SECTION I

ULTRASOUND imaging has been employed to measure quantitative muscle changes in morphology, such as muscle thickness [1], [2], [3], [4], [5], [6], muscle pennation angle [7], [8], [9], [10], [11], [12], [13], [14], muscle fascicle length [2], [7], [15], [16], [17], and muscle cross-sectional area [7], [18], [19], [20], [21] during contractions.

Muscle thickness is an important determinant of the muscle condition in many aspects. For example, Campbell *et al.* found that muscle thickness measured with ultrasound might be an indicator of lean tissue wasting in multiple organ failure in the presence of edema [1]. Miyatani *et al.* used muscle thickness to estimate the muscle volume of the quadriceps femoris based on ultrasonography [3]. Ohata *et al.* employed muscle thickness to quantify the muscle strength of people with severe cerebral palsy [5]. English *et al.*validated that ultrasound was a reliable measure of muscle thickness in acute stroke patients for some anatomical sites [22]. However, muscle thickness is traditionally measured by an experienced operator, and the process is not only subjective, but also time consuming. Recently, some researchers have presented several computer-aided methods, which could be used to estimate muscle thickness. Zheng *et al.* used sonomyography to describe the real-time change of muscle thickness detected using B-mode ultrasound images during its dynamic contraction [23] and proposed to use it for prosthetic control [20], [21]. Koo *et al.* used cross-correlation to track the locations of aponeuroses and measure muscle thickness on the ultrasound images [24]. They placed two windows on the superficial and deep aponeuroses and muscle thickness was calculated as the distance between the horizontal center lines of two windows. However, the line-shaped feature of aponeuroses would expose the cross-correlation tracking to risk of mismatches caused by feature self-similarity. It could be solved by constraining the cross-correlation tracking along the vertical direction only, supposing that the muscle hardly slides with respect to the ultrasound probe. Yet another more general problem with cross-correlation tracking is the choice of the matching template. The larger the template is, the more robust while less precise the tracking would be and vice versa. Wong *et al.* proposed a computerized system for measuring the thickness of the transverse abdominus, internal oblique, and external oblique muscles [6]. They performed ensemble registration using a sequential quadratic programming approach based on a log-Rayleigh likelihood function and then identified regions of interest based on the medial border of the transverse abdominus. Although the muscle thickness was quantified automatically, the methods were based on measurements at several subjectively selected locations by nature, rather than the entire muscle boundary in ultrasound images. All of the aforementioned methods are limited to estimation of muscle thickness at one or several specific locations, assuming that the muscle thickness is constant longitudinally along the aponeuroses. But strictly speaking, superficial and deep aponeuroses are not straight lines, nor always parallel to each other, which leads to changes of muscle thickness at different longitudinal locations of aponeuroses.

In this paper, to compute muscle thickness of gastrocnemius (GM) in ultrasound images along the entire contours of aponeuroses, we proposed to extract the aponeuroses boundaries and calculate the distance between the lower boundary of superficial aponeurosis and upper boundary of deep aponeurosis as the muscle thickness.

SECTION II

The flowchart of the proposed strategy for muscle thickness measurement via ultrasonography is shown in Fig. 1. For each frame from the studied ultrasonography sequence, the first step was to acquire two seed points (one on the superficial aponeurosis, and the other on the deep aponeurosis). Then, contours of aponeuroses were extracted and muscle thickness was achieved by calculating the distance between contours of superficial and deep aponeuroses.

In this paper, seed points in each frame are two points located on superficial and deep aponeuroses respectively, which will be used to generate initial contours in the aponeuroses segmentation stage. Given changes of muscle's shape and position in a sequence, locations of two seed points were adaptively adjusted in a frame-by-frame manner. However, considering the large number of images in a sequence, selecting seed points for every frame by hand is tedious and subjective. So we adopt an image registration method, named mutual information-based free-form deformation (MI-FFD) [25] tracking, to automatically track seed points in the ultrasound image sequence, since it produces local registration fields that are smooth, continuous and establish one-to-one correspondences. In this method, the transformation function which describes the deformation between two successive images is determined by minimizing a MI-based objective function. The derivation and details of the MI-FFD method can be found in Appendix I.

In summary, for each trial, seed points of the first frame are determined manually as two arbitrary points inside the contours of aponeuroses. For two consecutive frames of each sequence, the previous frame is selected as reference, and then, the MI-FFD method is applied to track the seed points in the subsequent frames automatically.

