Precise Euclidean distance transforms in 3D from voxel coverage representation

Highlights

• We propose a method for computing Euclidean distance transform (EDT) in 3D images.

• The method utilizes voxel coverage information to increase precision of EDT.

• The method can be used with any vector propagation based EDT in 3D.

• Synthetic tests confirm significant improvement in achieved precision.

• Both the related binary and the existing coverage based methods are outperformed.

Abstract

Distance transforms (DTs) are, usually, defined on a binary image as a mapping from each background element to the distance between its centre and the centre of the closest object element. However, due to discretization effects, such DTs have limited precision, including reduced rotational and translational invariance. We show in this paper that a significant improvement in performance of Euclidean DTs can be achieved if voxel coverage values are utilized and the position of an object boundary is estimated with sub-voxel precision. We propose two algorithms of linear time complexity for estimating Euclidean DT with sub-voxel precision. The evaluation confirms that both algorithms provide 4–14 times increased accuracy compared to what is achievable from a binary object representation.

Introduction

A distance transform (DT) assigns to each image element its distance to a selected subset of image elements. Often, the selected set is an observed object in the image, and every image element is mapped to its distance to the object. The concept of DT is closely related to many procedures and operations in image processing, such as computation of morphological operations and/or geometrical representations, image segmentation, template matching, image registration, and many others. Interest for further development and improvement of methods and algorithms for DT computation, as well as for finding new applications of DTs, remains high till nowadays. Even though the main idea of DT is rather straightforward, its efficient implementation is far from trivial. Many algorithms are proposed over the years, differing in terms of time complexity, accuracy, order of pixel processing, suitability for parallelization, etc.

The Euclidean distance is most often considered in DTs, due to its properties desired in many applications. The appealing property of rotational invariance of Euclidean DT (EDT) is, however, challenged by discrete binary object representations in rectangular grids, most often used in practice. We argue for efficient utilization of information contained in acquired images, as a way to address these challenges. Intensity values in original images often relate to the (partial) coverage of each image element by the imaged object. If this information is utilized to provide coverage object representations [13], object boundary position can be estimated with sub-pixel precision. This information can be successfully utilized in EDT estimation, as was shown in [4] for the 2D case. In this paper we propose to utilize the coverage approach and volume sampled image representations to improve the performance of vector propagation based EDT algorithms in 3D. We present two algorithms for computing approximate EDT in 3D which, as shown in the tests conducted, provide between 4 and 14 times lower error of distance values compared to a binary DT, at a reasonable (linear) computational cost, ($\mathcal{O}\left(N\right)$ where N is the number of the image voxels).

The paper is organized as follows: Section 2 provides a brief overview of related work on DTs and on the Coverage model. In Section 3 we present a novel method for estimating 3D EDT with sub-voxel accuracy. Section 4 contains results of performance evaluation of the proposed method. Section 5 summarizes and concludes the paper.