=Paper=
{{Paper
|id=Vol-3925/short09
|storemode=property
|title=Spline model of digital images based on moment characteristics (short paper)
|pdfUrl=https://ceur-ws.org/Vol-3925/short09.pdf
|volume=Vol-3925
|authors=Anhelina Zhultynska,Pylyp Prystavka
|dblpUrl=https://dblp.org/rec/conf/cmigin/ZhultynskaP24
}}
==Spline model of digital images based on moment characteristics (short paper)==
https://ceur-ws.org/Vol-3925/short09.pdf
Spline model of digital images based on moment
characteristics
Anhelina Zhultynska1,2,† and Pylyp Prystavka1,∗,†
1
National Aviation University, Liubomyra Huzara Ave. 1, Kyiv, 03058, Ukraine
2
Interregional Academy of Personnel Management, Frometivska Str., 2, Kyiv, 03039, Ukraine
Abstract
The paper proposes a spline model of a parametrized digital image based on two momentary texture
features, namely, the locally averaged illumination intensity and the root mean square. The model is an
estimate of the density function of the distribution of these characteristics and is an alternative to the model
of a mixture of normal distributions.
Keywords
computer vision, image recognition, image preprocessing 1
1. Introduction and problem statement
The problem of digital image segmentation is not new – there are numerous diverse and effective
approaches to solving it, both in purely mathematical methods and through the use of hardware-
software approaches. However, the segmentation of oversized realistic images or video data (e.g.,
from aerial or space observations) in real-time or near-real-time mode gains particular relevance.
By its nature, the segmentation (clustering) problem relates to unsupervised pattern recognition,
generally solved based on the model of feature distribution density defining certain characteristics
of digital images. In general, the problem of pattern recognition can be broken down into the
following main stages [1]:
1. Generating features of the recognized object.
2. Selecting features of the recognized object.
3. Building a classifier.
4. Assessing the quality of classification.
Feature formation is one of the most critical stages in constructing any information processing
system. The quality of the chosen feature system and algorithms for their computation depends on
the specific requirements of the applied system for processing and analyzing digital images.
Therefore, when prioritizing the requirement for segmentation speed, it is necessary to pose a
demand for both the high adequacy of the density function model on one hand and the efficiency of
obtaining and learning it on the other hand [2, 3].
Hence, the relevant research direction of this work will be to define a model of realistic digital
images that would address the segmentation task in a mode close to real-time within modern
computerized systems
CH&CMiGIN’24: Third International Conference on Cyber Hygiene & Conflict Management in Global Information Networks,
January 24–27, 2024, Kyiv, Ukraine
∗
Corresponding author.
†
These authors contributed equally.
angelinaremark1@gmail.com (A. Zhultynska); chindakor37@gmail.com (P. Prystavka)
0000-0001-9178-897X (A. Zhultynska); 0000-0002-0360-2459 (P. Prystavka)
© 2025 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
2. Literature review
To further the analysis, let's introduce the concept of texture. Texture refers to fragments of an image
that can be characterized by different properties. Texture can be homogeneous when the pattern or
structure repeats with the same intensity in all directions and at all points. For example, the surface
of sand on a beach or a field can have a homogeneous texture. On the other hand, texture can be
heterogeneous when the pattern or structure varies at different points or directions. For instance, an
image of a forest with different types of trees, shrubs, and other plants can have a heterogeneous
texture. Let's consider methods for forming textural features related to the segmentation and
classification of digital images, commonly encountered in tasks of processing aerial photography,
aerial, and space monitoring data.
The simplest and often-used segmentation method is based on dividing the image by a brightness
threshold [4, 5]. For this, the intensities of each image element are used. If the intensity of a certain
selected threshold value increases, the element is classified as an object component; otherwise, it's
considered a background element. However, such methods may be ineffective for complex image
structures like aerial photographs.
The application of segmentation methods based on textural characteristics holds great potential
in various image processing areas. Statistical methods, such as local dispersion analysis [6, 7],
histogram-based analysis [8], and using local binary patterns for feature extraction and region
merging [9], efficiently highlight textural areas of images by computing statistical characteristics.
They can be effective in recognizing different types of textures but may require substantial
computational resources for calculating a large number of characteristics.
Structural methods, like the multi-layer adaptive graph cut method [10], excel in segmenting
textures with complex structures by identifying primitive structures. However, they may be less
effective for irregular textures or textures without known standards.
Spectral methods, like deep learning-based methods and conditional random fields [11, 12], ensure
high segmentation accuracy by processing deep features and modeling context using conditional
random fields. Additionally, wavelet transform methods also prove effective for segmenting textural
images. Wavelet transformation [13, 14] enables the extraction of textural details of various scales in
an image using multi-scale analysis.
