The automatic detection of end-diastole and end-systole frames of echocardiograph images is the first step for calculation of the ejection fraction, stroke volume, and some other features which related to heart motion abnormalities. In this paper, manifold learning algorithm is applied on 2-D echocardiographic images to find out the relationship between the frames of one cycle of heart motion. By this approach the nonlinear embedded information in sequential images are represented in a two dimensional manifold by LLE algorithm and each image is depicted by a point on reconstructed manifold. There are three dense regions on manifold which are corresponded to the three phases of cardiac cycle, wherein there is no prominent change in ventricular volume. By the fact that the end-systolic and end-diastolic frames are in isovolumic phases of cardiac cycle the dense regions can be used to find these frames. By calculating the distance between consecutive points in the manifold the isovolumic frames are mapped on the three minimums of the distance diagrams which were used to select the corresponding images. The minimum correlation between these images leads to detection of end-systole and end-diastole. The results on 6 healthy volunteers have been validated by experienced cardiographist and depict the usefulness of presented method.
A number of features that have been developed to characterize cardiac motion abnormalities are based on detection of the end of the left ventricular systole and end diastole frames of echocardiographic image sequences. These frames are routinely used for the measurement of end systolic and end diastolic volumes, Stroke volume, global and local ejection fraction (EF) ratio , Cardiac output and wall thickening which are fundamental parameters for heart function assessment. Moreover, detection of these frames is necessary for various post processing methods like the computation of 2D strain rate  or color kinesis . Thus the automatic detection of the end diastole and end systole frames will lead to fully automatic calculation of these features.
The frame corresponding to the maximal cavity area that is achieved at the expansion phase of the heart is the end diastole frame and the frame corresponding to the minimal cavity area achieved at the contraction phase is the frame assumed to be the end of systole.
Presently, in the routine practice, determination of end systole frame and end diastole frames remains mostly visual through a slow animation of the loops with the trackball. In general, end-diastolic frame is identified in ultrasound sequences in three ways: (a) from the R-wave of the electrocardiogram (ECG); (b) as the instant after mitral valve closure; (c) as the instant of maximum cardiac dimension and end systole frame is selected by the instant preceding mitral valve opening or by the instant of minimum cardiac dimension.
Different image processing approaches have been proposed for automatic detection of end systole frame from 2-D echocardiagraphic sequences. The first method  took the advantage of the rapid mitral opening in early diastole to estimate the end systole from the time signal intensity variation in a cavity region defined thanks to three landmarks: the apex and each angle of the mitral annulus. A principal problem of this method is that requires the identification of three landmarks by the user and also some large differences between the method and reference visual reading were identified. The second method was based on the left ventricular deformation during the cardiac cycle. The correlation coefficients between the end diastolic image and each of the following images were calculated. The image with the lowest correlation value corresponded to the end systolic image. This method was fully automatic and did not require any manual intervention. However, in these cases the curve expressing the correlation coefficient versus image number showed a flat area around. This configuration led to uncertainties in the determination of the end systolic image. The third method was a combination of the two previous methods to overcome their limitations. This method was achieved in three steps. The first step was the application of first method to estimate the curve of intensity variation during the cardiac cycle within the previously defined ROI. After that, the second method was applied and the resulted correlation curve enabled the determination of a window of a few frames around end systolic image of second method. Finally, the research of the peak intensity on the curve was restricted to the defined window and the corresponding image was called end systolic image.
In this paper, a new method for an automatic detection of end diastole and end systole frame on apical four chambers and two chambers left ventricular echocardiographic image sequences using manifold learning algorithm has been proposed. Manifold learning is a recently introduced approach in nonlinear dimension reduction. Image manifolds have been shown to be useful representation tools for MRI sequences and facial images. In this work manifold learning algorithm is implemented on echocardiography images and representing relationship between frames is used to extract end diastolic and end systolic frames in one cycle of the heart. The paper is organized as follows. The Method section gives a brief background in cardiac cycle physiology and followed by a description of manifold learning and LLE algorithm which is applied in this research. Section 3 offers illustrative results on image manifolds of two and four chamber echocardiographic images. Discussion about extracting end systole and end diastole frames is presented.
