Gallbladder shape extraction from ultrasound images using active contour models.

Computers in Biology and Medicine 43 (2013) 2238–2255

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Gallbladder shape extraction from ultrasound images using active contour models Marcin Ciecholewski n, Jakub Chochołowicz Institute of Computer Science, Jagiellonian University, ul. prof. Stanisława Łojasiewicza 6, 30-348 Kraków, Poland

art ic l e i nf o

a b s t r a c t

Article history: Received 9 May 2013 Accepted 8 October 2013

Gallbladder function is routinely assessed using ultrasonographic (USG) examinations. In clinical practice, doctors very often analyse the gallbladder shape when diagnosing selected disorders, e.g. if there are turns or folds of the gallbladder, so extracting its shape from USG images using supporting software can simplify a diagnosis that is often difficult to make. The paper describes two active contour models: the edge-based model and the region-based model making use of a morphological approach, both designed for extracting the gallbladder shape from USG images. The active contour models were applied to USG images without lesions and to those showing specific disease units, namely, anatomical changes like folds and turns of the gallbladder as well as polyps and gallstones. This paper also presents modifications of the edge-based model, such as the method for removing self-crossings and loops or the method of dampening the inflation force which moves nodes if they approach the edge being determined. The user is also able to add a fragment of the approximated edge beyond which neither active contour model will move if this edge is incomplete in the USG image. The modifications of the edge-based model presented here allow more precise results to be obtained when extracting the shape of the gallbladder from USG images than if the morphological model is used. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Edge-based active contour Region-based active contour Shape extraction Self-crossings Gallbladder Medical image analysis Ultrasonography

1. Introduction Ultrasonography (USG) is the primary method of examining the gallbladder. However, USG images are much harder to analyse than those acquired using computed tomography (CT) or magnetic resonance imaging (MRI). An example image of the gallbladder acquired by USG is shown in Fig. 1. It is obvious that the USG image has uneven background (Fig. 1(a)), while the edges of the gallbladder shape may be blurred or missing, which makes it very difficult to draw the contour correctly with the available algorithms and computer methods. This situation is shown in Fig. 1(b). Fig. 1(b) presents an example in which the active contour method has “pilled” beyond the analysed shape of the gallbladder in the part of the image where this edge is missing because it has merged with the black background of the image. Fig. 1(c) shows an example with the missing fragment of the gallbladder edge drawn manually. Of the many methods used to segregate shapes in medical images, active contour models are attracting increasing interest because they self-adapt to the analysed shapes, frequently very complex, found in images with uneven contrast, such as USG

n

Corresponding author. Tel.: þ 48 791366871. E-mail addresses: [email protected], [email protected] (M. Ciecholewski), [email protected] (J. Chochołowicz). 0010-4825/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compbiomed.2013.10.009

images. In general, current active contour models can be divided into two main classes: edge-based and region-based. Edge-based models evolve the contour in the direction of edges with steep gradients of pixel intensity using an edge detector function. These models usually have an edge-based stopping term and a term representing the inflation or deflation force. Edge-based models feature a parametric form of the active contour curve. Starting with the publication of Kass et al. [1], many studies were done on edge-based models, for example [2–5]. Unfortunately, the basic version of the edge-based model has significant drawbacks which very significantly restrict the ability to use it. These drawbacks are as follows: (a) It is difficult to accurately define the inflation/deflation force. If the inflation force is too weak, the active contour may be unable to pass through narrows present in the analysed shape. If, on the contrary, the force is too strong, the active contour may pass through weak edges of the analysed shape. The appropriate examples illustrating these situations are presented in Fig. 2(a) and (b). (b) It is necessary to solve the problem of how to control the number of nodes during subsequent iterations, if the initial contour lies far from the edges of the analysed shape and, during subsequent iterations, respectively, increases (an expanding contour) or decreases (a contracting contour) its area in order to fit itself to the edges looked for.

M. Ciecholewski, J. Chochołowicz / Computers in Biology and Medicine 43 (2013) 2238–2255

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Fig. 1. An example of USG image of the gallbladder. (a) A description of the background of a USG gallbladder image. (b) USG image with improved contrast after histogram normalisation transformation. A situation in which the active contour has “spilled” beyond the analysed shape in the region in which the edge merged with the black background of the USG image. (c) A manual approximation of a fragment of the gallbladder edge – the continuous line (green in the colour image in the electronic format of the publication). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

Fig. 2. Limitations of the basic version of the edge-based model which prevent approximating the gallbladder shape in example USG images. (a) Difficulties with defining the inflation force. The value of this force was too low and prohibited the active contour from passing through the narrowing present in the lower part of the gallbladder shape. (b) Difficulties with defining the inflation force. Even though the folded fragment of the gallbladder has been approximated, the excessive inflation force caused the contour to pass through weak edges found in the upper fragment of the analysed shape and end up beyond it. (c) Looping of the active contour as a result of the self-crossing of nodes. Iteration 210. The analysed shape of the gallbladder in a USG image has not been entirely approximated. (d) Iteration 300 with a looped contour. The analysed shape of the gallbladder in a USG image has not been entirely approximated. (e) A USG image in which the active contour looped after approximating the edges of the gallbladder. Iteration 400. (f) A USG image in which the active contour looped after approximating the edges of the gallbladder. Iteration 420.

(c) Self-crossings of the active contour may occur, resulting in single or multiple loops, particularly in noisy images. It should be noted that self-crossings can ruin the entire process of extracting the analysed shape from a single image or when the object is tracked in e.g. a sequence of images coming from a camera. Publications [6,7] describe the possibility of self-crossings and loops occurring in the edge-based active contour model and

an attempt at solving this problem. Paper [6] describes applying the active contour to segment blood vessels in two-dimensional MRI images. Unfortunately, publication [6] focuses on just one case in which the contour expands during subsequent iterations and new nodes are automatically added to approximate the analysed shape. In addition, one should also consider situations in which the local looping and self-crossings of the active contour may occur in a contracting contour. Publication [7] presents algorithms

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for detecting self-crossings and an algorithm for loop removal in an active contour used for tracking moving objects in images from a video camera. In publication [7], self-crossings are removed based on calculated angle values between neighbouring nodes, calculated slopes of consecutive segments and with the use of four-connected line interpolation. In publication [7], a certain constant number of nodes of the active contour approximating the moving object in a video image sequence is assumed, and the removed nodes are replaced in the most sparse regions. When segmenting shapes in medical images, which are frequently very complex, it is difficult to assume a constant number of nodes, so there is unfortunately no single solution for all possible applications. In the analysis of shapes in medical images, the initial contour may be made up of just a few nodes, and then the algorithm should gradually increase their number to approximate the shape as accurately as possible and should also remove surplus nodes. Self-crossings, and the resultant loops of the contour may appear during subsequent iterations when the contour has not yet reached the maximum number of nodes because the analysed shape has not been entirely approximated yet. This situation is illustrated by examples from Fig. 2(c) and (d). Self-crossings may also appear after the edges of the analysed shape have been found, when the number of nodes is close to the maximum. Such examples are shown in Fig. 2(e) and (f). In summary, the algorithm most convenient for a user is one which, at subsequent iterations, supports adding and removing nodes both for an expanding and a contracting contour, while simultaneously preventing the formation of self-crossings and loops. Region-based approaches use statistical data from sub-regions and find the optimum energy where the model fits the image best. One of the first region-based methods was the model proposed by Mumford–Shah [8], in which the image is approximated using a smooth function inside every region. In publication [9], in turn, Chan and Vese proposed an active contour method which approximates the image with a certain constant function in each region. Region-based active contour models approximate weak edges better than edge-based models and are less sensitive to the location of the initial contour. In addition, the contour does not self-cross or loop in them, as the topological changes of the contour are automatically managed. Some of the best known and used region-based active contour models assume that homogenous regions of interest occur in the analysed image [9,10]. However, this is not always true for medical images. In general, the regions in medical images are usually statistically non-homogenous. In the morphological approach proposed in paper [11], differential operators used in a standard partial differential equation (PDE) are replaced with morphological operators on a binary level set. This approach is stable and numerically efficient because it does not make use of floating point operations, and what is more, the contour distance function does not have to be estimated in it. In addition, in the morphological model the solution of PDEs is obtained using only the following operators: inf–sup. There are publications to be found in the literature dealing with analyses of organ shapes in USG images, including papers on extracting the shape of the gallbladder. In publication [12], the authors presented two methods for determining the gallbladder shape which they stated were successful in about 70% of the cases. The first method comprises 3 steps: 1. Binarization. 2. Filtering the binary image using a rank filter. 3. Determining the contour between areas of various greyscales. The second method of detecting edges [12] consists in a histogram analysis of individual sections in USG images. The methods proposed in publication [12] yield imprecise results if there are

