1 Introduction

Heart failures have become some of the major causes of death in developed and developing countries [11]. Detection of cardiac diseases during early stages of illness can be fundamental to preserve the life of patients. In this sense, cardiac imaging techniques such as MR and CT have become essentials for heart evaluation. Both technologies consist of noninvasive tests which provide a set of image slices of the heart. Several cardiac affections and conditions can be diagnosed with heart imaging techniques, including coronary artery diseases, affections provoked by heart attacks, heart muscle diseases, congenital defects, left ventricle (LV) dysfunction, heart valve diseases, ischemia, myocardial mechanical problems and others [4].

LV, within the cardiac structures, is of great concern for heart function evaluation [15, 32]. LV size at both end-diastolic (ED) and end-systolic (ES) phase is commonly used to identify cardiac failures [21]. Moreover, LV mass can be used as a predictor of morbidity and mortality in some patients [30]. Segmentation of the LV is required to quantify these variables. The need for automatic or semiautomatic algorithms is justified by the fact that manual delineation is tedious, time consuming and observer-dependent. More objective measurements can be achieved by using robust computer-aided segmentation methods.

LV segmentation using cardiac images is a problem that researches have tried to solve for many years. Although many algorithms have been proposed, this problem still remains as an open issue [32].

Active contours (ACs) are some of the most accepted methods for medical image segmentation. They consist of evolving interfaces guided by internal and external forces which depend on geometrical and image features [14]. In these methods, the segmentation process is generally modeled as energy functionals whose terms are designed to stop the interface evolution at the object boundaries.

External forces in ACs need for efficient methods to extract image features which are used to guide the evolution of the interfaces. Although many methods have been proposed for this purpose (boundary-based techniques [9], region-based approaches [10], methods based on probabilistic interpretation [31] and others), all of them attempt to analyze image features using the original space. Multiscale and multiresolution analysis have demonstrated being very powerful strategies to extract features in medical image applications [1]. In this work, we use the 3D Hermite transform for data characterization. It consists of a powerful mathematical tool that projects a function onto the space composed by the Hermite polynomials [28]. Local and directional analysis [18], as well as multiscale and multiresolution analysis [43] can be competently performed with the HT.

In this work, a novel 3D AC model embedded into the HT space is proposed . Features such as intensities, boundary information, and geometrical constraints are combined using a hierarchical multiscale scheme. Intensity and boundary information are directly computed from coefficients of the HT while shape-based constraints are based on the AC properties. We exploit some advantages of the HT to provide local directional and multiscale analysis. We use elliptical shapes for geometrical constraint. An initialization stage using a fuzzy C-means algorithm, combined with morphological operations, is also employed. The proposed method is experimentally evaluated using MR and CT studies. We demonstrate that combining different energy models into a multiscale strategy based on the HT, cardiac volumes can be efficiently processed.

1.1 Related works

There have been numerous contributions about segmentation methods applied to LV in cardiac MR/CT images [15, 32, 47]. Implicit [5, 22, 26, 33, 34, 49, 52] and parametric [24, 37, 44, 50] ACs, probabilistic algorithms [13, 42], statistical shape models [2, 3, 8, 20], and atlas-based algorithms [6, 7] are the most typical methods used for LV segmentation.

Statistical shape models such as ASMs (active shape models) and AAMs (active appearance models) use a set of sample shapes to train a statistical model which is subsequently used for segmenting new data. Hans van Assen et al. [3] combined a 3D ASM with a C-means algorithm to segment the LV in MR volumes. Data with lack of information can be segmented with this method. Ecabert et al. [17] developed an approach to extract four cavities of the heart as well as the myocardium and great vessels using CT volumes. The method uses an ASM-based parametric model with prior information. Although these methods are efficient for segmentation of cardiac data, their performances depend on the number of training samples.

Parametric ACs have been mainly employed for segmentation of cardiac images. They consist of parametric curves which move with a velocity field depending on image features and geometrical information. Wu et al. [50] used a classical snake model for LV segmentation in MR images. Here, edges are detected using an external force called gradient vector convolution. Circular shape constraints are used. The problem with methods based entirely on edge detectors is the sensitivity to noise. Sliman et al. [44] proposed a method for 2D segmentation of the myocardium using a parametric AC which combines two terms: a first-order combination of discrete Gaussians and a second-order Markov-Gibbs random field. A distance measure was also incorporated to maintain a minimal separation between both contours.

