Abstract— There are several medical image modalities which enable a proper diagnosis of Heart Diseases. Magnetic Resonance imaging is one of the most non invasive techniques used by radiologist for diagnosis nowadays. Segmentation of Cardiac MRI is basically dividing the heart into left and right ventricle, where left ventricle plays a vital role. In this paper the segmentation of Left Ventricle using parametric model is done which helps to calculate geometrical parameters. These parameters further helps to distinguish between an abnormal and normal heart. The segmentation is carried out on multi slice MRI frame. Parametric model used in this paper is Active Contour Model which is also called as Snake Model or Deformable model. The geometrical analysis is shown graphically to show the difference between different patient databases. The clinical diagnosis is also made on the physiological parameters like volume of blood and ejection fraction which will be the future extension to this paper. Key Terms—Active contour models, deformable models, gradient vector flow, snakes, image segmentation.

I. INTRODUCTION Active contours popularly called as snakes which are geometrical curves described within an image that can progress under the influence of internal forces coming from the curve itself and external forces computed from the image data. The internal and external forces are defined so that the snake will lock itself to an object boundary or other desired features within an image. The active contour is applied to the preprocessed cardiac MR images. The preprocessing includes the smoothing and threshold operation. Snakes are used widely in many applications mainly in medical imaging which includes edge detection[6], shape modeling, segmentation , and motion tracking . Parametric active contours are therefore closed curves which are defined within or across an image domain and forces allow them to move toward desired features, usually edges. Typically, the curves are drawn toward the edges by these

forces, which are defined to lock the contour on the region of interest. The internal forces helps to grip the curve together (elasticity forces) and prevents it from bending too much (bending forces). Mathematically internal force of snake model is the summation of elasticity and bending force. The external forces which is used here is a type of a vector field, which are called as gradient vector flow (GVF) fields[4][5]. These are intense vector fields derived from images by minimizing certain energy functional so that the summation of internal and external force is always nullified. This minimization is accomplished by solving a pair of linear partial differential equations that diffuses the gradient vectors of a gray-level or binary edge map computed from the image. The active contour that uses the GVF field as its external force is called as GVF snake. II. METHODOLOGY

A. Parametric Model

A parametric model also called as snake [1][2] is a curve that moves through the spatial domain of an image to minimize the energy functional. During its efforts to minimize its energy, it gets attracted towards the object edges. The user must initialize the contour close to the object of interest. The energy of contour which makes up a snake as set of points is made up of 3 components: Esnake = Einternal + Eexternal (1) The internal energy controls the natural behavior of the snake and it is further summed as: Einternal = Eelastic + Ebending (2) Esnake = Eelastic + Ebending + Eexternal (3) The internal energy is further defined as the weighted summation of first and second -order derivatives.

Anupama Bhan Department of Electronics and Communication Engineering

Amity School of Engineering and Technology Amity University, Noida, India

anupama.bhan@gmail.com

Parametric Models for Segmentation of Cardiac MRI database with Geometrical Interpretation

2014 International Conference on Signal Processing and Integrated Networks (SPIN)

978-1-4799-2866-8/14/$31.00 ©2014 IEEE 711

Eelastic = X (s) 2 ds (4) Ebending =

X (s) 2 ds

The external energy of the snake is derived from the image so that it takes on its smaller values the region of interest such as boundaries.

Eexternal = Eimage X(s) ds (5)

The total energy of the contour is thus given by:

Etotal = [ X (s) 2 + X (s) 2 ] + Eext (X (s))ds (6) The aim here is to minimize this class of problem called as Euler-Lagrange differential equation. Using calculus of variations, it can be shown that a snake that minimizes Etotal must satisfy the Euler equation. Eenergy = X (s) X (s) Eext (7)

Where and β weighting parameters that controls the snake’s tension and rigidity, respectively. X’(s) and X”(s) denote the first and second derivatives of with respect to s. In the above equation, the internal force is given by Fin== X (s) X (s) (8) The internal force tries to stop the stretching and bending of contour or Snake. The external force is given by Fext = Eext (9) The external force tries to pull the snake towards the desired edge or boundary of the image. Here the boundary of the interest is the left ventricle of cardiac MR image. Then the equation should be balanced in such a way that: Fin + Fext=0 (10)

The external energy function derived from the image takes o smaller values at the features of interest, such as boundaries or edges because of its high gray level intensity. The two-dimensional Gaussian function is also used with standard deviation σ and ∇ is the gradient operator for smoothing purpose. Larger value of σ will cause the boundaries to become blurry and distorted. [6], [7] due to which thickness of the edges will increase and it will lead to inaccurate segmentation results. The GVF field points toward the object boundary when very near to the boundary, but varies smoothly over homogeneous image regions, extending to the image border. The main advantages of the GVF field are that it can lock itself to the desired area to be segmented with number of iterations which again depend on how big is the target to be segmented.

B. Edge Map

Once the input image of cardiac MRI is loaded, some preprocessing can be done if required. The second step is to define an edge map f(x, y) derived from the image I(x,y) is important having the property that it is larger near the image edges. Accordingly, we can use f as an edge map of the image I, with the property that it’s large near the image edges. The preprocessing is done using a Gaussian blur by convolving the image I with Gaussian filter using the equation: f = | (G * I ) | . Given a gray level image I(x, y) is viewed as a function of continuous position variables (x, y). From equation (1), the external energies which lead the active contour towards the desired edges are found as: E1

ext (x, y) = I(x, y) 2 and E2

ext (x,y) = G ( (x,y) I(x,y) ) 2 (11)

where G (x,y) is a two-dimensional Gaussian function with standard deviation and gradient operator. The convolution of image with Gaussian blur is more robust in presence of noise. It is easy to see from these equations that larger will cause the boundaries to become blurry but at the same time such large sigma are often necessary, in order to increase the capture range of the deformable contour. If the cardiac MR image is noisy then any noise removal methods like median filtering, morphological filtering and anisotropic diffusion can be used to improve the edge maps.[7][8] The noise removal technique used in this paper is Gaussian blur. The Cardiac MRI which is used in this paper is Short Axis MRI, Because of the fact that this type of MRI has better visualization of left ventricle. In this paper, the database for short axis MRI is taken from Auckland Medical Research centre which has the database for Normal Heart and Heart suffering from diseases like RVOT and Ischemia.

