pSnakes: A new radial active contour model and its application in the segmentation of the left ventricle from echocardiographic images

Accepted Manuscript Title: pSnakes: a new radial active contour model and its application in the segmentation of the left ventricle from echocardiographic images Author: Auzuir Ripardo de Alexandria Paulo C´esar Cortez Jessyca Almeida Bessa John Hebert da Silva F´elix Jos´e Sebasti˜ao de Abreu Victor Hugo C. de Albuquerque PII: DOI: Reference:

S0169-2607(14)00207-7 http://dx.doi.org/doi:10.1016/j.cmpb.2014.05.009 COMM 3806

To appear in:

Computer Methods and Programs in Biomedicine

Received date: Revised date: Accepted date:

10-12-2013 14-5-2014 15-5-2014

Please cite this article as: A.R. Alexandria, P.C. Cortez, J.A. Bessa, J.H.S. F´elix, J.S. Abreu, V.H.C. Albuquerque, pSnakes: a new radial active contour model and its application in the segmentation of the left ventricle from echocardiographic images, Computer Methods and Programs in Biomedicine (2014), http://dx.doi.org/10.1016/j.cmpb.2014.05.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

pSnakes: a new radial active contour model and its application in the segmentation of the left ventricle from echocardiographic images Auzuir Ripardo de Alexandria1, Paulo César Cortez2, Jessyca Almeida Bessa1, John Hebert

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da Silva Félix3, José Sebastião de Abreu4, Victor Hugo C. de Albuquerque5

1

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Programa de Pós-graduação em Engenharia de Telecomunicações, Instituto Federal de Educação, Ciência e Tecnologia do Ceará, Fortaleza, Ceará, Brazil.

2

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E-mail: [email protected], [email protected]

LESC, Departamento de Teleinformática, Universidade Federal do Ceará, Fortaleza,

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Ceará, Brazil.

E-mail: [email protected] 3

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Universidade da Integração Internacional da Lusofonia Afro-Brasileira, Fortaleza, Ceará, Brazil

4

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E-mail: [email protected]

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HUWC, Hospital Universitário Walter Cantídio, Universidade Federal do Ceará,

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Fortaleza, Ceará, Brazil.

E-mail: [email protected]

5

Programa de Pós-Graduação em Informática Aplicada, Universidade de Fortaleza, Fortaleza, Ceará, Brazil.

Email: [email protected]

Corresponding author: Prof. Victor Hugo C. de Albuquerque Graduate Program in Applied Informatics (PPGIA) Center of Technological Sciences (CCT) University of Fortaleza (UNIFOR) Washington Soares Av. 1321, Bl. J-30, CEP 60811-905 Fortaleza, Ceará, Brazil Phone: +55 85 81297776 / +55 85 81167329 e-mail: [email protected]

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pSnakes: a new radial active contour model and its application in the segmentation of the left ventricle from echocardiographic images Abstract

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Active contours are image segmentation methods that minimize the total energy of the

contour to be segmented. Among the active contour methods, the radial methods have

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lower computational complexity and can be applied in real time. This work aims to present a new radial active contour technique, called pSnakes, using the 1D Hilbert

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transform as external energy. The pSnakes method is based on the fact that the beams in ultrasound equipment diverge from a single point of the probe, thus enabling the use of polar coordinates in the segmentation. The control points or nodes of the active contour

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are obtained in pairs and are called twin nodes. The internal energies as well as the external one, Hilbertian energy, are redefined. The results showed that pSnakes can be used in image segmentation of short-axis echocardiogram images and that they were

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effective in image segmentation of the left ventricle. The echo-cardiologist’s golden standard showed that the pSnakes was the best method when compared with other

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methods. The main contributions of this work are the use of pSnakes and Hilbertian energy, as the external energy, in image segmentation. The Hilbertian energy is

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calculated by the 1D Hilbert transform. Compared with traditional methods, the

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pSnakes method is more suitable for ultrasound images because it is not affected by variations in image contrast, such as noise. The experimental results obtained by the left ventricle segmentation of echocardiographic images demonstrated the advantages of the proposed model. The results presented in this paper are justified due to an improved performance of the Hilbert energy in the presence of speckle noise. Keywords: radial active contour methods; pSnakes; echocardiogram image segmentation; Hilbertian energy.

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1 Introduction Heart diseases are one of the leading causes of death in the world. According to United Nations (UN) data, cardiovascular diseases are increasing in both developed and

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developing countries due to the changes in the way people eat and live,. In the United States 317 per 100,000 inhabitants died of heart related diseases in 2002. The highest

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index was registered in Ukraine, where 1,032.4 per 100,000 people died in the same year. The same index in Ethiopia, for example, was 148.1 inhabitants (WHO, 2004).

Único

de

Saúde)

the

national

Brazilian

Health

System

there

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(Sistema

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In Brazil the index per 100,000 people was 224.8 in 2014. According to SUS

were 548,138 hospitalizations of men and 563,916 of women in 2009 in Brazil

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for cardiovascular disease. As for deaths, there were over 300,000 deaths due to circulatory system problems. Given the cost of hospitalization and the number

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of deaths, medical diagnosis and treatment are important measures to reverse this morbid

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condition as well as reducing the costs.

Echocardiography has an important role in this context, consisting of a reliable evaluate

the cardiovascular health of

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tool to

patients,

promoting early

diagnosis. However, ventricle segmentation is a difficult task due to the relatively poor quality (speckle

noise) and discontinuous edges of the echocardiographic images

(Nandagopalan, 2010).

