Why Using the Alpha-stable Distribution in Neuroimage?

Diego Salas-Gonzalez

, Juan M. G´orriz

, Javier Ram´ırez

and Elmar W. Lang

Institut f¨ur Biophysik und physikalische Biochemie , University of Regensburg, Regensburg, Germany

Dpt. Signal Theory, Networking and Communications, University of Granada, Granada, Spain

Keywords:

α-stable Distribution, Biomedical Applications, Brain Image Processing.

Abstract:

The main goal and overall objective of this contribution is to attract the attention of the potentialities and wide

range of applications of the α-stable distribution in biomedical applications, speciﬁcally in neuroimaging. The

α-stable density is a heavy-tailed, non-symmetric distribution with similar desirable properties to the Gaussian.

Indeed, the Gaussian distribution is a particular case of the α-stable family. The Gaussian distribution is used

ubiquitously in brain image processing. For this reason, we believe that the α-stable density can be potentially

used as an alternative to the Gaussian distribution in several biomedical applications regarding brain imaging.

Some of the proposed applications of the α-stable distribution considered in this work are the development

of brain image processing approaches with applications to intensity normalization of SPECT images, MRI

segmentation and feature extraction for the diagnosis of Parkinsonian’s syndrome.

1 INTRODUCTION

Non-Gaussian statistical signal processing is impor-

tant when signals deviate from the ideal Gaussian

model. α-stable distributions are amongst the most

important non-Gaussian models. They share deﬁning

characteristics with the Gaussian distribution, such as

the stability property and central limit theorem, and

include the Gaussian distribution as a limiting case.

Heavy-tailed modeling using the alpha-stable dis-

tribution has been successfully applied in many ﬁelds

of research. In engineering related methods: radar

processing, telecommunications, acoustics, network

modeling, queuing theory or ICA/blind source sep-

aration. In economics and ﬁnance: modeling asset re-

turns, option pricing and commodity price modelling.

In computer science, physics, astronomy, chemistry,

geology, geophysics and genetics.

The history of research on this particular distri-

bution family is old starting with the work of Paul

L´evy in 1925. The applications of alpha-stable dis-

tributions have been limited however until much later

when (Mandelbrot, 1963) suggested they could be

used to model ﬁnancial time series data. There-

after, the alpha-stable distribution has been frequently

found in analysis of critical behaviour and ﬁnancial

data (Voit, 2003).

The α-stable distribution in engineering gained

popularity and attracted the attention of engineers

worldwide after the work published by Shao in Pro-

ceedings of the IEEE (Shao and Nikias, 1993). Appli-

cations of the α-stable distribution in engineering are

still a productive ﬁeld of research nowadays. Never-

theles, the α-stable distribution has been used just in

a few cases recently published in neuroimaging appli-

cations.

On contrast, the Gaussian distribution is used

ubiquitously in neuroimaging. For this reason, we be-

lieve alpha-stable density can be potentially used as

an alternative to the Gaussian distribution.

2 SOME PROPERTIES OF THE

α-STABLE DISTRIBUTION

The characteristic function ϕ(ω) of an α-stable distri-

bution f

α,β

(y|γ, δ) is given by:

ϕ(ω) =

(

−|γω|

[1−isign(ω)βtan(

πα

)]+iδω

, (α 6= 1)

−|γω|[1+isign(ω)

βlog(|ω|)]+iδω

, (α = 1)

(1)

where the parameters of the stable distribution are:

α ∈ (0, 2] is the characteristic exponent which sets

the level of impulsiveness, β ∈ [−1, +1] is the skew-

ness parameter, (β = 0, for symmetric distributions

and β = ±1 for the positive/negative stable family re-

spectively), γ > 0 is the scale parameter, also called

dispersion, and δ is the location parameter.

297

Salas-Gonzalez D., Górriz J., Ramírez J. and Lang E..

Why Using the Alpha-stable Distribution in Neuroimage?.

DOI: 10.5220/0005091102970301

In Proceedings of the 11th International Conference on Signal Processing and Multimedia Applications (SIGMAP-2014), pages 297-301

ISBN: 978-989-758-046-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

−5 0 5

0.2

0.4

0.6

0.8

α = 1.5

α = 1.0

α = 0.5

−5 0 5

0.1

0.2

0.3

0.4

β = −1.0

β = 0.0

β = 1.0

−5 0 5

0.2

0.4

0.6

0.8

1.2

1.4

γ = 1.0

γ = 0.5

γ = 0.25

−5 0 5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

δ = 0.0

δ = 1.0

δ = 2.0

Figure 1: α-stable probability density function with refer-

ence parameters α = 1.5, β = 0, γ = 1 and δ = 0 with chang-

ing: (a) Characteristic exponent α. (b) Skewness parameter

β. (c) Dispersion γ. (d) Location parameter δ.