The Gaussian Mixture Model (GMM) method for constructing digital image models is another
prevalent approach to data segmentation [15]. This method involves creating a complex image model
using a combination of several Gaussian functions with different parameters. The use of the
Expectation-Maximization (EM) algorithm adapts GMM to a dataset by considering the multimodal
data distribution as a combination of individual unimodal distributions. GMM assumes that data in
a certain cluster is generated by a specific Gaussian distribution. The method allows approximating
complex structures and details in an image, facilitating their further analysis. However, constructing
a complex model requires significant computational resources, which can complicate processing a
large amount of data.
Therefore, the analysis of research and sources shows that a computationally simple approach to
texture analysis is possible based on the construction of textural features using the distribution of
statistical characteristics, given the simplicity of estimating (modeling) the corresponding probability
density function. According to studies [16], two-dimensional splines based on B-splines meet similar
conditions, closely resembling interpolations on average. This approach may have advantages in
terms of computational efficiency and interpretability. However, its accuracy may be limited
compared to some deep learning models capable of solving more complex texture classification
tasks [17–21].
Therefore, the goal of this work is to formalize and justify the possibility of applying the
mentioned splines to the task of constructing a digital image model based on moment characteristics
for automated processing systems of digital images, including aerial observation data.
3. The study materials and methods
= , , = 0, − 1; = 0, − 1 ,
Let's consider a certain digital image (DI), represented by the illumination intensity of each pixel
(1)
where is for instance, any of the color components in the RGB color space or a component
representing DI in grayscale; , are the linear dimensions of DI.
If we consider a realistic DI, such as an aerial image, then any element of intensity (1) is a random
Let = , , , ,…, , ; = 1,
variable, so when choosing a DI model, this fact should be taken into account.
within a certain neighborhood determined by the size of a sliding window (2 + 1) × (2 + 1),
be a set of textures that exhibit similar characteristics
= 1,2, …, where − texture; ,! − characteristic of the -th texture; − number of textures;
" − number of characteristics of the texture. In fact, each of the features of a separate texture ,! is
defined as some random function of the pixel intensities in the local (2 + 1) × (2 + 1) area
around any ( , )-th pixel of DI:
,! = #! $,% ,& = − , + ,' = − , + , ( = 1, ",
= 0, − 1, = 0, − 1.
(2)
Therefore, we're discussing the parameterization of DI (2) based on texture characteristics. In such
a case, a DI model can be proposed by estimating the density distribution of parameters in the texture
characteristic space. Regarding each separate g-th group of "similar" textures, due to the randomness
of characteristics, it's possible to assume the existence of some parameter distribution density
* ( , , , , … , , ),
function
(3)
.
where the natural assumption is that the texture group distribution present in DI is a mixture
*(Θ) = , - *( , , , ,…, , ),
/
.
, - = 1.
(4)
/
For instance, assuming a normal distribution of densities (3), the task of evaluating model (4) boils
down to reproducing a mixture of normal distributions in the texture characteristic space. Solving
this latter task would allow for image segmentation (clustering), for example, based on dividing the
mixture followed by a naive Bayesian pixel classification (1). Despite all the advantages (low
sensitivity to "noise" and anomalous values, independence of the clustering result from the data entry
order, the ability to highlight clusters of different structures, etc.), the algorithms developed to solve
such a task have significant drawbacks – the assumption of a normal density distribution (3) is not
always valid, hence the limited applicability functional of the mixture model (4). Moreover, such
algorithms are characterized by relative computational complexity, thus not recommended for
processing large DIs or streaming video. In general, it can be assumed that the hypothesis of local
continuity and smoothness of the density distribution of texture characteristics holds true – for most
*( ) ∈ С2,…,2 , 3 ≥ 2 .
realistic images, this is quite acceptable. Therefore, without reducing generality, let's consider that
Further on, let's examine the case when the number of texture characteristics " = 2 is only two.
For example, these could be parameters characterizing the average color (illumination) of a group of
pixels and their variability. Publications [1, 4, 6] have shown that moment characteristics of DI are
invariant to scale and rotation with respect to the size of the local window, so for the
parameterization that defines the properties of textures, let's limit the choice to estimating the local
67 67
1
mean and standard deviation of pixel intensities in the area:
5 , = , , $,% ,
(2 + 1)
(5)
$/ 87 %/ 87
67 67
1
9, =: , , $,% − 5 , ,
(2 + 1)
(6)
$/ 87 %/ 87
= 3, − 3 − 1, = 3, − 3 − 1.
dimensional sample (5, 9) in the texture feature space ; ,< = (5! , 9! ); ( = 1, = , where = is the
Therefore, as a result of calculating the characteristics (5) and (6), we obtain a random two-
worth noting that the order of forming the sample Ω ,< is not critical in this presentation: the
number of sliding windows within which mean and standard deviation estimates are calculated. It's
calculation can be done for a sliding window of size (2 + 1) × (2 + 1) with a shift of 1 pixel or
any other shift of the central element of the area relative to the pixels of DI. Moreover, for the
With the sample Ω ,< , we will proceed to construct a two-dimensional variational series divided
following exposition, the magnitude of the positive integer is also not crucial.
into classes. For this, we'll define the division ?@A,@B of the plane of realizations of characteristics (5)
Algorithm for constructing the partition CDE ,DF .
and (6) according to the following algorithm of actions.