Thus, the use of digital image processing and image

segmentation techniques can contribute to provide a more precise quantitative evaluation of left ventricular volumes, contributing to more accurate diagnoses. For example, Kang et al. (2012) provided an overview of cardiac segmentation techniques, with useful advice and references from clinical applications, imaging modalities, and the validation methods used for cardiac segmentation. Petitjean and Dacher (2011) proposed an original categorization for cardiac segmentation methods with a special emphasis on

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what level of external information is required (weak or strong) and how it is used to constrain segmentation. Many cardiac left ventricle (LV) segmentation methods have been studied to

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calculate blood volume, myocardial volume, and ejection fraction using magnetic resonance imaging (MRI). These methods can be categorized as follows: traditional

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segmentation, graph-based segmentation, active shape models (ASMs), and level-set algorithms.

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Among these techniques are the active contour methods (ACM), also known

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as snakes, which were introduced by Kass, Witkin and Terzopoulos (1987) to solve problems of edge detection on real images. The ACMs are based on variational methods,

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which aim to minimize a function that represents the energy of a curve. Active contour methods present limitations, for example: the initial contour must

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be initialized close to the edge of the object of interest in order to avoid any incorrect

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convergence and also this method hase difficulties to move the curve along objects that have concave regions due to horizontal force vectors pointing in opposite directions

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and canceling each other out, and thus preventing their evolution (Xu and Prince, 1998a). The Gradient Vector Flow (GVF) active contour methods overcome some

restrictions in relation to the previously mentioned ACMs, but have limitations such as a higher computational cost of the gradient vector diffusion calculations for parameter dependent vector fields.

Therefore due to these limitations other approaches have been proposed. One of

these approaches is called radial active contours, which is a relatively new concept. Radial active contours were developed in order to decrease the computational complexity of the active contour methods and consequently be used for real-time applications (Gemignaniet et al., 2007). The energy calculation and its minimization are performed in

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one dimension (1D) (polar coordinates), thus making them faster. The motivation to develop new active contour methods is related to the lack of a tool capable of solving all kinds of analyzes due to the peculiarities of each type of

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image. Thus, the main aim of this work is to make a comparative study of methods based on the active contour methods, such as traditional snakes, GVF snakes, radial snakes

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with derivative, radial snakes with Hilbert energy, and, also to propose a method called

pSnakes. These methods were applied and evaluated for the segmentation of the left

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ventricle in echocardiographic images.

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This work is organized into five sections. Section 2 is a review of left-ventricle segmentation and the active contour methods, in particular the traditional ACM, the

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GVF active contour methods, radial snakes and pSnakes. The proposed approach is presented in Section 3. Section 4 presents the Results and Discussions. Finally, the

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Conclusions, highlighting the main contributions of this work, are shown in Section 5.

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2 Brief literature review

2.1. Left-ventricle segmentation

There are many difficulties to analyze and identify the edges of the left ventricle

(LV) in echocardiographic images due to factors such as noise and discontinuous edges. Consequently, the automatic identification of LV boundaries using computational methods has been the focus of attention of many researchers in order to help cardiologists in heart disease diagnosis. These methods have been used to calculate blood volume, myocardial volume, and ejection fraction. The techniques used can be categorized as follows: traditional segmentation, graph-based segmentation, active shape

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model (ASM), and level-set algorithms. A popular group of statistical models are Active Appearance Models (AAMs) (Cootes, Edwards and Taylor, 2001). AAMs show two features of the datasets, shape

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and texture. The model is created from a training set. These models were first presented by Mitchell et al. (2002) and Stegmann et al. (2001, 2003) and they were developed and

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tested only on a single cardiac phase, rather than on the entire cardiac cycle. Additional

studies combined the AAMs with Active Shape Models (ASM) to introduce a hybrid

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model (Mitchell et al.,2001; Zambal et al.,2006; Zhang et al.,2010).

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Several techniques have used image based segmentation methods, including thresholding, described in Goshtasby and Turner (1995), and/or dynamic programming

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(DP) (Yen et al.,2005). These methods apply specific features of the images such as texture, shape, area and they require manual user intervention.

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Other techniques have been developed for the segmentation of the LV in short

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axis views, e.g. wavelet-based enhancement (Fu, Chai and Wong, 2000) or gradient value methods (Cousty, Najman and Couprie, 2010), which include the papillary muscles

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within the myocardium.

Active contours (Kass, Witikin and Terzopoulos, 1987) or snakes have often been

used for the processing of medical images. A review of these approaches and their medical application is given in McInerney and Terzopoulos (1996). Cootes et al. (1994) introduced the Active Shape Model (ASM), described in the previous paragraph, similar to the snake model of Kass et al. (1987) but employed global shape constraints for locating structures in medical images.    The

contributions regarding ventricle segmentation using deformable models

have mainly dealt with the design of the external force term. Region-based terms, which are based on a measure of region homogeneity, were introduced by Paragios (2002). But

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these models are initiated using gradients and thus sensitive to noise. A segmentation algorithm for short-axis must comply with several requirements: First, it should be able to process, and overcome the problem of small structures, e.g., the

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papillary muscles. Second, a large dataset is required to validate the proposed segmentation algorithm. Third, a gold-standard, based on manual tracing of the LV

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boundaries, should be constructed in order to test the performance of the algorithm. Fourth, the proposed algorithm should not be sensitive to noise, in order to reduce error

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rates. At present, there is no segmentation method that complies with all 4 of these

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requirements. However here we present a method that attempts to comply with these

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requirements.

2.2 Active contours

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Active contours allow segmentation of images by edge detection. This method is

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applied successfully to problems of image processing and Computer Vision, such as edge detection and object tracking, among others. There is no perfect solution for all

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cases, because of the uniqueness of each problem characterized by specific images. Snakes were innovated to solve problems where edge detection by gradient was not successful due to low contrast contours, the presence of noise, among other reasons. The active contours method consists of drawing a curve, starting near or inside

the object of interest. This line is deformed by forces that move until the edges of the object are defined. This edge approach is performed through successive minimization iterations of a previously specified energy.