large, clear lesions like gallstones or polyps having the form of bright areas on the background of a ‘dark’ shape of the gallbladder and are practically ineffectual if the stones and polyps are located right next to the edge of the gallbladder and are blurred on the white background. The study [13] presents the use of a regionbased level set active contour method taken from paper [14] to extract stones from USG images. Unfortunately, the paper [13] omits very important details. Neither does the study [13] contain any information on the number of USG images of the gallbladder used in the research, nor any assessment by a physician or physicians of the results of the applied methods. In addition, there is no example showing how the contour was initiated to extract gallstones using a level set active contour. Publication [15] proposes using the traditional active contour method with B splines for the semiautomatic segmentation of the gallbladder shape in USG images. However, publication [15] does not present experiments with a large number of USG images of the gallbladder, and it only analyses examples of images showing no lesions. In paper [15] the active contour is initiated close to the edge of the gallbladder, whereas the active contour may have a very small surface and be located in any fragment of the gallbladder shape. The purpose of this work is to apply two active contour models: the edge-based model and the region-based model using the morphological approach – to extract the gallbladder shape from USG images. Another purpose is to compare the results obtained by segmenting the gallbladder shape in USG images using the two active contour models with results obtained in paper [12] using other methods. This work reuses the set of USG images used in publication [12]. This set has been expanded to 800 USG images, more than in the previous research described in publications [16,17]. Computer methods used in practice to support physicians' work must be stable and tested on a large database of images, containing various clinical cases. This is why this publication presents significant improvements to the edge-based model, which eliminate the limitations of its basic version listed in items (a), (b) and (c) in Section 1 and shown in Fig. 2. This publication describes a universal solution which automatically removes selfcrossings and loops of both inflating and deflating contours. What is more, apart from the automatic addition of nodes, excess nodes are removed. This publication also presents a solution for damping the inflation force when the contour is close to the edge of the analysed shape, thus blocking nodes from moving beyond the edge searched for. In this publication the authors have also made it possible to eliminate the situation shown in Fig. 1(b) by manually approximating the missing edge, as presented in Fig. 1(c), for both active contour models: the edge-based one and the region-based one using the morphological approach. This paper is structured as follows. Section 2 presents the methods used to extract the shape of the gallbladder from USG images. Section 3 presents the completed experiments and the obtained results of research on extracting the gallbladder shape from USG images. Sections 4 and 5 present the research currently in progress and the summary.

2. Extracting the gallbladder contour in USG images The overview of the proposed methods is presented in a diagram in Fig. 3. At the first stage, USG images of the gallbladder are pre-processed using the histogram normalisation transformation and the Gaussian filter smoothing. This pre-processing is aimed at improving the contrast of USG gallbladder images and removing noise from them. The next action is to start the initial contour, and one is also able to manually approximate a fragment of the edge(s) of the gallbladder in regions in which it/they is/are not found (the edges merge with the dark background of the USG image). Then, the gallbladder edge is determined using one of the


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presented below in this section. To simplify the calculations, the mass of the nodal point is set to zero (μi ¼ μ ¼ 0). In addition, the finite difference method can be used to approximate the derivative of v_ i ¼ ðvi ðt þ ΔtÞ vi ðtÞÞ=Δt, where Δt is a finite time step. The iterative equation which can be used to update the location of nodes after each iteration takes the following form: vi ðt þ ΔtÞ ¼ vi ðtÞ

Δt tensile ðαF i ðtÞ þ βF flexural ðtÞ F external ðtÞ F inflation ðtÞÞ i i i γ ð2Þ

F tensile ðtÞ i

represents the force counteracting the stretching of the contour in the nodal point with the index i at the moment t and is given by the following formula: F tensile ðtÞ ¼ 2vi ðtÞ vi 1 ðtÞ vi þ 1 ðtÞ i

ð3Þ

ðtÞ F flexural i

represents the force counteracting the contour bending and is expressed by the following formula: tensile F flexural ðtÞ ¼ 2F tensile ðtÞ F tensile i i 1 ðtÞ þF i þ 1 ðtÞ i

ð4Þ

ðtÞ F external i

Fig. 3. The general diagram of the proposed method for extracting the gallbladder shape from USG images.

two active contour methods: the edge-based model or the morphological model. The last operation which supports the process of extracting the gallbladder shape from USG images is running the convex hull algorithm to smoothen the identified edge of the gallbladder in USG images. 2.1. Histogram normalization transformation and Gaussian filtering A histogram is a one-dimensional statistical function which allows calculating the number of pixels having specific brightness values (grey levels in USG images). The histogram normalization transformation makes it possible to raise the contrast of images if the values of image brightness do not cover the entire range of possible values (see e.g. Fig. 1(a)). Both the histogram normalization transformation and Gaussian filtration are basic operations well known in the literature on digital image processing [18]. 2.2. Edge-based active contour model The edge-based model represents a deformable curve which, at subsequent iterations, can change its shape fitting itself to image elements such as edges and borders. In order to adjust the model to the data coming from the image and at the same time keep a specified smoothness of the contour, certain forces act on every node of the contour. These forces enable the contour to fit the object in the image in a way which minimises the total value of the energy of the moving contour. This energy depends on both the contour shape and the data coming from the image (i.e. the brightness function of the digital image Iðx; yÞ) and is defined by terms for the internal and external energy [1,19]. The basic equation used in this publication to deform the discrete model has the following form:

μi v€ i þ γ v_ i þ αF tensile þ βF flexural ¼ F external þ F inflation i i i i

ð1Þ

where fvi ðtÞ ¼ ðxi ðtÞ; yi ðtÞÞgi ¼ 0; 1; …; N 1 are the nodal points of the active contour. Parameters v_ and v€ represent the first and the second derivative v calculated in relation to the variable t. Parameters μ; γ represent, respectively, the mass and the damping coefficient, α and β are the weighing factors, while F are the forces

represents a force acting on the contour from the outside, which means that it is the attraction force of the image, which is caused by e.g. the brightness of the image. It takes such values that the active contour moves in the direction of these fragments of the image in which the gradient of the image brightness function is high. It is represented by the following relationship: F external ðtÞ ¼ ∇Pðxi ðtÞ; yi ðtÞÞ i

ð5Þ

The function Pðx; yÞ is defined so that it would reach minima in those points of the image in which object edges are found, i.e. in those points in which the gradient of the brightness function reaches high values. The function Pðx; yÞ may be defined as follows: Pðx; yÞ ¼ c J ∇ðGs nIðx; yÞÞ J . The expression Gs nIðx; yÞ represents a convolution of the brightness function with the smoothing filter, e.g. a Gaussian one, where the parameter s describes the degree of smoothing. F inflation ðtÞ is the inflation force which allows the i contour to be initiated far from the edge of the analysed object present in the image. It has the following form: F inflation ðtÞ ¼ FðI s ðxi ; yi ÞÞni ðtÞ i

ð6Þ

where ni(t) is a unit vector oriented in the direction normal to the contour at the node with the index i. The function FðIðx; yÞÞ is a binary function given by the following formula: ( þ1 if Iðx; yÞ Z T Fðx; yÞ ¼ ð7Þ 1 if Iðx; yÞ o T This links the inflation force with the data contained in the analysed image. The value T represents the brightness threshold set by the user. 2.2.1. Computer implementation of the edge-based active contour model The mathematical equations defining the movement of nodes of an edge-based model presented in the previous section are not sufficient for the active contour to work correctly. Iterative equations do not have a mechanism controlling the inflation force in a way that would simultaneously enable nodes to pass through the narrowing in the image and prohibit it from passing through weak edges if nodes are in the vicinity of the edge. What is more, iterative equations defining the moving of nodes do not assume that new nodes are automatically added and that superfluous nodes are deleted to approximate the edge more precisely. Neither do iterative equations help prevent the looping of the active contour. Fig. 2 contains examples illustrating the listed

