeberg [15, 20] argues that in the case of N -dimensional natural images the assumption
of a uniform energy distribution at all scales leads to a power spectrum proportional to |ω|
E −N .
With reference to Field [1], Lindeberg utilizes the assertion that the power spectrum has
equal energy at all scale-invariant frequency intervals. We find that this only coincides with
H = 1/2 for 2D images, which is the case where the images can be modeled by classical
Brownian motions. For Lindeberg’s proportionality to hold for other values of the scaling
exponent H the exponent must be H = N 2−1 , which only makes sense for N < 3, due to
the constraint 0 < H < 1. So in general we cannot assume that | f˜ H (ω)|
E 2 ∝ |ω|
E −N under the
assumption that N -D natural images can be modeled by fBms.
3. SCALE-SPACE SCALING OF DERIVATIVES OF FRACTAL IMAGES
In this section we will first give a short introduction to the linear Gaussian scale-space and
its normalized derivatives. Then we will state our proposal for an extension of Lindeberg’s
normalization method based on the fractal dimension.
3.1. Scale-Space and Normalization
Linear Gaussian scale-space of images was independently introduced by Iijima [10],
Witkin [11], and Koenderink [12]. The scale-space of an image L(E
x ) : R N 7→ R can be
defined as a solution to the diffusion equation and it is given by
L(E
x ; t) = G(E
x ; t) ∗xE L(E
x ),
4 The Hausdorff dimension can intuitively be viewed as a scaling exponent of the space filling of the graph in
question.
HAUSDORFF DIMENSION AND SCALE-SPACE NORMALIZATION
269
x ; t) : R N ×
where t is the scale parameter, the notation ∗xE denotes convolution in xE, and G(E
R 7→ R is the Gaussian
µ
¶
1
kE
x k22
G(E
x ; t) =
exp − 2 ,
(2π σ 2 ) N /2
2σ
where t = σ 2 . The nth order partial derivative of a scale-space image can be found by using
the following commutation relation,
∂ n G(E
x ; t)
∂n
x ; t) ∗xE L(E
x )) =
∗xE L(E
x ),
n (G(E
∂ xi
∂ xin
where xi denotes the ith element of xE. In this paper we will in general use tensor notation
and Einstein’s summation convention when writing image derivatives.
Normalization of image derivatives has been proposed by several authors [15, 21, 14].
The standard normalization of the nth image derivative in scale-space, based on dimensional
analysis [14], is
L i1 ···in ,norm = t n/2 L i1 ···in ,
which for the first order derivatives is the same as (∂/∂ x)norm = t 1/2 ∂/∂ x. Lindeberg proposes [15, 16, 17] another method of normalization of image derivatives. He proposes that
the nth order derivatives could be normalized by
L i1 ···in ,γn −norm = t γn L i1 ···in ,
where γn = nγ /2 and γ is a free normalization parameter. In conjunction with feature
detection Lindeberg has determined γ by an analysis of model patterns reflecting the features
under consideration.
3.2. Scale-Space Normalization Using the Fractal Dimension
We propose that γn can be stated as a relation of α (i.e., H ), the topological dimension N
of the image, and the order n of differentiation. This is based on an assumption that images
of natural scenes may be modeled as fBms and that normalized derivatives must have equal
energy at all scales.
We will investigate quadratic differential image invariants on the form
I (n) = L i1 ···in L i1 ···in .
We say that these kind of invariants are of the nth order of differentiation. In the following
we examine the L 1 -norm of such invariants, which corresponds to looking at the L 2 -norm
of image derivatives. That is, we examine scaling of the energy of image derivatives. Note,
furthermore, that the L 1 -norm of any other invariant quadratic in L of total order of derivation
2n also equals kI (n) k1 (see [22]).
x ) : R N 7→ R is an N -D fBm and L(E
x ; t) : R N × R 7→ R is the scaleTHEOREM 3.1. If f H (E
(n)
space of f H (E
x ), then the nth order invariants I (E
x ; t) in this scale-space can be normalized
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PEDERSEN AND NIELSEN
to equal energy on all scales by the following relation,
(n)
Inorm
(E
x ; t) = t γ n I (n) (E
x ; t),
where γn = −α/2 + n + N /2, and α = 2H + 1.