2.2.1 The traditional active contour methods The traditional active contour method, also called snakes, is a framework for

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delineating the outline of an object from an image. The goal is to minimize a function representing the snake energy. The curve evolves so that its energy decreases with each new interaction. The snake model is the 2D parameterization of a geometric curve,

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(Nixon and Aguado, 2002), defined as

⎧⎪ [0,1] → R 2 ⎨ ⎪⎩ s → c ( s ) = ( x ( s ) , y ( s ) ) .

cr

(1)

(Cohen, 1993), which is expressed as E (s) ⎧ ⎯ →R R 2 ⎯⎯ ⎪ ⎨ ' 2 '' 2 ⎪c → ∫[ 0,1] e1 c ( s ) | + e2 c ( s ) | + e3 Eext ⎡⎣ c ( s ) ⎤⎦ ds, ⎩

}

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{

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The model is called deformable because it is described by one energy function E ( s ) , 

(2)

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where the apostrophe represents the derivation, e1 , e2 and e3 are real constants and Eext is

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2.2.2 The Energies

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the energy term associated with external forces.

The energy of a snake depends on its shape

[0]and

its location. It usually breaks

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down into two energies: the internal energy and the external energy (Sonka, Hlavac and Boyle, 2007).

The first energy E1 = ∫e1 c '( s) ds is linked to the elasticity. This means that it 2

expresses the characteristic for each point of the snake to move away from its neighbors. The minimization favors the search for points, when c ' ( s ) is small. This means that these points tend to approach each other and the snake contracts. The second energy, E2 = ∫e2 c ''( s) ds is the bending energy or snake curvature. 2

The energy is minimal when c '' ( s ) = 0 , i.e., when c ( s ) = ks ( k is constant), that is, when it is a line. Thus, favoring the coefficient e2 during the minimization, it forces the snake

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to lose its curvature. The external energy allows the snake to shape itself around the edges of objects. It is the energy that compensates the other energies and thus prevents the snake from

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contracting in on itself, without perceiving the image contours. In general, external energy is calculated using the square of the image gradient at a point ( x, y ) of the snake,

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i.e.,

(3)

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Eext ( x, y ) = − | ∇[ I ( x, y )] |2 , and also,

(4)

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Eext ( x, y ) = − | ∇[Gσ * I ( x, y )] |2 ,

Where ∇ is the gradient operator, Gσ is a Gaussian filter centered at the point ( x, y ) with

2.3

d

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a variance σ 2 and I is the image to be segmented.

Balloon snake

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The balloon ACM (Cohen, 1991) adds a force, called the pressure force to the

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traditional ACM, simulating a behavior similar to a balloon, causing the curve to expand (inflate)

or retract (deflate) depending

on

the

position of

the

contour

initialization, and thus defining the direction of this force. The pressure force causes the ACM balloon to pass isolated noise points and low contrast (soft) edges and to stop at high-contrast edges (protruding). This pressure force is obtained by the equation G G P ⎞ G ⎛G F balloon = ⎜ F int + F ext ⎟ + F pressure , ⎝ ⎠

(5) GP

Gb

where the external potential force is Fext = Fext = − γ∇E iext with i =1, 2, 3or 4 . Thus, the balloon snake has the same internal forces which are present in the traditional ACM and this extra force called the pressure force.

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Gb

According to the above, the Fext balloon force follows the equation, Gb

Fext = −k

∇E ext ,  || ∇E||ext

 

 

 

 

 

(6)

 

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where k is the weight associated to its normalization. Thus, the pressing force G

F pressure is given by

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G G F pressure = k1n ( s )

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and G

(8)

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E balloon = || F pressure ||,  

(7)

G where n ( s ) is the vector normal to the curve at the point c(s) and k1 is the amplitude and

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direction of the force, indicating inflation, if positive, or deflation, if negative. k1 also influences the velocity of evolution (deformation) of the balloon snake (Dagher and

G

∇E ext G + kn ( s ) .   ∇E ext

(9)

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F balloon = F int − k

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G

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Tom, 2008). Thus, equation 5 can be rewritten as

where k and k1 are of the same order, smaller than a pixel. The value of k should be a little larger than k1 to finish the inflation or deflation of the balloon snake on the edge of the object of interest.

2.3.1

Total energy of the balloon snake

Based on the above definitions, the total energy of the balloon snake is given by Ebtotal = α E cont + β E curv + δ E balloon + γ E ext ,  

(10)

where α , β , γ and δ are real constants. Whenever a new energy value ( Ebtotal )  is  calculated, a new position is determined

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for the snake. In this case, the problem becomes an optimization of a numerical function with several variables (Bouhours, 2006). The optimal displacement is calculated for each vertex vi of the snake, however

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this depends on its position and the positions of its neighbors  vi−2 , vi−1 , vi+1 and vi+2 . This can be applied for all points of the snake, taking each one in turn and deciding which

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position the snake reaches to attain the minimum energy (Amini, Weymouth and Jain,

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1990). The complexity of the algorithm is much greater when a large number of locations are chosen for the snake.

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A balloon snake has superior characteristics to a traditional snake. However, there are limitations that are subject to external energy and pressure forces (Dagher and

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Tom, 2008). Some limitations of the balloon snake are: failure to capture the contours with discontinuities; inadequate identification of concave regions as this ACM

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uses the same external energy as the traditional snake (Xu, 1999). The magnitude of the

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balloon force is difficult to set considering the stop criteria. It is less reliable when the contour is pulled near an edge with discontinuity (Wang, 2012).

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The traditional ACMs cannot be applied to the segmentation of any kind of

image due to problems such as inappropriate runtime, requirements for automatic selection of the initial contour, non-convergence in images with low sharpness, borders with few definitions or without edges, and multiple images to be segmented at the same time. Thus, other approaches have been applied and evaluated in the literature.