Figure 1 shows the α-stable probability density

function for different values of the parameters. We

use the distribution with parameters α = 1.5, β = 0,

γ = 1 and δ = 0 as reference. This ﬁgure also explain

the name of the parameters: α controls the degree of

impulsiveness. When α decreases, the degree of im-

pulsiveness increases. β controls the skewness and its

sign, if the asymmetry is on the left or the right. γ con-

trols the concentration of the samples along the bulk

of the distribution. Lower values of γ correspond with

higher concentration of the samples. Lastly, differ-

ent values of δ produce the same probability density

function but shifted in the x-axis.

3 α-STABLE DISTRIBUTION IN

NEUROIMAGE

3.1 Intensity Normalization of Brain

FP-CIT SPECT Images

Previous studies have demonstrated that when

[123I]FP-CIT SPECT reaches equilibrium binding in

the brain, a simple unitless ratio of regional radioac-

tivities is proportional to the binding potential (Scher-

ﬂer et al., 2005; Aarts et al., 2012). Nevertheless, this

binding ratio leads to intersubject differences in the

histogram of intensity values. As the histogram of in-

tensity values is unimodal, skewed and heavy-tailed,

one can model this histogram using an alpha-stable

distribution in order to perform intensity normaliza-

tion of the brain images.

The histogram of intensity values in FP-CIT

SPECT images shares some of the properties of the

α-stable distribution. Some of these common proper-

ties are:

• Heavy probability tails due to the existence of a

few regions of the brain with high intensity values.

• Peaked bulk, because most of the voxels in the

brain, the non-speciﬁc area, have very similar in-

tensity values, except the striatum, which is the

area with greater variability depending on the type

of image (Parkinson’s syndrome (PD) or Normal

Control patients (NC)).

• Positive asymmetry, because intensity values are

always greater than 0 and the bulk of the distribu-

tion reaches lower intensity values compared with

the values obtained in the striatum.

The predicted α-stable parameters and the location-

scale property can be used to transform the intensity

values in each voxel linearly. This transformation en-

sures that the new histograms in each image have a

pre-speciﬁed α-stable distribution with desired loca-

tion and dispersion values.

These features have been exploited in (Salas-

Gonzalez et al., 2013b). The histogram of a vector

of intensity data with α-stable distribution f with pa-

rameters [α, β, γ, δ], denoted by X ∼ f

α,β

(y|γ, δ) can

be easily transformed to another α-stable distribution

with parameters Y ∼ f

α,β

(y|γ

∗

, δ

∗

) by using the fol-

lowing expression (Samoradnitsky and Taqqu, 1994):

Y = aX + b (2)

where a =

∗

and b = δ

∗

−

∗

δ.

Figure 2 depicts the histogram of intensity val-

ues in FP-CIT SPECT brain images showing the spe-

ciﬁc and non-speciﬁc regions and their locations in a

transaxial slice.

3.2 Modelization of the Distribution of

Brain Matter

The principal goal of a segmentation process is to

partition an image into different regions which are

homogeneous with respect to one or more features.

Segmentation is an important tool in medical image

processing and has been useful in many applications.

In MRI, segmentation is performed to divide the en-

tire image into sub-regions such as white matter, gray

matter and cerebrospinal ﬂuid spaces of the brain

(Balafar et al., 2010). The Gaussian mixture model

has been widely applied in brain MRI segmentation,

nevertheless, distribution of white and grey matter is

more similar to the α-stable distribution, As it has

been pointed out in (Salas-Gonzalez et al., 2013a).

Figures 3 and 4 show the histogram of intensity

values for a magnetic resonance brain image.

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0 10 20 30 40 50

2000

4000

6000

8000

10000

12000

14000

16000

18000

20 40 60 80 100 120 140

100

120

140

Nonspecific binding regions

Specific binding regions

PD NC

Figure 2: Continuous red line: predicted α-stable density.

50 100 150 200 250

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Image1

Figure 3: Histogram of intensity values. Red dashed line:

predicted alpha-stable mixture density. Blue solid line:

Gaussian mixture model with two components.