ACs based on level sets acquired much relevance for segmentation of medical images. Unlike parametric ACs, curves and interfaces are represented implicitly using level sets. Easy implementation, extension to higher dimensions, and possibility to combine different types of energies in a transparent way are the most typical advantages of level set methods. Changes of configuration are also assumed naturally. In [22], the LV endocardium was segmented using a Local Binary Fitting (LBF) energy functional [27] and a convex hull algorithm while a dynamic programming method was used for the epicardium. However, the method does not use shape energies. A variational method is presented in [5] in which two curves are evolved to extract the endocardium and epicardium. A prior term is used to model the overlapping intensities in all regions of the LV. Shape-based energies are not used which may hinder the segmentation of papillary muscles. In contrast, Pluempitiwiriyawej et al. [34] proposed a level set functional which includes a region term, a boundary term, a regularization term and a shape term. In the latter, they assume the LV can be modeled using an elliptical shape. However, inhomogeneity problems of cardiac images are not considered. Attempting to model similarity between the LV cavity and epicarial boundary, Woo et al. [49] designed a multiphase level set approach to segment the endocardium and epicardium. The method includes a shape prior based on the discrepancy between both regions, which is assumed to follow a Gaussian distribution. Zhu et al. [51] presented a combination of thresholding, mathematical morphology, geometrical operations and classical AC methods [9] to segment the endocardium and epicardium in 3D cardiac CT studies. The poor contrast between the myocardium and surrounding tissues in cardiac CT studies affects the performance of classical AC models.

Other researches have recently opted for dynamic programming models [23, 35, 41], registration-based methods [48], random walk [19], and sparse modeling [36]. In spite of all the described methods attempt to settle the segmentation problem of the LV, they use image features obtained from the original space of the input data. Nonetheless, most of them do not use shape prior, they need of a huge dataset to train a model and image inhomogeneities are not considered. Even more, segmentation of cardiac images has been mainly focused on analyzing either MR or CT images without generalization to both types of modalities.

1.2 Challenges in LV segmentation

Although many works have been proposed, the segmentation of the LV remains as an active field of research. Different types of problems are found when segmenting the LV in cardiac MR/CT data. Variations of contrast, irregular shapes of the endocardial and epicardial boundaries, diffuse edges, inhomogeneities, lack of information at apex and base of the heart and noise are the most common difficulties to be handled by segmentation algorithms in these types of images [47]. There is also a main consensus about the assumption that papillary muscles are considered part of the LV cavity for measurement purposes, which additionally imposes a great challenge for segmentation approaches [15].

Other common trouble we can find is this type of application is the difficulty to address extensive and objective comparisons among many algorithms. Some collaborative works and workshops have significantly contributed to the field of LV segmentation using MR studies because they have provided public datasets and evaluation protocols for use of the researcher community. The MICCAI 2009 challenge [46], designed to perform valuable competitions of LV segmentation algorithms, is perhaps the most known workshop organized for this purpose. Nonetheless, a more recent initiative called the cardiac atlas project [45] was created to provide a huge database of annotated cases with the objective that researchers prove their algorithms. These collaborative works are also useful because they allow the researchers to evaluate the performance of several methodologies for this specific application. Despite the effort made, the LV segmentation using MR data remains as an open issue. In many cases, authors have decided to use their own set of data, and they also used different metrics for evaluation. Regarding cardiac CT studies, there are not public datasets which imposes a bigger challenge for evaluating segmentation algorithms.

2 Method

The developed method consists of a 3D local AC model embedded into a hierarchical multiscale scheme provided by the HT applied to LV segmentation in cardiac MR/CT volumes. Coefficients of the HT guide the AC evolution. The LV cavity is modeled using elliptical functions which are incorporated into the shape energy. An endocardial-based shape energy is employed for segmentation of the epicardium based on the assumption that epicardial and endocardial shapes are similar. The process follows a coarse-to-fine segmentation. The endocardium is segmented first followed by the epicardium. An initialization method is proposed using a fuzzy C-means algorithm combined with mathematical morphology operations. The general diagram of the designed method is depicted in Fig. 1. The 3D HT is calculated for the input volume using several scales. Coefficients are posteriorly steered. The initialization algorithm is then runned in order to obtain the initial surface at the largest scale. Here, the AC model is used iteratively. Once the AC has converged, its result is interpolated and used as initialization for the next scale. These steps are repeated until reaching the finest scale. Each stage of the proposed framework will be described in this section.

Fig. 1
figure 1

Scheme of the proposed segmentation method

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2.1 The 3D Hermite transform (HT)

The goal of the 3D HT is to project the input volume onto the space of Hermite polynomials in order to extract relevant information [28]. Let f(x, y, z) be a 3D input function, its 3D HT is computed as follows:

$$\begin{array}{@{}rcl@{}} L_{n-m, m-l,l}(p,q,r)\!\! &=&\!\! {\int}_{\!\!\!\!-\infty }^{\infty}f(x,y,z) V^{2}(x\,-\,p,y\,-\,q,z\,-\,r)\\ &&\!\! \cdot\, G_{n-m,m-l,l}(x\,-\,p,y\,-\,q,z\,-\,r)\mathit{dxdydz}\\ \end{array} $$
(1)