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shortcomings of the edge-based active contour model. This subsection presents solutions for eliminating its limitations. The most important force acting on the contour is the inflation force. It causes the active contour to expand or shrink in places in which the searched-for edge of the analysed shape is present in the digital image. In the computer implementation, several parameters associated with this force have been added. If the values of these parameters are not right, they can frequently lead to the active contour not finding the edge of the analysed shape (or it can even fail to start moving from its initial position). In the edgebased model, the expanding (or shrinking) force acts along a vector normal to the curve. In the discrete case, this is a vector perpendicular to the vector expressed by the difference between nodes iþ1 and i 1. The value of the inflation force and also its direction are calculated using several parameters set by the user, who defines the following parameters: Threshold (T): defines the brightness threshold T. Bright to dark/Dark to bright: the selection of the value of the brightness and background function for nodes pushed by the inflation force. Pixels with a brightness greater than the value T are considered bright, those with a brightness below the T – dark. Damping factor (DF): the factor of inflation force damping Fiinflation from Eq. (2). Inflation reversals (IR): the number of iterations which identifies the number of ‘reversals’ after which the value of the inflation force is dampened. Reversal history (RHi): the number of iterations used to count the ‘reversals’ for the vi node of the contour. In USG images of the gallbladder, the active contour is positioned within the area of the gallbladder shape consisting of ‘dark’ pixels and is to move towards ‘bright’ pixels defined by the brightness threshold T. In the experiments conducted on USG gallbladder images, the active contour increases its area by fitting itself to the edge of the analysed gallbladder shape. Examples of the initialisation and the last iteration of the active contour are shown in Fig. 5 (a) and (b), respectively. An active contour can initially be very small in area, e.g. consist of at least three nodes and lie in any fragment of the gallbladder shape in images showing no lesions or those in which the gallbladder is turned or folded. If USG images of the gallbladder show stones and polyps, then the initial contour should be initiated outside the lesion(s) present. In the computer implementation, Eq. (2) was modified to control the damping of the inflation force and has the following form: vi þ 1 ðtÞ ¼ vi ðtÞ ðα^ F tensile ðtÞ þ β^ F flexural ðtÞ i i ðtÞ τðiÞF inflation ðtÞÞ ηF external i i

ð8Þ

The parameter τðiÞ controls the damping of the inflation force for the node with the index i. This parameter is calculated during the execution of Algorithm 1. For every nodal point vi, the Algorithm 1 is executed. Algorithm 1 is to maintain a high inflation force for every node: until the node reaches the edge searched for; in regions in which there are narrows. If the threshold T has been exceeded for the first time, i.e. when the node ends up in a place brighter than T, then the normal vector is reversed and makes the node return to the neighbourhood of the place where it was before. If the node moves like this several times (more than specified by the inflation reversals (IR) parameter), then its inflation force Fiinflation will be reduced by the

damping factor (DF) parameter. This way, the nodes are effectively moved to the neighbourhood of the edge looked for. If a deflating contour is used, the reverse condition is checked, i.e., Iðvi Þ o T. If the user has drawn a fragment of the approximated edge, the distance between the active contour nodes and points approximating the edge fragment (which belong to set V^ in Algorithm 1) is checked, and if the distance is equal to 1 or smaller, the inflation force on these nodes is reduced using the DF parameter, so the further movement of these nodes is stopped. In Fig. 5(a), a fragment of the edge which merges with the dark background of the USG image is approximated with a continuous line. The edgebased model stops after reaching this manually drawn line, as shown in Fig. 5(b) for the last iteration. The purpose of Algorithm 1 is also to automatically add new nodes, remove superfluous ones and prevent the local looping of the active contour, i.e. a situation in which two adjacent sections (as identified by the index i) which are parts of the polyline forming the active contour intersect. Constants Dmin, Dmax, θmin, θmax are defined by the user. These provide local control over the moving active contour. For example, let there be a condition that jvi vi 1 j 4Dmax . This means that the distance between two subsequent nodes vi and vi 1 exceeds the maximum distance. Algorithm 1 replaces the node vi with two new nodes ðv1i Þ ¼ 23 vi 1 þ 13 vi and ðv2i Þ ¼ 23 vi þ 13 vi þ 1 . As a result of this operation, the specific fragment of the active contour is made up of the following subsequent nodes: vi 1 , ðv1i Þ, ðv2i Þ and vi þ 1 . It should be noted that 1 ðv Þ vi 1 ¼ 2 vi 1 þ 1 vi vi 1 ¼ 1 jvi vi 1 j: i 3 3 3 Algorithm 1. Modified edge-based model. 1:

Input: V – a set of N nodes; V^ – a set of K nodes defined by the user, approximating gallbladder edge; and T – a set containing the values of inflation force damping parameters for the appropriate nodes from the set V, initialized with a constant value of τ for each nodal point; and α^ , β^ , η, T, IR, init

DF, Dmin, Dmax, θmin, θmax – param. set by the user; and I – the brightness function 2: Output: V – a set of N modified nodes; T a set containing modified values of inflation force damping parameters 3: for i ¼ 1-N do 4: NodePosition←0 5: n←NormalVector (vi þ 1 vi 1 ) 6: if Iðvi Þ 4T 7: n← n {vector reversal} 8: Store the reversal for node vi 9: else 10: Store the lack of a reversal for node vi 11: end if 12: if RH i 4 IR then 13: τi ←DF nτi 14: end if ^ Jv v J 15: d← i

i1

16: 17: 18: 19:

θi ←∠ðvi 1 vi ; vi þ 1 vi Þ if (d 4 Dmax and θ 4 θmin ) or (d 4Dmin and θ 4 θmax ) do

20: 21: 22: 23: 24: 25:

Add node, 23 vi þ 13 vi þ 1 NodePosition←1 else if d o Dmin then Remove node, vi NodePosition←1 end if if V^ a ∅ then

26: 27:

Remove node vi Add node, 23 vi 1 þ 13 vi

for j ¼ 1-K do


28: 29: 30: 31: 32: 33: 34: 35: 36: 37: 38: 39: 40:

^ J v v^ J d← i j if d^ r 1 then

τi ←DF nτi end if end for end if

if NodePosition ¼ 0 OR d^ Z 1 then Calculate Fitensile Calculate Fiflexural Compute the value of the image gradient for the vi point and the Fiexternal Calc. the new coord. of the node vi (8) end if end for

Thus the distance between adjacent nodes has been reduced threefold. In contrast, if two nodes are too close to each other, one of them is deleted (Fig. 4(a) and (b)). In addition, Algorithm 1 calculates the angle θi between pairs of adjacent nodes: ðvi 1 ; vi Þ and ðvi ; vi þ 1 Þ. If this angle is too small, then even if the distance between the nodes is long and exceeds the permissible constant value of Dmax, no new node will be created. Examples from Fig. 4(c) and (d) demonstrate the elimination of self-crossings consisting in removing one node and adding another two. There, nodes are added if the distance between the two nodes considered exceeds Dmin, while the angle is big (bigger than the declared value of θmax). Fig. 5(a) and (b) shows the initialisation and the last iteration of an expanding active contour inside the gallbladder shape. Fig. 5(c) shows the change of the number of nodes during consecutive iterations, while Fig. 5(d) presents the measurement of the time it takes to execute Algorithm 1 at consecutive iterations. This time was measured using a PC with an Intel Core i7 2 GHz processor. The software is written in C# with the use of multithreading programming techniques and runs on Windows 7/8. Graph 5(d) indicates that 250 subsequent iterations were executed in 11.2 s, while the first physician taking part in the research needed 1 min and 7 s to manually draw the gallbladder contour, and the second needed 1 min and 25 s. The active contour parameters for example from Fig. 5 are shown in Table 1. 2.3. Region-based active contour model – morphological approach The morphological approach was first presented in the publication [11]. Assuming that the digital image is expressed by the following function I : R2 -R, v A R is a parameter and g(I) is a function of image influence, u is the function u : R þ R2 -R2 which makes it possible to determine the location of the curve at the moment t [11], the following equation is given: ∂u ∇u ¼ gðIÞj∇uj div þ υ þ ∇gðIÞ∇u ð9Þ ∂t j∇uj Differential operators in Eq. (9), which allow the active contour to evolve, are implemented using discrete morphological operators, while the solution of the PDE is approximated by a composition of these morphological operators on a binary level set. The right side of Eq. (9) is a sum total of three components: the balloon force term; the smoothing term; the attraction force term. Each of these components will be presented separately.

