A color image reduction based on fuzzy transforms

doi:10.1016/j.ins.2014.01.014

Information Sciences

Volume 266, 10 May 2014, Pages 101-111

https://doi.org/10.1016/j.ins.2014.01.014 Get rights and content

Abstract

We present a new method for color image reduction based on the concept of fuzzy transform. Any image in a single band can be considered as a fuzzy matrix which is subdivided into submatrices called blocks. Each block is compressed with various_compression rates by means of a fuzzy transform in two variables. We compare our method with recent three algorithms due to G. Beliakov, H. Bustince and D. Paternain based on the minimizing penalty functions defined over a discrete lattice. The quality of the reduced image is measured by the Mean Square Error (MSE) and Penalty function (PEN) obtained by comparing both magnified and original images. We also point out a threshold of the compression rate beyond which the MSE follows a linear trend and the corresponding loss of information is still acceptable.

Introduction

A fuzzy transform (shortly, F-transform) [16], [17] is an operator which transforms a continuous function into a n-dimensional vector. Applications of the F-transforms were made in data analysis [7], [8], [14], image analysis [3], [4], [5], [6], [9], [17], [18], [19] and comparisons with the fuzzy relation equations method and JPEG appear in [10], [11], [12], [13]. In [1] three new color images reduction algorithms are presented and based on the optimizing penalty functions [2] defined over discrete product lattices. Furthermore the authors in [1] proved that these algorithms are better than other reduction algorithms based on appropriate re-sampling and F-transforms. Here we show that our algorithm based on decomposition of blocks reduced via F-transforms [3], [4], [5] gives better results than those obtained with the algorithms from [1]. In other words, as in [3], [4], [5], any image is divided into submatrices of equal dimensions, called blocks. Every block is reduced under a specific compression rate with a F-transform and reconstructed via a simple algorithm. The re-composition of these decompressed and magnified blocks gives an overall magnified image comparable with the original image. From the point of view of Granular Computing [15], we can also say that these blocks are the information granules which are then re-composed in accordance to some suitable criteria for giving the overall final information.

The quality of the reduced image is measured by the Mean Square Error (MSE) and the error based on Penalty function (PEN) obtained by comparing both magnified and original images. In addition, we develop a process to establish a compression rate threshold, through the analysis of the trend of the MSE with respect to the compression rates. Beyond this threshold the MSE follows a linear trend and the corresponding loss of information, due to reduction, is still acceptable. In Sections 2 , 3 we provide the definition of the F-transform in one and two variables, respectively. In Section 4 we present our reduction method. In Section 5 we present the results of our experimental study. Section 6 is conclusive.

Section snippets

F-transforms in one variable

Following the definitions and notations of [16], let [a, b] be a closed interval, n ⩾ 2, and x₁, x₂, … , x_n be points of [a, b], called nodes, such that $x_{1} = a < x_{2} < \dots < x_{n} = b$ . We say that an assigned family of fuzzy sets A₁, … , A_n: [a, b] → [0, 1] is a fuzzy partition of [a, b] if the following conditions hold:

(1)
$A_{i} (x_{i}) = 1$ for every $i = 1, 2, \dots, n$ ;
(2)
$A_{i} (x) = 0$ if $x \notin (x_{i - 1}, x_{i + 1})$ , where we assume $x_{0} = x_{1} = a$ and $x_{n + 1} = x_{n} = b$ by convenience of presentation;
(3)
A_i(x) is a continuous function on [a, b];
(4)
A_i(x) strictly increases on [x_i−1, x_i] for $i = 2,$

F-transforms in two variables

We can extend the above concepts to functions in two variables In the discrete case, let f the function assume determined values in some points (p_j, q_j) ∈ [a, b] × [c, d], where i = 1, … , N and $j = 1, \dots, M$ . Moreover, let the sets $P = {p_{1}, \dots, p_{N}}$ and $Q = {q_{1}, \dots, q_{M}}$ be sufficiently dense with respect to the chosen partitions, i.e. for each $i = 1, \dots, N$ there exists an index k ∈ {1, … , n} such that $A_{i} (p_{k}) > 0$ and for each $j = 1, \dots, M$ there exists an index $l \in {1, \dots, m}$ such that $B_{j} (q_{l}) > 0$ . In this case we define the matrix [F_kl] to be the

