Contribution to the Characterization of the Kidney Ultrasound Image Using Singularity Levels †

: The aim to improve diagnosis decision systems of kidneys led to the concept of many methods of texture characterization of kidneys from ultrasound images. Here, as a ﬁrst main contribution, we propose a texture characterization method based on the singularity deﬁned by the Hölder exponent, which is multifractal local information. Indeed, the originality of our contribution here is to build a singularity-level matrix corresponding to different levels of regularities, from which we extract new texture features. Finally, as a second main contribution, we will evaluate the potential of our proposed multifractal features to characterize textured ultrasound images of the kidney. Having more reproducibility of the texture features ﬁrst requires a good choice of Choquet capacity to calculate the irregularities and a selection of a more representative region of interest (ROIs) to analyze by carrying out an adapted virtual puncture in the kidney components. The results of the supervised classiﬁcation, using three classes of images (young, healthy, and glomerulonephritis), are interesting and promising since the classiﬁcation accuracy reaches about 80%. This encourages conducting further research to yield better results by overcoming the limitations and taking into account the recommendations made in this article.


Introduction
Ultrasound imagery is a useful, convenient, and safe diagnosis modality of kidneys, and the analysis of the information contained in the ultrasound image is within the competence of the clinician.Unfortunately, the quality of ultrasound images is considered poor due to a large amount of noise, mainly speckles that are due to the interference of backscattered signals, which distorts the information sought in the image.This is why the clinician's perception must be complemented by automatic recognition based on texture characterization using image processing tools, such as multifractal analysis, which is becoming increasingly efficient.Indeed, most of those techniques are using statistical texture analysis approaches based on the extraction of texture features that allow the kidneys to be classified as a normal or abnormal organ.Clinicians mainly use statistical texture analysis methods [1][2][3][4][5].Generally, these research works have two disadvantages, namely that they not only remain, unfortunately, manual in the selection of the region of interest to be studied but also have difficulty carrying out a complete classification accounting for different pathologies in their several stages.These disadvantages are probably the reason for the lack of papers aimed at classifying the kidney from ultrasound images via texture analysis based on the multifractal approach, unlike the characterization of other human 2 of 6 organs such as the liver or bone.This is why we attempt in our work to characterize the kidney with an adapted multifractal method based on the extraction of texture features from an ultrasound image of the kidney.Indeed, all texture feature extraction methods based on multifractal analysis reported in all fields of image processing exist in two types: those based on local information given by the singularity [6][7][8] and others based on global information given by the multifractal spectrum [9][10][11][12][13].
The aim of this work is to propose a new multifractal approach for texture characterization by introducing new texture features extracted from a singularity in the first stage.Afterward, the second goal is to perform texture characterization of kidney ultrasound images using the proposed set of texture features.

Image Analysis Based on Local Information
The study of the local regularity at each point of the texture image can be a means of texture analysis because it can simply be seen as a particular combination of Hölder exponent values α(i, j) [6,14].This amounts to saying that a texture is made up of a set of singularities and that the value and the spatial arrangement of these singularities can provide features to characterize the texture.

The Image of Singularities
The Hölder function calculated on an image m × n is stored in an array of the same size m × n, where each point (i, j) carries the value of the Hölder exponent α(i, j) estimated on the pixel (i, j) (Figure 1).The realization of the image of the singularities can be carried out from a sampling of the values of singularities, which consists of calculating the maximum value of the singularities α max and covering the interval [α min , α max ] with the set of intervals I i defined by the following: those based on local information given by the singularity [6][7][8] and others based on global information given by the multifractal spectrum [9][10][11][12][13].
The aim of this work is to propose a new multifractal approach for texture characterization by introducing new texture features extracted from a singularity in the first stage.Afterward, the second goal is to perform texture characterization of kidney ultrasound images using the proposed set of texture features.

Image Analysis Based on Local Information
The study of the local regularity at each point of the texture image can be a means of texture analysis because it can simply be seen as a particular combination of Hölder exponent values  ,  [6,14].This amounts to saying that a texture is made up of a set of singularities and that the value and the spatial arrangement of these singularities can provide features to characterize the texture.

The Image of Singularities
The Hölder function calculated on an image  ×  is stored in an array of the same size  × , where each point ,  carries the value of the Hölder exponent  ,  estimated on the pixel ,  (Figure 1).The realization of the image of the singularities can be carried out from a sampling of the values of singularities, which consists of calculating the maximum value of the singularities  and covering the interval ; α u is the reference step defined, in a normalized way, to establish irregularity level classes within the image of the singularities.Indeed, α a,max (resp.α a,min ) is the mean maximum (resp.mean minimum) calculated on N images.