2.4 GVF snake

Due to the limitations of the traditional and balloon snakes, particularly in their lack

of

ability

to capture edges,

developed. The GVF

snake

the is

Gradient Vector Flow formed

(GVF)

snake was

by a vector

field

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VGVF ( x, y ) = ⎡⎣uGVF ( x, y ) , vGVF ( x, y ) ⎤⎦

calculated by

fusion

of

the

gradient vectors, which are defined as the edge maps of a binary image or gray levels. This vector field replaces the potential external force, which defines the condition of

the

equation:

vt ( s, t ) = α ⎡⎣ v '' ( s, t ) ⎤⎦ − β ⎡⎣v ''' ( s, t ) ⎤⎦ + VGVF ( s, t ) ,  

 

 

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VGVF ( x, y ) = ⎡⎣uGVF ( x, y ) , vGVF ( x, y ) ⎤⎦ . Rewriting this equation produces

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equilibrium

 

 

      (11)

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where s can be replaced by ( x, y )   and VGVF ( x, y )   is defined in such a way that it

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minimizes the energy

EGVF = ∫∫ [ μ (u 2GVFx + u 2GVFy + v 2GVFx + v 2GVFy )+ | ∇fGVF |2 | VGVF − ∇fGVF |2 ]d x d y ,  

      (12)

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where fGVF is the map derived from the edge image; uGVFx , uGVFy , vGVFx , vGVFy are the

regularization parameter.

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first partial derivatives of uGVF and vGVF in relation to x and y , respectively, and μ is a

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The regularization parameter μ , in equation 12, controls the relationship between

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the first and the second term of the integral; this relationship can reduce the influence of noise. When the value of ∇fGVF is small, the energy is predominant in the first term, resulting in a field with little variation. On the other hand, when ∇fGVF   presents a high value, the

second

term dominates

the integral

and

is minimized considering ∇fGVF = VGVF . This condition produces the desired effect of keeping VGVF similar to the gradient of the edge maps, where this value is high, but forcing the field to vary little in homogeneous regions.

2.5

Radial Snakes Review

One of the most well known and relevant radial snakes is known as Active Rays

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(Denzler and Niemann, 1999; 1996). The technique is applied to trace the outline of objects in real time. The idea is to define a point of origin inside the boundary and find points that characterize it, searching along rays that diverge from a central source m . For

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this, active contour equations are adapted. The contour   c(s) shall be defined as ⎧⎪ [0,1] → R 2 ⎨ ⎪⎩ s → c ( s ) = Cm (θ ( s ) , r ( s ) ) ,

cr

(13)

where c(s) is the active contour and Cm(s) is the contour defined from the origin

, in

2

d d2 r (θ ) + β (θ ) r (θ ) dθ dθ 2

2

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Ei ( cm (θ ) ) = α (θ )

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polar coordinates (θ , r ) . The internal energy from the contour is calculated by

(14)

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where   α (θ ) and  β (θ ) are real constants to the determined angle. The first equation term is the definition of the energy continuity and the second term, the curvature energy

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for the Active Rays. The above equation may be calculated along the beam. Then, the

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calculations are made only in one dimension. Another relevant work is known as optimal radial active contours (Chen, Huang and Rui,

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2001) that use dynamic programming to search for contour energy optimization and are mainly used for object tracking. The definition for total contour energy E is given by 2π

{

}

E ( rm (θ ) ) = ∫ Ei ⎡⎣ rm (θ ) ⎤⎦ + Ee [rm (θ )] dθ ,

(15)

0

where rm (θ ) is the distance from the origin

, considering a given angle   θ ;   Ei is

internal energy and Ee the external. The external energy Ee from the active contour is the gradient function from the image to be segmented

{

}

2 ⎡ d ⎤ 2 ρ m (θ , r ) ⎥ = α e .g − ⎡⎣ ρ m (θ , r + 1) − ρ m (θ , r ) ⎤⎦ , Ee [ rm (θ )] = α e g ⎢ − ⎢⎣ dr ⎥⎦

(16)

where the r and θ are polar coordinates from one control point (the node itself) of the

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active contour;   g is a nonlinear monotonically increasing function and ρ m is the active ray. Internal energy is the sum of the continuity and curvature energies. The continuity energy Eicont in the i th node from the active contour is calculated by the expression

Eicont (rm (θ )) = α i rm ( θi ) − rm (θi −1 ) , 2

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(17)

cr

where α i is a real constant. The curvature energy Eicurv is given by the equation

}

^

(18)

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The Hilbert Transform

,

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where β i is a real constant.

2.6

2

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{

Eicurv (rm (θ )) = βi ⎡⎣ rm ( θi ) − rm (θi −1 ) ⎤⎦ − ⎡⎣ (rm ( θ i −1 ) − rm ( θi − 2 )) ⎤⎦

The Hilbert transform   f (t ) is a real function   f ( t ) defined by Johansson (1999) and

π

∞

P∫

−∞

f (τ ) dτ , t −τ

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f (t ) =

1

te

^

d

Cizek (1970) as

(19)

where   P denotes the main value of Cauchy, since there is a singularity in the integral for   t =τ . Another way to represent the Hilbert transform is through integral convolution as ^

f (t ) =

1 * f (t ). πt

The function

(20)

1 1 Fourier transform is given by   F = − j.sgn(ω) , and the signal πt πt

function is given by

⎧ +1 se ω > 0; ⎪   sgn ( ω ) = ⎨0, se ω = 0 and ⎪ −1, se ω < 0. ⎩

(21)

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In this way, based on the above data, the Hilbert transform is normally implemented through the application of the inversed Fourier transform on the result of the multiplication of − j.sgn(ω) by the Fourier transform of   f (t ) , in other words, ^

f ( t ) = F −1 (− j.sgn ( ω ) F ( ω )).

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(22)

One important characteristic of the Hilbert transform is its capacity to detect

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3

cr

edges well, even in signals with noise.

Radial active contours method: pSnakes

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The pSnakes radial active contours method (Alexandria et al., 2009) makes use of polar coordinates (r , θ ) where

is the radius and θ is the angle that represents the

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image pixels in the ultrasound mode. The origin of the reference axis is located at the

probe is located.

