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methods, Markov random fields, active contours, model-based, and deformable model methods. As this domain has been studied extensively, there exist many published reviews of medical image segmentation [7, 39, 40], with specialized surveys on deformable models [7,8], vessel extraction [9,10], or brain segmentation [10, 11].

This chapter is organized as follows. In Sect. 4.2, we review the main concepts of the deformable model and its role in image segmentation. We present several types of deformable models according to how they are represented, followed by a description of their energy definition and possible optimization methods. In Sect. 4.3, we compare the performance of these deformable models according to properties such as initialization, topological change handling, stability, etc. In Sect. 4.4, we discuss several cases of developed applications of medical image segmentation and conclude in Sect. 4.5.

4.2 Deformable Models

The basic idea of active contours for image segmentation is to embed an initial contour (or surface in the three-dimensional case) into the image, and to subsequently let it evolve while being subjected to various constraints. In order to detect objects in the image, the contour has to stop its evolution on the boundary of the object of interest. Although the term deformable models first appeared in the work by Kass et al. [12] in the late eighties, the idea of deforming a template for extracting image features dates back much further, with work on spring-loaded templates [13] and on the rubber mask technique [14]. In image processing literature, deformable models are also variously known as snakes, active contours or surfaces, balloons, and deformable contours or surfaces. An extensive review of the current research in this area can be found in [1, 38].

In the following subsection, we present the different types of deformable models classified according to contour representation, and some energy optimization strategies.

4.2.1 Point-Based Snake

This classical snake was firstly proposed by Kass et al. [12] and represents the contour using discrete points. The behavior of this classical snake is usually modeled by a weighted linear combination of: internal energy calculated from the contour that determines the regularity of the curve; external energy which attracts the contour towards the significant features in the image; and often an additional user energy allowing the operator to better interact with the model.

In the first snake-type, one applies the simplest way to represent the model: a set of discrete points as snake elements (C(s)). Using this representation, closed

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contours can be formed by connecting the last “snaxel” (snake element) to the first one. For segmentation, the snake has to minimize the energy as follows:

E(C(s)) =	Econtour(C(s)) + Eimage(C(s)) + Euser(C(s)) ds.									(4.1)
	Ω
		internal energy				external energy

Applying discrete-point representation, the internal energy can be approximated by accommodating the elasticity and rigidity terms:

Econtour(C(s)) = w1	(s)Eelasticity (C(s)) + w2(s)Erigidity(C(s))
= w1	(s)Cs(s)2 + w2(s)Css(s)2,	(4.2)

where the subscripts s and ss denote the first and second derivatives with respect to the curve parameter. The model behavior is controlled by constants w1 and w2, respectively, weighting the curve elasticity and rigidity.

Internal Energy Definition: In the first term in equation (4.2), Cs(s) represents the elastic energy and makes the snake behave like a membrane. The second term Css(s) represents the contour’s bending energy that makes the model act like a thin plate. Decreasing elasticity allows the contour to increase in length, while increasing elasticity increases the tension of the model by reducing its length. Decreasing rigidity allows the active contour model to develop corners, while increasing rigidity makes the model smoother and less flexible. Setting either of the weighting coefficients to zero permits firstand second-order discontinuities, respectively.

This energy equation can then be discretized using the finite difference method as:

Eelasticity ≈ (xs − xs−1)2 + (ys − ys−1)2 .

(4.3)

This term will minimize the distance between the points on the snake, causing shrinking during optimization of the energy process in the absence of external energy. In a similar way, the rigidity term is discretized as:

Erigidity ≈ (xs+1 − xs − xs−1)2 + (ys+1 − ys − ys−1)2 .

(4.4)

The elasticity definition using finite differences discretization scheme is valid for the condition that the model’s snaxels are evenly spaced [15]. In other cases, a continuity term can be defined that subtracts the average distance of the snaxels, otherwise the energy value will be larger for points which are further apart. This constraint forces the points to be more evenly spaced, and avoids possible contraction of the snake.

