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In recent years, discrete orthogonal moments have attracted the attention of the scientific community because they are a suitable tool for feature extraction. However, the numerical instability that arises because of the computation of high-order moments is the main drawback that limits their wider application. In this article, we propose an image classification method that avoids numerical errors based on discrete Shmaliy moments, which are a new family of moments derived from Shmaliy polynomials. Shmaliy polynomials have two important characteristics: one-parameter definition that implies a simpler definition than popular polynomial bases such as Krawtchouk, Hahn, and Racah; a linear weight function that eases the computation of the polynomial coefficients. We use IICBU-2008 database to validate our proposal and include Tchebichef and Krawtchouk moments for comparison purposes. The experiments are carried out through

Over the past few years, discrete orthogonal moments (DOMs) have attracted the attention of the image analysis community because they possess the attribute of describing local and global features in images efficiently. The DOMs are optimal in the sense that they represent images with minimal information redundancy. The characteristic is originated by the orthogonality condition of the polynomial basis where the moments are computed [

Families of DOMs such as Racah [

An important impulse that has placed them in the spotlight is the discussion of the numerical instability problem that occurs in high-order moments. Since the DOMs are calculated as the projection of the image on a weighted kernel or set of orthogonal polynomials, their computation is usually linked to the size of the image. The higher the order of the moment, the higher the numerical error.

Traditionally, the computation of the DOMs is carried out by the use of recursive equations. However, the methodology tends to accumulate and propagate errors that degrade the representation of the image. Other methodologies have tackled the issue. In [

In addition to the aforementioned normalization methods, other approaches such as partitioning and sliding window have been also proposed. The former divides the image into smaller nonoverlapped sub-images where low-order moments are calculated independently, while the latter uses a rectangular window that slides usually from left to right and from top to down. On every window region low-order DOMs are computed.

A new class of discrete orthogonal moments derived from Shmaliy polynomials [

So far, the experiments conducted by Shakibaei Asli and Flusser have shown the capability of the DSMs as feature descriptors in one dimension. In this paper, we explore the use of the discrete Shmaliy moments as 2D texture descriptors and present an extensive comparative study using the IICBU-2008 database [

In Section

In 1980, Teague defined the orthogonal moments in the continuous space [

Generally speaking, the orthogonal moments are a set of scalar quantities that are not correlated among them. They are useful for characterizing local and global features in images [

Discrete Shmaliy polynomials were developed as unbiased finite impulse responses for predictive FIR filters [

The

In practice, the weighted discrete Shmaliy polynomials [

The weighted discrete Shmaliy polynomials satisfy the following orthogonality condition:

According to Shakibaei Asli and Flusser, it is possible to define multivariate polynomials (2D and 3D) as products of the univariate polynomials (1D) in each dimension.

Therefore, using (

According to Shakibaei Asli and Flusser [

In order to conduct a comparative study, we also use two well-known discrete orthogonal moments to compare with. A brief introduction of the basic theory of the discrete Tchebichef moments [

The discrete Tchebichef polynomials are defined in terms of hypergeometric functions as follows:

where

The property of orthogonality of the discrete Tchebichef polynomials is satisfied by

where

For a practical and stable implementation, the normalized discrete Tchebichef polynomials (see Figure

Using (

where

Discrete Krawtchouk Polynomials [

where

For stable computation, the weighted discrete Krawtchouk polynomials determine the basis as follows:

where

The weighted discrete Krawtchouk polynomials satisfy the orthogonality condition.

From (

where

DKMs are able to extract local features when the parameter

Unidimensional discrete orthogonal polynomials. (a) Tchebichef and (b) Krawtchouk polynomial bases from order 0 to 4.

We have mentioned that a key issue of the orthogonal polynomials is the numerical instability when the order of the polynomial becomes large. However, another important problem is the computational cost that tends to be high and limits the use of the orthogonal moments when an on-line computation is required [

There are many different ways to compute the coefficients of the orthogonal polynomials, i.e., by the hypergeometric definition [

Morales-Mendoza et al. [

We conducted time complexity experiments to compare the recursive implementation of Shmaliy, Tchebichef, and Krawtchouk polynomials. Specifically, the implementation of the weighted discrete Shmaliy polynomials is based on the

Figure

Mean computation time in milliseconds of the discrete Shmaliy, Tchebichef, and Krawtchouk polynomials.

Asli and Flusser [

Texture classification involves two main assignments: feature extraction and similarity measure. This paper is focused on feature extraction; however, the target is the correct classification of the image database described in Section

The discrete orthogonal moments, described previously in Section

The polynomial basis

Marcos and Cristóbal [

The textural features could be computed using any of the polynomial basis mentioned on Section

In [

Our proposal includes a modification based on overlapping square windows that shift over the image. This modification avoids the computation of high-order moments and assures numerical stability. Since overlapping windows generate an over-description of the image, we calculate the corresponding first three central moments (see Figure

The description of the textures is based on sliding windows that move top-to-down and left-to-right. On every window position, the corresponding orthogonal moments are computed.

The statistical textural feature,

This description scheme is useful in spite of the differences of size of inter- and intra-classes. The vector

We can say that (

General scheme of our proposal to classify texture images. After computing local DOMs using sliding windows, the statistical textural feature vectors are computed and classified.

The computation of the statistical textural features involves large vectors because the textural features (Section

In order to avoid that issue, Fisher proposed the discriminant analysis [

The between-scatter matrix is

On the other hand, the within class scatter matrix is defined by

Finally, the optimal projection matrix

IICBU-2008 benchmark was presented by Shamir [

Characteristics of the datasets included in the IICBU-2008 benchmark.