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point of divergence from the ultrasound beams, i.e., at the point where the ultrasonic

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Systems based on the pSnakes method might be embedded in the ultrasound

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equipment itself, filtering and segmenting the image elements. In the case of echocardiography, the segmentation of the left ventricle is the object of interest, since its measurement readings, in real time, provide important information for the doctor’s diagnosis. This new method might also be applied to images with Cartesian coordinates, and ultrasound images using a linear probe. Besides ultrasonic images, other images such as those from sonar and radar, which are also naturally represented in polar coordinates, can be segmented using the pSnakes method. The size of an input digital image such as I r ,θ , in polar coordinates, in a matrix form, is represented by the quantity of lines   R and columns   T , data which define the resolution of the image   R × T . The lines correspond to the angles of the ultrasonic beams and the columns to the rays, i.e., radial distances. Thus, the input image is denoted as a

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matrix I r ,θ .

3.1 pSnakes definition

used to segment objects from digital images, and is defined by

cr

⎧⎪ [0,1] → R 2 ⎨ ⎩⎪ s → c ( s ) = ⎡⎣ (r1 ( s ) ,θ ( s ));(r2 ( s ) ,θ ( s )) ⎤⎦ ,

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The active contours, pSnakes, is a radial active contour method which can be

(23)

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where (r1 , θ ) and (r2 , θ ) are polar coordinates of the control points (node) of the polar

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active contour (pSnake). It should be pointed out that r1 , r2 ⊂ [ 0, rmax ]θ ( s ) ⊂ [θ min , θ max ] , where rmax is the ray with the greatest range of the ultrasonic beams; θ min and θ max are the

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limit of the angles for the radial coordinates and the limit of the deflection angles of the beam. However, in this definition, only two points (r1 , θ ) and (r2 , θ ) are considered. This has two nodes in the positions r1 and r2 ,

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d

implies that each angular coordinate

denominated twin nodes. Dual radial snakes is a special case when θ min = 0 and

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θ max = 2π . An illustration showing the main geometric differences among snakes, radial snakes, pSnakes and dual radial snakes is presented in Figure 1. The pSnakes method uses a system of polar coordinates, defined in Figure 2,

which are used in the location of the control points (node or spline   s ), where r (radius) is the distance from a control point to the referred point   O , while   θ refers to the angle between the horizontal axis (θ ≥ 0) and the segment formed by the node and the point O . The pSnakes method is demonstrated in a flowchart (Figure 3). Initially the image to

be segmented is acquired. The image may have been converted from Cartesian coordinates to polar coordinates, may be a synthetic image or the image may have been obtained directly from the echocardiograph in polar coordinates. From this input image

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the initial contour is defined manually or automatically. Then, external and internal energies are calculated. Next, new nodes are added and deleted according to the predetermined criteria. The total energy from the present contour, made up of the internal

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and external energy, is calculated. After this calculation, new positions for the nodes along the edges are sought to minimize the total energy of the contour using specific

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optimization algorithms. If a stopping criterion is reached, the method comes to the end of its application. Otherwise, a new iteration is carried out. A number of iterations or the

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total acceptable minimum energy are commonly used as stop criteria.

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The dynamic of the active contour methods, in general, including the radial ones, such as pSnakes is based on the minimization of the total energy, which is composed of

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the internal and external energies.

d

3.2 Internal energies

te

Energy is the main descriptive characteristic of the snake and the internal energies depend on the shape and the location of the active contour points and, in

Ac ce p

general, are related to the curvature and contour continuity. The polar curvature energy Ecurvr ,θ is calculated by [20]

Ecurvr ,θ = ⎡⎣( rθ − rθ +1 ) − ( rθ +1 − rθ + 2 ) ⎤⎦ , 2

(24)

where rθ is the radial position of one given node along the beam with angle   θ , rθ +1 is the radial position of another coordinate adjacent node   θ +1 and rθ + 2 is the radial position for the following node of θ +2 . Therefore, when the contour tends to a straight line, this energy tends to zero. In addition, when there is one node with a radial distance greater than its neighbor (left or right), the curvature energy value tends to increase in this region.

Page 17 of 61

The polar continuity energy Econtr ,θ is calculated by [20]

Econtr ,θ = ( rθ − rθ +1 ) . 2

(25)

The Equations 24 and 25 are similar to the Equations 17 and 18, defined by Chen

ip t

et al. [20], respectively. The difference is in the origins of the polar coordinates, for

example the origin of the radial snakes point of polar axis is in the region to be targeted,

cr

whereas in the pSnakes this point is usually outside the region. Equation 24 is based on

us

the second derivative of the ray in relation to the angle   θ . In Equation 25, the continuity energy is calculated based on the difference from the node radial coordinate

an

rθ and its neighbor on the right, rθ +1 . These equations are chosen as the definitions of

M

internal energies as they are used in the literature.

3.3 Hilbert transform as external force in radial snakes

d

First the 1D Hilbert transform must be applied along the radial beams, which are

te

the image rays represented in polar coordinates, after which it should be normalized and

Ac ce p

its absolute value can then be used as the external energy. Thus, the external energy in radial active contours is given by ^

Eext ( r ,θ ) = f ( r ) ,

(26)

that might be normalized by the expression ^

n Eext ( r ,θ ) = 1 −

f (r )

⎛ ^ ⎞ max ⎜ f (r ) ⎟ ⎝ ⎠

,

(27)

or ri ∈ [0, rmax ] , where rmax is the greatest ray reached by the beam in which the external ^

energy is being calculated and f is the Hilbert transform.

