Application of the fractal Perlin noise algorithm for the generation of simulated breast tissue

Abstract

Software breast phantoms are increasingly seeing use in preclinical validation of breast image acquisition systems and image analysis methods. Phantom realism has been proven sufficient for numerous specific validation tasks. A challenge is the generation of suitably realistic small-scale breast structures that could further improve the quality of phantom images. Power law noise follows the noise power characteristics of breast tissue, but may not sufficiently represent certain (e.g., non-Gaussian) properties seen in clinical breast images. The purpose of this work was to investigate the utility of fractal Perlin noise in generating more realistic breast tissue through investigation of its power spectrum and visual characteristics. Perlin noise is an algorithm that creates smoothly varying random structures of an arbitrary frequency. Through the use of a technique known as fractal noise or fractional Brownian motion (fBm), octaves of noise with different frequency are combined to generate coherent noise with a broad frequency range. fBm is controlled by two parameters - lacunarity and persistence - related to the frequency and amplitude of successive octaves, respectively. Average noise power spectra were calculated and beta parameters estimated in sample volumes of fractal Perlin noise with different combinations of lacunarity and persistence. Certain combinations of parameters resulted in noise volumes with beta values between 2 and 3, corresponding to reported measurements in real breast tissue. Different combinations of parameters resulted in different visual appearances. In conclusion, Perlin noise offers a flexible tool for generating breast tissue with realistic properties.

Full text

Application of the fractal Perlin noise algorithm for the generation of
simulated breast tissue
Magnus Dustler*1, Predrag Bakic2, Hannie Petersson1, Pontus Timberg1, Anders Tingberg1 and
Sophia Zackrisson3
1Medical Radiation Physics, Department of Translational Medicine, Lund University, SE 205 02
Malmö, Sweden; 2Department of Radiology, University of Pennsylvania, Philadelphia, PA 19104,
USA; 3Diagnostic Radiology, Department of Translational Medicine, Lund University, SE 205 02
Malmö, Sweden
ABSTRACT
Software breast phantoms are increasingly seeing use in preclinical validation of breast image acquisition systems and
image analysis methods. Phantom realism has been proven sufficient for numerous specific validation tasks. A
challenge is the generation of suitably realistic small-scale breast structures that could further improve the quality of
phantom images. Power law noise follows the noise power characteristics of breast tissue, but may not sufficiently
represent certain (e.g., non-Gaussian) properties seen in clinical breast images. The purpose of this work was to
investigate the utility of fractal Perlin noise in generating more realistic breast tissue through investigation of its power
spectrum and visual characteristics. Perlin noise is an algorithm that creates smoothly varying random structures of an
arbitrary frequency. Through the use of a technique known as fractal noise or fractional Brownian motion (fBm), octaves
of noise with different frequency are combined to generate coherent noise with a broad frequency range. fBm is
controlled by two parameters – lacunarity and persistence – related to the frequency and amplitude of successive octaves,
respectively. Average noise power spectra were calculated and beta parameters estimated in sample volumes of fractal
Perlin noise with different combinations of lacunarity and persistence. Certain combinations of parameters resulted in
noise volumes with beta values between 2 and 3, corresponding to reported measurements in real breast tissue. Different
combinations of parameters resulted in different visual appearances. In conclusion, Perlin noise offers a flexible tool for
generating breast tissue with realistic properties.
Keywords: Mammography, Tomosynthesis, Anatomical noise, Software breast phantoms, Perlin noise
1. INTRODUCTION
Realistic-looking software breast phantoms have many applications in breast imaging research, especially for
characterization of the properties of new or emerging technologies. Using a software phantom – with or without a
simulated lesion – provides knowledge of the ground truth in observer studies, gives the researchers control over
breast anatomy and allows the generation of an arbitrary amount of images without concerns for radiation doses.
Many groups thus have worked on generating anthropomorphic phantoms that are suitably realistic for specific tasks
[1-15]. The challenge is to implement a random generation of realistic patterns of breast tissue using statistical
methods.
1.1 Power-law noise