Active contour models have been widely used in image segmentation with promising results since its introduction by Kass *et al.* [26]. In this paper, we adopt a region-based model, named the local and global intensity fitting (LGIF) model [27], to extract contours of aponeuroses, with the advantages of accuracy and robustness. In this model, an energy function is defined with a local intensity fitting term, which induces a local force to attract the contour and stops it at object boundaries, and an auxiliary global intensity fitting term, which drives the motion of the contour far away from object boundaries. The combination of these two forces can handle intensity inhomogeneity and allows for flexible initialization of the contours. Denoting $\phi$ as the level set function, the energy functional of the model is defined as
TeX Source
$$E\left(\phi \right) = \left({1 - \omega } \right)E_{{\rm local}} \left(\phi \right){\rm + }\omega E_{{\rm global}} \left(\phi \right){\rm + }\nu L\left(\phi \right){\rm + }\mu P\left(\phi \right) \eqno{\hbox{(1)}}$$ where $\omega$ is a positive constant which determines the weight of the local force $E_{{\rm local}}$ and the global force $E_{{\rm global}}$; $\nu > 0$ and $\mu \, {\rm > }\, 0$ are constants as the weights of the length term $L\left(\phi \right)$ and the level set regularization term $P\left(\phi \right)$, respectively.

In this study, contours of each frame are initialized as two circles (radii = 2 pixels) centered at two seed points, which have been tracked by mean of MI-FFD described previously. Then, entire contours of aponeuroses will be extracted by evolving the level set iteratively using the LGIF model. The details and implementation of the LGIF model are introduced in Appendix II.

After segmentation of the aponeuroses, muscle thickness at each point along the aponeuroses is then estimated by calculating the distance between two corresponding points on superficial and deep aponeuroses, as shown in Fig. 2(a). Traditionally, muscle thickness is measured as the distance between two points, selected by an experienced operator, on the lower edge of superficial aponeurosis and the upper edge of deep aponeurosis respectively, as shown in Fig. 2(b). Therefore, to make comparison to the manual method, we additionally estimate the mean muscle thickness for each frame by calculating the mean distance between contours of superficial and deep aponeuroses, though our method outputs a curve of muscle thickness along the aponeuroses for each frame of ultrasound image.

SECTION III

Five healthy male subjects (mean ± SD, age = 28.5 ± 0.6 years; body mass 67.3 ± 1.7 kg; height = 171.8 ± 0.6 cm) volunteered to participate in this study. No participant had a history of neuromuscular disorders, and all were aware of experimental purposes and procedures. The human subject ethical approval was obtained from the relevant committee in the authors’ institution, and informed consents were obtained from subjects prior to the experiment.

The testing position of the subject was in accordance with the Users Guide of a Norm dynamometer (Humac/Norm Testing and Rehabilitation System, Computer Sports Medicine, Inc., Stoughton, MA, USA). Each subject was required to put forth his maximal effort of isometric plantar flexion for a period of 3 s with verbal encouragement provided. The maximal voluntary contraction (MVC) was defined as the highest value of torque recorded during the entire isometric contraction. The MVC torque was then calculated by averaging the two recorded highest torque values from the two tests. The subject was instructed to generate isometric plantar-flexion movements in prone position with angle of knee joint at 180°. The torque was measured by the aforementioned dynamometer and muscle contracting in a range from 0% to 90% MVC is imaged by ultrasonography. 90% MVC is set as the highest value to avoid muscle fatigue.

A real-time B-mode ultrasonic scanner (EUB-8500, Hitachi Medical Corporation, Tokyo, Japan) with a 10-MHz electronic linear array probe (L53L, Hitachi Medical Corporation, Tokyo, Japan) was used to obtain ultrasound images of muscles. The long axis of the ultrasound probe was arranged parallel to the long axis of the GM and on its muscle belly. The ultrasound probe was fixed by a custom-designed foam container with fixing straps, and a very generous amount of ultrasound gel was applied to secure acoustic coupling between the probe and skin during muscle contractions, as shown in Fig. 3. The probe was adjusted to optimize the contrast of muscle fascicles in ultrasound images. Then, the B-mode ultrasound images were digitized by a video card (NI PCI-1411, National Instruments, Austin, TX, USA) at a rate of 25 frames/s for later analysis.

