We use a dataset consisting of 22 3-D T2 fat-suppressed MR images of the prostate from individuals with BPH. T2 fat-suppressed MRI provides good contrast not only between the prostate and its surrounding tissue but also between the prostatic anatomical zones. The images were acquired using a 1.5 T Philips Gyroscan ACS MR scanner. After their acquisition, all images were manually cropped close to the prostate (Fig. 3). The ground truth for each image is a binary volumetric mask produced after averaging the manual delineation of two radiologists on the cropped images. Prior to the experiments, the intensities of all images were normalized to lie in the bounded interval [0, 255]. Lastly, all images were resampled to allow for an iso-voxel resolution and volumes of equal sizes to be created.

The framework’s first stage addresses the localization of the anatomy of interest. We evaluate four different registration methods and AGC. In the following paragraphs, we provide the necessary background information with respect to these techniques.

Registration is a process that establishes a point-to-point correspondence between two images. The images are considered to be identical, but one of them is treated as being corrupted by spatial distortions; therefore, they cannot be aligned in their current form. The two images are known as target and floating image. The terms reference or fixed and template or moving image respectively are also encountered in the literature [38, 41]. The aim of registration is to compute the exact geometrical transformation that the floating image needs to suffer, in order to match the target image. For a more comprehensive review of registration and its constituent components as a framework, readers can refer to the relevant literature (e.g., [38, 41, 42]).

In the context of our framework, the example image plays the role of the template image (Fig. 2). The registration scheme computes the spatial transformation, which best aligns this image to the image to be segmented, denoted as New Image in Fig. 2. Subsequently, the spatial transformation is applied to the template image’s binary mask. The deformed (warped) binary mask constitutes the segmentation suggestion of the framework’s first stage. It also represents the prior knowledge with respect to the segmentation outcome, in the new image’s coordinate system.

The registration methods that we assess are the B-Spline-based registration method of Rueckert et al. which employs nonrigid free-form deformations [43]; Thirion’s demons registration method [44]; the deformable registration method of Glocker et al. [45], which employs MRFs and discrete optimization [46]; and the groupwise registration method of Cootes et al. [47], utilized here in a pair-wise fashion. The implementations of Kroon and Slump [48] were used for the first two registration methods, whereas the authors’ implementations were provided for the latter two methods. In the next paragraphs, we highlight briefly the main components of these registration methods.

The B-Spline method of Rueckert et al. [43] employs a hierarchical transformation model, which combines global and local motion of the anatomy of interest. Global motion is described by an affine transformation, whereas local motion is described by a free-form deformation, which is based on cubic B-Splines. The overall transformation is performed within a multi-resolution setting, which reduces the likelihood of occurrence of a deformation field with invalid topology due to folding of the control points (grid points). The similarity metric employed in this method is normalized mutual information, and the optimization component is based on a gradient descent approach [49].

The underlying concept of the original demons method [44] is that every voxel of the template image is displaced by a local force, which is applied by a demon. The demons of all voxels specify a deformation field, which describes fluidlike free-form deformations; when this field is applied to the template image, the latter deforms so that it matches the reference image. The algorithm operates in an iterative multi-resolution fashion for increased robustness and faster convergence. The original demons algorithm, as presented in [44], is data driven and demonstrates analogies with diffusion models and optical flow equations. Since its original conception, several variants have emerged in the literature [50]. In our study, we use and evaluate the implementation of Kroon and Slump [48], which employs a variant suggested by Vercauteren et al. in [51]. In this variant, the authors follow an optimization approach to demons image registration; more specifically, they employ a gradient descent minimization scheme and operate over a given space of diffeomorphic spatial transformations. Diffeomorphic transformations can be inverted, which is often desirable in image registration. Lastly, Kroon and Slump employ the joint histogram peaks as the similarity metric of their implementation, which allows for the computation of local image statistics [48].

The registration method of Glocker et al. [45], denoted as DROP (deformable image registration using discrete optimization), follows a discrete approach to deformable image registration. Similarly to the method of Ruckert et al., local motion of the anatomy of interest is modeled by cubic B-Splines. The difference, however, is that image registration is reformulated as a discrete multi-labeling problem and modeled via discrete MRFs. In addition, their optimization scheme is based on a primal-dual algorithm, which circumvents the computation of the derivatives of the objective function [46, 52]. This is due to its discrete nature. In their approach, they also follow a multi-resolution strategy, which is based on a Gaussian pyramid with several levels. Moreover, diffeomorphic transformations are guaranteed through the restriction of the maximum displacement of the control points. The authors’ implementation, which is available online [53], features a range of well-known similarity metrics. In this study, the sum of absolute differences was employed.