Proof. The proof is inspired by a similar analysis of the power spectrum of images of
natural scenes by Lindeberg [15, 20]. By using Parseval’s identity we find
Z
¯
¯ 2
(n)
¯G̃
E t) f˜2H (ω)
E ¯ dω
E
kI k1 =
i 1 ···i n (ω;
Z
=
ω
E ∈R
N
ω
E ∈R
N
ω
E ∈R
N
Z
=
¯2n E 2 t 2
¯
f˜ H (ω)
| j n |ω
E ¯ e−|ω|
E ¯ dω
E
E t
e−|ω|
|ω|
E −α+2n d ω,
E
2
E and G̃ i1 ···in (ω;
E t), respectively, are the Fourier transformed image
where j 2 = −1, and f˜α (ω)
and the Fourier transformed nth order differentiated Gaussian. Using the relation
Z ∞
0((m + 1)/2)
2
x m e−ax d x =
2a (m+1)/2
0
and introducing N -D spherical coordinates, we find
Z
2
ρ −α+2n+N −1 e−ρ t · dρ dϕ1 · · · dϕ N −1
ρ∈[0,∞[;ϕ1 ,...,ϕ N −1 ∈[0,2π ]
= (2π)
N −1
¡
0 − α2 + n +
α
2t − 2 + n+ 2
N
N
2
¢
α
= K · t 2 −n− 2 = K · t −γn ,
N
where K is an arbitrary constant, and hereby we arrive at γn = −α/2 + n + N /2.
j
The normalization relation of Theorem 3.1 implies a special case for the 0th order derivatives, meaning the case of the undifferentiated scale-space. For this case we should, according to Theorem 3.1, scale-normalize the scale-space by an exponent γ0 = −α/2 + N /2
introduced by the fractal dimension of the original image. Normalization of the nth order
derivatives can then be seen as just the normalization based on dimensional analysis, because γn = γ0 + n. This special case of the 0th order derivatives comes from the fact that
the fBm is the fractional derivative or integral of the Brownian motion.
A benefit of the proposed normalization method is that the normalization relation can be
used as a method for estimation of the fractal dimension of images. This can be done by
calculating the L 1 -norm of a collection of differentiated scale-space images and then fitting
the logarithmic norm values to a straight line. We use this method in Section 4 to estimate
the fractal dimensions of synthetic and real images. We will not conduct a comparative
study of this method and other methods for estimation of the fractal dimension of images
(see [8] for a study of other methods), but merely point out the existence of the method.
In conjunction with feature detection, we must use the fractal dimension of the image
in a neighborhood of the feature of interest. This suggests a simple relation between the
topological dimension of the feature and a suitable choice of fractal dimension. In Table I
we have listed Lindeberg’s [17] suggested normalized measures of feature strength. For
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HAUSDORFF DIMENSION AND SCALE-SPACE NORMALIZATION
TABLE I
The Measure of Feature Strength Used by Lindeberg for Feature Detection
with Automatic Scale Selection Using His γ-normalization
Feature type
Normalized strength measure
γ
H
DH
T
Edge
Ridge
Corner
Blob
t γ /2 L v
t 2γ (L pp − L qq )2
t 2γ L 2v L uu
tγ ∇2 L
1/2
3/4
1
1
1
1
1/2
1/2
2
2
2.5
2.5
1
1
0
0
Note. We have calculated the corresponding values of H and D H using our definition of γ . This table is a
reproduction of a table from [17] extended with columns for H , D H and the topological dimension T of the
features. Note that a simple relation exists between the fractal dimension D H and the topological dimension
T . The relation between γ and T is not as straightforward.
each feature we have calculated the H and D H values that correspond to his suggested γ
values. It is interesting to notice that corners and blobs have a fractal dimension of 2.5 and
edges and ridges only have a fractal dimension of 2. The topological dimension of corners
and blobs is 0, while edges and ridges have a topological dimension of 1. Near a corner
or a blob, we would expect the void hypothesis of H = 12 . This is not expected to be true
in a neighborhood of 1D features owned to the spatial extend and we see that Lindeberg’s
choice of γ leads us to the hypothesis of H = 1 for both 1D features.
4. EXPERIMENTS
We have conducted several experiments on synthetic and real 2D images in order to
study the normalization of digitized images. We can, as stated earlier, use the normalization
method to find the fractal dimension of images by computing unnormalized derivatives
of the scale-space of the considered image. From this unnormalized scale-space we can
estimate the value of γ and calculate the Hausdorff dimension D H . In the same manner one
can get estimates of the local fractal dimension at a point in the original image. The fractal
dimension of a point could be viewed as a contradiction in terms, but it is nevertheless
possible to assign some meaning to this concept due to the intrinsic property of scale-space:
A point in scale-space corresponds to a neighborhood in the underlying image.