Step 1. The central point of the (G, H)-th partition element ?@A,@B is taken as the variant row
(5I , 9I )
5I = 5JIK + (G + 0,5)ℎJ , G = 0, NJ − 1,
9O = 9JIK + (H + 0,5)ℎP , H = 0, NP − 1,
5JWX − 5JIK
5JIK = minT5! U ; 5JWX = maxT5! U ; ℎJ = ;
where
! ! NJ
9JWX − 9JIK
9JIK = minT9! U ; 9JWX = maxT9! U ; ℎP = ;
! ! NP
where NJ , NP − the number of partition elements (classes) along the respective axes.
Step 2. The frequency [I,O of the variant row is calculated as the number of sample points
Ω ,< falling within the limits of the (G, H)-th partition element ?@A,@B , whereby the indices G, H
determining to which of the partition elements ?@A,@B any point (5! , 9! ); ( = 1, = belongs, are
5! 9!
G = \ ], H =\ ],
found using the expressions
ℎJ ℎP
where ^∙` is the integer part.
Step 3. The randomness aI,O of the variants (5I , 9I ) is determined by the relation
bA 8 bB 8
[I,O
aI,O = , , , aI,O = 1.
=
I/c O/c
Let us consider the estimation of the feature density function of 5 and 9 as a parameterised image
model and evaluate it on the basis of the sample ; ,< = (5! , 9! ); ( = 1, = and the two-dimensional
5I , 9O , aI,O ; G = 0, NJ − 1, H = 0, NP − 1
variation series built on its basis
of this random variable. Let the density function be smooth. If (5I , 9I ) is the centre point of the (G, H)-
th element of the partition ?@A,@B , then
1 Jh 6c,g@A Pf 6c,g@B
̅ =
*I,O e e *(5, 9)'5'9
ℎJ ℎP Jh 8c,g@A Pf 8c,g@B
is the average value of the probability density function of the features (5, 9) in the specified area
and the following connection with the randomness of the option takes place
̅ ℎJ ℎP =
aI,O = *I,O
= i j: 5I − 0,5ℎJ ≤ m(j) < 5I + 0,5ℎJ , 9O − 0,5ℎP ≤ o(j) < 9O + 0,5ℎP
Thus, up to the constant ℎJ ⋅ ℎP , the randomness is an estimate of the average value of the density
function *(5, 9). Then, as a model for density approximation, we propose a model in the form of a
two-dimensional polynomial spline based on q-splines that is close to the interpolation average.
That is, the array of values a̅ = a̅I,O , G ∈ s, H ∈ s will be matched with the following local
t ,c (a, 5, 9) = , , q ,@A (5 − GℎJ )q ,@B (9 − HℎP )aI,O
polynomial spline [18]:
I∈u O∈u
(7)
0, 5 ∉ ^− 3ℎ⁄2 ; 3ℎ⁄2`,
where (with the notation accuracy up to the split step) [12]
⎧
(3 + 25⁄ℎ) ⁄8 , 5 ∈ ^− 3ℎ⁄2 ; − ℎ⁄2`,
q ,@ (5) =
⎨3⁄4 − (25⁄ℎ) ⁄4 , 5 ∈ ^− ℎ⁄2 ; ℎ⁄2`,
⎩ (3 − 25⁄ℎ) ⁄8 , 5 ∈ ^ℎ⁄2 ; 3ℎ⁄2`.
1
or in expanded form
t ,c (a, 5, 9) = (1 − •) (1 − €) aI8 ,O8 + (1 − •) (6 − 2€ )aI8 ,O +
64
+(1 − •) (1 + €) aI8 ,O6 + (6 − 2• )(1 − €) aI,O8 + (6 − 2• )(6 − 2€ )aI,O +
+(6 − 2• )(1 + €) aI,O6 + (1 + •) (1 − €) aI6 ,O8 + (1 + •) (6 − 2€ )aI6 ,O +
(8)
+(1 + •) (1 + €) aI6 ,O6
• = @ (5 − GℎJ ), |•| ≤ 1 ; € = @ (9 − HoP ), |€| ≤ 1.
where
A B
precise estimation of the approximation error of the density distribution *(5, 9) indirectly through
By means of the explicit analytical representation (8), the models (7) can obtain an asymptotically
the estimation of the function a(5, 9), the values of which are the relative frequencies of the
aI,O = a̅I,O + ‚I,O ,
respective two-dimensional variational series:
1 (Jh 6c,g) @A Pf 6c,g @B
where
āI,O = e e a(5, 9)'5'9 ,
ℎJ ℎP (Jh 8c,g) @A Pf 8c,g @B
‚I,O − an arbitrary error.