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2.3.1. The balloon force term The balloon force term is expressed by the following equation: ∂u ¼ gðIÞ υ j∇uj ∂t

ð10Þ

Component g(I) modifies the force moving the active contour towards the edge of the analysed shape. When g(I) is high, the location of the active contour is far away from the target (i.e. the fragment of the edge searched for) and the force that makes it possible for the active contour to move must be strong. On the contrary, if the value of g(I) falls, the active contour is getting closer to the target, i.e. the force causing it to move becomes unnecessary. The component g(I) from Eq. (10) can be presented in the discrete form with a single threshold T. If g(I) is greater than T, the location of the appropriate active contour point is updated in accordance with the current value of the shifting force, otherwise this location remains unchanged. Depending on the value of the parameter v, the remaining components (υ j∇uj) cause a dilatation and erosion. Taking the iteration n and the active contour location for this iteration as un : R2 -f0; 1g, and also accounting for the discretisation of the component g(I), Eq. (10) can be solved using morphological operators as follows: 8 n > < Dðu ðxÞÞ if gðIÞðxÞ 4 T and υ 4 0 n ð11Þ un þ 1 ðxÞ ¼ Eðu ðxÞÞ if gðIÞðxÞ 4 T and υ o 0 > : un ðxÞ otherwise where D and E denote, respectively, the discrete operations of dilatation and erosion. The structural element is a disk having the radius of 1 with eight neighbouring pixels. In the computer implementation, Eq. (11) may be expressed using intervals of grey levels in the analysed image, and then it takes the following form: 8 n > < Dd u ðxÞ if IðxÞ A ½I 0 ; I 1 and υ 4 0 n nþ1 u ðxÞ if IðxÞ A ½I 0 ; I 1 and υ o 0 E u ð12Þ ðxÞ ¼ d > : un ðxÞ otherwise The grey level interval ½I 0 ; I 1 is taken from the neighbourhood of the area of the moving contour for the current iteration n. 2.3.2. The smoothing term The smoothing or regularization term is expressed by the following equation: ∂u ∇u ¼ gðIÞj∇uj div ð13Þ ∂t j∇uj The solution of Eq. (13) can be presented using linear morphological operators. Let B represent a set of all sections in the space R2 with the length of 2 and the centre in the point ð0; 0Þ. Continuous linear morphological operators are defined as ðSI h uÞðxÞ ¼ supB A B inf ða;bÞ A x þ hB uða; bÞ and ðISh uÞðxÞ ¼ inf B A B supða;bÞ A x þ hB uða; bÞ where h A ½0; 1. These linear morphological operators can be used to obtain an approximate numerical solution of Eq. (13): ( ðSI d ○ISd un ÞðxÞ if gðIÞðxÞ 4 T nþ1 u ðxÞ ¼ ð14Þ otherwise un ðxÞ where SId and ISd are discrete versions of continuous linear morphological operators defined above. Graphic examples illustrating the operation of the composition of SI d ○ISd are presented in publication [11]. 2.3.3. The attraction force term The last part of Eq. (9): ∂u ¼ ∇gðIÞ∇u ∂t

ð15Þ

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Fig. 4. Adding and deleting nodes and removing loops and self-crossings of the contour based on the action of Algorithm 1. (a) Exceeding the Dmin distance of node v2. (b) Removing the node v2. (c) Exceeding the θmax angular value of the node v3, the assumption is satisfied that d 4Dmin . (d) The location of contour nodes in the next iteration iþ 1 if Algorithm 1 was not applied. (e) The location of nodes in the next iteration iþ1 after Algorithm 1 was applied. Removing the node v3 and adding two new nodes ðv13 Þ and ðv23 Þ. (f) Removing nodes v13 and v23 in iteration iþ 2 as their distance is lower than Dmin: J v2 ðv13 Þ J o Dmin and J v4 ðv23 Þ J o Dmin .

Fig. 5. Application of edge-based model (Algorithm 1) approximating the shape of the gallbladder in an example USG image free of lesions. A continuation of the example from Fig. 1. The fragment of the edge merging with the dark background of the image was manually approximated with a continuous line (green in the electronic version of this publication). (a) Initiating the contour composed of four nodes connected with sections inside the shape of the gallbladder. (b) The approximate shape of the gallbladder edge (iteration no. 250, the contour has 117 nodes). (c) Graph showing the change in the number of nodes at subsequent iterations. (d) A graph of the measured time in milliseconds for subsequent iterations. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

is related to the attraction by the image. The solution of this equation can be approximated by 8 > : un ðxÞ

if ∇un ðxÞ∇gðIÞðxÞ 4 0 if ∇un ðxÞ∇gðIÞðxÞ o 0 if ∇un ðxÞ∇gðIÞðxÞ ¼ 0

ð16Þ

The equation u represents the active contour in such a way that uðxÞ ¼ 1 for all points located inside the area which it delineates and uðxÞ ¼ 0 for all points located outside the active contour. Thus all x points satisfying the equation uðxÞ ¼ 1 for which there exists such a neighbouring point y that uðyÞ ¼ 0 constitute the nodes. It is easy to see that, with this definition, there is no risk of the contour looping.


2.3.4. Computer implementation of the morphological active contour model The morphological solution allowing the active contour to be moved is obtained by the composition of the three partial solutions (11) or (12) as well as (14) and (16) presented in Sections 2.3.1–2.3.3. Knowing the nth solution un, the solution un þ 1 is obtained by executing three steps: 8 n > < Dðu ðxÞÞ if jυjgðIÞðxÞ 4 T and υ 4 0 n n þ 1=3 u ðxÞ ¼ Eðu ðxÞÞ if jυjgðIÞðxÞ 4 T and υ o 0 > : un ðxÞ otherwise 8 1 if ∇un þ 1=3 ðxÞ∇gðIÞðxÞ 4 0 > < if ∇un þ 1=3 ðxÞ∇gðIÞðxÞ o 0 un þ 2=3 ðxÞ ¼ 0 > : n þ 1=3 ðxÞ if ∇un þ 1=3 ðxÞ∇gðIÞðxÞ ¼ 0 u ( ðSI d ○ISd un þ 2=3 ÞðxÞ if gðIÞðxÞ 4T un þ 1 ðxÞ ¼ ð17Þ otherwise un þ 2=3 ðxÞ The attraction force has been presented in three ways, one of which the user of the program selects (below referred to as ‘snake type’): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1. Edge border attractor, gðIÞ ¼ 1= 10 20 þ j∇Gs nIj. 2. Dark center line attractor, gðIÞ ¼ jGs nIj. 3. No attractor defined, gðIÞ ¼ 1, where Gs is a Gaussian operator with a sigma parameter. The value of the T variable is estimated using the normalised histogram value for the function g(I). This means that the user enters a number from the interval (referred to as ‘Balloon Threshold’), and for this number, such a T value is determined that: jx A Ω : gðIÞðxÞ Z Tj=jΩj ¼ p, where Ω represents the image domain. On the other hand, the threshold value from the interval p A ½0; 1, below which value the g(I) attraction force acts is called ‘Edge threshold’. In the case of Eq. (12), a parameter is used which defines the grey level interval radius ½I 0 ; I 1 inside which a force expanding the active contour acts (referred to as the ‘Interval Radius’). The middle of the grey level interval ½I 0 ; I 1 is calculated as the median of the grey levels computed for the image and the parameter p entered by the program user. The v variable from Eq. (17) is called ‘Snake balloon’ parameter: ‘1’ means expanding contour, ‘ 1’ shrinking contour and ‘0’ lack of expanding or shrinking force. In the morphological model, the approximate gallbladder contour is determined in USG images using Algorithm 2. Such parameters as: Snake Balloon, Interval Radius, Edge Threshold and Balloon Threshold are set by the user to ensure control over the moving active contour. Just as in Algorithm 1, the user can manually approximate a fragment of the edge to prevent the contour from “spilling” beyond the analysed shape in regions in which the edge merges with the background of the image. In Algorithm 2, the distance between the active contour nodes and points approximating the edge fragment (which points belong to the V^ set in Algorithm 2) is checked, and if this is distance equal to 1 or smaller, further moving of nodes will be stopped. Fig. 6(a) and (b) presents the initialisation and the last iteration of an expanding active contour inside the gallbladder shape. Fig. 6 (c) presents the change in the number of nodes at subsequent iterations. Algorithm 2. A morphological approach using the binary level-set model. 1:

Input: V – a set of N nodes; V^ – a set of K nodes defined by the user, approximating gallbladder edge; Snake balloon, Interval Radius, Edge Threshold, Snake type parameters set by

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the user; and I – the brightness function of the analysed image 2: Output: V – a set of N modified nodes 3: Initialisation of the binary level set using Eq. (17) based on the initial contour V 4: if V^ a ∅ then 5: for i ¼ 1-N do 6: for j ¼ 1-K do ^ J v v^ J 7: d← i

8: 9: 10:

11: 12: 13: 14: 15: 16:

17: 18: 19:

j

end for if d^ Z1 then Compute the dilatation operator in the 3 3 neighbourhood of point vi to update the binary level set model Update the contour using (17) end if end for else for i ¼ 1-N do Compute the dilatation operator in the 3 3 neighbourhood of point vi to update the binary level set model Update the contour using (17) end for end if

The active contour parameters for example from Fig. 6 are shown in Table 2. Unlike in the edge-based model, in the morphological model there is no distance between adjacent nodes, so the active contour is made up of connected nodes which are therefore very numerous, as is obvious in the graph from Fig. 6(e). The initial contour presented in Fig. 6(a) is identical and located in the same place as the initial contours for the edge-based model: Fig. 5(a). The initial values of time turn out to be similar for both methods: edge-based model from Fig. 5(d) and morphological one from Fig. 6(d). In the morphological model, it is necessary to initialise the binary level set and compute the starting binary values in the image. With the passing of subsequent iterations, it becomes obvious that calculating morphological operators is more efficient than computations in the edge-based model. This is due to the lack of floating point operations, and only executing inf–sup operations and dilatations, which are performed only in the vicinity of nodes and not in the entire image. 2.3.5. Applying the convex hull algorithm The convex hull algorithm has been used to eliminate concavities from contours produced by the edge-based and morphological models, as shown in the diagram in Fig. 3. This algorithm, described in the literature [20], makes it possible to find the smallest convex polygon represented by a set of points (i.e. a set of pixels located inside the determined contour) such that every point from this set lies on the edge of the polygon or inside it. The convex hull algorithm makes it possible to raise the precision of the approximated gallbladder shapes, particularly when a lesion lies right next to the edge of the gallbladder in the analysed USG image. Example of the use of the convex hull algorithm is presented in Fig. 8.