where L n−m, m−l, l (p, q, r) are the 3D cartesian Hermite coefficients, V (x, y, z) is an isotropic 3D Gaussian window with standard deviation σ and \(G_{n-m,m-l,l}= W H_{n-m}(\frac {x}{\sigma })H_{m-l}(\frac {y}{\sigma })H_{l}(\frac {z}{\sigma })\) correspond to the normalized 3D Hermite polynomials which are orthogonal with respect to V 2(x, y, z), and \(W=\frac {1}{\sqrt {2^{n}(n-m)! (m-l)! l!}}\). Coefficients are calculated for subscripts n = 0,1,…, N; m = 0,1,…, n and l = 0,1,…, m where n is the order of the transform. Implementation of the 3D HT can be performed by convolving the input function with the set of Hermite filters defined as:

$$\begin{array}{@{}rcl@{}} D_{n-m, m-l,l}(x,y,z) \!\!&=&\!\! V^{2}(x,y,z)G_{n-m,m-l,l}(x,y,z) \\ &=&\!\! W \frac{d^{n-m}}{d\!\left( \frac{x}{\sigma}\right)^{n-m}} \frac{d^{m-l}}{d\!\left( \frac{y}{\sigma}\right)^{m-l}} \frac{d^{l}}{d\!\left( \frac{z}{\sigma}\right)^{l}} V^{2}(x,y,z)\\ \end{array} $$
(2)

and subsequently subsampling at positions (p, q, r). For simplicity, we will use T = p = q = r as the subsampling variable. The list of coefficients obtained for the 3D HT until order n = 2 are depicted in Table 1.

Table 1 List of coefficients, 3D HT until order n = 2
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The subsampling T and the standard deviation σ are the free parameters to be configured. The latter determines the scale of the transformation. In this work, we use a hierarchical multiscale framework in which the 3D HT is computed using several values of σ and T. Here, a specific value of T is used for each value of σ. Then, the multiscale scheme is defined by the couple (σ j , T j ), with j = 1,2,…, p where p is the maximum number of scales. The first coefficient L 0,0,0 consists of a smoothed version of the input volume while coefficients corresponding to order n = 1 are edge maps. Coefficients of higher order, n > 1, are very suitable to analyze texture of the input data. Figure 2a illustrates an example of the 3D HT until order n = 1 applied to a cardiac MR volume using two scales (σ 1 = 1, T 1 = 1;σ 2 = 1.5, T 2 = 2). Slices show the middle of the LV.

Fig. 2
figure 2

Coefficients of the 3D HT applied to a cardiac MR volume. Slices show the middle of the heart. a Cartesian coefficients. b Steered coefficients

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In general, one of the main advantages of the HT is its ability to perform local directional analysis [18, 43]. Cartesian coefficients of order n ≥ 1 obtained using Eq. 1 can be steered using a linear combination of them. Patterns such as edges and textures of the input data are better described by performing directional analysis. Two angles, 𝜃 and ϕ, are needed in order to perform the steering process. The first angle defines the rotation w.r.t. x y − p l a n e and the second angle w.r.t. z − a x i s. Therefore, coefficients are steered as follows:

$$ LR_{n-m, m-l,l}(\theta,\phi) \,=\, \sum\limits_{k=0}^{n}\sum\limits_{s=0}^{k} C^{(n)}_{m,l,k,s}(\theta,\phi)L_{n-k,k-s,s} $$
(3)

where \(C_{m,l,k,s}^{(n)}(\theta ,\phi )\) represents a set of coefficient values used for the steering process, L n−k, k−s, s are the cartesian Hermite coefficients and L R n−m, m−l, l (𝜃, ϕ) are the steered Hermite coefficients. Figure 2b illustrates the steered Hermite coefficients obtained from the cartesian coefficients shown in Fig. 2a. It can be seen that most of the energy is concentrated in coefficient L R 1,0,0. It is a frequent practice to use only the steered coefficients L R 1,0,0(𝜃, ϕ), L R 2,0,0(𝜃, ϕ), … , L R n,0,0(𝜃, ϕ) meanwhile the rest of them are discarded, which means that a denoising process is indirectly being applied.

2.2 Proposed initialization

The initial shape must be carefully selected when using local AC models [25]. For this purpose, we have designed a novel semi-automatic initialization scheme which combines a fuzzy C-means method [12] with morphological operators. Figure 3 graphically outlines this scheme. Firstly, we manually define limits for base and apex of the LV. The histogram of the image corresponding to middle of the LV is processed to obtain initial average values for the LV cavity and myocardium. A fuzzy C-means algorithm is then applied to segment the volume in three classes: (1) background, (2) myocardium and similar tissues, and (3) LV cavity and similar tissues. Figure 3 briefly describes the initial process.

Fig. 3
figure 3

First stage of the initialization based on a fuzzy C-means algorithm

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Once the input volume is processed using the fuzzy C-means algorithm, we need to compute the initial surfaces for the endocardium and epicardium. This proccess is performed using two separate algorithms.

