Scale-adjusted distance transform and its applications to segmentation of multimodal images

Distance transform (DT) is widely used for structural analysis of multi-dimensional (mainly 2-D and 3-D) objects. Association of DT values with local structure scale, often, adds challenges and limits the scope of applications of DT in relative structural analysis among multiple objects with varying scales. In this paper, we introduce a new notion of scale-adjusted distance transform (SADT), conceptually different from traditional DT, which is independent of object scale and offers DT values of scale varying objects on a uniform scale with the value of '1' at ridges. It has been shown that scale-adjusted distance is a metric function in a continuous Euclidean space, and SADT generates a normalized field that is invariant under translation, rotation, and isotropic scaling. The computational method for digital objects traces gradient flow paths on a conventional DT field and uses the change in velocity along a digital path to detect local ridges, which are then used to generate a scale-adjusted density (SAD) field. Finally, SADT is computed using the SAD value. The results of applying the method on 2-D and 3-D multimodal image datasets are presented. Two real-life applications of SADT are shown: 1) segmentation of conjoined nuclei from 2-D microscopic images, and 2) multi-scale separation of conjoined artery-vein in 3-D pulmonary CT image of a pig lung phantom. SADT outperforms the traditional marker-controlled watershed algorithm in conjoined nuclei segmentation from 2-D images and achieves highly accurate multi-scale artery-vein separation in the pig lung phantom experiment. The performance of SADT is invariant to image dimension and imaging modality. Unlike modern deep learning methods, the proposed fuzzy method is transparent and data modality independent. The source code and sample data are freely available at: https://github.com/CMATERJU-BIOINFO/Scale-Adjusted-Distance-Transform.