In particular, for ℎJ → 0, ℎP → 0 for ∀a(5, 9) ∈ • , is true ^10`:
ℎJ ” ℎP
’a(5, 9) − t ,c (a, 5, 9)’С ≤ ’aJ“ (5, 9)’С + ’aP”“ (5, 9)’С +
6 6
ℎJ ℎP (–)
+ •aJ“ P“ (5, 9)• + ‚‖a(5, 9)‖˜ + ™(ℎ– )
(9)
36 С
‚ = max ‚G.H , ℎ = maxTℎJ , ℎP U.
where
G,H
4. Results of the study
a(5, 9), for instance, allows solving the segmentation problem. With this approach, individual
Analyzing the local peculiarities of the constructed model of the two-dimensional normalized density
segments correspond to clusters determined by local maxima (modes) of the estimate (7).
As an application example of model (7), consider an image representing several textures: a grass
lawn, a runway, airplanes (Figure 1). We'll illustrate how these textures look locally and what
distribution they have. We'll consider the two-dimensional density functions depending on the mean
and standard deviation for each of the textures.
The lawn is a simple texture (Figure 2a). By “simple”, we mean relatively low variability of light
(Figure 3) based on model (7) is unimodal and has a mean 44 ≤ 5 ≤ 84 and standard deviation 9 ≤
intensity within a local window. The estimation of the normalized density function of the “lawn”
14.
Figure 1: Image with textures: grass lawn, runway, airplanes.
a) b) c)
Figure 2: Examples of textures: a) grass lawn; b) runway; c) airplane.
intensity 134 ≤ 5 ≤ 170 and variability 9 ≤ 12 is also unimodal (Figure 4).
Similarly, the runway is a simple texture (Figure 2b). The distribution density of the average
a) b)
window: а) = 4; b) = 8.
Figure 3: Two-dimensional normalized density function of the lawn (Figure 2a) with a sliding
a) b)
window: а) = 4; b) = 8.
Figure 4: Two-dimensional normalized density function of the runway (Figure 2b) with a sliding
Complex textures are characterized by high variability in light intensity due to the presence
within a local window of pixels from various 'simple' textures, for example, the 'airplane' texture
(Figure 2c). The normalized density function (Figure 5) is multimodal, indicating the presence of
pixels from the “grass lawn”, “runway” and airplane.
a) b)
window: а) = 4; b) = 8.
Figure 5: Two-dimensional normalized density function of the airplane (Figure 2c) with a sliding
parameterised by (5, 9) (Figure 1). There are three local groups of pixels defined by the proposed
Let us consider (Figures 6 and 7) the density function of the entire original digital image
model (7). According to a natural analysis, the first group of pixels belongs to the lawn, the second
to the runway, and the third to the airplanes (Figures 6a and 7a). Visually, it's not difficult to identify
local maxima and intuitively divide the model into separate segments (Figures 6b and 7b), which
determine the corresponding segmentation of textures in the original image.
= 4: а) spline model; b) 1 –
a) b)
Figure 6: Spline model of the image (Figure 1) with a sliding window
local maxima defining separate image segments.
a)
= 8: а) spline model; b) 1 –
b)
Figure 7: Spline model of the image (Figure 1) with a sliding window
local maxima defining separate image segments.
5. Conclusions
In the study, a spline model (7) for parameterized digital imagery (DI) based on two momentary
characteristics of textures was proposed, namely: the locally averaged illumination intensity (5) and
the root mean square (6). This model represents an estimation of the probability density function of
the mentioned characteristics and serves as an alternative to a mixture model of normal distributions.
Thanks to the analytical presentation (8), the spline model demonstrates low computational
complexity, substantiated acceptable approximation errors, and can be recommended for
implementation in automated DI processing systems operating in near real-time regimes.
Future research involves developing a DI segmentation method based on the introduced spline
model and procedures for detecting low-probability textures that provide enhanced informativeness
in the analysis of aerial observation data. Additionally, the results of this work are expected to be
used in developing neural network models and exploring the hyperparameters of deep learning
algorithms.
Declaration on Generative AI
The author(s) have not employed any Generative AI tools.
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