Endocardium initialization

The schematic diagram of this algorithm is illustrated in Fig. 4a. For each slice of the pre-segmented volume with fuzzy C-means, we proceed as follows:

  • Each object corresponding to third class is labeled.

  • Circularity C r, area A and center of mass C m measurements are used to determine the object corresponding to the LV cavity which is selected according to three conditions: C r > T c, A > T a and C m being the most central object satisfying the other two conditions. T c and T a are two thresholds.

  • When all volume slices have been processed, the initial endocardial surface is constructed.

Fig. 4
figure 4

a Endocardial initialization scheme, b Epicardial initialization scheme

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Epicardium initialization

Segmentation results of the endocardium are also incorporated as input information to initialize the epicardium. Figure 4b describes the general scheme. Similarly, for each slice of the volume resulted from the fuzzy C-means stage, we proceed as follows:

  • Take the second class as the region of interest. The rest of the image is assumed to be background.

  • Small objects (smaller than the LV cavity) of the background are removed. This operation helps to remove, from the myocardial region, small artifacts resulted from the fuzzy C-means stage.

  • An edge detector algorithm is used. The objective is to detect edge points of the epicardium.

  • Using the center of mass of the LV cavity previously segmented, we divide the complete slice in four regions.

  • The contour points closest to the center of mass for each region are selected. Then, the initial shape is built at a distance d from the endocardial boundary using as reference the four contour points.

  • Finally, the initial surface of the epicardium is built when all slices have been processed.

2.3 Proposed active contour model

We have designed an AC model with five energy functions: two region terms (one local and one global), an edge-based term and two geometrical terms. The first three energies incorporate coefficients of the HT which guides the AC evolution. The designed energy functional is used to segment the endocardial and epicardial boundaries. The AC has been represented using level sets [29]. Let C = C(X), X ∈ R 3 be a moving interface which fragments the volume space Ω into two regions. It can be represented as the zero level set of a higher dimensional function, C = {X|φ(X) = 0}. The following convention has been adopted for the level set function: φ(X) > 0 for X outside C, φ(X) < 0 for X inside C, and φ(X) = 0 for X ∈ C. The general proposed energy functional is written as follows:

$$ E = \lambda_{1}E_{L}+\lambda_{2}E_{G}+\lambda_{3}E_{B}+\lambda_{4}E_{S}+\lambda_{5}E_{R} $$
(4)

where E L and E G are the local and global energy functions, E B is the boundary-based energy, E S is the shape constraint and E R is the regularization term. Weight parameters λ m , m = 1,…,5 control the contribution of each energy term. All energy terms will be described in this section.

2.3.1 Local region energy

In general, inhomogeneities are frequent issues found in MR/CT images. Classical region-based AC models [10, 31] fail to segment objects with intensity inhomogeneities because they are based on global features. Several previous works based on local level sets have attempted to solve the problem of image inhomogeneity [25, 27]. In this work, we used a local energy term E L to characterize each point p of the input volume using information of its neighborhood. Taking a squared window W centered on each voxel, two regions are selected: inside (Ω i n ) and outside (Ω o u t ) the current interface. We then compute the averages μ 1 and μ 2 for both regions respectively. Figure 5 outlines this modeling process. For visualization purposes the scheme is presented in 2D, however it is similar in 3D. The Hermite coefficient of order n = 0 is used as input volume.

Fig. 5
figure 5

Data modeling in the local region energy

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The energy term is defined as:

$$\begin{array}{@{}rcl@{}} E_{L}(\mu_{1},\mu_{2},\varphi) \!&=&\! {\int}_{\Omega}(L_{0,0,0}(X)\!-\!\mu_{1}(X))^{2}H(\varphi(X))\mathit{dX} \\ &&+{\int}_{\Omega}(L_{0,0,0}(X)\!-\!\mu_{2}(X))^{2}(1\!-\!H(\varphi(X)))\mathit{dX}\\ \end{array} $$
(5)

where μ 1 and μ 2 are calculated as follows:

$$\begin{array}{@{}rcl@{}} \mu_{1}(X) &=& \frac{{\int}_{W}I(X)H\left( \varphi(X)\right)\mathit{dX}}{{\int}_{W}H\left( \varphi(X)\right)\mathit{dX}} \\ \mu_{2}(X) &=& \frac{{\int}_{W}I(X)\left( 1-H\left( \varphi(X)\right)\right)\mathit{dX}}{{\int}_{W}\left( 1-H\left( \varphi(X)\right)\right)\mathit{dX}} \end{array} $$

where I(X) = {L 0,0,0(X)|X ∈ W}. The level set is regularized using Heaviside \(H_{\epsilon }(z)=\frac {1}{2}\left (1+\frac {2}{\pi }tan^{-1}(\frac {z}{\epsilon })\right )\) and Delta Dirac \(\delta (z) = \frac {d}{dz}(H(z))\) functions for minimization purposes [10]. The terms H(φ(X)) represents the region inside the interface, and (1 − H(φ(X))) is the outer region. Because the zero-order Hermite coefficient is used as input data, a noise reduction process is indirectly introduced in this energy term.

