Combining a morphological interpolation approach with a surface reconstruction method for the 3-D representation of tomographic data

Abstract

In this paper, a new interpolation scheme, based on Mathematical Morphology and a modified Marching Cubes (MC) Algorithm to reconstruct 3-D anatomical structures is presented. The proposed interpolation technique is implemented using morphological operations and incorporates a distance function to improve the computational effectiveness of the technique. The morphological interpolation technique is compared to an existing shape based interpolation method and its advantages include superiority capability on handling various cases such as the branching and holes problem (appearance and disappearance of information) and more accurate volume estimation. Furthermore, the morphological technique is companied with a 3-D reconstruction algorithm capable of representing any anatomical structure from real 3-D medical data. Introducing a novel general rule, the algorithm triangulates all standard cube configurations introduced from the standard MC algorithm, without producing topologically incoherent surfaces or holes. Finally, the technique is implemented in JAVA and its output is in VRML 1.0 format; therefore it can be executed over the internet and implemented for telemedicine applications.

Introduction

Medical image interpolation is a process often required for discrete image analysis, manipulation, and visualization. Various interpolation methods have been introduced for the image generation and more frequently for the image post-processing. In the latter, there are cases where interpolation techniques have been applied in order to reduce errors during monomodal or multimodal medical image registration (Meijering et al., 2001), to improve image quality due to an implementation of lossy image compression algorithm (Beucher, 1998), or to improve volumetric medical image representation from various medical imaging devices (Ostuni et al., 1997). In particular, three-dimensional (3-D) anatomical structures are commonly obtained from a sequence of cross-sectional slices (equally spaced 3-D samples) by contiguously scanning a 3-D region of the body. However, in clinical practice, it happens very often to collect a limited number of slices in order to reduce the patient’s exposure and the examination time. Additionally, there are cases where the spacing between slices is not uniformly distributed, resulting in an anisotropic data set. Since most of the visualization techniques operate on equally spaced 3-D samples, a medical image interpolation technique becomes inevitable.

Broadly, interpolation techniques can be divided in two categories: gray level interpolation and binary interpolation methods. In gray level based methods, the interpolated values are determined directly from the gray values of neighboring image elements. Examples of methods belonging to this category are: nearest neighbor, linear, quadratic, cubic, spline family, Lagrange polynomials, truncated sinc, etc. (Meijering et al., 2001; Lehmann, 1999; Thevenaz et al., 2000). The gray level interpolation is based generally on the assumption that if a small change of the shape of object occurred then it would result a small change of the chromatic density of the object. This is sufficient when the images acquired from a modality are dense (e.g., MRI of the brain with resolution 1 × 1 × 3 mm). Medical data are often quite sparse and the application of the gray level interpolation in these data creates blurred intermediate slices and/or it assigns to area intermediate colors that corresponds to artifacts.

Shape based interpolation techniques are object based and applied on structures of interest according to their geometrical characteristics, after a segmentation procedure is applied. In object based methods, some object information is extracted from the given data and is used in guiding the interpolation process. An example of method belonging to this category is that developed by Goshtasby et al. (1992), where correspondence between feature points is used in directing the interpolation process. An efficient binary interpolation algorithm was introduced in (Raya and Udupa, 1990). The algorithm consists of first segmenting (using thresholding) the given image data into a binary image and then converting the binary image back into a gray image wherein the gray value of a point represents its shortest distance from the cross-sectional boundary (positive for the points of object and negative for those outside) (Rausin, 1995). After acquiring the signed distance maps of the two sequential contours, a linear interpolation is performed generating a set of intermediate distance maps according to the required resolution. Then, the inverse procedure is performed creating form the distance maps the binary contour interpolating slices. This algorithm, proposed in (Raya and Udupa, 1990), is also implemented for comparison purposes and it will be referred as shape based interpolation in the rest of the paper. Other interpolation methods on binary objects are based on elastic dynamic interpolation (Burr, 1981; Chen et al., 1990) and direction interpolation, called staircase (Werahera et al., 1995). These algorithms have difficulties when dealing with non-convex objects and the computational requirements are enormously increased.

Interpolation techniques have been also used on binary, mosaic, gray level, and color images, based on mathematical morphology (Serra, 1983, Serra, 1988. These methods were consisted by creating intermediary two-dimensional images between two given ones by morphological filtering. Morphological interpolation approaches were based on the application of either a morphological median (Meijering et al., 2001; Iwanowski and Serra, 1999), or using an interpolation function which describes the relative distance between the objects (Meyer, 1996). In terms of interpolating binary images, a major drawback of these methods is that the intersection of the input objects must be non-empty. The Hausdorff distance was also incorporated for interpolating images in (Serra, 1998), which allows interpolation between disjoint input objects but it creates large interpolated objects compared with the input ones. Furthermore, an interpolation method, trying to correct the disadvantage of the previous method, was introduced implementing in (Iwanowski, 2002). According to this method, a geodesic set is used covering both input objects as well as the gap between them. Then, interpolation sets are created by the implementation of morphological dilations of the geodesic set controlled by a distance function. The algorithm does not deal with interpolation of on non-convex objects where irregularities or holes and concavities are presented, as they often occurred in clinical medical data.

Another shape based interpolation technique was performed for two adjacent slices using mathematical morphology operations and conditional erosions applied iteratively (Joliot and Mazoyer, 1993). After segmentation, the two slices are scanned and three regions are identified containing common, non-common and intermediate points, respectively. Since the intermediate slice must contain the common points and part of intermediate points, iterative erosions of the intermediate points by 3 × 3 structuring element are applied until a required number of points is achieved. The algorithm fails if there is a cavity present in start slice, which disappears in the goal. To correct this drawback the algorithm requires the identification and skeletonization of these components and the removal of the skeleton points. Even with these corrections, the algorithm loses its attraction because the skeletonization is a complex morphological procedure and is often difficult to recognize areas that require the above transformation. Furthermore, this interpolation technique is applied in order to interpolate only MRI brain data and it was not extensively tested to other general cases that occur in medical imaging.

In this paper, a 3-D medical imaging representation scheme is presented. The scheme consists of the novel implementations of two algorithms for interpolating and reconstructing medical data from different imaging modalities. The first algorithm is an interpolation algorithm based on mathematical morphology capable of handling various cases occurring on medical data, including the branching and holes problem as well as the accurate volume representations. The second algorithm, as a reconstruction algorithm, consists of a modification of the standard MC algorithm and it is capable to fast and efficient reconstruct 3-D medical structures. The medical representation scheme is successfully validated on several synthetic data as well as on real medical data providing 3-D medical data to further used for various telemedicine applications.