Five sequences of musculoskeletal ultrasound images were acquired and the number of frames per sequence was 100 images. All images were cropped to remove imaging tags and kept the image content only using home-made software. All data were processed offline using programs written in MATLAB R 2010b (Math Works, Natick, MA, USA) on a Windows-based computer with a P4 (3 GHz) processor and 8-GB memory.

SECTION IV

A representative example of the procedure for seed points tracking is shown in Fig. 4. Fig. 4(a) shows the current frame of one image sequence whose seed points has been determined, as represented by the two blue points in the figure. Seed points of the subsequent frame in the sequence are shown in Fig. 4(b), where red points are true seed points determined by MI-FFD method and blue points are at the same position as Fig. 4(a). It can be seen that the method can update positions of seed points back onto aponeuroses when the muscle contracted and location of aponeuroses changes accordingly.

In our numerical experiments for aponeuroses segmentation using the LGIF model, we empirically adopted parameters as follows: $\lambda _1 = \lambda_2 = 1.0, \nu = 0.001 \times 255 \times 255, \mu = 1, \omega = 0.01, \sigma = 3.0$, and time step $t = 0.23$, where $\lambda _1$ and $\lambda {}_2$ are parameters of the intrinsic energy function, and $\sigma$ is the standard deviation (SD) of the Gaussian kernel. Detailed information about these parameters can be found in Appendix II. Simply speaking, the larger σ or $t$ is, the sooner the methods would converge, meanwhile more exposed to risk of trapped by less accurate or wrong features.

A representative example of aponeuroses segmentation process using LGIF evolution model is shown in Fig. 5.

Our strategy proposed in this paper has the ability to measure muscle thickness along the entire contours of aponeuroses, so we can get a curve of muscle thickness for each frame of ultrasound image. Muscle thickness of a representative frame estimated by our method is shown in Fig. 6.

Estimation of the mean muscle thickness for each frame was additionally made by calculating the mean distance between contours of superficial and deep aponeuroses, for the purpose to make the comparison to the manual method. A representative result of muscle thickness of the whole image sequence, measured by the proposed technique and the manual method, is displayed in Fig. 7. To further quantitatively evaluate our algorithm, we define the thickness error rate (TER) to compare muscle thickness measured by the proposed strategy (PT) and the manual method (MT) as TeX Source $${\rm TER = }\left\vert {{\rm (PT} - {\rm MT)/MT}} \right\vert \times 100{\rm \% }{\rm.} \eqno{\hbox{(2)}}$$the average thickness and TER measured for all subjects are listed in Table I.

To further investigate the performance of the proposed strategy, interframe differences of MT and PT curves for a representative subject are compared in Fig. 8. Detailed quantitative results are also listed in Table II. It should also be noted that, in addition to measurement of the muscle, the thickness of superficial and deep aponeuroses can also be estimated readily by calculating the width of their contours. Fig. 9 shows the thickness measurements of superficial aponeuroses in a representative image sequence and thickness measurements of superficial aponeuroses from all subjects are listed in Table III.

SECTION V

Previous studies for muscle thickness measurement using either manual or computer-aided methods, estimate muscle thickness at one or several locations, ignoring the fact that muscle thickness changes longitudinally along aponeuroses. It can be seen from Fig. 6that muscle thickness is not constant along the aponeuroses. Therefore, a new strategy was proposed to estimate muscle thickness for each point along the longitudinal axis of the ultrasound image based on extracting contours of aponeuroses, which would be very useful when interested in muscle thickness at more than one or several specific locations.

To evaluate the validity of our strategy, we additionally averaged muscle thickness of the frame by calculating the mean distance between superficial and deep aponeuroses, and made comparison with the measurement result of manual method. As shown in Fig. 7, it is found in the experiments that this mean muscle thickness, derived from the results generated using the proposed strategy, agrees well with the manual results. Meanwhile, it can be observed in Table Ithat the proposed strategy shows good performance with maximal TER of 1.35%. The average TER value for all images is 0.87%. The differences are inevitable since MT calculates only the distance between two selected points, while PT calculates the mean distance between contours of aponeuroses. In practice, the muscle thickness can change slightly along longitudinal axis of the ultrasound image, as shown in Fig. 6.

