![](bg3.png)
First, from a CT slice, the region containing the
tissue to be analyzed must be isolated; a matrix
containing values of each pixels shade is obtained
(the value can vary between 0 and 255
corresponding to different shades of grey; 0 stands
for black and 255 for white). The time (spatial)
series is generated in the following manner: the
matrix resulting from the original image is cut in
horizontal strips of 1, 2, 4, 8, … pixels, with respect
to the initial image dimension and precision; all
strips are put together one after another and generate
one single strip associated to the image; the time
(spatial) series - x(t) - is generated by computing
either the mean value or the maximal (dominant)
value of each column of pixels within the strip.
As result of this procedure, the time (spatial)
series associated to the section of the analyzed tissue
is obtained. For this study, since the analyzed CT
regions are not extremely large, a 1-pixel strip was
associated to each original image, this way not
altering the information provided by the image.
Having the associated series, the next step of the
procedure implies calculating the correlation
dimension of the attractor. This value is the
discrimination criterion.
However, in practical applications, in order to
determine the dimension of an attractor, we cannot
directly use the above formulae for d
C
due to the
following aspects: limited time series; noisy time
series; unknown fractal dimension of the attractor;
for different s - delay values different results due
autocorrelations; unknown d
E
– leading to time
correlations when reconstructing the series in a
embedding space with unsuitable dimension; time
series with the first part of data not on the attractor.
The delay or lag value -s- used to create the
delayed embedding must be properly chosen (Kantz
and Schreiber, 2003). A small value of the delay
generates correlated vector elements, while large
delay values yield to uncorrelated data and a random
distribution in the embedding space. The delay can
be chosen with good results as the moment of time
where the autocorrelation function of the
reconstructed series decays to 1/e of its initial value:
() (1)(1 1/)
NRN e
<−
.
(4)
Generally, the lag value is found between 4 and
10, while the used search interval is [1, 20].
The minimum allowed embedding dimension is
the dimension where the number of so called false
nearest neighbours drops under a certain percent. A
false neighbour is a point that under a certain higher
dimensional embedding is projected near a point that
in the previous embedding is not in its vicinity.
In order to implement this procedure, each point
of the delayed series is tested by taking its closest
neighbour in d
E
dimensions, and computing the ratio
of the distances between these two points in d
E
+1
dimensions and in d
E
dimensions. If this ratio is
larger than a certain threshold th, the neighbour is
false (this threshold is taken large enough to take in
consideration points that exponential diverge due to
deterministic chaos):
11
,,
,,
EE
EE
id jd
id jd
yy
th
yy
++
−
>
−
(5)
where ||.|| is the Euclidian distance.
The percentage of false neighbours is computed
over a range of embedding dimensions (d
E
between 2
and 15) until it reaches a value less than a specified
limit; otherwise it considers the minimal obtained
value.
Once a proper delay and a minimum allowed
embedding dimension are determined, the
correlation dimension is calculated over a range of
different
- values and embedding dimensions
higher than the first assuring a decreased number of
false neighbours.
The d
C
differs from one embedding dimension to
another due to the noise in the data, but there is a
particular region, usually called the scaling region
where d
C
stabilizes (Kantz and Schreiber, 2003).
This is the interval where a mean value for the
correlation dimension of an attractor is calculated.
2.2 The Box-Counting Dimension
Estimation Method
Fractal analysis methods are used for the description
and quantization of geometric features of irregular
forms and patterns. Its most known tool is the fractal
dimension used to provide information on the
irregularity of an object contour or self-similarities
of a texture, which associates to some pathology as
well. It was applied for the study of medical systems
and subsystems at microscopic and macroscopic
scale, fracture analysis or texture classification
(Peitgen, 1992). The simplest medical application
consists in the morphological analysis of a structure
(for example, the lung network of arteries and
veins). This analysis of irregularities can be applied
in a similar manner on different forms, like the
delimitation between normal and affected tissue,
lesions, and tumors.
Here are some examples of fractal analysis
results in medicine and biology: classification in
pathology (Bassingthwaighte, 1994; Dobrescu si
NonlinearDeterministicMethodsforComputerAidedDiagnosisinCaseofKidneyDiseases
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