Abstract in the Setting of the Automatic Analysis of Mamographic Images, the Detection

Morphological Filtering of the Mammographic Image

Hamdi Mohamed Ali

Ecole Nationale d’Ingénieurs de Tunis

Laboratoires Des Systèmes et Traitement du Signal

BP 37 Belvédère 1002 Tunis Tunisie

Tel:( 216.1)874700, Fax: ( 216.1)872729

Abstract— In the setting of the automatic analysis of mamographic images, the detection of opacities of the breast is a complex well-known problem whose feasibility remains to be proven again. We present in this article a technique of filtering in order to detect the microcalcifications in clinical mammograms while achieving a more effective algorithm.

I- Introduction

The goal of mammography exam is the research of possible suspected radiological signs, that is translating lesions of the mammary gland. Among them, opacities correspond to the anomalies on-density beaches. Opacities distinguish themselves from the normal structures of the mammary gland by a certain number of more or less variable features according to the topic and the nature of the opacity: the shape, the contrast, the contour, the texture...[1]. In this context, the problem of the implementation of an artificial vision system resides globally in the difficulty to create a model of the problem and to automate a complex analysis based on experience.

The aim of this work is the elimination of the impulsive noise resulting of the image segmentation of mammogram. With the purpose of comparison we carried out some experiments with different morphological operators, arriving to the final conclusion that the best results are obtained with generalized morphological operators.

Digital images enhancement, specifically noise reduction and smoothing, have been an important topic of researching in digital image processing (DIP). Many applications associated with this thematic have acquired great importance in different fields of DIP, among others, in digital image restoration, image reconstruction and segmentation and transmission of images [2],[3],[4].

In many image processing tasks, segmentation is an important step toward image analysis. It allows quantification and visualization of the objects of interest. Recently, image segmentation methods were extensively reviewed by Sijbers et al [5]. They concluded that segmentation of medical images is so far a difficult task and fully automatic segmentation procedures are far from satisfying to specialists in many realistic situations. Nevertheless, efforts towards the solution of the segmentation problem are motivated by the variety of applications wherein segmentation plays a crucial role [6].

The main problem in the reduction of noise is to get a clean image, that is without noise, but keeping all attributes of the original image as could be the shape, size, color, disposition and edges, among others.

Many methods have been proposed so far for the elimination of impulsive noise, among them, the median filter [8], rank-order filter [9], and more recently the so-called morphological filters [10].

Mathematical morphology (MM) is a powerful tool inside the DIP, since that science can rigorously quantify many aspects of the geometrical structure of the images in a way that agrees with the human intuition and perception [11]. This method developed in the 1960s mainly by Serra [12],[13],[14],[15],[16] and Matheron [17],[18] has its mathematical origins in concepts of the set theory, convex analysis, integral geometry, stereology and geometrical probabilities [19],[20],[21],[22].

Techniques of MM have been used widely in the analysis of biomedical and electron microscopy images, and in many computer visions applications as well, specially in the area of automated visual inspections [11]. However, in some applications, it is not possible to eliminate completely the noise by using morphological filters alone, but it is also necessary to combine them with others techniques.

The paper is organized as follows: Section II reviews the basic concepts of the mathematical morphology; section III shows the experimental results using ordinaries and generalized filters, section IV the most important conclusions are given.

II. THEORETICAL ASPECTS

2.1 Morphological filters

In mathematical morphology, signal transformations are called morphological filters, which are nonlinear operators that locally modify the geometrical features of multidimensional signals . In MM images are described by sets. Let F  E be the set which represents an image, where E is the Euclidean space. Let B  E be a simple compact set of small size called structuring element.

Definition 1

The translation of a set F by a point x denoted by F + x is defined as:

(1)

Definition 2

The fundamental operation of MM is erosion. The erosion of set F by set B is denoted by Fθ B and is defined by:

(2)

where < denotes the subset relation.Erosion can be formulated in other ways besides the fitting characterization of eq.(2). Its representation by an intersection of image translates is of particular importance:

(3)

The second most basic operation of binary mathematical morphology is dilation. It is a dual operation to erosion, meaning that it is defined via erosion by set complementation.

Definition 3

The dilation of set F by B is denoted by and is defined by :

(4)

where FC denotes the set-theoretic complement of F.

Dilation can be formulated in other ways, that is, it can be found by translating the input image by all points of the structuring element and then taking the union, namely,

(5)

There are two secondary operations that play a key role in morphological image processing, these being opening and its dual, closing.

Definition 4

The opening of image F by image B is denoted by and is defined as an iteration of erosion and dilation, namely,

(6)

Definition 5

The closing of F by B is denoted by and is defined by:

(7)

2.2 Generalized morphological filter.

In the following, we review the generalized morphological operators proposed by Agam [23].Dilation of F byB is considered as the union of all the possible shifts for which the reflected and shiftedB interceptsF, that is:

(8)

where is the reflection of B given by:

Definition 6

The generalized dilation of F by B with strictness s is defined by:

; (9)

where the symbol # denotes the cardinality of a set.So, the erosion of F by B may be obtained as the union of all the possible shifts for which the shifted B does not intersect FC:

(10)

Definition 7

The generalized erosion of F by B with strictness s is defined by:

; (11)

where it is assumed that

Lemma

Given two sets F, the cardinality of the intersection between F and the reflected B shifted by (k, l) may be obtained as the value at location (k, l) of the linear convolution between the respective images F and B:

(12)

Proposition 1

Given two sets F, , the generalized morphological dilation (erosion) of F by B may be obtained by thresholding the linear convolution between the respective binary images F and B :

(13)

(14)

Expressions (13) and (14) used in this work can be programmed easily.

III. EXPERIMENTS AND DISCUSSION

3.1. Use of ordinary morphological filters

Figure 1 shows a filtering example carried out by means of an opening operation whose SE is a circle with 21 pixel . a greatest of the noise was eliminated, however, narrow and sharp parts of the breast was eliminated too. Experiments with other types of large structuring elements give the same results. For this reason, we can conclude that to realize filtering by using ordinary morphological filters (OMF) with large kernels, segmented mammography are exposed to lost many useful information.

Fig.1: mammographic image before treatment.

Fig.2: Segmented mammographic imageaccording to OMF method

3.2 Use of generalized morphological filters.

With generalized morphological filters (GMF) it is possible to use SE of greater size, since they permit to control its strictness.

Fig.3: Segmented mammographic imageaccording to GMF method

In this case, a circle with 21 pixels is used with strictness . It can visually be appreciated from this figure that, although it presents a little more noise compared with that of the figures 2, the breast is less affected. However, comparing this with the result obtained with OMF, it can clearly be appreciated a superior performance, moreover it is a lesser time-consuming task.

IV. CONCLUSIONS

In conclusion, we proposed an alternative scheme using ordinary morphological filters and generalized morphological filters to eliminate the impulsive noise resulting of the mammographic image segmentation . We can conclude that the use of generalized morphological filters was better than ordinary ones, although they do not eliminate completely the impulsive noise. Also the effectiveness of this strategy for filtering of impulsive noise. We demonstrated by extensive experimentation, using real image data, that proposed scheme is fast and robust in the environment of a personal computer.

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