Database | Classes | Num. of img. | Format | Microscopy |
---|---|---|---|---|

Binucleate | 2 | 40 | 1280 × 1024 16bit | Fluorescence |

Celegans | 4 | 252 | 1600 × 1200 16bit | Fluorescence |

Liver aging | 4 | 850 | 1388 × 1040 32bit color | Brightfield |

CHO | 5 | 340 | 512 × 382 16bit | Fluorescence |

HeLa | 10 | 860 | 382 × 382 16bit | Fluorescence |

Lymphoma | 3 | 375 | 1388 × 1040 32bit color | Brightfield |

The dataset was acquired for classification purposes of binucleate cellular phenotype and normal cells. Binucleate phenotype is associated with failures in cell division and is a target when specialists look for genes that have effects on cell division. Many agents used on chemotherapy also alter cell division. The cells included in this dataset belong to

The two classes defined in the

The full name of the dataset is

The chronological ages of the

The Chinese Hamster Ovary (

Classes of the

Classes of

The Atlas of Gene Expression in Mouse Aging Project (AGEMAP) is a study performed by the National Institute of Aging and the National Institute of Health in the United States. This study involves 48 mice, males and females, with

The images belong to

The hematologic diseases include different types of leukemia and lymphomas. Although these diseases are not usual, they have aroused medical interest in recent years. For that reason, this dataset is one of the most popular databases because it has been used on several studies for classification [

Classes of

During the preprocessing stage, color images in

In addition, the backgrounds of the images in

For local analysis, every image is divided into overlapping square windows. We varied the size of the windows from

The overlapping windows are processed as Figure

Linear discriminant analysis, described in Section

The statistical textural features are transformed with linear discriminant analysis. Note that the projections obtained improve the quality of the clusters and the boundaries among them.

The classification of the datasets is performed through

The mean accuracy results for each dataset are shown in the following figures:

Mean accuracy results for

Mean accuracy results for

Mean accuracy results for

Mean accuracy results for

Mean accuracy results for

Mean accuracy results for

Every subfigure in Figures

ANOVA compares the means between groups and resolves if any of them has significant statistical difference from each other. ANOVA tests the null hypothesis,

In our experiments, the accuracy is the dependent variable. The window size, classifier, and discrete orthogonal moment are the three nominal or independent variables; it means

WSize & | WSize & | Class. | ||||
---|---|---|---|---|---|---|

Database | WSize | Class. | Mom. | Class | Mom. | Mom. |

Celegans | | | | | 0 | |

CHO | | | | | 2.7 | 98.7 |

HeLa | | | | | | |

Liver aging | | | | | 0 | 98.2 |

Lymphoma | | | | | 3.3 | 99 |

Based on Table

First, the window size is important because the larger the size, the better the overall accuracy classification results. The classifier is also statistically relevant; it shows that better results are obtained, in general, with both naïve Bayes and SVM (linear and Gaussian) classifiers. DOMs have also a statistical significance. The best classification results for

It is important to mention that

The best results from our experiments are compared with the state-of-the-art works: Shamir et al. [

Shamir et al. presented IICBU-2008 benchmark in [

Siji et al. proposed to apply principal component analysis (PCA) and Fisher analysis score based on feature selection for image retrieval applications. They used the same features computed by WND-CHRM for

Meng et al. proposed a classification model called collateral representative subspace projection modeling (C-RSPM) for

Table

Comparison results between discrete orthogonal moments and state-of-the-art methods.

Shmaliy | Tchebichef | Krawtchouk | WND- | Shamir | Siji | Meng | |
---|---|---|---|---|---|---|---|

Database | Moments | Moments | Moments | CHRM | | | |

Binucleate | 100 | 100 | 100 | 100 | - | 100 | - |

Celegans | 100 | 100 | 100 | 53 | - | 75 | - |

Liver aging | 99.81 | 100 | 99.05 | 51 | - | - | 95.09 |

CHO | 100 | 100 | 100 | 93 | 94 | 99 | - |

HeLa | 98.25 | 97.79 | 98.15 | 84 | 83 | 74 | - |

Lymphoma | 100 | 99.74 | 99.74 | 85 | - | - | 84.79 |

Since their recent introduction, Shmaliy moments have not been treated or tested on image texture analysis. In this paper, we show for the first time the capability of Shmaliy moments for computing texture features and compare them with the most popular families of discrete orthogonal moments: Tchebichef and Krawtchouk moments. Moreover, the results are compared with a set of methods that use the IICBU-2008 benchmark.

The proposed classification method is sufficiently general and useful in many different types of images. We included a preprocessing stage that consists in color space transformation and, in some cases, resizing the datasets to accelerate the description and classification processes. Our proposal is robust and it may be useful in many different cases that not necessarily belong to the biomedical field.

The classification results obtained by discrete Shmaliy moments are similar to the ones achieved with Tchebichef and Krawtchouk moments; therefore, a 3-way ANOVA was applied in order to verify if the parameter selection (descriptor, window size, and moment) was significant. ANOVA suggests that the descriptor combination is the most important parameter because the null hypothesis was rejected for all databases while the discrete orthogonal moment and the window size elections show equal low significance since the null hypothesis was rejected only on three of five databases.

On the other hand, comparison among discrete orthogonal moments and other techniques found in the literature shows competitive results. Accordingly, Shmaliy moments have shown the same capability for texture description as other classical discrete polynomial bases; therefore, its consideration is pertinent for texture analysis, and probably a combination of Shmaliy texture features and other discrete moments could be the topic for a future work.

To download the IICBU-2008 benchmark, use

The authors declare that they have no conflicts of interest.

This work has been sponsored by UNAM Grants PAPIIT IN116917 and SECITI 110/2015. Germán González thanks CONACYT263921 Scholarship.