Material and Method
Since the basic motivation of this study is obtained from the cardiac cycle's physiology, in the following subsection, cardiac events occurring in the cardiac cycle is explained. The method and data acquisition procedure is presented in the next.
2.1 Cardiac Cycle
Cardiac cycle is the term referring to all or any of the events related to the flow or blood pressure that occurs from the beginning of one heartbeat to the beginning of the next. Figure 1 illustrates the Cardiac events occurring in the cardiac cycle. A single cycle of cardiac activity can be divided into two basic stages. The first stage is diastole and the second stage is systole.
To analyze these two stages in more detail, the cardiac cycle is usually divided into seven phases. First, "Atrial Contraction"; it is initiated by the p wave of the electrocardiogram (ECG), which represents electrical depolarization of the atria. Atrial depolarization then causes contraction of the atrial musculature. Second, "Isovolumetric Contraction"; this phase of the cardiac cycle begins when the atrioventricular AV valves close and end with the opening of the aortic and pulmonic valves. During this time, ventricular pressure rises rapidly without a change in ventricular volume (i.e., no ejection occurs).Ventricular volume does not change because all valves are closed during this phase. Contraction, therefore, is said to be "isovolumic" or "isovolumetric". At this time, the ventricular volumes are maximal, which is termed the end-diastolic volume (EDV).Third, "Rapid Ejection"; this phase represents the initial and rapid ejection of blood into the aorta and pulmonary arteries from the left and right ventricles. Fourth, "Reduced Ejection", Approximately 200 msec after the QRS and the beginning of ventricular contraction ventricular repolarization occurs as shown by the T-wave of the electrocardiogram. Fifth, "Isovolumetric Relaxation"; this phase begins when the aortic and pulmonic valves close and ends with the opening of AV valves. Although ventricular pressures decrease during this phase, volumes remain constant because all valves are closed. The volume of blood that remains in a ventricle is called the end-systolic volume. Sixth, "Rapid Filling"; as the ventricles continue to relax at the end of phase 5, the intraventricular pressures will at some point falls below their respective atrial pressures. When this occurs, the AV valves rapidly open and ventricular filling begins. Seventh, "Reduced Filling". As the ventricles continue to fill with blood and expand, they become less compliant and the intraventricular pressures rise thus change in ventricular volume is very little.
2.2 Manifold Learning and LLE Algorithm
Algorithms for manifold learning are based on the intuition that the dimensionality of data sets may be artificially high and each sample can be described as a function of only a few underlying parameters. Hence, algorithms attempt to uncover intrinsic parameters in order to find a low-dimensional representation of the data. In other words, the data points are actually samples from a low-dimensional manifold which is embedded in a high-dimensional space. Isomap , Locally Linear Embeddings (LLE) [13,14], Laplacian Embeddings  and Local Tangent Space Alignment (LTSA) are instances of manifold learning algorithms.
In echocardiography imaging, the dimensionality of frames from a video sequence is usually considered as the number of pixels in a frame, which can be very large; but frames of a particular patient may vary due to a small number of factors such as, deformation caused by the patient's heartbeat, imaging geometry, noise and time-varying effects of contrast agents. When there are only a few comprising causes of the variation, these images have a natural low dimensional structure which can be extracted by manifold learning algorithms . In [18, 19 and 20] manifold embedding of cardiac MRI images offers an opportunity to segmentation, interpolation and noise reduction.
In this study the LLE method is selected, because it has the advantage of "nearby points in the high dimensional space remain nearby and similarly co located with respect to one another in the low dimensional space" .
LLE algorithm transforms N data samples (observations) of X data set with dimensionality D into a new dataset Y consisting of N points with dimensionality d, while retaining the geometry of the data as much as possible.