3. Completed experiments and research results Images from the Department of Image Diagnostics of the Regional Specialist Hospital in Gdańsk, Poland, were used in the research on USG image analysis. The USG images were acquired

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Table 1 Edge-based model – list of active contour parameters. α^

β^

η

τinit

List of parameters for example from Fig. 5 Parameter value 300

1

0.5

40

8

The best set of parameters Parameter value 500

1

0.5

40

8

Parameter name

No. of iterations

T

0.11 [0.1, 0.5]

Dmin

Dmax

θmin

θmax

DF

IR

4 pixels

9 pixels

211

301

0.8

3

4 pixels

9 pixels

211

301

0.8

3

Fig. 6. Application of the active contour – the morphological model (Algorithm 2) approximating the shape of the gallbladder in an example USG image free of lesions. A continuation of the example from Fig. 1. The fragment of the edge merging with the dark background of the image was manually approximated with a continuous line (green in the electronic version of this publication). (a) Initiating the contour made up of 40 connected nodes inside the shape of the gallbladder. (b) The approximate shape of the gallbladder edge (iteration no. 145, the contour has 392 nodes). (c) Graph showing the change in the number of nodes at subsequent iterations. (d) A graph of the time measured in milliseconds for subsequent iterations. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

with a GE Healthcare Logiq C3 apparatus. The research was based on a set containing 800 grey-level USG images of the gallbladder, including

400 images without lesions; 120 images showing polyps; 150 images containing stones; 130 images where the gallbladder is folded or turned.

Each image has the resolution of 512 512 pixels and every pixel is represented by 8 bits.

3.1. Validation The precision of marking the approximate edge of a gallbladder in USG images was estimated by comparing the results obtained using active contour models with contours hand-drawn by two doctors specialising in radiology. During the measurements, two methodologies were employed which made it possible to assess the active contour methods used. First, measures of similarities of identified areas in USG images were used, including Dice's similarity coefficient.

Similarity measures, including Dice's similarity coefficient [21], are values that enable the similarity of sets to be compared. These sets can be defined as areas with specific pixel numbers in the analysed digital image. Let Lvaccon Z 2 be the fragment of the image obtained using the active contour method and Lvexpert Z 2 signify the image fragment extracted manually by a radiologist. It was assumed that jLvaccon j is the number of pixels found in the area delineated using the active contour method, while jLvexpert j is the number of pixels in the area drawn by the radiologist. The number of pixels found in the overlapping area is jLvaccon \ Lvexpert j. The measurements made can lead to an error – under or overestimating the number of pixels identified in the area enclosed by the contour. Dice's similarity coefficient is defined as follows: DSI ¼ 2 jLvaccon \ Lvexpert j=jLvaccon j þ jLvexpert j. The overlap fraction: OF ¼ jLvaccon \ Lvexpert j=jLvexpert j. The overlap value: OV ¼ jLvaccon \ Lvexpert j=jLvaccon [ Lvexpert j. The extra fraction: EF ¼ jLvaccon \ Lvexpert j=jLv expert j. If the values of DSI, OF and OV are close to 1 and the value of the EF ratio is close to 0, this means that the calculated contour is close to the one drawn manually by the expert. In this project, a second error measurement methodology based on the publication [22] was also used. This method is aimed at identifying the variability of location of two contours, i.e. the


Table 2 Morphological model – list of active contour parameters. Parameter name

No. of iterations

Snake type

Snake balloon

Interval radius

Edge Balloon threshold threshold

List of parameters for example from Fig. 6 Parameter 300 1 1 value

21.4

0

0

The best set of parameters Parameter 500 1 value

[18, 42]

[0, 0.3]

[0, 0.5]

1

one manually drawn by an expert involved in the research and the calculated active contour. To assess the error, the mean absolute distance (MAD) is computed between the contours drawn by two radiologists and the ones calculated using the two active contour methods. Two sets of contour points are given: a ¼ fa1 ; a2 ; …; an g and b ¼ fb1 ; b2 ; …; bm g. Every ai and bi point has specific coordinates (x, y) defining the location of the contour. The distance from the closest point (DCP) of contour b to the point ai is expressed as follows: dðai ; bÞ ¼ min1 o j o m ðdistanceðbj ai ÞÞ where the distance operator ( ) represents an Euclidean norm. The distance from the closest point is calculated for all points from sets a and b. The position error is expressed as the mean of absolute values of distances between two contours and is expressed as follows: ( ) 1 1 n 1 m ∑ dðai ; bÞ þ ∑ dða; bj Þ Ep ðb; aÞ ¼ 2 ni¼1 mj¼1

3.2. Selection of parameters used in active contour models Values of parameters used in the research on two active contour models, i.e. the edge-based and morphological models, have been set so that factors DSI, OF and OV would take maximum values, while the EF factor and the Ep position error – their minimum values. Table 1 shows the best set of parameters for the edge-based model and Table 2 – for the morphological model. The maximum number of iterations for both active contour models used and all the experiments conducted is 500. Tables 1 and 2, containing the best set of parameters for the edge-based and morphological models, show an interval of threshold values (threshold, edge threshold, balloon threshold) because it is impossible to set the one best threshold value for every image, particularly for pairs of USG gallbladder images in which edges of the analysed gallbladder shape merge with the dark background (pixels have grey levels close to black, i.e. with values close to 0); in the region where the gallbladder edges are found, the grey levels of pixels are highly variable (from low values to high values of grey levels). Once the user has initiated the starting contour, the active contour expands in subsequent iterations to approximate the gallbladder shape in USG images. In the edge-based model, a positive value of the inflation force (τ) is set (see Table 1), while in the morphological modes dilatation operations are executed after selecting the parameter of the ‘snake balloon’ ¼1 (see Table 2). In the morphological model, some improvement in finding the approximate gallbladder edge can be achieved by changing values within the interval [18, 42]. Data from Tables 1 and 2 justifies a claim that in the edge-based model there are significantly more parameters to be set (12) than in the morphological one (6).

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However, in the case of the edge-based model, it is enough to change the value of just one parameter – the brightness threshold – for subsequent images and this will improve the results of estimating the gallbladder contour, while the remaining 11 parameters stay the same. In the morphological model, the values of three parameters (Interval Radius, Edge threshold, Balloon threshold) should be changed to optimise the results of gallbladder shape extraction. The remaining parameters, such as the set number of iterations, the type of attraction function (snake type) and the choice of the expanding contour (snake balloon ¼1) do not change for subsequent USG images of the gallbladder that are analysed.

3.3. Gallbladder shape extraction At the first stage of analysing USG images of the gallbladder, the histogram normalisation transformation is executed. It can improve the contrast of images. The next step in image preprocessing is the Gaussian smoothing which eliminates noise from digital images. At the next step, the missing fragments of the gallbladder edge can be manually approximated if these gallbladder edge fragments merge with the dark background of the USG image. Then, the gallbladder shape is extracted using the active contour morphological model and edge-based model. Some examples of the differences between areas determined using: the basic version of the edge-based model (abbreviation: BME); the modified edge-based model without the capability of manually approximating the edge (abbreviation: ME); the modified morphological model without the capability of manually approximating the edge (abbreviation: MO); the modified edge-based model including the manual approximation of the edge (abbreviated: MEap); the morphological model with the capability of manually approximating the edge (abbreviated: MOap), have been presented in Figs. 7 and 8. In the presented examples from Figs. 7 and 8 the initial contour was located inside the gallbladder shape and its location was identical for the five models applied: BME, ME, MO, MEap, MOap. Fig. 7(b) shows a situation in which the BME model “spilled out” in a region where edges were poorly visible in the USG gallbladder image. In the case of the ME and MEap models from Fig. 7(c), dampening the inflation force prevented the contour from spilling beyond the analysed gallbladder shape. The MO and MOap morphological models whose operation is presented in Fig. 7(d) also approximated the gallbladder shape. The results obtained using the ME and MEap models as well as the MO and MOap models are identical for examples from Fig. 7 (c) and (d). Fig. 7(f) and (g), on the other hand, shows problems by defining the inflation force in the BME model. An insufficient inflation force meant that the active contour did not pass through the narrowing, as illustrated in Fig. 7(f). The opposite situation is presented in Fig. 7(g), because although the contour passed through the narrowing and approximated the fragment of the area of the USG gallbladder image which had not been approximated in example shown in 7(f), the excessive inflation force caused the contour to pass through the week edges in the upper part of the gallbladder and end up beyond the gallbladder. In the example from Fig. 7(h), the application of the inflation force damping mechanism made it possible to approximate the gallbladder shape, and models ME and MEap work in the same way here. The application of the morphological models MO and MOap also allowed the gallbladder shape to be approximated.