For consecutive ultrasound images captured during muscle contraction, location of aponeuroses in a specific frame should not change much from neighboring frames. Therefore, it is expected to see that muscle thickness only change slightly across frames. It can be seen that PT exhibited much smaller fluctuations than MT in Fig. 7. As shown in Fig. 8 and Table II, for the results from a representative subject, PT provided a higher level of measurement stability with lower interframe difference. Given the fact that change of aponeuroses width in a sequence is trivial, the SD of thickness of superficial aponeurosis itself in a sequence could indirectly reflect the error level of our strategy. From Fig. 9 and Table III, it can be observed that the thickness of superficial aponeurosis is almost constant and SDs from all subjects are 0.1 ± 0.0 mm or 0.3 ± 0.1 pixels, which can serve as an indirect proof that the error level of the proposed method was no larger than 0.1 mm.

The proposed methods work well under isometric contraction because muscle sliding under this condition should be minimal. However, during joint motion, the relative motion of the muscles to the skin surface could change the portion of the muscle being imaged. This may affect the muscle thickness measurement. In addition, if the transducer is not positioned such that the aponeuroses run along the horizontal direction of the ultrasound image, the thickness measurement could increase substantially, therefore in our study, the ultrasound probe was fixed by a custom-designed foam container with fixing straps to force the probe remain in the same vertical imaging plane. However, when the muscle belly bulged, the probe could be pushed away slightly. Both problems are also general for all sonography studies (either manual or automatic) on muscle contractions. We would like to point out that all data in current report are all collected from young and healthy subjects. More data are planned to be collected from subjects with musculoskeletal disorders covering a wider age range to test and further improve the methods.

SECTION VI

In this paper, we present a new strategy for the measurement of muscle thickness along the entire length of aponeuroses in ultrasonography, which is useful when curvature of aponeuroses and longitudinal variance of muscle thickness are unnegligible. Results of the experiments suggest that the proposed strategy can be used for objective estimation of muscle thickness in musculoskeletal ultrasound images.

APPENDIX I

The MI-FFD method, originally proposed in [25] to do shape registration based on the MI between two images, was applied in this paper for frame registration and seed points tracking. The similarity criterion and transformation model of the method adopted in this method were MI and B-spline FFD, respectively.

The essence of FFD is to deform the shape of an object by manipulating a regular control lattice $P$ overlaid on its volumetric embedding space. A dense deformation field for every pixel in the embedding space can be acquired through interpolation using an interpolating basis function, such as Bezier spline or B-spline functions [25]. In this method, the deformation function T is defined by a 2-D spline function, which is parameterized by displacement vectors at a uniform grid of control points.

Let $P$ be a uniform grid of control points TeX Source $$P \!=\! \left\{{P_{m,n} } \right\} \!=\! \left\{{\left({P_{m,n}^x, P_{m,n}^y } \right)} \right\},m \!=\! 1, \ldots, M,n \!=\! 1, \ldots, N \eqno{\hbox{(A1)}}$$ overlaid to an image domain $A$, and the size of $A$ is $X \times Y$. The spacing of the grid P in the $\vec x$ and ${\rm \vec y}$ directions are $n_x = {X \mathord{\left/ {\vphantom {X M}} \right. \kern-\nulldelimiterspace} M}$ and $n_y = {Y \mathord{\left/ {\vphantom {Y N}} \right. \kern-\nulldelimiterspace} N}$. Then, FFD parameters are deformation improvements of control points in both $\vec x$- and $\vec y$-directions TeX Source $$\theta = \delta P = \left\{{\left({\delta P_{m,n}^x, \delta P_{m,n}^y } \right)} \right\};\left({m,n} \right) \in \left[{1,M} \right] \times [1,N].\eqno{\hbox{(A2)}}$$ So, the transformation function $T\left({x,y} \right)$ is defined by a tensor product of cubic B-splines TeX Source $$T\left({x,y} \right) = \sum_{l{\rm = }0}^3 {\sum_{k{\rm = }0}^3 {B_l \left(u \right)B_k \left(v \right)P_{i + l.j + k} } } \eqno{\hbox{(A3)}}$$ where $i = \lfloor {{x \mathord{/ {\vphantom {x {n_x}}} \kern-\nulldelimiterspace} {n_x}}} \rfloor - 1, j = \lfloor {{y \mathord{/ {\vphantom {y {n_y}}} \kern-\nulldelimiterspace} {n_y}}} \rfloor - 1, u = {x \mathord{/ {\vphantom {x {n_x}}} \kern-\nulldelimiterspace} {n_x}} - \lfloor {{x \mathord{/ {\vphantom {x {n_x}}} \kern-\nulldelimiterspace} {n_x}}} \rfloor, v = {y \mathord{/ {\vphantom {y {n_y}}} \kern-\nulldelimiterspace} {n_y}} - \lfloor {{y \mathord{/ {\vphantom {y {n_y}}} \kern-\nulldelimiterspace} {n_y}}} \rfloor$, and $B_l$ is the $l{\rm th}$ cubic B-spline basic function.