The embedding is optimized to preserve the local configurations of nearest neighbors. The algorithm, summarized in Figure 2, can be divided into three different steps.
In the first step, the K nearest neighbors for each data sample is determined as measured by Euclidean distance. The results of LLE are typically stable over a range of neighborhood size, in this study the stable range is and we chose k=10 nearest neighbors per data point.
The second step of LLE is to reconstruct each data point from its nearest neighbors. This can be expressed in minimizing the reconstruction error, or the cost function (2):
The embedding cost function can be minimized by solving a sparse eigenvalue problem of cost matrix M. The bottom d +1 non-zero eigenvectors of the sparse, symmetric matrix M provide an ordered set of d embedding coordinates. Note that the dth coordinate output by LLE always corresponds to the (d+1)st smallest eigenvector of the matrix M, regardless of the total number of outputs computed or the order in which they are calculated. Thus, the bottom eigenvectors of Equation (5) can be computed one at a time, yielding a "nested" set of embeddings in successively higher dimensions.
2.3 Data acquisition
The apical two-dimensional gray scale sequences of 10 healthy volunteers were acquired using a Vivid 3 Ultrasound Machine (GE, USA) with a 2.0 MHz blind CW probe and 53 frame rates and stored in AVI format with resolution, including the ECG display. Four-chamber and two-chamber views were used. For conducting the present study, sequences of three successive cycles were stored, with care being taken to ensure that there was no probe or respiratory movement during the data acquisition. Separated cycles were identified by selecting the onset of the QRS complex. To evaluate the proposed method, for each view, the image sequences were visually analyzed by experienced echocardiographers and the end systolic and diastolic frames were visually determined.
In this paper, we presented a new, fully automatic procedure to identify end systole and end diastole frame from black and white 2-D echocardiographic image sequences corresponding to a cardiac cycle using manifold learning technique. This method is fast and does not require any supplementary manual intervention and can be easily implemented. Also no need to ECG references is one of the most advantages of the proposed method.
Since echocardiographic images have natural and non-linear deformations, linear dimension reduction methods cannot recognize the correct relationships between frames. LLE algorithm is one of the most popular manifold learning and non-linear dimension reduction methods. The main advantage of LLE algorithm is that similar frames in high-dimensional space retain their neighborhoods in low-dimension space too. Results of using manifold learning, demonstrated its power in analyzing data sets of echocardiographic images. Since in cardiac cycle there are three phases which ventricular volume does not have any sensible changes, the difference between frames of these phases are very small. It was our principal motivation that we merged it by LLE algorithm which computes low-dimensional, neighborhood-preserving embeddings of high-dimensional images. By embedding images in a 2 dimensional space and then calculating the distance between consecutive points in manifold, three significant frames have been extracted. Finally Correlation between them leads to automatic detection end systole and diastole frames.
Nevertheless, it may seem that the same result can be derived by computing Euclidean distance between consecutive original frames instead of consecutive points in manifold. But, because of inherent noisy nature of echocardiographic images, directly calculating distances between frames does not give such distinguishable result compare to proposed method. Figure 10 shows the calculation of Euclidean distances between points of the manifold and between original frames of echocardiographic images.
The combination of this detection with an efficient edge analysis of the endocardium might be helpful for an automatic and fast calculation of the ejection fraction and stroke volume from echocardiography images.
The setup of this study was limited in several aspects. Only normal subjects were included, whereas for a good and confident result, other pathologies should be included. Moreover the number of data is little. The technical solutions used in this work could be equally applicable to stress echoes and 3D echocardiography which are receiving a lot of interest from the cardiology community at the current time also the same approach could be tested on long-axis and short-axis images. Moreover the methodology is not specific to ultrasonic modalities and could be equally applied to cardiac CT or MR.
Research in image manifold of normal and abnormal cases and effort to relate similar manifolds to special diseases is the future work. We believe that manifold learning algorithm can make a new horizon in analyzing the medical images and particularly echocardiography images.
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