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Fig. 7. The use of models BME, ME, MO, MEap, MOap to identify the approximate gallbladder shape in examples of USG images. (a)–(d) A USG image with a gallstone. (e)–(i) A USG image showing a folded gallbladder. (a) The starting contour is initiated outside lesions. (b), (f), (g) The use of the BME model. (h) The application of ME and MEap models. (i) The use of MO and MOap models.

In the BME model, it was impossible to set the parameter of the inflation force in such a way that the contour would approximate the whole gallbladder shape, i.e. with the narrowing, the fold and the upper part of the shape where the edges are the weakest. In the example from Fig. 8(b), the BME model has “spilled out” in two regions in which the gallbladder edges merge with the dark background of the image and where weak edges are found. Also when the ME and MO models were used, the fragments of the border where no edge is present caused the contours to spill out, as shown in Fig. 8 (c) and (d). It was only the manual approximation of the edge that allowed the gallbladder shape to be correctly approximated, as illustrated in Fig. 8(e) and (f) for models MEap and MOap, respectively. The example from Fig. 8(h) shows a looping of the model while the analysed gallbladder shape in the USG image has not been wholly approximated yet, so the further evolution of the contour is stopped. Because the evolving, looped contour will not correctly approximate the edge of the gallbladder, the convex hull algorithm which removes contour concavities will not be applied.

Fig. 8(i) presents the application of models ME and MEap in which the gallbladder shape has been approximated without one polyp, while Fig. 8(j) shows the use of the MO and MOap models, in which the approximated gallbladder edge has two concavities in the area where two polyps are found. Using the convex hull algorithm for the last iteration eliminated the concavity of the contour and allowed the shape of the gallbladder to be approximated together with the polyp: this is demonstrated in Fig. 8(k) and (l). Compared to previous publications [16,17], the set of USG images has been extended to 800 various clinical cases, of which some 200 analysed USG images are cases in which the gallbladder edge merges with the background of the image as in examples from Figs. 1 and 7(a), and then basic active contour models which do not include the manual edge approximation spill beyond the analysed shape as shown in Figs. 1(b), 8(b) and (d). Figs. 7(b) and 8(b) show examples in which the excessive area occupied by the active contour is greater than the area of the


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Fig. 8. The use of models BME, ME, MO, MEap, MOap to identify the gallbladder contour in examples of USG images. (a), (g) Contour initialisation. (b), (h) The use of the BME model. (c) The application of the ME model. (d) The application of the MO model. (e) Using the MEap model in a USG image with approximated edge fragments marked with a continuous line (green in the electronic version of the publication). (f) The use of the MOap model for a USG image with approximated edge fragments marked with a continuous line. (i) The application of ME and MEap models. (j) The use of MO and MOap models. (k) and (l) Applying the convex hull algorithm to the appropriate contours from figures (i) and (j). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

2250


gallbladder in the USG image. Consequently, in the experiments carried out for this paper, MEap and MOap models with manual edge approximation were used and their results were compared with those of the ME and MO models without manual edge approximation. Unfortunately, the repeating situations in which single and multiple loops formed in the basic, edge-based active contour model (BME) made it impossible to approximate the gallbladder shape, so the BME model was not included in the measurements presented below in this publication. On the average, self-crossings and loops appeared in oneeighth of the analysed USG images, which translates into 100 cases for the 800 analysed USG images. In addition, changes of the parameters of the basic edge-based active contour model, such as determining a new maximum number of nodes, changing the value of the inflation force or establishing a new location of the initial contour, caused loops to appear for a given USG image in the basic model even though this model had behaved in a stable way for the parameters set earlier. What is more, it is technically difficult to make measurements if single or multiple internal loops form, particularly before the entire gallbladder shape has been approximated, and they increase in size at subsequent iterations. Tables 3–6 show results for contours extracted using the edgebased model and the morphological model compared to contours drawn by two physicians (Measurement 1 and Measurement 2). These tables contain specific statistical parameters like: the maximum value (max), the minimum value (min), the mean value (mean) and the standard deviation (sd) for: the calculated indices: DSI, OF, OV, EF, PE; the calculated measure of the position error. The position error is expressed in mm (labelled in tables as PEmm). Tables 3–5 present, respectively, the measurements of indices and of position errors for USG gallbladder images portraying no lesions and lesions (polyps, stones and of anomalous shape, such as folded or turned ones). Table 6, in turn, presents a comparison of measurements of contours drawn by two physicians. Table 7 summarizes the mean values and standard deviations of the areas occupied by the gallbladders expressed in pixels for the analysed set of 800 USG images. Table 7 also compares results of average DSI, OF, OV, EF and PE indices.

Fig. 9 presents mean values and standard deviations of DSI, OF, OV and EF parameters for the analysed set of 800 USG images. Measurement data presented in Tables 3–7 and Fig. 9 justify the claim that: The MEap and MOap methods produced better results of the analysed indices DSI, OF, OV, EF and a smaller position error than the ME and MO methods. The former methods (MEap and MOap) also achieved a high convergence with manual examinations of the gallbladder shape for the analysed set of 800 USG images. The average values of the indices DSI, OF, OV, EF vary within the following ranges: 80% and 90% (DSI), 75% and 84% (OF), 73% and 83% (OV), 1% and 12% (EF). Average values of the position error range from 1.1 mm to 2.5 mm. In the case of the applied ME and MO methods, the average values of indices DSI, OF, OV, EF vary within the ranges: 70% and 85% (DSI), 68% and 81% (OF), 66% and 80% (OV), 10% and 32% (EF) while the average values of the position error fall within the following range from 1.5 mm to 3.8 mm. Results presented in Table 7 indicate that there is no significant difference in the averages for the areas determined by the active contour methods applied and those delineated by two radiologists. The position error is the lowest for USG images of gallbladders containing polyps, and for the mean value in the first measurement it amounts to: 1.03 80.92 mm (for the MOap model) and 1.6781.01 mm (for the MEap model). However, the set of USG gallbladder images showing polyps is the least numerous (120 images) and the difference in the position error and other indices compared to stones (150 images) is insignificant. In comparison, publication [12] reports obtaining 46% of correctly detected gallbladder edges in 41 images showing stones and 70% in 11 images with polyps. When the applied methods are compared in the following pairs: (MEap, MOap) and (ME, MO) it can be stated that for all sets of USG images, better results of approximating the gallbladder edge were obtained using an edge-based model than a morphological one. However, the differences in identifying contours using these two methods are relatively minor. The obtained values of (DSI), (OF) and (OV) indices prove that the morphological model leads to a slightly greater

Table 3 Measurements of indices (DSI, OF, OV, EF) and of position errors (PE) expressed in millimetres for 400 USG gallbladder images showing no lesions. Parameter

DSI

OF

OV

EF

PE

ME: Measurement 1 Min 0.7441 Max 0.9854 Mean 0.8020 sd 0.1178

0.7007 0.9390 0.7704 0.1243

0.6954 0.9127 0.7682 0.1221

0.0000 0.3025 0.1938 0.0905

0.1478 4.7913 2.8670 1.8433

MO: Measurement 1 Min 0.7143 Max 0.9577 Mean 0.7804 sd 0.0988

0.6612 0.9090 0.7336 0.1045

0.6567 0.9023 0.7201 0.9778

0.0000 0.1805 0.1091 0.0405


0.7233 0.9453 0.8011 0.1340

0.7184 0.9127 0.7905 0.1271


0.6744 0.9132 0.7503 0.1121

0.6654 0.9102 0.7477 0.1005

DSI

OF

OV

EF

PE

MEap: Measurement 1 Min 0.7945 Max 0.9855 Mean 0.8671 sd 0.0725

0.7231 0.9522 0.8186 0.0827

0.7367 0.9024 0.8111 0.0817

0.0000 0.0736 0.0577 0.0394

0.1326 3.3791 2.3451 1.4189

0.2362 3.9208 2.0992 1.5880

MOap: Measurement 1 Min 0.7567 Max 0.9681 Mean 0.8329 sd 0.0645

0.6889 0.9147 0.7842 0.0724

0.6922 0.9131 0.7817 0.0694

0.0000 0.0345 0.0227 0.0097

0.1120 2.4408 1.2240 1.1235

0.0000 0.2567 0.1644 0.0831

0.2137 4.2067 2.6091 1.6078


0.7425 0.9662 0.8439 0.0854

0.7412 0.9130 0.8372 0.0826

0.0000 0.0562 0.0318 0.0312

0.2012 2.8215 1.3819 1.1622

0.0000 0.1579 0.0894 0.0458

0.3112 3.8670 1.8624 1.4091


0.6912 0.9256 0.8057 0.0745

0.6923 0.9207 0.8032 0.0728

0.0000 0.0225 0.0134 0.0112

0.1031 2.3270 1.1244 1.0588


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Table 4 Measurements of indices (DSI, OF, OV, EF) and of position errors (PE) expressed in millimetres for 120 images of gallbladders containing polyps. Parameter