Page 18 of 61

On the assumption that the divergence point of the beam in radial snakes is within the edges of the image to be segmented, the Hilbert transform tends to hit negative values close to the rise edge and positive values after the fall edge. Therefore,

ip t

the following expression may be used, rejecting the values in the region which are

^

⎞ ⎟ ⎠

, se f ( r ) < 0 and

(28)

us

^ ⎧ f (r ) ⎪ ⎪1 − ^ ⎪ n Eexr ( r , θ ) ⎨ max ⎛⎜ f (r1 ) ⎪ ⎝ ⎪ ^ ⎪⎩ 0, se f

cr

outside the object contour:

( r1 ) ≥ 0.

an

The module (or the absolute value) of the normalized Hilbert transform from the

M

pixels intensity along the beams is denominated as the Hilbertian field or Hilbertian energy and is used as external energy in the radial snake in this work. Thus, the external

d

Hilbertian energy is simply defined as the module (or the absolute value) of the

te

normalized Hilbert transform applied along the beam signal, that is, along the ray. One typical case of Hilbertian energy behavior is illustrated in Figure 4. The

Ac ce p

valley shown in Figure 4(b) corresponds to the object edge shown in Figure 4(a). According to Figure 4, the field intensity increases to the left of the mentioned valley, serving as guide to search for the minimum snake energy, corresponding to the desired segmentation. It means that there is a border expansion along the radius. Thus, on using Equation 27 or 28 to calculate the external energy, it can be seen that a node (control point) of the radial snake, under the effect of this energy, tends to approach the edges of the object. The external Hilbertian energy of pSnakes is defined as the absolute value of the normalized Hilbert transform applied along the signal beam, i.e., along the ray, according to Equation 27.

Page 19 of 61

3.4 Total Energy The total energy from the pSnakes is given by pEtotalr ,θ = α Ecurvr ,θ + β Econtr ,θ + γ Eextr ,θ , ,

ip t

(29)

cr

where   α ,   β and   γ are real constants and   Eext is the Hilbertian external energy.

us

3.5 Application of the ACMs in echocardiogram images

This section presents the tests performed to assess the performance of the

an

pSnakes in comparison to traditional snakes with balloon (STB), GVF, radial snakes with derivative without mean filtering (SRDSFM), radial snakes with derivative mean filtering

(SRDCFM),

and

radial

snakes

with

Hilbert

energy

M

with

(SRH). The ACMs are assessed using measures of error, and computational cost in the

d

segmentation of echocardiogram images and compared to the gold standard obtained

te

by a medical expert. The tests were conducted using a computer with an Intel Core i5 processor, 3GB memory and Windows 7 system. The simulations were carried out using

Ac ce p

MATLAB, version 7.6 2008a.

The programs used in the evaluation of traditional and GVF snakes were

obtained at the site of the authors Xu and Prince (2011). The remaining programs were implemented as part of this work. The test images were obtained from the Clinical Prontocárdio

(Prontocárdio

echocardiograph,

model

Clinic)

Vivid 7 Pro

(Fortaleza, belonging

Ceará, to

the

Brazil) using

a

GE

Clinic. A

total

of

17 echocardiogram tests were used, making a total of 34 images, considering enddiastole

and end-systole

images. An echo-cardiographer (ECO1) recorded

the

17 examinations synchronized by electrocardiogram (ECG) in the position of the short axis of the left ventricular (LV).

Page 20 of 61

For the calculation of error measurement, the results of ECO1 were used as the gold standard. Thus, for each echocardiographic image, the LV is segmented using the methods

and variations mentioned,

the segmentation

error), RMSE (root

mean

errors were

square)

quantified

and ADPV (average

ip t

by RMS (radial maximum

and

deviation of pixel values). Also the influence of startup, performed manually, was taken

cr

into account for the results obtained. Thus, the initial contour is obtained at four levels of edge distance, starting from the initial contours used in the earlier step, but inside the LV

us

cavity. These levels are: (i) close to the edge, (ii) 30%, (iii) 50% and, finally, (iv) 70%

an

nearer the center of the initial contour used in the previous phase, as shown in Figure 5.

3.6 Evaluation based on difference measures

segmented

if

difference

the

segmentation is

measures used are

based

in on

an acceptable the

distance

te

tolerance range. Some

image defined

d

image and

M

The difference measures obtained between the reference (gold standard)

between pixels, position, shape, extracted features from objects such as area and

Ac ce p

perimeter, among others. In this work the measures of distance between pixels were used. For the calculation of error two contours are considered: (i) the contours of the reference image cr and (ii) the contour of the segmented image cs , as is shown in Figure 6. From a point Pi , located in the center of the object N equally spaced fi beams that cross both contours are drawn. Each fi beam crosses the contour cr at the point cri and also crosses the contour cs at the point csi . The distance di is computed as the difference between cri and csi in the radial direction. The measures used in this work are: radial maximum error, average of pixel values, and mean squared error.

Page 21 of 61

3.7 Radial maximum error (RMS) For the radial maximum error measures the maximum distance error between cr and cs , given by Davatzikos and Prince (1995) is considered to be   RMS = max ( di ) , i ∈ [1, N ] .

illustration

error measure that

reflects on

ip t

An

(30) the

axis

us

cr

echocardiograph image is shown in Figure 7.

RMS short

3.8 Average deviation of pixel values (ADPV)

  ADPV =

N

di

∑N. i =1

illustration

of error

measures that

correspond

(31) to ADPV

in

short

d

An

M

Barrett et al. (1980) and Pope et al. (1985) as

an

ADPV is calculated as the average distance between cr and cs , defined by

te

axis echocardiographic images is shown in Figure 8.

3.9 Root mean square error (RMSE)

Ac ce p

The RMSE is defined as the difference between the distances of cr and cs given

by Adam et al. (1987)   RMSE =

An

1 N d i2 ∑ . N i =1 N

illustration

of error

(32) measures that

reflect

the RMSE short

axis echocardiographic images is presented in Figure 9. These measures are used to obtain the results of comparative analysis of segmentation between the traditional balloon, GVF, radial with derivative, radial with Hilbert energy ACMs and pSnakes.