Our method

Let P be a grey image divided in N $\times$ M pixels. We normalize P into an image S, with $S (i, j) = P (i, j) / (G_{level} - 1)$ , where $G_{level}$ is the length of the grey scale, for instance, $G_{level} = 256$ , where S: $(i, j) \in {1, \dots, N} \times {1, \dots, M} \to$ [0, 1]. In [4] S is compressed by using the discrete F-transform in two variables [F_kl] (cfr., formula (4)) defined for each $k = 1, \dots, n$ and $l = 1, \dots, m$ , as $F_{kl} = \frac{\sum_{j = 1}^{M} \sum_{i = 1}^{N} S (i, j) A_{k} (i) B_{l} (j)}{\sum_{j = 1}^{M} \sum_{i = 1}^{N} A_{k} (i) B_{l} (j)},$ where by simplicity, we have $p_{i} = i$ and $q_{j} = j$ (then a = c = 1, b = N, d = M), {A₁, … , A_n} and {B₁, … , B_m

Simulation results

We have extracted 100 images from the color image dataset at the URL “http://decsai.ugr.es/cvg/dbimagenes/index.php”. For reasons of brevity, here we only present the results for the images of Fig. 2.1–2.11 $(N = M = 256)$ .

For our comparisons we use a compression rate $ρ = 0.111$ corresponding to a block with $N (B) = M (B) = 9$ into a reduced block with $n (B) = m (B) = 3$ . We compare our results with those obtained by using only F-transforms $(ρ = 1)$ and the three reduction algorithms from [1]. Fig. 3.1–3.11 show the

Conclusions

We have considered 100 images downloaded from the color image database “http://decsai.ugr.es/cvg/dbimagenes/index.php”. For brevity, we have shown the results on a sample of 11 images and we have observed that the reduction of color images based on the decomposition of blocks (submatrices) via F-transforms of the original images gives better MSE and PEN than those obtained by using the algorithms from [1] except few cases. Among others, we have evaluated over the same sample of images that the

Acknowledgements

The authors I. Perfilieva and P. Hurtik acknowledge support by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070). The authors S. Sessa and F. Di Martino perform this work in the context of the project FARO 2010–2013 under the auspices of the “Polo delleScienze e delleTecnologie” of Università degli Studi di Napoli Federico II, Italy.

References (19)

T. Calvo et al.
Aggregation functions based on penalties
Fuzzy Sets Syst.
(2010)
F. Di Martino et al.
An image coding/decoding method based on direct and inverse fuzzy transforms
Int. J. Approx. Syst.
(2008)
F. Di Martino et al.
Compression and decompression of images with discrete fuzzy transforms
Inform. Sci.
(2007)
F. Di Martino et al.
A segmentation method for images compressed by fuzzy transforms
Fuzzy Sets Syst.
(2010)
F. Di Martino et al.
Fuzzy transforms method in prediction data analysis
Fuzzy Sets Syst.
(2011)
F. Di Martino et al.
Fragile watermarking tamper detection with images compressed by fuzzy transform
Inform. Sci.
(2012)
H. Nobuhara et al.
Efficient decomposition methods of fuzzy relation and their application to image decomposition
Appl. Soft Comput.
(2005)
H. Nobuhara et al.
A motion compression/reconstruction method based on max t-norm composite fuzzy relational equations
Inform. Sci.
(2006)
I. Perfilieva
Fuzzy transforms
Fuzzy Sets Syst.
(2006)

There are more references available in the full text version of this article.

Cited by (30)