2.3.2 Global region energy

A global energy term is incorporated to strengthen the AC evolution, particularly in homogeneous regions. We adopted the classical energy functional based on probabilistic models [14, 31]. Assuming the input volume V is the result of a stochastic process, the partition P(Ω) of V can be obtained by optimizing the a posteriori probability p(P(Ω)/V ) [31]. Considering the binary case of two independent regions, Ω1 and Ω2 (object and background), the global region energy is defined as follows:

$$\begin{array}{@{}rcl@{}} E_{G}(\varphi) \!&=&\! - {\int}_{\Omega}log (p(L_{0,0,0}(X)/{\Omega}_{1})) H(\varphi(X))\mathit{dX} \\ &&\! +{\int}_{\Omega}log (p(L_{0,0,0}(X)/{\Omega}_{2}))(1\,-\,H(\varphi(X)))\mathit{dX}\\ \end{array} $$
(6)

where p(L 0,0,0(X)/Ω i ) with i = 1,2 is the probability function assumed for each partition. Similarly, the zero-order Hermite coefficient is used as input volume.

2.3.3 Boundary-based energy

The boundary-based energy aims at stopping the moving interface when edges are reached. Due to its simplicity and efficiency, we used the following functional:

$$ E_{B}(\varphi) = {\int}_{\Omega}gH\left( \varphi(X)\right)\mathit{dX} $$
(7)

where \(g=\frac {1}{(1+LR_{1,0,0})}\). The steered coefficient L R 1,0,0 is used as input data in this energy term. Since it consists of a local directional boundary detector, the edge map of the analyzed volume is captured efficiently for all orientations. This energy term takes advantage of the directional properties of the HT.

2.3.4 Shape constraint

Generally speaking, it is a common practice to employ geometrical shapes to model structures in medical images. LV segmentation in cardiac volumes becomes challenging due to presence of the papillary muscles which are considered to be part of the LV cavity. Moreover, subtle edges, low contrast as well as lack of information in some regions of the endocardium and epicardium impose more challenges for segmentation algorithms. Some authors have adopted elliptical [34] or circular [50] shapes to model the LV in cardiac MR/CT data with short axis view. In this work, we also consider that the LV cavity can be represented using elliptical shapes in all volume slices from base to apex. Our goal is to maintain a regular shape of the interface by controlling its deformation. This process is performed for each slice. The current 3D zero level set is then mapped to 2D zero level sets, see Fig. 6. Afterwards, an ellipse for each plane is estimated using each contour slice. Finally, we build a stack of signed distance functions from contours of the estimated ellipses.

Fig. 6
figure 6

Ellipses estimation process. The initial surface is mapped to 2D contours and ellipses are estimated for each x y − p l a n e

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The ellipses estimation process is performed using the simple but effective method described in [39]. The stack of ellipses-based signed distance functions φ s is compared with the original level set which allows penalizing the AC deformation when it differs from the elliptical shapes. For this purpose, we used the following energy functional [40]:

$$ E_{S}(\varphi_{s},\varphi) = {\int}_{\Omega}\left( \varphi(X)-\varphi_{s}(X) \right)^{2} H\left( \varphi(X)\right)\mathit{dX} $$
(8)

The shape constraint consists of a geometrical function that only depends on the interface.

For the epicardial boundary, we assume that shape of the epicardium is similar to shape of the endocardium. The final segmentation of the endocardial boundary is then coded as φ s . Therefore, when segmenting the epicardial boundary this energy term penalizes the deviation of the AC from the endocardial shape.

2.3.5 Regularization energy

We also include a regularization term which is normally used to privilege smooth interfaces in the segmentation process [9, 10, 14, 29, 31]. It is defined as:

$$ E_{R}(\varphi) = {\int}_{\Omega}\left| \nabla H\left( \varphi(X)\right) \right|\mathit{dX} $$
(9)

where ∇ is the gradient operator.