After the transformation of the floating image, we will compute MI between the transformed image and reference image. Then, parameters $\theta$ of the transformation function $T\left(\cdot \right)$ will be optimized by maximizing MI.

In probability theory and information theory, the MI of two random variables is a quantity that measures the mutual dependence of the two random variables. Here, we introduce two random variables A and B denoting reference image and floating image respectively. Image B is then deformed by mean of a transformation function $T\left(\cdot \right)$ with parameters to be determined by maximizing the MI between $A$ and $T\left(B \right)$. The MI between the two images can be defined as [25] TeX Source $$\eqalignno{&MI\left({A;T\left(B \right)} \right) = H\left(A \right) + H\left({T\left(B \right)} \right) - H\left({A,T\left(B \right)} \right) \cr & \quad = \sum_{b \in T\left(B \right)} {\sum_{a \in T\left(A \right)} {P\left({a,b} \right)\log \left({{{P\left({a,b} \right)} \mathord{\left/ {\vphantom {{P\left({a,b} \right)} {\left({P\left(a \right)P\left(b \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left({P\left(a \right)P\left(b \right)} \right)}}} \right)} } \qquad &\hbox{(A4)}}$$ where $H\left(A \right), H\left({T\left(B \right)} \right)$, and $H\left({A,T\left(B \right)} \right)$ are the entropies of reference image $A$, transformed image $T\left(B \right)$, and their joint entropy, respectively; $P\left(a \right)$ and $P\left(b \right)$ represent intensity probability distribution functions of $A$ and $T\left(B \right)$, respectively; and $P\left({a,b} \right)$ is their joint probability distribution function.

Therefore, by minimizing the objective function $E{\rm = } - {\rm MI}$, using parameters of transformation function $T\left(\cdot \right)$, the two original images can be registered and initial seed points can be tracked continuously.

In this paper, we adopted the Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm [28] as the optimization method to search optimal transformation parameters which could be minimize the object function $E$. An advantage of the L-BFGS algorithm is that the explicit evaluation of the Hessian matrix is not required, so it can be recursively estimated. Moreover, the L-BFGS algorithm is much faster than the conventional level set method for solving the nonlinear optimization problem.