Page 22 of 61

4

Results and Discussion The results obtained are presented according to the sequence presented in the

methodology. The images used are called Case xy, where x varies from “A” up to

at the

end

of

systole. The

results are

presented step

by

according

to

us

4.1

step,

cr

the methodology presented.

ip t

“Q” (17 patients) and y can be “D” for images obtained at the end of diastole and “S”

Evaluation of ACMs in echocardiogram images for different initialization

an

curve distances

Initially, the ACMs were applied to 34 images from 17 tests and their error

evaluations

are repeated using

M

measures calculated, as shown in the description of the tests for this step. The different manual

initializations:

close

to

the

on

the

average and

standard

deviation

with

te

The consolidated results

d

edges, 30%, 50% and 70% nearer the center of the initial contour, see Figure 5.

initializations close to the edges, 30%, 50% and 70% of distance for error measures,

Ac ce p

RME, ADPV, RMSE are shown in Tables 1, 2 and 3, respectively. In the first column of Table 1 each ACM has been evaluated. In the second and third columns, the RME results for initialization close to the edges as average and standard deviation, respectively, are shown. In the fourth and fifth columns the results for the initialization of 30% distance as average and standard deviation, respectively, can be seen. The sixth, seventh, eighth and ninth columns show the results for the average and standard deviation, respectively considering the initialization of 50% distance, and, finally, initialization of 70% distance for the same statistical measures. The same analysis is used in Tables 2 and 3. The RME results were very similar to the ACMs evaluated with initialization close to the edges. When the initialization is far from the edges, the radial methods were

Page 23 of 61

more

efficient, especially

the

proposed

method

(pSnakes),

obtaining the

lowest RMS errors: 16 ± 3, 19 ± 7, 20 ± 6 and 21 ± 7 for initialization close to the edges; 30%, 50% and 70% of distance, respectively. After pSnakes, the best results were

ip t

obtained by SRH, SRDCFM and SRDSFM. The STB and GVF methods had higher error values with greater distances from the edge of the LV.

cr

For RMSE, STB snake has minor errors close to the edges, followed closely

by pSnakes and SRH, SRDCFM and GVF. The SRDSFM method presented the largest measures. A

sequence is

the initialization

obtained

of 50%

with

the

distance, the

initialization

performance

of STB

an

of 30% distance. For

similar

us

error

and GVF were reduced, with error values greater than for STB snake. The pSnakes

M

proposed method, the SRH and SRDCFM methods were more efficient for initialization far from the edges. The low performance of the GVF and STB snakes is associated to the

d

use of the Gaussian filtering and derivative or gradient because they are susceptible

te

to speckle noise, which is always present in echocardiographic images. The results for ADPV have the same tendency as those for RMSE. This is expected due to the

Ac ce p

similarities in the expressions of each error measures. The results of the average processing time to segment each echocardiogram

image using all snakes tested here are presented in Table 4. There were 140 iterations for each method. The execution of each case was repeated 100 times. The results in Table 4  were expected, since GVF and STB snakes use 2D space to find the minimum value of total energy of the active contour, and this space is larger than the radial snakes space (1D). Also, GVF has a high processing time to calculate the edge maps. These processing times can be improved by optimizing the algorithms used and by the use of faster computational tools such as C/C++ embedded in dedicated processors. The results presented in this section are justified due to an improved performance

Page 24 of 61

of the Hilbert energy in the presence of speckle noise. The GVF, STB, and SRDSFM and SRDCFM snakes are based on derivatives, gradients and Gaussian filters or their variations, and do not give the same performance as Hilbert energy when this type of

cr

5 Conclusions, contributions and future work

ip t

noise is present.

us

This work proposed a new radial active contour method, called pSnakes, and was based on 1D Hilbert transform, which can be used for echocardiogram image

an

segmentation of the left ventricle. The pSnakes method was presented and evaluated with short axis echocardiogram images. The pSnakes was compared to other methods, such

M

as: STB, GVF, SRDSFM, SRDCFM and SRH, considering statistical parameters of RMS, RMSE and ADPV to quantify the performance of each method.

d

The results for the RMS were similar when initialization was close to the edges of

te

the LV. For distances further away from the edge, the radial methods presented better

Ac ce p

results than the others, especially the pSnakes proposed in this work. For RMSE, the STB maintained good performance close to the edges, but for initialization of 50% distance from the LV, the STB and GVF snakes were not efficient. pSnakes, SRH and SRDCFM are more stable for this type of initialization (away from the edges). As the correlation between ECO1 and ACMs showed, pSnakes was the most accurate for all situations.

The use of the Hilbert transform to calculate the external energy of the proposed pSnakes method, and the radial snakes using correlation data, average and standard deviation, was shown to be very robust, fast and accurate for the LV segmentation of echocardiogram images. This is mainly due to the characteristics of Hilbert energy to reach a minimum energy at the edges of objects and undergo only a small increase of its

Page 25 of 61

value, on moving away from the edges. It also performed well when speckle noise was present. The results of short axis echocardiogram image segmentation using the pSnakes

ip t

method were very satisfactory where compared to manual segmentation carried out by a medical specialist, and better than the other active contour methods evaluated. Thus, the radial pSnakes method

proposed here can be used with confidence

cr

new

in echocardiogram image segmentations.

us

The main contributions of this work are: the use of the pSnakes method in

an

the image segmentation of the LV and the use of Hilbert energy to calculate the external force of the radial pSnakes method though Hilbert transform. A study of the automatic

M

initialization of the pSnakes method is a possible future work to be carried out as well as the application of long axis echocardiographic images (4 cavities). Beside this, a more

d

diverse dataset with ultrasound images from different devices and resolutions will be

Ac ce p

Acknowledgements

te

considered to prove the potential of our proposed method. 

The authors would like to offer their thanks to the Laboratório de Sistemas de

Computação - LESC from Departamento de Engenharia de Teleinformática and to the Hospital Universitário Walter Cantídio - HUWC, both at the Universidade Federal do Ceará, in Brazil, and also to Clínica Prontocárdio. Thanks also to LEM from the Instituto Federal de Educação with Prof. Dr. André Luiz Souza Araújo and Prof. Dr. Willys Aguiar.