DSI

OF

OV

EF

PE


0.7537 0.9167 0.8168 0.1004

0.7401 0.8823 0.8009 0.0977

0.0031 0.2248 0.1477 0.0645

0.3613 4.3672 2.5017 1.7903


0.6857 0.8605 0.7633 0.0977

0.6739 0.8416 0.7529 0.0938

0.0018 0.1567 0.1108 0.0433


0.7422 0.8745 0.7780 0.1033

0.7330 0.8685 0.7688 0.0985


0.6730 0.8453 0.7539 0.1027

0.6635 0.8370 0.7422 0.1013

DSI

OF

OV

EF

PE


0.7349 0.9632 0.8343 0.0588

0.7419 0.9651 0.8321 0.0572

0.0012 0.0541 0.0338 0.0245

0.4721 2.8921 1.6781 1.0132

0.3881 3.7144 1.4076 1.1963


0.7331 0.9473 0.8257 0.0489

0.7302 0.9426 0.8246 0.0411

0.0015 0.0253 0.0171 0.0127

0.1296 2.2572 1.0342 0.9227

0.0040 0.2526 0.1783 0.0771

0.4222 4.6675 2.8330 1.8603


0.7212 0.9418 0.8123 0.0643

0.7334 0.9423 0.8087 0.0612

0.0050 0.0723 0.0718 0.0283

0.5493 3.6327 2.2752 1.2372

0.0058 0.1733 0.1203 0.0530

0.4129 3.7984 1.6705 1.0976


0.7155 0.9226 0.8042 0.0553

0.7113 0.9103 0.8002 0.0537

0.0055 0.0305 0.0252 0.0196

0.2793 2.7252 1.4752 1.0318

Table 5 Measurements of indices (DSI, OF, OV, EF) and of position errors (PE) expressed in millimetres for: (a) 150 USG images of gallbladders containing stones and (b) 130 USG images of gallbladders that are folded or turned. Parameter

DSI

OF

OV

EF

PE

(a) ME: Measurement 1 Min 0.7793 Max 0.9772 Mean 0.8273 sd 0.1322

0.7403 0.9343 0.7907 0.1427

0.7374 0.9260 0.7844 0.1378

0.0032 0.3059 0.1588 0.0833

0.3522 4.5600 2.7708 1.8163

(a) MO: Measurement 1 Min 0.7538 Max 0.9728 Mean 0.7913 sd 0.1214

0.6708 0.8635 0.7533 0.1248

0.6632 0.8426 0.7441 0.1155

0.0012 0.1756 0.1246 0.0479

(a) ME: Measurement 2 Min 0.7704 Max 0.9731 Mean 0.7944 sd 0.1366

0.7308 0.9115 0.7664 0.1482

0.7245 0.9044 0.7522 0.1430

(a) MO: Measurement 2 Min 0.7363 Max 0.9690 Mean 0.7677 sd 0.1288

0.6696 0.8515 0.7229 0.1314

(b) ME: Measurement 1 Min 0.6702 Max 0.9723 Mean 0.7626 sd 0.2318

DSI

OF

OV

EF

PE

(a) MEap: Measurement 1 Min 0.8512 0.7221 Max 0.9821 0.9558 Mean 0.8532 0.8279 sd 0.0715 0.0692

0.7176 0.9474 0.8211 0.0633

0.0068 0.0627 0.0496 0.0311

0.5338 3.1675 1.5679 1.3233

0.3226 3.9550 1.6442 1.4390

(a) MOap: Measurement 1 Min 0.8193 0.7112 Max 0.9736 0.9276 Mean 0.8217 0.8083 sd 0.0624 0.0620

0.7022 0.9153 0.8067 0.0605

0.0022 0.0313 0.0167 0.0118

0.4534 2.6568 1.2483 1.0020

0.0033 0.3402 0.1820 0.0905

0.4027 4.9072 3.0058 1.9110

(a) MEap: Measurement 2 Min 0.8306 0.7089 Max 0.9744 0.9347 Mean 0.8324 0.8023 sd 0.0742 0.0728

0.7043 0.9306 0.7987 0.0678

0.0120 0.0945 0.0862 0.0523

0.7457 3.8847 2.0768 1.4267

0.6591 0.8402 0.7138 0.1277

0.0046 0.2107 0.1526 0.0522

0.3478 4.0221 1.8021 1.2032

(a) MOap: Measurement 2 Min 0.7968 0.7089 Max 0.9711 0.9033 Mean 0.7971 0.7828 sd 0.0654 0.0609

0.7007 0.8945 0.7803 0.0574

0.0087 0.0356 0.0217 0.0176

0.5437 3.0756 1.4884 1.0275

0.6314 0.8767 0.7329 0.2329

0.6225 0.8723 0.7218 0.2307

0.0031 0.4513 0.2434 0.1724

0.3613 6.6711 3.5406 3.1204

(b) MEap: Measurement 1 Min 0.7966 0.7116 Max 0.9450 0.9238 Mean 0.8178 0.7930 sd 0.0875 0.0822

0.7042 0.9101 0.7871 0.0803

0.0088 0.0827 0.0592 0.0450

0.7420 3.7081 2.2211 1.4022

(b) MO: Measurement 1 Min 0.6426 Max 0.9711 Mean 0.7342 sd 0.2215

0.6219 0.8605 0.7021 0.2263

0.6027 0.8416 0.6953 0.2142

0.0018 0.2735 0.1657 0.1156

0.4336 6.7683 2.7337 2.2751

(b) MOap: Measurement 1 Min 0.7540 0.6903 Max 0.9207 0.8705 Mean 0.7913 0.7698 sd 0.0770 0.0702

0.6817 0.8644 0.7514 0.0670

0.0041 0.0567 0.0220 0.0182

0.4540 3.2041 1.7280 1.0330

(b) ME: Measurement 2 Min 0.6515 Max 0.9628 Mean 0.7345 sd 0.2426

0.6140 0.8745 0.7056 0.2471

0.6051 0.8685 0.6945 0.2360

0.0040 0.5547 0.3231 0.1835

0.4222 7.4232 3.7314 3.4156

(b) MEap: Measurement 2 Min 0.7760 0.6921 Max 0.9322 0.8822 Mean 0.7924 0.7702 sd 0.0945 0.0906

0.6903 0.8725 0.7609 0.0829

0.0120 0.0945 0.1207 0.0680

0.7743 4.0225 2.5671 1.4251

(b) MO: Measurement 2 Min 0.6341 Max 0.9527 Mean 0.7038 sd 0.2249

0.6174 0.8453 0.6726 0.2237

0.5987 0.8370 0.6649 0.2116

0.0058 0.3146 0.1959 0.1176

0.5457 6.2178 2.9458 2.3852

(b) MOap: Measurement 2 Min 0.7423 0.6822 Max 0.9108 0.8613 Mean 0.7771 0.7405 sd 0.0800 0.0792

0.6702 0.8511 0.7343 0.0748

0.0057 0.0681 0.0492 0.0235

0.5803 3.5706 2.0055 1.0882

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Table 6 A comparison of measurements taken by two doctors: Expert1 and Expert2. Parameter

DSI

OF

OV

EF

PE

DSI

Expert1 vs Expert2: No lesions Min 0.8659 Max 0.9952 Mean 0.9337 sd 0.0542

0.8536 0.9873 0.9028 0.0643

0.8523 0.9641 0.8961 0.0614

0.0000 0.0562 0.0245 0.0267

0.1123 1.4126 0.7628 0.5834

Expert1 vs Expert2: Stones Min 0.8723 Max 0.9950 Mean 0.9065 sd 0.0618

0.8615 0.9727 0.8868 0.0546

0.8538 0.9724 0.8834 0.0527

0.0036 0.0611 0.0347 0.0223

0.2134 1.6486 0.8627 0.7461

OF

OV

EF

PE

Expert1 vs Expert2: Polyps Min 0.8845 0.8727 Max 0.9961 0.9754 Mean 0.9143 0.8865 sd 0.0536 0.0447