2.3.6 Energy minimization

The complete energy functional is a weighted combination of the described energy terms, see Eq. 4. It is well known that minimizing the complete functional is equivalent to minimizing each energy term separately and then performing the corresponding combination [49]. Fixing μ 1, μ 2 and φ s , using the calculus of variation and the steepest descent method, the differential equation which minimizes E with respect to φ is obtained:

$$\begin{array}{@{}rcl@{}} \frac{\partial \varphi}{\partial t} &=& \left[\lambda_{1} \left( -\left( L_{0,0,0}-\mu_{1} \right)^{2} + \left( L_{0,0,0}-\mu_{2} \right)^{2} \right) \right. \\ && +\, \lambda_{2}\left( \frac{log \left( p\left( L_{0,0,0}/{\Omega}_{1} \right) \right)}{log \left( p\left( L_{0,0,0}/{\Omega}_{2} \right) \right)}\right) \,+\, \lambda_{3}g \\ &&+ \left. \lambda_{4}Div\left( \frac{\nabla \varphi}{\left|\nabla\varphi \right|}\right)\!\right]\!\delta(\varphi) \,-\, 2 \lambda_{5}H(\varphi)(\varphi\,-\,\varphi_{s}) \end{array} $$
(10)

2.4 Implementation details

Regarding the 3D HT, coefficients were steered using a criterion of maximum energy. Two or three scales were used depending on the input data.

For the level set implementation, we used a narrow band algorithm since the final segmentation is only defined by the zero level set. In the global term E G , we assume Gaussian distributions whose parameters were computed from results of the initialization algorithm. All energy terms have been normalized within the range [0,1]. In order to stop the AC evolution, we used the following criterion:

$$ \left| {\int}_{\Omega}H\left( \varphi_{k_{1}}(X)\right)\mathit{dX} - {\int}_{\Omega}H\left( \varphi_{k_{2}}(X)\right)\mathit{dX} \right|<\tau $$
(11)

where k 1 and k 2 represent two different iterations with k 2 = k 1 + q, q ∈ Z + and τ is a threshold.

The shape energy maintains a regular deformation of the AC during evolution. Since it does not depend on image features, it cannot guide the contour toward the boundaries of the object. For this energy term, we have used a double configuration: a small weight value is assigned to this energy at the beginning of the evolution and for a specific number of iterations. This weight is subsequently incremented. With this configuration, the AC is freely deformed during the first iterations and then adjusted to maintain the geometrical shape defined for the processed object.

3 Results

In this section, we present the results obtained with the proposed algorithm using MR/CT studies. Comparisons with other methods are also presented. Several metrics were used for evaluation: point-to-curve distance (PCD), point-to-surface distance (PSD), modified Hausdorff distance (MHD) [16] and the Dice similarity coefficient (DSC). Moreover, clinical indices were also used for evaluating the algorithm.

3.1 Materials

Two sets of MR and one of CT cardiac volumes were used for validation. The first set of MR data consists of 15 patients selected from the MICCAI challenge database [46]. These images were acquired with a 1.5T GE Signa MRI during 0–15 second breath-holds with 20 cardiac phases as temporal resolution over the complete cardiac cycle. Images were acquired in short axis view. The number of slices for each volume varies between 6 and 12 from the atrioventricular ring to the apex (thickness = 8 mm, gap = 8 mm, FOV = 320 mm 320 mm, matrix = 256 ×256).

The second set of MR data is composed by 15 patients selected from the database shared by [2]. Similarly, images were acquired with a GE Genesis Signa MR scanner. For each patient, 20 volumes describe the entire cardiac cycle. The number of slices ranges between 8 and 15. Each image slice consists of 256 ×256 pixels with pixel-spacing values between 0.93 and 1.64 mm. Spacing-between slices ranges from 6 to 13 mm.

We also used a cardiac CT dataset consisting of 15 patients. Tomographic studies were acquired with a SIEMENS 16-slice CT system. The scanner is composed of 128 detectors and synchronized with the ECG signal. Each image consists of 512 ×512 pixels, quantized to 12 bits × pixel. There are 10 volumes for each patient describing the entire cardiac cycle form diastole to systole.

Figure 7 shows image examples taken from each dataset. It can be noted the high variability regarding the image features.

Fig. 7
figure 7

Image examples used for validation. Images show the LV at base (left), middle (center) and apex of the heart (right)

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3.2 Parameters selection

In this work, we used experimental tests in order to determine the best set of parameters to configure the algorithm. Free parameters in the Hermite-based multiscale stage are the scales and subsampling values controlled by σ and T. In the level set functional, we must assign weight values for each energy term. In our experiments, parameters of the HT were fixed while parameters of the AC functional were configured for each test. Table 2 describes the algorithm configuration. Values were assigned depending on the dataset.

Table 2 Parameters configuration of the proposed algorithm
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3.3 MR volumes

Results using the first set of MR data using the DSC metric are presented in Table 3. The DSC was computed for all slices at ED and ES phase for each patient. Results are finally averaged for all patients and presented separately for the endocardium and epicardium. The best result for the endocardium was 0.9387 and the worst 0.8323. Similarly, for the epicardium we obtained a maximum DSC of 0.9441 and a minimum of 0.8824. Comparisons with other methods which have used the same dataset are presented in Table 3 as well. It can be noted that our method achieved better results as those reported in the state of art.