Page 26 of 61

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ip t

Adam, D., Hareuveni, O., & Sideman, S. (1987). Semiautomated border tracking of cine

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Allili, M. S., & Ziou, D. (2008). Object tracking in videos using adaptive mixture

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Barrett, W. A., Clayton, P. D., & Warner, H. R. (1980). Determination of left

te

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Cohen, L. D. (1991). On active contour models and balloons. CVGIP: Image Understanding, 53(2): 211–218. Cohen, L. D., & Cohen, I. (1993). Finite-element methods for active contour models

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Dagher, I., & Tom, K. E. (2008). WaterBalloons: A hybrid watershed Balloon Snake

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Gemignani, V.; Faita, F.; Ghiadoni, L.; Poggianti, E.; Demi, M., "A System for RealTime Measurement of the Brachial Artery Diameter in B-Mode Ultrasound Images," Medical Imaging, IEEE Transactions on , vol.26, no.3, pp.393,404, March

ip t

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d

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FIGURE CAPTION

Figure 1: Differences among a) snakes, b) radial snakes, c) pSnakes and d) dual radial snakes.

ip t

Figure 2: Polar coordinate system for pSnakes. Figure 3: Functional flowchart for the pSnakes method.

cr

Figure 4: Hilbertian energy along a beam passing through a cavity - typical situation, a) beam passing through the object’s edge and b) Hilbertian energy along the beam.

us

Figure 5: Illustrations of manual initialization close to edge, distance of 30%, 50% and 70%.

, with

and the

representing the beam.

M

segmented image contours

an

between reference image contours

Figure 6: Distance measures

Figure 7: Illustrations of RME for error values equal to a) 7.9 b) 11.9 c) 21.8, and

d

d) 35.6.

d) 54.32.

te

Figure 8: Illustrations of ADPV for error values equal to a) 5.00 b) 10.40 c) 18.25 and

Ac ce p

Figure 9: Illustrations of RMSE for error values equal to a) 5.43 b) 10.13 c) 19.03 and d) 47.51.

Page 33 of 61

TABLE CAPTIONS

Table 1:

Average of error

measures

for

RMS obtained

from

tested parameter

configurations for all ACMs considering different distances for curve initialization. Average of error

measures

for

RMSE obtained

from tested parameter

ip t

Table 2:

configurations for all ACMs considering different distances for curve initialization. Average of error

measures

for

ADPV obtained

from tested parameter

cr

Table 3:

configurations for all ACMs considering different distances for curve initialization.

us

Table 4: Average results of processing time for each type of snake considering 140

Ac ce p

te

d

M

an

iterations.

Page 34 of 61

Table 1:

Average of error

measures

for

RMS obtained

from

tested parameter

configurations for all ACMs considering different distances for the initialization curves. RMS 30% distance

50% distance

16

8

18

7

GVF

22

8

23

6

SRDSFM

23

7

25

6

SRDCFM

19

6

19

6

SRH

19

7

20

psnake

16

3

19

39

10

us

STB

cr

deviation

deviation

deviation

Average

Average

Average

Standard

Standard

Standard

Standard Average

70% distance

ip t

Edge ACMs

deviation

57

13

11

51

15

29

8

32

7

24

8

27

6

6

20

8

24

7

7

20

6

21

7

Ac ce p

te

d

M

an

38

Page 35 of 61

Table 2:

Average of error

measures

for

RMSE obtained

from tested parameter

RMSE Edge

30% distance

50% distance

ACMs Standard Average

Standard Average

deviation

Standard

Average

deviation

us

deviation

70% distance

cr

Standard Average

ip t

configurations for all ACMs considering different distances for the initialization curves.

deviation

7.04

2.37

8.91

2.62

27.29

5.74

36.99

13.96

GVF

9.80

2.50

10.25

2.77

21.69

5.50

32.83

13.11

SRDSFM

12.84

3.46

14.05

3.50

17.56

4.14

20.55

3.73

SRDCFM

9.49

3.23

9.40

3.19

12.66

3.62

15.12

3.82

SRH

9.48

3.50

9.74

3.19

9.96

3.59

12.59

4.47

psnake

7.61

1.07

9.54

2.47

9.79

2.48

10.18

2.35

Ac ce p

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Page 36 of 61

Table 3:

Average of error

measures

for

ADPV obtained

from tested parameter

configurations for all ACMs considering different distances for the initialization curves. ADPV 30% distance

50% distance

2.07

7.65

2.44

26.67

GVF

8.08

2.46

8.54

2.71

19.77

SRDSFM

11.67

3.65

12.91

3.71

16.40

SRDCFM

8.06

3.05

8.05

3.11

SRH

7.97

3.14

8.24

psnake

6.53

1.44

8.30

             

17.80

5.30

30.00

14.93

4.48

19.50

4.08

11.21

3.70

13.53

4.12

3.01

8.57

3.47

11.10

4.59

2.48

8.54

2.45

8.80

2.38

M

d te

 

32.64

Ac ce p

 

5.69

deviation

us

5.83

an

STB

cr

deviation

deviation

deviation

Average

Average

Average

Standard

Standard

Standard

Standard Average

70% distance

ip t

Edge ACMs

         

Page 37 of 61

Table 4: Average results of processing time for each type of snake considering 140 iterations.

GVF

131.64

SRDSFM

0.56

SRDCFM

1.57

SRH

0.56

psnake

0.69

cr

1.79

te

d

M

an

STB

ip t

Average processing time [s]

us

ACMs

Ac ce p

 

Page 38 of 61

Highlights:  To propose a new method, called pSnakes, applied and evaluated in the segmentation of  the left ventricle from echocardiographic images;  To make a comparative study of methods based on active contour methods; 

ip t

To apply the Hilbert transform to calculate the external energy of the proposed  pSnakes method. 

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