APPENDIX II

The LGIF model [27] combines advantages of the Chan–Vese (CV) model [29] and the local binary fitting (LBF) model [30] by taking the local and global intensity information into account. The gradient decent flow equation of the LGIF energy function is defined as [27] TeX Source $$\eqalignno{{{\partial \phi } \mathord{\left/ {\vphantom {{\partial \phi } {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}} &= \delta \left(\phi \right)\left({F_1 + F_2 } \right) + \nu \delta \left(\phi \right)div\left({{{\nabla \phi } \mathord{\left/ {\vphantom {{\nabla \phi } {\left\vert {\nabla \phi } \right\vert}}} \right. \kern-\nulldelimiterspace} {\left\vert {\nabla \phi } \right\vert}}} \right) \cr &\quad + \mu \left({\nabla ^2 \phi - div\left({{{\nabla \phi } \mathord{\left/ {\vphantom {{\nabla \phi } {\left\vert {\nabla \phi } \right\vert}}} \right. \kern-\nulldelimiterspace} {\left\vert {\nabla \phi } \right\vert}}} \right)} \right) &\hbox{(B1)}}$$ where $\delta$ is the 1-D Dirac function. $F_1$ is the component of LBF model which was proposed to segment images with intensity inhomogeneity, and $F_2$ is the component of the CV model. Given the image $I\left({\bf x} \right)\left({{\bf x}:\left({x,y} \right),1 \le x \le X,1 \le y \le Y} \right)$ to be segmented, $F_1$ and $F_2$ can be formulated as [27] TeX Source $$\eqalignno{F_1 &= \left({1 - \omega } \right)[- \lambda _1 \smallint K_\sigma \left({{\bf y} - {\bf x}} \right)\left\vert {I\left({\bf x} \right) - f_1 \left({\bf y} \right)} \right\vert ^2 d{\bf y} \cr &\quad + \lambda _2 \smallint K_\sigma \left({{\bf y} - {\bf x}} \right)\left\vert {I\left({\bf x} \right) - f_2 \left({\bf y} \right)} \right\vert ^2 d{\bf y}] &\hbox{(B2)}\cr F_2 &= \omega \left[{- \lambda _1 \left\vert {I\left({\bf x} \right) - c_1 } \right\vert ^2 + \lambda _2 \left\vert {I\left({\bf x} \right) - c_2 } \right\vert ^2 } \right]&\hbox{(B3)}}$$ where $\lambda _1$ and $\lambda _2$ are nonnegative constants, and $\omega$ is a positive constant (0 ≤ $\omega$ ≤ 1). When images are corrupted by intensity inhomogeneity, the parameter value $\omega$ should be chosen small enough. $K_\sigma$ is a Gaussian kernel with SD $\sigma$. $c_1$ and $c_2$ are two constants that approximate the image intensity in the regions outside and inside the contour which are expressed in the following way [29]: TeX Source $$\eqalignno{c_1 &= \int {I}({\bf x})H({\phi ({\bf x})})\hbox{d}{\bf x}\Big/ \!\! \int{H}\left({\phi \left({\bf x} \right)} \right)\hbox{d}{\bf x} \cr c_2 &= \! \int{I}\left({\bf x} \right)\left({1 \!-\! H\left({\phi \left({\bf x} \right)} \right)} \right)\hbox{d}{\bf x}\Big/ \!\!\! \int \left({1 - H\left({\phi \left({\bf x} \right)} \right)} \right)\hbox{d}{\bf x} \quad \qquad &\hbox{(B4)}}$$ where $H$ is 1-D Heaviside function. In contrast to the two constants $c_1$ and $c_2$, functions $f_1$ and $f_2$ are spatially varying fitting functions [30] TeX Source $$\eqalignno{f_1 \left({\bf x} \right) &\!=\! K_\sigma \left({\bf x} \right)\left[{H\left({\phi \left({\bf x} \right)} \right)I\left({\bf x} \right)} \right]/{\left[{K_\sigma \left({\bf x} \right)H\left({\phi \left({\bf x} \right)} \right)} \right]} \cr f_2 \left({\bf x} \right) &\!=\! {K_\sigma \left({\bf x} \right)\left[{\left({H\left({\phi \left({\bf x} \right)} \right)} \right)I\left({\bf x} \right)} \right] /\left[{K_\sigma \left({\bf x} \right)\left({1 \!-\! H\left({\phi \left({\bf x} \right)} \right)} \right)} \right]}.\cr & &\hbox{(B5)}}$$ In practice, Heaviside function $H$ is approximated by a smooth function $H_\varepsilon$ defined by TeX Source $$H_\varepsilon \left({\bf x} \right) = {{\left[{{{1 + \left({2\arctan \left({{{\bf x} \mathord{\left/ {\vphantom {{\bf x} \varepsilon}} \right. \kern-\nulldelimiterspace} \varepsilon}} \right)} \right)} \mathord{\left/ {\vphantom {{1 + \left({2\arctan \left({{{\bf x} \mathord{\left/ {\vphantom {{\bf x} \varepsilon}} \right. \kern-\nulldelimiterspace} \varepsilon}} \right)} \right)} \pi}} \right. \kern-\nulldelimiterspace} \pi}} \right]} \mathord{\left/ {\vphantom {{\left[{{{1 + \left({2\arctan \left({{{\bf x} \mathord{\left/ {\vphantom {{\bf x} \varepsilon}} \right. \kern-\nulldelimiterspace} \varepsilon}} \right)} \right)} \mathord{\left/ {\vphantom {{1 + \left({2\arctan \left({{{\bf x} \mathord{\left/ {\vphantom {{\bf x} \varepsilon}} \right. \kern-\nulldelimiterspace} \varepsilon}} \right)} \right)} \pi}} \right. \kern-\nulldelimiterspace} \pi}} \right]} {\rm 2}}} \right. \kern-\nulldelimiterspace} {\rm 2}} \eqno{\hbox{(B6)}}$$ where $\varepsilon$ is a positive constant. The derivative of $H_\epsilon$ is the smoothed Dirac delta function TeX Source $$\delta _\varepsilon \left({\bf x} \right) = {\varepsilon \mathord{\left/ {\vphantom {\varepsilon {\left[{{\rm \pi (}\varepsilon ^2 + {\bf x}^2 {\rm)}} \right]}}} \right. \kern-\nulldelimiterspace} {\left[{{\rm \pi (}\varepsilon ^2 + {\bf x}^2 {\rm)}} \right]}}. \eqno{\hbox{(B7)}}$$the parameter $\varepsilon$ in $H_\varepsilon$ and $\delta _\varepsilon$ is set to 1.0. This parameter can be fixed as this value in practice.