Table 3 DSC obtained using segmentation results of four methods, including the proposed approach. Evaluation made using images of MR dataset 1
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Results obtained for the first and second dataset using the PCD and MHD metrics are presented in Table 4. A Box-Plot graph (see Fig. 8) shows the behavior of the proposed segmentation method for all patients using the PCD metric. It is accepted that a good segmentation is achieved when the computed distance is less than 5 mm [38].

Table 4 Average PCD, MHD and PSD (mm) obtained for the endocardial and epicardial segmentation using MR data
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Fig. 8
figure 8

Box-Plot graph using the average PCD (mm) metric evaluated with MR data

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Similarly, results obtained for the second MR dataset are also compared with two different methods of the state of art which used the same database of cardiac MR studies, see Table 5.

Table 5 DSC computed for the proposed segmentation approach and two methods of the state of art using MR dataset 2
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Visual performance of the proposed segmentation method is evaluated in Fig. 9. Volume slices selected from each MR dataset are illustrated. The LV is visualized at base, middle and apex of the heart. The red contour is the manual annotation and the green contour is the segmentation obtained with the proposed algorithm.

Fig. 9
figure 9

Visual results for LV segmentation at base (first column), middle (second column) and apex (third column) of the heart for diastole and systole phases using MR volumes

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We also addressed evaluations using a 3D metric. For this purpose, we employed the PSD metric whose results are presented in Table 4 for all MR data. Surface examples obtained from the segmentation are visualized in Fig. 10. Examples are presented for the endocardium at ED and ES phase, and for the epicardium at ED phase.

Fig. 10
figure 10

Surface examples obtained with the segmentation. The red surfaces correspond to the manual segmentation. The green surfaces were obtained using the proposed algorithm

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Bland-Altman analysis was also performed to evaluate the segmentation using clinical indices. The main objective of the segmentation in this application is to provide an efficient mechanism to aid in the cardiac function assessment. End-Diastolic Volume (EDV), End-Sysolic Volume (ESV), Stroke Volume (SV) and Ejection Fraction (EF) are typical measurements used for LV evaluation [4, 11, 15, 21, 30, 32]. These metrics can be easily computed from the segmentation results. Bland-Altman analysis applied to these clinical indices is depicted in Fig. 11. Here, all volumes gathered from both MR datasets were used.

Fig. 11
figure 11

Bland-Altman graph computed for EDV, ESV, SV, and EF indices using all MR volumes, with μ being the mean and std corresponding to standard deviation. Green lines define the limits of agreement

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3.4 CT volumes

We also addressed evaluations of the algorithm using cardiac CT volumes. Results are presented for the LV cavity. Unlike MR studies, the contrast is lower in cardiac CT and boundaries which separate heart structures are more difficult to identify. However, cardiac CT studies present images with bigger size and the number of slices substantially increases, ranging from 80 to 170. Similarly, the LV cavity is one of the brightest regions in cardiac CT images. On the other hand, cardiac CT images are normally acquired using the original axial view with respect to the human body. This implies that a rotation process is needed in order to provide the short axis view used for LV evaluation. Figure 12 illustrates a segmentation example using the proposed approach. Results for contours are illustrated. The PCD, MHD and PSD obtained are presented in Table 6 for ED and ES phase. Likewise, obtained surfaces are presented in Fig. 12 as well. A Box-Plot graph using the PCD metric is illustrated in Fig. 13 which was also configured to show results for the end-diastolic and end-systolic phases. In addition, Bland-Altman and linear regression analysis using the cardiac clinical indices (EDV, ESV, SV, and EF) are also employed for evaluation whose results are presented in Figs. 14 and 15, respectively.

Fig. 12
figure 12

Segmentation of the LV cavity obtained with the proposed approach using a cardiac CT study. First row: volume and slices at ED phase. Second row: volume and slices at ES phase. Slices are presented for base, middle and apex of the heart

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Table 6 Average PCD and MHD (mm)
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Fig. 13
figure 13

Box-Plot graph using the average PCD (mm) metric evaluated with CT data

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Fig. 14
figure 14

Bland-Altman graph computed for EDV, ESV, SV, and EF indices using CT volumes

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Fig. 15
figure 15

Linear regression analysis using the manual and automatic segmentation for EDV, ESV, SV and EF

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4 Discussion

The objective of this work is to develop a segmentation algorithm for LV analysis in cardiac MR/CT volumes. The performed experiments demonstrate that the proposed approach is able to extract the LV cavity in CT studies, and the endocardial and epicardial boundaries in MR data. The Box-Plot graph in Fig. 8 and results presented in Table 4 show a PCD less than 2 mm achieved for all evaluated examples using both MR datasets. This value is within the acceptation error which must be less than 5 mm [38] in order to be considered as a good segmentation. The MHD is an alternative distance metric used for performance evaluation. Although this metric presents higher values than PCD, it can be noted that results are still within the acceptation error. The PSD metric was also used for evaluation. Results were exposed in Table 4 demonstrating the good performance achieved.