It is the authors’ opinion that in principle all theory of fractal measures may be reformulated in the inherently well-posed framework of linear scale-space theory, thereby easing
operationalization of fractal measures.
In Table II we show some results for small synthetic images (see the images in Fig. 1). The
synthetic images used for these experiments were constructed in the frequency domain and
were given a power spectrum proportional to |ω|
E 2 and a random phase. We have calculated
the L 1 -norm of different images from two scale-space invariants L i L i and L i j L ji . On this
basis we have estimated the γ values for the synthetic images and compared them to the
theoretical values from the continuous domain theory.
The method of estimating the fractal dimension that we propose is fairly accurate on
synthetic images of known fractal dimensions. From Table II we see that our method delivers
an inaccurate result for increasing α values. The reason for this inaccuracy is that when the
α value is increased, the synthetic image will have structure on an increasing scale and when
the α value of the image becomes large enough the outer scale of the large structures will
exceed the outer scale of the image, thereby misleading our method. Furthermore, our results
are biased by spectral leakage, because artificial periods are introduced into the images by
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PEDERSEN AND NIELSEN
FIG. 1. The synthetic images with different α values and their corresponding graphs of the L 1 -norm of
L i L i and L i j L ji which we used for estimation of the γ values in Table II. The synthetic images all have 256 ×
256 pixels and α = 1, 2, 2.5, 3 from the top down. The L 1 -norm graphs were produced by calculating a scale-space
for the two set of derivatives. This scale-space has 10 different scales between σ = 2 and σ = 30 with exponential
growing increments. On the graphs it can be seen that high scales make the estimate of the L 1 -norm inaccurate,
i.e., the estimate becomes too small. The reason for this inaccuracy is discretization effects introduced by the outer
scale of the image.
HAUSDORFF DIMENSION AND SCALE-SPACE NORMALIZATION
273
TABLE II
Estimated γ Values for Synthetic Images
α
Estimated γ values
Actual γ values
1
2
2.5
3
a. Values for a 2D image differentiated n = 1 times (L i L i )
−1.57
−1.5
−1.15
−1
−0.96
−0.75
−0.78
−0.5
−4.46%
−15.0%
−28.0%
−56.0%
1
2
2.5
3
b. Values for a 2D image differentiated n = 2 times (L i j L ji )
−2.55
−2.5
−2.11
−2
−1.89
−1.75
−1.68
−1.5
−2.0%
−5.5%
−8.0%
−12.0%
Relative error
Note. The synthetic images used, and the corresponding graphs of the L 1 -norm of L i L i
and L i j L ji , are depicted in Fig. 1. In the table there are four synthetic images with different
α values. The γ values are estimated by first constructing the synthetic image with the
specified α parameter and then computing a series of 10 differentiated scale-space images
of ascending scales. For each of these 10 images we have calculated the L 1 -norm, which
can be seen in the graphs of Fig. 1. These graphs reveal an inaccuracy of the estimated
L 1 -norms at high scales, which is why we chose to use only the L 1 -norms of the first five
images of the scale-space (σ ∈ [2.0; 6.7[) for our estimation of γ . The γ value is estimated
by calculating the logarithmic slope of the L 1 -norms of the scale-space images. The slope
is the estimated γ value. The reason why the estimated values of the synthetic images are
not exact is because the images we used were small. That is, the span from the inner scale to
the outer scale of the images is not sufficient to establish γ as a single global average over
all scales. Note that α = 2 corresponds to a classical Brownian motion.
the Fourier transformation. It can also be seen from Table II that when we increase the order
of differentiation we also increase the accuracy of the method. The reason for this is that
when we derive our image we enhance the fine structure of the image by effectively looking
at a scale interval which has been moved toward smaller scales. In real examples, image
FIG. 2. We have computed the scale-space invariants L i L i , L i j L ji , and L i jk L k ji of the garden image from
Fig. 3. These graphs show the L 1 -norm of I (n) t n+1 and (I (n) )1/n of these three scale-space invariants. The solid
lines correspond to L i L i , the dashed lines to L i j L ji , and the dotted lines to L i jk L k ji . The estimated slopes are
for I (n) t n+1 , γ = 1.13, 1.10, 1.12 (n = 1, 2, 3), and for (I (n) )1/n , γ = −0.87, −0.95, −0.96 (n = 1, 2, 3).