Breast tissue, when projected onto a 2D-plane, have a Fourier domain frequency spectrum which follows a power
law distribution [16]. Based on these results, 2D-power law noise with β parameter values somewhere between 2
and 3 have been used to simulate the anatomical noise seen in clinical breast images [2, 16-18].
Power law noise alone, however, cannot capture the continuous anatomical structures which make up the breast and,
also, may not sufficiently represent certain (e.g., non-Gaussian) properties seen in clinical breast images [16, 17, 19,
20]. Recent papers by Abbey et al. have used a novel measure known as Laplacian fractional entropy (LFE), which
*Corresponding author. magnus.dustler@med.lu.se
Medical Imaging 2015: Physics of Medical Imaging, edited by Christoph Hoeschen, Despina Kontos,
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quantifies the difference in entropy between the image histogram and a best-fit Gaussian distribution, normalized by
the entropy of a best-fit Laplacian distribution [19, 20]. They showed significant differences in LFE between
different simulated breast phantoms [20].
1.2 Appearance of dense tissue
Breast tissue is far from homogenous and the dense tissue that needs to be realistically simulated to obtain a good
software breast phantom actually consists of several types of distinct tissues with distinct appearances and patterns of
growth. According to the well-known Tabar classifications the four types of tissue within the breast are 1: nodular, 2:
linear, 3: homogenous and 4: radiolucent [21, 22]. As breasts consist of a mixture of these types of tissue, it is important
that any simulated noise share characteristics of all four.
1.3 Aim
The aim of this paper was to investigate the utility of fractal Perlin noise – a noise-generating method which creates
continuous, smoothly varying, random shapes – to simulate breast tissue by investigating its power spectrum and visual
appearance.
2. METHODS AND MATERIALS
2.1 Perlin noise
The fields of medical imaging and computer graphics share a need to realistically simulate natural structures and
textures. One of the most well-known methods to generate such structures in computer graphics is the Noise-algorithm
[23-25], which was the subject of a special Academy Award in 1997. The algorithm is commonly known as Perlin noise
after its creator and is employed to generate e.g. landscapes and clouds. Unlike a Gaussian noise distribution, Perlin
noise consists of coherent structures that are connected through smoothly varying gradients [26]. These basic shapes
form the basis of various more complex textures.
Perlin noise is a wavelet-based method where an n-dimensional set of points is evaluated on an n-dimensional evenly-
spaced grid, where every grid-point has a random gradient associated with it. Each point is set to a weighted average
between the scalar products between the gradients of the 2n closest grid-points and the vectors between it and those grid-
points. This creates structures of a scale (or, in other words, frequency) that is band limited by the spacing of the grid-
points and the spacing of the evaluated points. Figure 1 shows Perlin noise of various scales.
Figure 1. Examples of Perlin noise with different scale, i.e. number of grid points along each dimension. All images are
projections of 128x128x128 voxel 3D volumes. From left to right: scale = 2, 4, 8, 16, 32, 64, 128.
2.2 Fractal noise
As noted earlier, the spectral density of breast tissue obeys the power law distribution with β parameter values between 2
and 3 [16, 17, 27, 28]. As such, it includes tissue structures of many sizes with diminishing amplitudes, spanning a wide
range of spatial frequencies. Noise generated by the Perlin Noise-algorithm is itself limited to a narrow range of
frequencies.
Fractal noise is a process through which noise is generated at different scales (or frequencies) and added together to
create noise with a broader frequency range [29-32]. Successively smaller scales (higher frequencies) of noise are known
as octaves. Fractal noise is used both in signal processing and in 2D- and 3D-imaging applications. Due to its similarities
with Brownian motion, it is sometimes called fractional Brownian motion (fBm) [31, 32, 35]. A substantially different
implementation of fractional Brownian motion has also been used to simulate breast tissue using a random-walk
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.
approach [2]. Many different noise algorithms can be used to create the basis for fractal noise, including so-called
cellular or Worley noise, another widely used noise-generating algorithm [26].
Fractal Perlin noise can simulate many different structures and processes and has been used to render landscapes, water
and similar systems [35-39]. The fractal noise distribution f of starting frequency u can be expressed as
()=( ∙)