In this study, to reduce the number of iterations, contours are initialized as two circles centered at two seed points. In our implementation, assuming the coordinates of the two seed points are $\left({x_1, y_1 } \right), \left({x_2, y_2 } \right)$, respectively, then contours are initialized as $\phi _0 = \left[{\phi _{1,0}; \phi _{2,0} } \right]$, where TeX Source $$\eqalignno{&\left\{{\matrix{{\phi _{1,0} \left({\bf x} \right) = \sqrt {\left({x - x_1 } \right)^2 + \left({y - y_1 } \right)^2 } - 2,} \cr {\phi _{2,0} \left({\bf x} \right) = \sqrt {\left({x - x_2 } \right)^2 + \left({y - y_2 } \right)^2 } - 2,} \cr}} \right.\cr &\qquad\qquad\qquad\left({0 \le x \le X,0 \le y \le Y/2} \right).&\hbox{(B8)}}$$ Finally, the principle steps of the LGIF algorithm are as follows.

- Set parameters $\lambda _1, \lambda _2, {\rm \omega }, \nu, \mu$, and time step $\Delta t$;
- Initialize $\phi ^0$ by $\phi _0, n$ = 0;
- Compute $F_1 \left({\phi ^n } \right)$ and $F_2 \left({\phi ^n } \right)$ by (A4) and (B1);
- Obtain $\phi ^{n + 1} = \phi ^n + \Delta t{\rm (}{{\partial \phi } \mathord{\left/ {\vphantom {{\partial \phi } {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}}{\rm)}$, according to (A3);
- Check whether the solution is stationary. If not, $n = n$ + 1 and repeat.

This work was supported in part by the Hong Kong Innovation and Technology Commission under Grant GHP/047/09, the National 863 Program of China under Grant 2012AA02A604, the National 973 Program of China under Grant 2010CB732606, the next-generation communication technology Major project of National S&T under Grant 2013ZX03005013, the “Low-Cost Healthcare” Programs of Chinese Academy of Sciences, the Guangdong Innovative Research Team Program under Grant 2011S013, GIRTF-LCHT), and the International Science and Technology Cooperation Program of Guangdong Province under Grant 2012B050200004. Dr. Yongjin Zhou is the corresponding author.

S. Ling is with the Shenzhen Key Laboratory for Low-cost Healthcare, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China, and also with the School of Geosciences and Info-Physics, Central South University, Changsha 410000, China.

Y. Zhou is with the Shenzhen Key Laboratory for Low-cost Healthcare, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China, and also with the Interdisciplinary Division of Biomedical Engineering, The Hong Kong Polytechnic University, Hong Kong (e-mail: yj.zhou@siat.ac.cn), phone: 086-134-2866-6265; fax: 086-755-8639-2299.

Y. Chen is with the Shenzhen Key Laboratory for Low-cost Healthcare, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China, and also with the School of Information Engineering, Nanchang Hangkong University, Nanchang 330000, China.

Y.-Q. Zhao is with the School of Geosciences and Info-Physics, Central South University, Changsha 410000, China.

L. Wang is with the Shenzhen Key Laboratory for Low-cost Healthcare, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China.

Y.-P. Zheng is with the Interdisciplinary Division of Biomedical Engineering, The Hong Kong Polytechnic University, Hong Kong.

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

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