• The individual histograms of white and grey mat-

ter have a shape which is similar to the α-stable

distribution.

• Even if in most cases the global histogram of in-

tensity values are unimodal, because their WM

and GM components are very mixed, the mixture

of α-stable model is robust enough to ﬁnd the hid-

den components successfully.

• Many MRI segmentation procedures are based on

a preliminary modelling of the histogram of WM

and GM, using a Gaussian mixture model, we be-

lieve that the α-stable distribution is a good alter-

native due to its theoretical and practical proper-

ties.

• The α-stable mixture model allows us to calculate

the relative likelihood for each voxel in the image.

50 100 150 200 250

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Image1

Figure 4: Red and blue lines: predicted alpha-stable com-

ponents. Dotted black line: Gaussian components. Solid

histogram: Ground truth (according to the Internet brain

segmentation repository IBSR- data).

Figure 5: Transaxial slices represented using a linear

grayscale map. First column: probability to belong to GM

). White voxels, p

= 1. Black voxels, p

= 0.

Second column: GM voxels according to the manual seg-

mentation procedure. Third column: p

. White voxels,

= 1. Black voxels, p

= 0. Fourth column: WM

regions according to manual segmentation procedure.

These quantities are very useful and could be used

in more complex Bayesian segmentation models

in future works.

WhyUsingtheAlpha-stableDistributioninNeuroimage?

299

3.3 Feature Extraction for Parkinson’s

Disease Diagnosis

After intravenous injection, 123I-FP-CIT SPECT

binds to the dopamine transporters in the striatum. It

has been found that patients with Parkinson’s disease

exhibit a decreased uptake of the tracer (Bhidayasiri,

2006; Hauser and Grosset, 2012). An accurate di-

agnosis of Parkinson’s disease is important because

it enables us to monitor disease progression and the

therapeutic effects of the treatment. This motivates

the development of automated techniques for quan-

tiﬁcation which neither depend on time consuming

operator-intensive work, nor expert skills in manu-

ally locating the regions of interest in the brain (Pa-

pathanasiou et al., 2006). We believe the α-stable

distribution and its properties can be used to develop

a computer aided diagnosis system as a decision-

making aid for Parkinson’s syndrome diagnosis for

automatic classiﬁcation.

The discriminative area to perform diagnosis of

Parkinson’s syndrome in FP-CIT SPECT brain im-

ages is located in a speciﬁc region of the brain, the

striatum. According to the histogram of the intensity

values, this area also controls the degree of impulsive-

ness in the histogram. The degree of impulsiveness

is related to the characteristic exponent, which also

models the Paretian behaviour of the tails in alpha-

stable distributions. This property can be exploited

to assess differences between images belonging to

normal controls and patients with Parkinsonian syn-

drome, measuring the Paretian behaviour in the tail

of the distribution to extract the discriminant features.

Then, these features could be used for statistical clas-

siﬁcation using support vector machines or classiﬁca-

tion trees.

4 NOVELTY, ADVANTAGES AND

DISADVANTAGES

The main advantage of the α-stable methods is that

they are generalizations of the Gaussian distribution

which is widely used in neuroimagingmethods, there-

fore they are expected to perform better than Gaus-

sian, or equally when the Gaussian assumption holds.

The main disadvantage of the α-stable distribution

is, mainly, the non existence of a closed form for

its probability density function, therefore, numerical

methods needs to be used to evaluate it.

The originality of the goals and methods envis-

aged in this paper is demonstrated by the fact that the

α-stable distribution has not been previously used in

neuroimaging apart from two very recent works pub-

lished in 2013 (Salas-Gonzalez et al., 2013a; Salas-

Gonzalez et al., 2013b). We believe is timely to ex-

tend these recently published methods, exploiting ad-

ditional and useful properties of the α-stable distri-

bution in the study of signal processing methods for

brain tomographic applications.

5 CONCLUSION

The Gaussian distribution and mixture of Gaussian

model are ubiquitous in brain imaging literature; nev-

ertheless, the Gaussian distribution, and the mixture

of Gaussians are particular limiting cases of the alpha-

stable distribution, and the mixture of alpha-stable

model. Sometimes, brain-imaging data present a cer-

tain degree of asymmetry and/or impulsiveness and

therefore, it can be modelled more accurately using

alpha-stables. For this reason, the alpha-stable dis-

tribution is expected to work better than those ap-

proaches in the literature assuming Gaussian distri-

bution of the data.

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