where g(u) is the noise function (Perlin noise, in this case), s is the persistence, w is the lacunarity and n is the number of
octaves included. Persistence controls the amplitude of the noise, with the amplitude at each successive octave being
equal to the amplitude at the previous octave multiplied by the persistence. Lacunarity, a value related to the fractal
dimension, defines the frequency gap between successive octaves, with the frequency of each octave equaling the
frequency of the previous one multiplied by the lacunarity. Varying these parameters creates structures with different
appearance and power spectrums. Examples of Perlin noise with fractional Brownian motion for various combinations of
persistence and lacunarity are shown in Figures 2.
Figure 2. Examples of fBm Perlin noise volumes with different combinations of lacunarity and persistence, projected onto
the 2D-plane. Lacunarity ranges from 2-4 with increments of 1 and persistence from 1-2 with increments of 0.5. On the left,
noise values are continuous and range from -1 to 1. On the right, noise values are binary, i.e. either set to 1 (dense tissue) or
0 (fatty tissue).
There are also many other ways of adding together octaves of noise to create interesting textures, such as modifying the
input noise function g(u) with some other function h(u), sometimes called a turbulence function, i.e.
()=ℎ( ∙)


Various turbulence functions can be used to change the appearance of the simulated noise. One example is Perlin’s
original implementation of marble-like textures using a sinusoidal function [23].
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2.3 Noise
ge
Fractal 3D P
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arameters c
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.
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2.5 Qualita
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Five additio
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shown in Fig
u
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s
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n
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w
e
r
r
s
.
y
r
c
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Figure 4. UPenn breast phantom modified with Perlin noise. Bottom projection image is the unmodified
phantom, with compartments designated as either dense or fatty tissue. The top image is the same phantom
modified with binary Perlin noise, with dense compartments being assigned a denser distribution of Perlin noise
than fatty ones.
3. RESULTS
3.1 Quantitative analysis
Estimates of β from linear regressions are shown in Tables 1 and 2. Binary volumes had a substantially lower β value,
indicating a lesser dampening of high frequencies. As expected increasing lacunarity and persistence both decrease the β
value. Using high lacunarity however means that the number of octaves will be low, making the spectrum discontinuous.
High persistence on the other hand emphasizes high-frequency noise, deemphasizing the continuous lower frequency
structures. Despite this, both assuming binary dense/non-dense tissue, and non-binary dense tissue fraction, values of β
between 2 and 3 were found for a range of persistence/lacunarity combinations.
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Table 1. Measured β for projected 3D-volumes
Persistence
1/4 ½ ¾ 1
Lacunairty
2 6.85±0.06 4.80±0.08 3.64±0.05 2.93±0.06
3 5.27±0.10 3.79±0.07 3.03±0.08 2.51±0.09
4 4.34±0.09 3.29±0.04 2.76±0.08 2.40±0.07
Table 2. Measured β for projected binary 3D-volumes
Persistence
1/4 ½ ¾ 1
Lacunairty
2 2.73±0.83 3.21±0.08 2.70±0.06 2.20±0.08
3 3.27±0.17 2.86±0.11 2.33±0.08 1.99±0.04
4 3.23±0.18 2.66±0.08 2.17±0.08 1.89±0.07
3.2 Qualitative analysis
Radiologists’ rankings of the plausibility of the five simulated tissue are presented in Table 3.
Table 3. Radiologists’ rankings of simulated tissue
Simulation 1 Simulation 2 Simulation 3 Simulation 4 Simulation 5
Radiologist 1 2 1 2 1 1
Radiologist 2 3 1 1 2 3
Radiologist 3 3 4 1 2 4
Radiologist 4 4 3 1 4 5
Mean ranking 3.0±0.8 2.3±1.5 1.3±0.5 2.0±1.3 3.3±1.7
For the phantom study, the four participating radiologists consistently indicated that the UPenn phantom with added
Perlin noise was more realistic than the unmodified phantom.
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4. DISCUSSION
Perlin noise has been used to simulate many different natural textures and it shows promise in also being able to
accurately simulating breast tissue with some tweaking of parameters. Measured β values indicate that fractal Perlin
noise can be used to provide tissue volumes with power spectrums that match realistic values, both for continuous and
binary volumes. Whether the appearance of the noise is realistic is another question.
Different turbulence functions provide substantially different structures. Nodular density looks similar to unmodified
fractal Perlin noise. Linear density could be represented by ridged noise, i.e. the absolute value of the Noise-function.
Homogenous densities can perhaps be simulated by aggressively thresholded noise volumes. Directionality of structures
could also be added by e.g. dilating selected noise octaves – as shown in one of the sample simulations in Figure 3 – with
ellipsoidal structuring elements. In the subjective opinions of the four radiologists who took part in the study, several of
these approaches provided relatively plausible facsimiles of breast tissue. Although the sample sizes were too small and
the task too broadly defined for any statistical analysis, the radiologists agreed that both unmodified fractal Perlin noise
and ridged fractal Perlin noise (i.e. absolute value) were plausible representations of tissue, while binary noise was less
plausible. Nevertheless, the modified UPenn phantom (containing binary fractal Perlin noise) was consistently ranked as
plausible; indicating that results from simulations viewed in isolation might not be representative of results from viewing
the entire phantom.
Directionality of structures could also be added by e.g. dilating selected noise octaves – as shown in one of the sample
simulations in Figure 3 – with ellipsoidal structuring elements or by using anisotropic grid spacing and matching the
direction to e.g. the principal direction of the compartments and sub-compartments of the UPenn breast phantom.
Further investigation of the non-Gaussian properties of the simulated tissue is required, such as the Laplacian fractional
entropy mentioned before. Also in order to match the density of real breasts, volumetric density assessment software
such as e.g. Volpara (Matakina Technology Ltd., Wellington, New Zealand) could be used to get baseline measurements
of variously dense and fatty breasts.
In conclusion, fractional Perlin noise offers a promising and flexible way of simulating plausible breast structures.
However, much work remains to be done in order to better align its appearance to real breast tissue.
ACKNOWLEDGMENTS
The authors would like to acknowledge the contribution of Dr. A.D.A Maidment of the University of Pennsylvania. The
research at Lund University was supported by the Swedish Breast Cancer Foundation – Bröstcancerfonden. The digital
breast phantom research at the University of Pennsylvania was supported by the NCI/NIH under award R01CA154444
and the Susan G. Komen Foundation Grant IIR13262248.
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