The registration method of Cootes et al. [47], denoted as GWR, belongs to the groupwise approaches to image registration. Groupwise registration methods aim to establish dense correspondence among a set of images [47], as opposed to pair-wise approaches. In the context of groupwise registration, every image in the set is registered to the mean image, which evolves as the overall process advances. Groupwise registration is often employed in the context of automatic building of shape and appearance models from a group of images, given few annotated examples (e.g., [54]). Conversely, models of shape and appearance can assist registration, when integrated in the process, by imposing certain topological constraints in the spatial deformations. As a result of this integration, Cootes et al. follow a model-fitting approach in their registration method; for each image that is registered to the reference (mean) image, the parameters of the mean texture model are estimated, so that the texture model fits the target image. The overall aim in the process is to minimize the residual errors of the mean texture model with respect to the images of the set. This is achieved via an information theoretic framework, which is based on the minimization of description length (MDL) principle, described in [55]. Piecewise affine transformations are employed for the spatial transformations, which guarantee invertibility of the deformation field. A simple elastic shape model is employed to impose shape constraints to the transformation. Similarly to the previous methods, the registration technique of Cootes et al. follows a multi-resolution approach [47]. In our work, we utilize this method in a pair-wise fashion. We achieve this by employing the reference image to play the role of the mean image. Consequently, the texture and shape model are derived from this single image.

AGC [36] exploits an approximate segmentation as initialization, in order to compute the optimal final segmentation outcome via max-flow/min-cut algorithms. The authors pose no restriction on the nature of images that this algorithm may process and they claim that convergence is achieved even when the approximate segmentation is very different from the desired one. In the context of our study, this approximate solution is provided by the surface of the segmented example image. While it does not perform registration, AGC does provide a method of propagating an initial segmentation and is a likely candidate method for the first stage of our framework. For the needs of our experimental work, we realized our own implementation of the method in MATLAB^®. To the best of our knowledge, this is the first independent implementation and evaluation of this technique in the context of 3-D medical image segmentation. In the following paragraphs, we provide further details about the method.

AGC is a graph-based method; therefore, in the context of this technique, an image is represented as a graph. Its initialization, termed “initial cut”, is a set of contiguous graph edges, which separates the overall graph into two subgraphs that form two independent flow networks. The vertices adjacent to the initial cut are connected to the source graph terminal. Their t-link weight is equal to the capacity of the adjacent graph edge, which is part of the initial cut. In order to solve the max-flow/min-cut problem, different algorithms may be employed. In [36], the authors suggest a preflow-push [56] approach to tackle this problem on the subgraphs. Preflow-push strategies operate locally and thus flood the network gradually. This in turn generates intermediate cuts as the algorithm progresses. However, in our work, we are rather interested in the final min-cut instead of the intermediate cuts. Therefore, we employ an implementation of a max-flow/min-cut algorithm [57], which follows the augmenting paths approach as described in [58]. The final segmentation outcome is provided by the aggregation of the solutions of the max-flow/min-cut problem on the two subgraphs.

The authors provide no instruction about the positioning of the sink graph terminal on the two subgraphs in [36]. In our work, we observed that different choices may significantly affect the segmentation outcome. For the sake of consistency, with respect to this problem, in all our experiments, the voxels at the image borders were connected to the sink graph terminal of the subgraph that lies outside the initial cut, whereas for the subgraph that is contained by the initial cut, the voxels at a fixed distance, using a distance transform, from the centroid of the initial cut were connected to the sink.

In the previous sections, we highlighted the candidate methods for the first stage of our framework: localization of the anatomy of interest in the “New Image” of Fig. 2. The second stage of our framework aims for the delineation of an accurate boundary of the anatomy of interest, given the output of the first stage as initialization.

In a previous study [57], we demonstrated that graph cuts segmentation [39, 40] offers significant advantages over several other methods in the context of interactive segmentation. In the work presented in this chapter, we employ graph cuts in a semi-automated fashion to refine the boundary of the anatomy of interest. As shown in Fig. 2, graph cuts (GC) operates in a zone that surrounds the initial boundary defined by successive erosions and dilations of the initial binary segmentation. Erosion and dilation are morphological operations, which result into contraction and expansion of a binary object respectively [59]. The width of the zone is controlled via a user-defined parameter and depends on the number of erosions and dilations performed on the segmented image example. This is the only interaction that takes place during delineation of the anatomy of interest.