An Image Space Algorithm for Morphological Contour Interpolation

An image space algorithm for morphological interpolation between contours is presented. Image space interpolation avoids the need to represent or store contour data using intermediate data structures. The algorithm makes use of basic morphological transforms such as dilation and erosion and interimage operations such as XOR and union. Morphological interpolation is applied successfully to a variety of synthetic contours as well as naturally occurring contours such as those found in medical images or topographic maps [17]. The algorithm interpolates between nested, overlapping, nonoverlapping, or branching contours in a general way although nonoverlapping or minimally overlapping contours require initial registration. The algorithm is particularly appropriate for generation of digital elevation maps or whenever the original contour data is derived from a regular sampling grid. Image space morphological interpolation exploits pipeline architectures allowing simultaneous generation of interpolated contour values while making essential use of neighboring contour morphology. In addition, there is a logarithmic gain in the number of interpolated points when processing a contour interval exhaustively. keywords: contour interpolation, morphological transforms, parallel, height grid, DTM, DEM, cartography 1 . Introduction Many real world objects are effectively and succinctly represented by contours. For example, geologic terrain surfaces can be represented by nested, usually nonintersecting, isocontours found in topographic maps. Isocontours often are used to generate Digital Terrain Models (DTMs) or (discrete) Digital Elevation Models (DEMs) in automated cartography [1-4]. Contours also may be extracted from and used to represent closed three-dimensional objects such as medical anatomy. For example, contours of anatomical boundaries may be detected automatically from serial transaxial cross sections in CT or MRI scans [5]. In the case of closed three-dimensional objects, contours from two spatially adjacent slices frequently intersect or overlap when superimposed. Because it may not be economically or physically practical to densely sample the object of interest, contours usually provide only a sparse representation of the object(s) from which they were extracted. As a result, interpolation schemes [6-10, 18] often are necessary to recover the original three-dimensional surface geometry. Thus, connection of, or equivalently, interpolation between contours is a general problem in computer graphics. The most significant advancements in approaching this problem have come through algorithms which address the correct mapping or correlation of contours points at one level with those at an adjoining level [6, 18]. Some of the greatest difficulties associated with a general solution to the problem of contour interpolation arise due to topological changes and/or overlap between adjacent contours, such as when contours differ in both position and number from one level to an adjacent level. Such is the case when there is a natural branching of the surface geometry. Even without branching, if there is a striking disparity in contour shape or position between two adjacent levels, interpolation algorithms may fail to provide a smooth transition of object geometry at intermediate levels. Thus, efficient and robust contour interpolation algorithms which make essential use of contour morphology at both the local and the global level are still needed. A new image space contour interpolation algorithm which exploits both local and global contour morphology has been developed. The algorithm makes use of morphological transforms (such as dilation and erosion) and other image-level logical operations (AND, OR, or XOR), all of which operate directly in image space. The main idea of the algorithm is to find the midline between two contours and use it to split the intercontour space into two halves, each of which can be processed in the same way. This process is repeated recursively until the intercontour space is exhausted (i.e. filled with midlines). If the initial two contours overlap, the first midline simply passes through the point of intersection and the algorithm proceeds as usual. When compared to existing techniques, image space morphological contour interpolation offers advantages in robustness, accuracy, and computation. These include 1. Robustness: Handles any number of contours of any shape including branching or overlapping geometries. Nonoverlapping contours must be registered. 2. Accuracy: Makes essential use of contour morphology; local shape is extracted using dilation and erosion operations while global shape is represented by midlines. 3. Computation: All operations are performed directly in image space which avoids intermediate contour representation and storage while exploiting pipeline architectures. For nested contours this results in massively parallel speedup since all contour intervals as well as entire families of interpolated contour points within each interval are processed in parallel. In particular, for m contour intervals of width n the parallelism increases O(2m-1) with each recursion while the number of operations decreases O(log2n). This is because contour intervals are essentially split in two with each iteration. Morphological contour interpolation is very well suited to generation of height grid DEMs from discrete isocontours as is illustrated in this paper. However, the algorithm is also extensible and applicable to other 3D objects whose contours originate on the grid such as heart contours from CT scans [5]. A brief introduction to mathematical morphology is given in Section 2. This is followed in Section 3 with a definition of and a distinction between the midline and the medial axis of a region. Section 4 presents the algorithm for morphological contour interpolation followed by results from both simulated and real world contours in Section 5. Section 6 contains a summary of algorithm features with suggestions for future work. 2 . Mathematical Morphology One of the strengths of mathematical morphology lies in its ability to decompose complex shapes into their meaningful parts. In fact, some morphological transforms result directly in structures, such as the medial axis, which have powerful intrinsic shapedescribing content. The intent here is to present briefly some morphological operations which are integral to the problem of contour interpolation. For a more detailed treatment of mathematical morphology see references [11-12]. 2.1 Dilation and Erosion The most basic morphological transforms are dilation and erosion. Dilation and erosion transforms exist for both binary and grayscale images. We first define binary dilation, rb, and erosion, sb. Let X be a binary-1 object such as shown in the image in Figure 1a. (Black dots indicate binary-1 pixels; empty squares have value 0.) Let B be a structuring element of binary-1 pixels such as shown in Figure 1b. (A structuring element is similar to a convolution kernel in image processing.) Let Bx be the translation of B so that the origin of B is located at position x in the object image. The binary dilation of X by B (X rb B, Figure 1c) is obtained by passing B over the object image and ORing B to the (initially 0) output image whenever the origin of B is over a binary-1 pixel x ∈ X. Formally, X rb B = {x | Bx ∩ X ≠ Ø} (1) The binary erosion of an object X by B, (X sb B) is obtained by passing B over the object image and plotting the origin of B in the (initially 0) output image whenever B is completely contained in X. Formally, X sb B = {x | Bx ⊂ X} (2) If X were represented by the binary-1 pixels in Figure 1c, binary erosion of X by B would result in the object in Figure 1a. Thus, binary erosion is the dual of binary dilation.