In our experiments, we also addressed quantitative comparisons with other methods of the state of art. The DSC metric was used for this purpose. This is one of the most standard metrics used for segmentation evaluation. Most of the authors use this metric for performance assessment. Tables 3 and 5 describe results obtained with the proposed method and others found in the literature. It can be noted that results of our algorithm outperform in most cases the segmentation achieved with the other methods [23, 33, 35, 36, 41]. Unlike the compared methods, we used a generalized scheme which can be configured to be applied satisfactorily to both regions, endocardium and epicardium.

In general, the epicardial boundary is more difficult to detect due to inhomogeneities surrounding the myocardium in these types of cardiac studies. Variability of contrasts and intensities in this region is a direct consequence of different types of tissues connected with the myocardium. Since we are using an AC, we need to prevent leaking problems using a higher weight value for the shape energy (see Table 2).

Figures 9 and 10 illustrate visual results of the segmentation applied to MR data. They are compared with the manual annotation. The obtained contours and surfaces are very close to the manual segmentation.

Bland-Altman graph in Fig. 11 shows results using four typical cardiac indices. They have been calculated using all MR volumes. These indices are the final objective of the segmentation since they allow physicians to assess the LV function. It can be seen that most of the evaluated examples lie inside the limits of agreement for all cases.

The algorithm was evaluated using cardiac CT studies as well. Figure 12 shows visual segmentation results for the LV cavity. Figures 14 and 15 illustrate the performance achieved using the clinical indices. Linear regression analysis demonstrates the effectiveness of this work. A high correlation between the manual and automatic segmentation was obtained. Quantitative performance is exposed in Table 6. In general, the LV cavity is more difficult to segment at ES phase. Naturally, the cavity at this stage is more irregular and the size of the papillary muscles appears to be increased with respect to the region covered by the blood pool. Conversely, the latter characteristic is even problematic for physicians when performing manual segmentations because external boundaries of the papillary muscles are not visible in some parts (see Fig. 12).

Regarding the designed method, the selection process of the algorithm parameters has been made experimentally. It can be noted from Table 2 that more scales were used with the HT when using cardiac CT volumes. In addition, the window size used for the local region energy is bigger. This is logical because the size of volumes in CT is bigger than in MR. Perhaps, the most critical issue regarding the parameter configuration is to assign the weight values for the shape energy. If this value is too high during the first iterations, the AC does not evolve properly because the algorithm tries to maintain the elliptical shape of the interface on each x y − p l a n e. Figure 16 shows the effect of varying the weight value for the shape parameter. Images were taken from a CT volume. It can be noted that increasing the weight value helps to maintain the elliptical shape of the contour which avoids the segmentation of papillary muscles.

Fig. 16
figure 16

Segmentation results using different weight values for the shape parameter. The first row illustrates a basal image. The second row corresponds to an image in the middle of the LV

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On the other hand, the local region energy can easily handle illumination changes and inhomogeneity problems. It can also be adapted to any type of images. Figure 17 illustrates a comparison between the proposed algorithm and two classical methods (the Chan-Vese model and a Fuzzy C-means algorithm). This comparison was made using a MR volume with inhomogeneity problems. Due to the local region energy, the proposed active contour can handle these issues present in the LV cavity. Even though it is very difficult to provide parameter values without having some knowledge of the input data, it should be very interesting as future work to design an automatic mechanism for this task.

Fig. 17
figure 17

Results of three methods applied to segmentation of the LV cavity in a MR volume with inhomogeneity problems. Slices at base (first row) and middle (second row) of the heart are presented

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The 3D HT has been used as model for image data coding. The main advantages of this transform are the ability to provide local direction analysis, multiresolution and multiscale analysis, different types of texture features, and it is also easy to implement in any dimension. It can be verified in Fig. 2 how the useful information is compressed with the steering process, concentrating most of the energy in a small number of coefficients. In this work, we are employing coefficients until first order but the algorithm can be easily adapted to incorporate more coefficients for this and other applications if needed.

Finally, we are interested in developing an application for evaluating the heart mechanical function using MR and CT studies. As future work, the segmentation will be combined with motion estimation methods with the aim of providing a robust and complete tool which integrates these algorithms and computes different types of clinical indices used for physicians to evaluate the LV function. Some synchronization problems of the LV muscle, and patients with heart failure and cardiac insufficiency can be identified through a clinical quantification of heart structures analyzed with these methods.

5 Conclusion

We have proposed a robust hierarchical multiscale scheme which uses an AC model and the Hermite transform to segment cardiac MR/CT volumes. Experimental results demonstrate the implemented method can be efficiently used for LV evaluation. Several studies taken from two cardiac MR databases and our own CT collection of data were used for validation. As future work, we intend to adapt the algorithm for segmentation of other heart structures and other cardiac views in order to provide a more general framework useful for cardiac evaluation. Given the properties of the HT and the implicit AC, the proposed method can be extended to any dimension.