A novel Covid-19 and pneumonia classification method based on F-transform
2021, Chemometrics and Intelligent Laboratory Systems
Citation Excerpt :
In this method, a fuzzy transform (F-transform) [41] based on triangle fuzzy sets is proposed and a novel fuzzy tree is constructed using the triangle-based F-transform. As we know from literature, F-transforms have been used for image and noise reduction [42]. The proposed fuzzy tree is utilized as a novel operator like convolution.
Nowadays, Covid-19 is the most important disease that affects daily life globally. Therefore, many methods are offered to fight against Covid-19. In this paper, a novel fuzzy tree classification approach was introduced for Covid-19 detection. Since Covid-19 disease is similar to pneumonia, three classes of data sets such as Covid-19, pneumonia, and normal chest x-ray images were employed in this study. A novel machine learning model, which is called the exemplar model, is presented by using this dataset. Firstly, fuzzy tree transformation is applied to each used chest image, and 15 images (3-level F-tree is constructed in this work) are obtained from a chest image. Then exemplar division is applied to these images. A multi-kernel local binary pattern (MKLBP) is applied to each exemplar and image to generate features. Most valuable features are selected using the iterative neighborhood component (INCA) feature selector. INCA selects the most distinctive 616 features, and these features are forwarded to 16 conventional classifiers in five groups. These groups are decision tree (DT), linear discriminant (LD), support vector machine (SVM), ensemble, and k-nearest neighbor (k-NN). The best-resulted classifier is Cubic SVM, and it achieved 97.01% classification accuracy for this dataset.
Modifying the gravitational search algorithm: A functional study
2018, Information Sciences
Cubic B-spline fuzzy transforms for an efficient and secure compression in wireless sensor networks
2016, Information Sciences
Citation Excerpt :
In particular, in [30] new types of F–transforms were presented, based on B–splines, Shepard kernels, Bernstein basis polynomials and Favard-Szasz-Mirakjan type operators for the univariate case. There are many applications of the F–transforms in image processing ([15–21]) and some others in time series analysis [31–35], also by integrating the F–transform and the fuzzy tendency modeling [32] or the F–transform and fuzzy natural logic [33]. In particular, the paper by Novak et al. [35] focuses on application of fuzzy transform (F-transform) to the analysis of time series under the assumption that the latter can be additively decomposed into trend-cycle, seasonal component and noise.
Joining data compression and encryption is a way to keep secure data, as discussed by the current literature. While data compression responds to the great demand on data storage and transmission techniques, the encryption allows to handle some important parameters in a secure way. In wireless sensor networks the usual transform–based compression is the Discrete Wavelet Transform. In a previous paper we showed the good perfomance of the fuzzy transform (or F–trasform for short) based compression with respect to it. In this work, we propose a cubic B–spline F–transform in order to have a higher accuracy, even when data are not correlated, and a lower computational cost. Besides, in order to show the efficiency of the proposed approach, we compare it with the most recent lossless compression scheme in the field. We discuss these issues formally and numerically by using publicly available real–world data sets. The parameters required to decompress data are encrypted by means of a suitable existing encryption algorithm. We show that even if an illegal user had access to one of these parameters, our scheme would be still secure.
Lattice fuzzy transforms from the perspective of mathematical morphology
2016, Fuzzy Sets and Systems
The compositions of direct and inverse fuzzy transforms constitute powerful tools in knowledge extraction and representation that have been applied to a large variety of problems in computational intelligence as well as in image processing and computer vision. Fuzzy transforms (FTs) have linear as well as lattice-based versions. In this paper, we extend the latter FTs, known as lattice FTs, and relate these operators and their underlying mathematical structures to the ones of mathematical morphology (MM), in particular to the ones of MM on complete lattices and $L$ -fuzzy MM.
Development of autofocusing algorithm based on fuzzy transforms
2016, Fuzzy Sets and Systems
Citation Excerpt :
We propose the localized version of the image variance focus measure operation by using Fuzzy Partition Techniques [10] and Fuzzy Transform [9]. Nowadays, Fuzzy Transform has been used in various fields such as image processing [12,13], and data analysis [14–16]. In the proposed focus measure approach, we are not interested in the intensities of all pixels but our interest is in the intensities determined over the sub-space defined by using fuzzy partition techniques such as Fuzzy C-means clustering and conditional Fuzzy C-means clustering technique.
In this paper, we propose a new passive auto focusing algorithm to acquire images with enhanced quality through a digital camera. The focus measure operations play a key role in a passive auto focusing method. The proposed focus measure operation is based on a localized version of image variance method. The localization of the image variance focus measure operation can be implemented with the use of fuzzy partition and Fuzzy Transform. In order to validate our approach, experimental comparative studies are conducted on test images with the use of different focus measure operators. The selected operators have been chosen from an extensive review of the state-of-the-art approaches. A comparative analysis demonstrates that the proposed algorithm is superior to the conventional methods.
A new method for reduction of color in a carpet map using a deep belief network
2024, Multimedia Tools and Applications

View all citing articles on Scopus

View full text

A color image reduction based on fuzzy transforms

Abstract

Introduction

Section snippets

F-transforms in one variable

F-transforms in two variables

Our method

Simulation results

Conclusions

Acknowledgements

Fuzzy Sets Syst.

Int. J. Approx. Syst.

Inform. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inform. Sci.

Appl. Soft Comput.

Inform. Sci.

Fuzzy Sets Syst.