C. Gradient Vector Flow snake(GVF) We define the gradient vector flow (GVF) field to be the vector field V(x,y)= (u(x,y) v(x,y))that minimizes the energy functional E = ( ux

2 + uy2+ vx

2 +vy2 ) + f 2 V f 2 dx dy

(12) where f is an edge map of the image I, with the property that it’s large near the image edges. To define Eext from equation 6, firstly edge map is obtained of the image and Fext is calculated as the vector field that minimizes the integral of equation 12. In particular, we see that when ∇f is small, the energy is dominated by partial derivatives of the vector field, yielding a smooth field. The parameter μ is a regularization parameter which should be

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set according to the amount of noise present in the image (more noise, increase μ). [3]. The GVF field can be found by solving the following Euler equations: μ (u fx) (fx

2+ fy2) = 0

(13) μ (v fy) (fx

2+ fy2) = 0

where is the Laplacian operator. These equations are solves iteratively and in a homogeneous region where I(x, y) is constant the second term in each equation is zero because the gradient of f(x, y) is zero. The above solution of GVF is valid for gray level Magnetic Resonant Images of left ventricle.

D. Flow Diagram of Algorithm

III. RESULTS

The GVF snake is applied first on a single cardiac image. The initialization is done using the approximation of circle and values of elasticity and bending parameters are adjusted so that we achieve proper segmentation results. The results of GVF snake algorithm with final locked left ventricle is shown in Figure6. Finally the algorithm is applied to entire cardiac cycle provided by Auckland Medical Research centre database. The values of alpha, beta and gamma are 0.05, 0.8 and 0.4 respectively.

Figure1. Input Image of a Normal Heart

Figure2. Preprocessed Smoothed image with gaussian filter

Figure3.Edge map of the input image

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Figure4.Gradient vector Flow Field

Figure5. Segmented output image

No of iterations: 80

Figure6.Segmenting Cascaded Cardiac Cycle

Figure 7. Graphical Analysis Area Vs frames

Red-Normal Blue- Ischemia Green- RVOT

Figure8. Perimeter Vs frames

Figure9. Cross correlation vs. frames.

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IV. GEOMETRICAL INTERPRETATION

As mentioned earlier that clinical diagnosis of cardiac function is evaluated based on physiological measurements such as End-Diastolic Volume (EDV): the volume of blood in a left ventricle at the end of diastole, or filling stage and End-Systolic Volume (ESV): the volume of blood in a ventricle right after systole, or ejection stage. Though this will be future extension to my work. At present I have concentrated on the calculation of geometrical parameters which will enable to distinguish between three patterns of different patients. The diagnosis achieved on the basis of geometrical parameters related to the left ventricle is: Area: Number of pixels inside the contour of the left

ventricle Perimeter: Number of pixels on the contour of the left

ventricle 2D Cross correlation: Computes the correlation between two

cardiac images

From the graphical analysis, it is concluded that the plot in red color shows symmetry between consecutive cardiac images in one frame. In one Cardiac MRI frame, there are 32 slices or images. This indicates the proper function of heart during filling and ejection stage. Whereas other two cases shown in blue and green color indicates irregularity or unevenness between consecutive slices. This further indicates the abnormality in cardiac function.

V. CONCLUSION

The Parametric models are found to be better segmentation approach than traditional edge based and region based methods. The Gradient vector flow model is based on mathematical formulations which creates a vector field that forces the contour towards the desired boundary. The main challenge in this paper was to achieve segmentation results during each diastolic and end systolic phase of heart. GVF has been proved efficient segmentation method as it gives flexibility to adapt itself despite the fact that the shape of left ventricle changes during filling and ejection period. It can be concluded that proper segmentation is achieved and geometrical interpretation of each database is obtained which indicated the difference between the phases of normal versus abnormal heart.

VI. FUTURE WORK The future extension to my work is evaluation of clinical parameters like ejection fraction and volume of blood. And more challenging research work in the future is aimed at segmenting both left and right ventricle in same cardiac MRI frame which will reduce the problem of time consumption.

REFERENCES

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[2] C. Xu and J. L. Prince, “Snakes, Shapes, and Gradient Vector Flow.” IEEE Transactions on Image Processing, 7(3): 359-369, 1998.

[3] Chenyang X, Prince J L (1997): Gradient vector flow: A new external force for snakes, IEEE Proc. Conf. on Computer Vision and Pattern Recognition, pp 66–71.

[4] F. Leymarie and M.D. Levine, "Tracking deformable objects in the plane using an active contour model." IEEE Transactions on Pattern Analysis and Machine Intelligence.15(6). 617-634, 1993.

[5] Cohen L D (1991): On active contour models and balloons, CVGIP Image Understand, vol. 53, pp. 211–218.

[6] K Canny. “A computational approach to edge detection.” IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7): 629-639, 1990.

[7] V Caselles, R Kimmel, G Sapiro. "Geodesic Active Contours." International Journal of Computer Vision. 22(1): 61-79, 1997

[8] Surendra Ranganath(1995), “Contour Extraction from Cardiac MRI studies using Snakes,” IEEE Trans .on Medical Imaging,vol. 14, no. 2.

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