0.8649 0.9739 0.8832 0.0425

0.0007 0.0573 0.0281 0.0156

0.1832 1.5248 0.7314 0.6768

Expert1 vs Expert2: Turn / Fold Min 0.8073 0.7969 Max 0.9874 0.9756 Mean 0.8534 0.8315 sd 0.0742 0.0724

0.7905 0.9711 0.8234 0.0719

0.0012 0.0817 0.0525 0.0397

0.2868 2.1032 1.1376 0.9859

Table 7 Statistical comparisons of average values of indices DSI, SF, OV, EF, PE and areas occupied by the gallbladder expressed in pixels for the computerized delineation and radiologists' manual delineation for the analysed set of 800 USG images. Measurement 1

Measurement 2

Expert1 vs Expert2

Indice

ME

MO

MEap

MOap

ME

MO

MEap

MOap

DSI OF OV EF PE

0.8082 0.7777 0.7688 0.1859 2.9200

0.7773 0.7380 0.7281 0.1275 2.0702

0.8553 0.8184 0.8128 0.0500 1.9530

0.8278 0.7970 0.7911 0.0196 1.3086

0.7992 0.7627 0.7515 0.2119 3.0448

0.7671 0.7249 0.7171 0.1395 1.9711

0.8426 0.8071 0.8013 0.0776 2.0752

0.8154 0.7833 0.7795 0.0273 1.5233

0.9019 0.8769 0.8715 0.0349 0.8736

Areas occupied by the gallbladders: mean8 standard deviation Parameter

ME

MO

MEap

MOap

Expert1

Expert2

Mean sd

19 672 12 121

19 481 11 535

18 523 11 412

18 219 11 135

19 176 11 138

18 945 10 822

underestimation in extracting the gallbladder shape than the edge-based model. An analysis of the (EF) index and position error (PE) show that for all sets of USG images the edge-based model produced greater excess areas in the course of determining the gallbladder shape. The greatest observable error in measurements occurred for USG images showing gallbladder folds/turns during the second measurement. The position error for this measurement relative to the mean value amounts to: 2.56 81.42 mm for the MEap model, 2 81 mm for the MOap model, 3.37 83.41 mm for the ME model, 2.94 82.38 mm for the MO model. The mean values of the DSI, OF, OV, EF indices are as follows: 79%, 77%, 76%, 12% for the MEap model, 77%, 74%, 73%, 4% for the MOap model, 73%, 70%, 69%, 32% for the ME model and 70%, 67%, 66%, 19% for the MO model. In this measurement, also the minimum and maximum values of the above factors are the lowest for all active contour models. In comparison, the publication [12] reported 51% of correctly detected gallbladder edges in 23 images showing gallbladder folds/turns. The greatest differences between contours manually drawn by two physicians are observed in the case of folds/turns. The position error for this measurement relative to the mean value amounts to the following 1.13 mm. Mean values of indices DSI, OF, OV, EF are as follows: 85%, 83%, 82%, 5%. For the 400 USG images of gallbladders without lesions, a smaller position error is recorded in the second measurement. It is also clear that for this set of USG images, higher mean values of the SI, OF and OV indices and a lower value of EF were obtained in the second measurement. The opposite is true for every set of images containing polyps, gallstones or turns (400 in total, which is equal to the number of images free of lesions), because the first measurement yielded a lower mean value of the position error, higher mean values of SI, OF and OV, and a lower value of EF.

These results indicate a significant variability of manual shape extraction. For images with the same size of 512 512 pixels and for 500 iterations performed, the average duration of executing Algorithm 1 (the edge-based model) ranges from 24 to 60 s, while in the morphological model the average duration of executing Algorithm 2 ranges from 14 to 40 s, with the average difference between these two methods amounting to 20 810 s. It should be noted that these times were measured using a PC with an Intel Core i7 2 GHz processor. Let us stress that manually drawing the gallbladder edge is tedious and more time-consuming that using either active contour model. In comparison, the average time needed by physicians participating in the study to manually draw the gallbladder contour amounted to, respectively: (112, 165) seconds (Measurement 1) and (131, 213) seconds (Measurement 2).

4. Current research: extracting gallbladder lesions from USG images using selected active contour models The edge-based and the morphological active contour models, apart from extracting the gallbladder shape, also enable lesions like gallstones and polyps to be extracted from USG images. In order to extract a lesion, the fragment of the USG image in which it is found should be marked first. Then, the active contour will decrease its area in order to approximate the lesion or lesions. Two options of the use of active contour models are possible: First, the approximate gallbladder shape is determined using the active contour method, and then the heterogeneous background is eliminated by assigning the colour black to pixels located outside the gallbladder shape, Fig. 10(a). Subsequently,


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Fig. 9. Graphs of the mean value and the standard deviation based on measurements of four indices DSI, OF, OV, EF for active contour methods: ME, MO, MEap, MOap compared to contours drawn by two physicians (Expert1, Expert2).

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Fig. 10. Extracting lesions (gallstones) from fragments of USG images of the gallbladder. The marked initial area is illustrated by examples in the first column. The following models were applied: edge-based ME (second column) and morphological MO (third column).

a fragment of the USG image containing the lesion is marked, and after initiating one of the active contour models, the edges of lesions will be approximated. Fig. 10(b) and (c) shows an example application of the edge-based model (ME) and the morphological one (MO) with the same initial contour presented in Fig. 10(a) (the lesion is found right next to the gallbladder edge). Eliminating the uneven image background from the USG image makes it possible to extract lesions using an active contour regardless of their location (inside the gallbladder shape or right next to its edge). This approach, however, requires additional time for segmenting the gallbladder shape and eliminating the background of the USG image. The fragment of the image containing the lesion has been marked. Fig. 10(e) and (f) as well as 10(h) and (i) present the results of extracting lesions using the edge-based (ME) and the morphological model (MO). The location of initial contours for

both models is the same in example shown in 10(g) as in example shown in 10(a). For examples portrayed in Fig. 10(d)– (f), the location of the active contour was initiated in an area inside the gallbladder shape. Fig. 10(f) and (g) are examples showing the initialisation of active contours in an area containing a lesion located right next to the edge of the gallbladder in the USG image for, respectively, the edge-based and the morphological model. Clear differences in the operation of both active contour models are visible here. In the case of the edge-based model (Fig. 10(f)), the fragment of ‘white’ background adjoining the lesion causes this model to be unable to correctly approximate the lesion. Better results were obtained using the morphological model. Currently, the segmentation of lesions using active contour models is at a development phase. Various clinical cases of gallbladder


USG images must be taken into account. Work is in progress on optimising the operation of the edge-based model to approximate lesions and eliminate the situation presented in Fig. 10(f). 5. Summary and conclusions This publication presents a method of extracting the gallbladder shape from USG images using two active contour models: the edge-based model and the morphological model. The software under development now is a prototype and works independent of the ultrasound scanner recording the entire USG gallbladder examination. The most important functions implemented for gallbladder examinations are as follows: Processing films recorded during the ultrasound examination into a sequence of images, following which the gallbladder shape can be analysed in single images, as illustrated by the diagram shown in Fig. 3. Sequences of processed USG images can be combined into a short film. For the edge-based model, methods have been presented which significantly improve its operation, namely: A method which automatically supports adding and removing nodes both for an expanding and a contracting contour, while simultaneously preventing the formation of self-crossings and loops. A method allowing a high inflation force to be maintained for every node until the moment when the node reaches the edge searched for, and also in the areas of the image in which narrowings are present. A method allowing the automatic change of the location of nodes including a mechanism for damping the inflation force. If gallbladder edges are missing in fragments of the USG image, the edge can be manually approximated, and this approximation is taken into account when the gallbladder shape is approximated using the edge-based and the morphological model. Although the morphological model is more efficient, it underestimated the gallbladder area in all analysed sets of USG images slightly more, and also produced lower maximum values when approximating the gallbladder shape. What is most desirable in practice are automatic methods for segmenting the shape from digital images, but if the gallbladder shape is segmented from USG images with a uneven contrast in which the edges frequently merge with the dark background of the image, one cannot avoid using parameters or of user interaction to better to adjust the model better to the analysed situation. Software supporting the work of a physician must be stable and must permit obtaining repeatable, correct results for various clinical cases, and in this paper 800 clinical cases have been considered. A comparison with results described in publication [12] shows that in the approach proposed in this publication there are much fewer parameters to be set by the user

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and the process of extracting the shape is smoother. The approximation of gallbladder shapes in USG images produced better results and for more images than in the publication [12].

Conflict of interest statement None declared.

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