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PEDERSEN AND NIELSEN
FIG. 3. The images for which we have estimated the fractal dimension (see Table III). All images are gray
level images and the first six images all consist of 256 × 256 pixels and the last two consist of 512 × 512 pixels.We
have called the images from the top left corner going in the reading direction: Garden, X-rayed bone, Water Lily,
Seaweed, Grains of sand, Satellite clouds, Landscape, and Trees.
HAUSDORFF DIMENSION AND SCALE-SPACE NORMALIZATION
275
TABLE III
Estimated γ Values and the Corresponding Scaling Exponent
H and Hausdorff Dimension D H
Title
Estimated γ values
H
DH
Garden
X-rayed bone
Water lily
Seaweed
Grains of sand
Satellite clouds
Landscape
Trees
−1.90
−1.62
−1.53
−1.98
−2.09
−1.89
−1.75
−1.82
0.60
0.88
0.97
0.52
0.41
0.61
0.75
0.68
2.40
2.12
2.03
2.48
2.59
2.39
2.25
2.32
Note. The values of H were computed through H = n+ N2 − γ − 12 and the
values of D H were computed through D H = N + (1 − H ). The dimensions
of the images were N = 2 and they were differentiated by L i j L ji (n = 2).
The γ values were estimated in the same fashion as in Table II and the
images used can be seen in Fig. 3. The estimated values of D H indicated
the same results as Bialek [6] where the Hausdorff dimension of images of
natural scenes are not necessarily close to D H = 2.5. Unfortunately we have
no way of determining the error on the results in this table.
noise from the capture device will exhibit a structure other than the random process of the
scene. In general this is more uncorrelated noise and a scale interval of smaller scales will
exhibit structure merely from the capture device. That is, we must choose an appropriate
scale if we wish to measure scale characteristics.
We expect a logarithmic relation between the scale and kI (n) k1 . From Table II and Fig. 1
we can see that the method for normalization proposed here is quite reasonable for synthetic
images. In order to examine our method on real images, we have calculated the L 1 -norm of
invariants of increasing order of differentiation of the garden image from Fig. 3. We have
normalized the computed invariants by I (n) t n+1 , which corresponds to our normalization
method, and (I (n) )1/n , which corresponds to the standard normalization method, in order to
examine the scaling property of the image independent of the order of differentiation. The
slope of the logarithmic plot corresponds to γ and we would expect that this slope should be
approximately the same for all orders of differentiation only for our normalization method
I (n) t n+1 . The results can be viewed in Fig. 2. From this figure it can be concluded that
our normalization method seems to be a reasonable choice, but we can also see that the γ
of the standard normalization method for this image is fairly independent of the order of
differentiation. This inconclusive experiment therefore calls for a thorough evaluation of
the scaling properties of a large ensemble of images of natural scenes.
We have also tried to estimate the Hausdorff dimension of some 2D images of natural
scenes. The results can be viewed in Table III.
5. CONCLUSION
We have related Lindeberg’s [15–17] scale-space normalization method to the notion
of fractal dimension, assuming that images of natural scenes can be modeled by fractal
Brownian motions and we propose that feature strength measures are normalized using the
Hausdorff dimension of the local image.
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We have found a normalization expression that has the Hausdorff dimension as a parameter. This expression reveals a possible relation between the topological dimension of
the feature of interest and the fractal dimension of the local image around the feature (see
Table I). We conjecture (for future experimental testing): The topological dimension of the
feature uniquely determines the scale-space normalization parameter.
We propose a further investigation into the relation between different features and their
Hausdorff dimension. It would be interesting to see whether it is possible to generalize the
results described in Table I and further establish a general relation between the topological
dimension of features and the fractal dimension locally in the image. Furthermore, we
suggest a thorough investigation of the scaling properties of images of natural scenes using
a large ensemble of images.
ACKNOWLEDGMENTS
We thank Tony Lindeberg for inspiring us to do this work. Furthermore, we thank Peter Johansen for his
comments on some of the theory of this paper.