Singularity-Level Run-Length Matrix
This method, using the singularity level matrix H containing irregularity information α(x, y) at each point (x, y), is based on the definition of statistical parameters expressing the distribution of the levels of singularities existing in the image.A run of irregularity level is a consecutive set of collinear pixels having the same level of irregularity.The run-length matrix p θ (i, j) is given via counting the runs of irregularity levels i and size j.
The run-length matrix involves two textural information, namely the direction and the coarseness in terms of irregularities.The direction generally used to define run-length matrix are 0 • , 45 • , 90 • , and 135 • .
The reduced number of singularity levels will make it possible to avoid a large number of runs of irregularities, which contain only one point.This is how we consider the coding of singularity levels on 16 levels sufficient for texture analysis with the run-length matrix.
Thus, the generalized parameter defining the statistical distribution of the various runs of irregularity levels is given by the following: where GRLSLF is Generalized Run-Length Singularity-Level Feature; n θ is the total number of runs existing in the singularity-level matrix H along a θ direction; G is the number of singularity-level; N θ is the size of singularity-level matrix H along the θ direction.Several texture features can be defined from the GRLSLF parameter by giving k i and k j different values.The principle in the extraction of the features is to exclusively measure the accentuation of different types of singularity-level and run-length via the coefficients k i and k j .We propose some features, namely: -Short-Run Low-Level Singularity Emphasis (SRLLSE) given by k i = −2 and k j = −2; -Short-Run High-Level Singularity Emphasis (SRHLSE) given by k i = 2 and k j = −2; -Long-Run High Singularity Emphasis (LRHSE) given by k i = 2 and k j = 2; -Long-Run Low Singularity Emphasis (LRLSE) given by k i = −2 and k j = 2; Other run length statistics are defined, namely: -Run-Length Non-Uniformity (RLNU): -Singularity Level Non-Uniformity (SLNU): -Runs Percentage (RP):

First-Order Statistics
First-order statistics are estimated on singularity levels without taking into account their relative distribution.The consideration of these amounts calculated on the Hölderian image allows an evaluation of the different irregularities without taking into account their relative distribution in the image.However, we will consider the most used first-order statistics, namely: The average singularity level value: The skewness (centered space moment of order): The Kurtosis (centered space moment of order 4): h is the histogram of the singularity levels of the image of the singularities H.
Other first-order statistics extracted from the singularities image H will be used, such as energy, entropy, etc.

Application for Characterization of Kidney Ultrasound Images
To evaluate our proposed texture characterization approach, by applying it to the ultrasound image, we use the image slice representing the longitudinal section of the right kidney that provides a large amount of information on the kidney and achieves pseudo-reproducibility (Figure 2).Thus, the region of interest ROI(s) to study the kidney images will be windows selected over the three following regions: parenchyma, central, and "whole" kidney [15].
relative distribution in the image.However, we will consider the most used first-order statistics, namely: The average singularity level value: Variance: The skewness (centered space moment of order): The Kurtosis (centered space moment of order 4): h is the histogram of the singularity levels of the image of the singularities .
Other first-order statistics extracted from the singularities image  will be used, such as energy, entropy, etc.

Application for Characterization of Kidney Ultrasound Images
To evaluate our proposed texture characterization approach, by applying it to the ultrasound image, we use the image slice representing the longitudinal section of the right kidney that provides a large amount of information on the kidney and achieves pseudoreproducibility (Figure 2).Thus, the region of interest ROI(s) to study the kidney images will be windows selected over the three following regions: parenchyma, central, and "whole" kidney [15].To perform a local multifractal description of the ultrasound images of the kidney, we will use texture features calculated from the singularity level matrix (see Sections 2.2 and 2.3) of the three following selected regions of interest: parenchyma, central, and representative region of the "whole" kidney.Indeed, we will use the supervised K-nn classifier, which allows both the selection of all the most representative features and the evaluation of the discriminating power of the approach that will be given by the correct classification accuracy rate obtained by the selected subset of features.To perform a local multifractal description of the ultrasound images of the kidney, we will use texture features calculated from the singularity level matrix (see Sections 2.2 and 2.3) of the three following selected regions of interest: parenchyma, central, and representative region of the "whole" kidney.Indeed, we will use the supervised K-nn classifier, which allows both the selection of all the most representative features and the evaluation of the discriminating power of the approach that will be given by the correct classification accuracy rate obtained by the selected subset of features.

Results and Discussion
The results obtained by considering the representative window of each of the three regions are noted in Table 1.Analyzing the results allows us to make the following remarks:

•
Singularity-level run-length matrix methods, compared to the method using the first order statistics, provide a better classification accuracy rate, whereas the best result is obtained using the combined method applied on the ROI representative of the "whole" kidney, and the classification rate reaches 80% obtained with eight features and a test accuracy classification of about 78%.

•
Increasing the size of the database so that it contains other classes of images corresponding to different pathologies while carrying out image acquisitions in the most standard conditions; • Setting the parameters that most influence the reproducibility of the texture features;

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Combining the multifractal with other texture classification approach to improve the results; • Applying this approach for the characterization of texture images of other human organs such as the liver; • Improving a selecting puncture for considering all the kidney images.

Conclusions
This work was devoted to the presentation of a new multifractal approach based on the calculation of the levels of singularity, which we used for the characterization of texture images.The results obtained with this method applied for the characterization of ultrasound images of the kidney are quite interesting and promising.Indeed, our perspectives are mainly to optimize this approach by developing other more discriminating multifractal features and also taking into account the limits of this work.

Figure 2 .
Figure 2. The kidney US image and the three ROI windows.

Figure 2 .
Figure 2. The kidney US image and the three ROI windows.