Multi-object segmentation framework using deformable models for medical imaging analysis

Abstract

Segmenting structures of interest in medical images is an important step in different tasks such as visualization, quantitative analysis, simulation, and image-guided surgery, among several other clinical applications. Numerous segmentation methods have been developed in the past three decades for extraction of anatomical or functional structures on medical imaging. Deformable models, which include the active contour models or snakes, are among the most popular methods for image segmentation combining several desirable features such as inherent connectivity and smoothness. Even though different approaches have been proposed and significant work has been dedicated to the improvement of such algorithms, there are still challenging research directions as the simultaneous extraction of multiple objects and the integration of individual techniques. This paper presents a novel open-source framework called deformable model array (DMA) for the segmentation of multiple and complex structures of interest in different imaging modalities. While most active contour algorithms can extract one region at a time, DMA allows integrating several deformable models to deal with multiple segmentation scenarios. Moreover, it is possible to consider any existing explicit deformable model formulation and even to incorporate new active contour methods, allowing to select a suitable combination in different conditions. The framework also introduces a control module that coordinates the cooperative evolution of the snakes and is able to solve interaction issues toward the segmentation goal. Thus, DMA can implement complex object and multi-object segmentations in both 2D and 3D using the contextual information derived from the model interaction. These are important features for several medical image analysis tasks in which different but related objects need to be simultaneously extracted. Experimental results on both computed tomography and magnetic resonance imaging show that the proposed framework has a wide range of applications especially in the presence of adjacent structures of interest or under intra-structure inhomogeneities giving excellent quantitative results.

Appendix: Snake formulations

Snakes are explicit deformable models that can evolve toward an object boundary within the image under the influence of internal forces and external forces. After the initial proposal of Kass et al. [10], several formulations have been proposed [9]. In this paper, we consider two well-known DM techniques as examples. The first method is based on the T snake formulations proposed by McInerney and Terzopoulos [13], and the second is the gradient vector flow snakes (GVF snakes) presented by Xu and Prince [23].

1.1 T snake formulation

T snake-based methods are discrete approximations to a conventional parametric snakes model while retaining many of its properties. The model is geometrically represented by a closed polygonal for 2D problems and by a triangular surface mesh for 3D. The deformation of the snake model is governed by discrete Lagrangian equations of motion, and each element $\mathbf {x}(t)$ evolves according to the following motion equation:

$$\mathbf {x}(t+1) = \mathbf {x}(t) - {\varDelta } t \left( a\mathbf {\alpha }(t) + b\mathbf {\beta }(t) - p\mathbf {\rho }(t) - q\mathbf {f}(t) \right)$$

(4)

where $\mathbf {\alpha }$, $\mathbf {\beta }$ are the internal forces (tension and flexion), $\mathbf {f}$, $\mathbf {\rho }$ are the external forces (balloon and image gradient forces), and $a,b,p\, \text{ and }\, q$ are the force weighting parameters.

The T snake formulations modify the original external force adding an adaptive inflation force $\rho (t)$ term depending on the image intensity features.

$$\rho (t) = F(I(x(t))) \cdot \mathbf {n}(x(t))$$

(5)

where $\mathbf {n}$ is the unitary normal vector to the model, and F is a binary function relating $\rho$ to the intensity field of the image I:

$$F(I(x(s,t))){\text{ }} = \left\{ {\begin{array}{*{20}l} { + 1} & {{\text{if}}\;\frac{{|I(x(s,t)) - CI(r)|}}{{k\sigma (r)}} \le 1} \\ {- 1,} & {{\text{otherwise}}} \\ \end{array} } \right.$$

(6)

where CI(r) is the characteristic intensity of the ROI, $\sigma (r)$ is the ROI intensity deviation, and k is an input parameter.

1.2 GVF snake formulations

Gradient vector flow (GVF) snakes introduce a new external force for active contour models. The difference between traditional snakes and GVF snakes consists in that the latter converge to boundary concavities and they do not need to be initialized close to the boundary [23].

To improve the original snake formulation, the authors introduced a non-irrotational external force field $v(x,y) = [u(x,y),v(x,y)]$ known as GVF field. The field is calculated as a diffusion of the gradient vectors of a gray-level or binary edge map:

$$GVF = \iint \mu (u_x^2 + u_y^2 + v_x^2 + v_y^2) + | \nabla f |^2 |v - \nabla f|^2 \hbox {d}x \hbox {d}y$$

(7)

where $\mu$ is an input parameter.