REFERENCES
1. D. J. Field, Relations between the statistics of natural images and the response properties of cortical cells,
J. Opt. Soc. Am. 4, 1987, 2379–2394.
2. J. Gårding, A Note on the Application of Fractals in Image Analysis, Technical Report TRITA-NA-P8716
CVAP 49, Royal Institute of Technology, Dec. 1987.
3. D. C. Knill, D. Field, and D. Kersten, Human discrimination of fractal images, J. Opt. Soc. Am. 7, 1990,
1113–1123.
4. B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982.
5. A. P. Pentland, Fractal-based description of natural scenes, IEEE Trans. Pattern Anal. Mach. Intell. 6, 1984,
661–674.
6. D. L. Ruderman and W. Bialek, Statistics of natural images: Scaling in the woods, Phys. Rev. Lett. 73, 1994,
814–817.
7. N. L. Fazzalari and I. H. Parkinson, Fractal dimension and architecture of trabecular bone, J. Pathology 82,
1996, 100–105.
8. J. F. Veenland, J. L. Grashius, F. van der Meer, A. L. Beckers, and E. S. Gelsema, Estimation of fractal
dimension in radiographs. Med. Phys. 82, 1996, 585–594.
9. E. Ott, Chaos in Dynamical Systems, Cambridge University Press, Cambridge, UK, 1993.
10. T. Iijima, Basic theory on normalization of a pattern, Bull. Electrical Lab. 26, 1962, 368–388. [In Japanese]
11. A. P. Witkin, Scale space filtering, in Proc. International Joint Conference on Artificial Intelligence, Karlsruhe,
Germany, 1983, pp. 1019–1023.
12. J. J. Koenderink, The structure of images, Biol. Cybernet. 50, 1984, 363–370.
13. D. Marr, Vision, Freeman, San Francisco, 1882.
14. L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink, and M. A. Viergever, Linear scalespace. J. Math.
Imaging Vision 4, 1994, 325–351.
15. T. Lindeberg, Scale Selection for Differential Operators, Technical Report ISRN KTH/NA/P-9403-SE, Royal
Institute of Technology, January 1994.
16. T. Lindeberg, Edge detection and ridge detection with automatic scale selection, Int. J. Comput. Vision 30,
1998, 117–156.
17. T. Lindeberg, Scale-space: A Framework for Handling Image Structures at Multiple Scales, Technical Report
CVAP-TN15, Royal Institute of Technology, Sep. 1996.
18. B. B. Mandelbrot and J. W. van Ness, Fractional brownian motions, fractional noises and applications, SIAM
Rev. 10, 1968, 422–437.
HAUSDORFF DIMENSION AND SCALE-SPACE NORMALIZATION
277
19. R. F. Voss, Random fractal forgeries, in Fundamental Algorithms for Computer Graphics (R. A. Earnshaw),
Ed., Vol. 17, 805–835, Springer-Verlag, Berlin, 1985.
20. T. Lindeberg, Scale-Space Theory in Computer Vision, Kluwer Academic, Dordrecht/Norwell, MA, 1994.
21. J. H. Elder and S. W. Zucker, Local scale control for edge detection and blur estimation, in ECCV’96
(B. Baxton and R. Cipolla, Eds.), pp. 57–69, Springer-Verlag, Berlin, 1996.
22. M. Nielsen, L. Florack, and R. Deriche, Regularization, scale-space, and edge detection filters, J. Math.
Imaging Vision 7, 1997, 291–307.
KIM STEENSTRUP PEDERSEN received in 1999 his M.Sc. in computer science at the Department of Computer Science, University of Copenhagen, and he also holds a B.Sc. in physics from the University of Copenhagen.
His primary interest of research are computer vision and image analysis, especially scale-space theory, scaling
properties of natural image structures, and medical image analysis.
MADS NIELSEN received a M.Sc. in 1992 and a Ph.D. in 1995 both in computer science from DIKU,
Department of Computer Science, University of Copenhagen, Denmark. During his Ph.D. studies he spent one
year (1993–1994) at the Robotvis Lab at INRIA, Sophia-Antipolis, France. In the second half of 1995 he was
postdoc at the Image Sciences Institute of Utrecht University, The Netherlands. In 1996 he was joint postdoc
at DIKU and 3D-Lab, School of Dentistry, University of Copenhagen, where he served as assistant professor
for 1997–1999. In 1998–1999 he served as external associate professor at Institute of Mathematical Modelling,
Technical University of Denmark. In April 1999 he became the first associate professor at the new IT University
in Copenhagen.