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External Energy Definition: The external energy term in equation (4.1) represents the image potential derived from image data and guides the contour in finding the desired object. Ideally, this energy has a minimum value at the feature subject to detection. However, due to the presence of noise, there is often a convergence problem at object concavities. The snake can also integrate the constraint energy to interactively guide itself towards or away from particular features. This energy helps the contour to overcome the initialization problem or the sensitivity to noise.

Kass [12] proposed a weighted sum of the following energy terms in order to detect image features:

Eext(C(s)) = αline · I(s) + αedge · − I2 + αterm · Eterm.

(4.5)

The most common image functional in this model is the image intensity function I. The first term will simply attract the contour to lower or higher intensity values depending on the αline value. Large positive values of αline tend to make the snake align itself with dark regions in the image I(s), whereas large negative values of αline tend to make the snake align itself with bright regions in the image.

The edge energy that attracts the contour towards high gradient values is squared to narrow the edge-gradient response. Similarly, large positive values of αedge tend to make the snake align itself with sharp edges in the image whereas large negative values of αedge make the snake avoid the edges. Eterm is defined to find termination of line segments and corners. Kass proposed to use the curvature of iso-contours in a Gaussian smoothed image to attract the contours towards line termination.

4.2.1.1 User Constraint Energy

Constraint energy is applied to interactively guide the snake towards or away from particular features. This energy helps the contour to overcome the initialization problem or the sensitivity to noise.

Constraint energy was first proposed for the classical snake by allowing the user to attach springs between points of the contour and fix their position in the image plane. Kass define an energy in terms of spring (to attract the snake towards specified points) and volcano (to repulse the snake from specified points) within the image (Fig. 4.1).

The spring term attracts a contour point towards a spring point in the image plane, with a given constant, spring constant. The active contour model is attracted or repelled by the spring depending on this spring constant sign and value. The volcano term acts as a repulsive force between a point on the image at a inverse value of the distance from a point on the snake.

Balloon Force: Cohen [16,17] proposed an additional force that pushes the contour in the direction normal to the contour. Since it can either inflate or deflate the contour, this force is known as a balloon force, defined as:

	(4.6)
Fballoon(s) = kN(s),

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Fig. 4.1 Classical Snake [after [12]]

( )

where N s is the normal unit vector, k is the weighting parameter representing the strength of the balloon force, and the sign of k determines whether the model inflates or deflates.

Incorporating an additional balloon force helps the user with the classical problem of contour initialization when it is not close enough to the desired solution, as shown in Fig. 4.2b. This force also reduces the model sensitivity to noise, and can also push the model into object concavities. It should be noted that this force only has a single direction for all of the evolved deformable models. Hence, to capture the desired object perfectly the initial contour should be totally inside (or outside) the contour. A snake without any additional force and in the absence of image energy tends to shrink into a point to minimize the internal energy. Figure 4.2c shows this condition.

Gradient Vector Flow: The basic idea of this method [18, 19, 41] is to replace the external force term Eext(I) = P(I) with a gradient vector field (ν ), which can be derived from the equilibrium state of the following partial differential equation:

νt = g(| f |) 2ν − h(| f |)(ν − f ).

(4.7)

The first term in (4.7) is referred to as the smoothing term since it produces a smoothly varying vector field. The second term involving (ν − f ) is referred to as the data term, since it encourages the vector field ν to be close to f . The weight functions g(.) and h(.) are applied to the smoothing and data terms.

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Fig. 4.2 Deformable model with additional balloon force
The authors in [5] proposed the following weight functions:
g( f ) = e− | Kf |
h( f ) = 1 − e− | Kf | .	(4.8)

Using these weight functions, the gradient vector flow field will conform to the distance map gradient near the relevant features, but will vary smoothly away from them. The constant K determines the extent of the field smoothness and the conformity gradient.

Figure 4.3 Illustrates the Gradient Vector Flow performance when a thin concavity or sharp corner is present in the segmented object. The Gradient Vector Flow force is able to attract the snake towards the desired contour and towards concave areas. This would not be the case with conventional external energy, since no external force attracts the snake towards that part of the image.