SIMD hypercube algorithm for complete Euclidean distance transform

Anisotropic three dimensional (3-D) Euclidean distance transform (EDT) has important applications in medical image processing, but few algorithms have been reported so far. An anisotropic 3-D EDT algorithm is designed and realized based on Eggers's 2-D algorithm. The characteristic of the algorithm is that all voxels in the propagation contour have the same chessboard distance from the feature voxels. The voxels in the surface of the feature voxel set constitute the initial propagation contour. The voxels in the contour propagate the distance information to the neighbors in special directions, which constitute the new propagation contour. The voxels in the contour are classified into three types, and propagate to seven, three, and one neighbors respectively. The proposed algorithm can generate exact singed and unsigned EDT. The validity of the algorithm is proved theoretically. The speed and efficiency of the algorithm is evaluated, and compared to other algorithm. Methods to reduce the memory requirements are discussed.

A fast and exact Euclidean distance transformation using decomposed grayscale morphological operators is presented. Applied on a binary image, a distance transformation assigns each object pixel a value that corresponds to the shortest distance between the object pixel and the background pixels. It is shown that the large structuring element required for the Euclidean distance transformation can be easily decomposed into 3/spl times/3 windows. This is possible because the square of the Euclidean distance matrix changes uniformly both in the vertical and horizontal directions. A simple extension for a 3D Euclidean distance transformation is discussed. A fast distance transform for serial computers is also presented. Acting like thinning algorithms, the version for serial computers focuses operations only on the potential changing pixels and propagates from the boundary of objects, significantly reducing execution time. Nonsquare pixels can also be used in this algorithm. An example application, shape filtering using arbitrary sized circular dilation and erosion, is discussed. Rotation-invariant basic morphological operations can be done using this example application. >

In this article, we present the Euclidean distance transformation (EDT) of three-dimensional (3D) binary pictures and applications, with stress on Dr. Toyofumi Saito's work in collaboration with one of the authors. First, EDT, skeletons, and reverse EDT are defined, and then the algorithms to perform them are presented. Next, a sequential-type 3D thinning algorithm that preserves the topology of an input object is introduced. This algorithm is derived by combining an ordinary two-dimensional thinning algorithm with EDT. Finally, applications to medical image processing are presented such as automated path finding in virtual endoscopy, as well as analysis of 3D pathological sample images. Dr. Toyofumi Saito, an associate professor who was one of the most active researchers in the field of image processing in Japan and a young leader of the author's laboratory, and who was initially scheduled to be the first author of this paper, passed away on 26 October 2000. We have lost a most reliable and most promising colleague, an experienced supervisor, and a very sincere friend. He is deeply missed by all in his family, his friends, and his colleagues. On behalf of the laboratory the authors wish to dedicate this short note in memory of Dr. Toyofumi Saito and to express our sincere condolences to his loved ones.

Euclidean distance transformation (EDT) is used to convert a digital binary image consisting of object (foreground) and nonobject (background) pixels into another image where each pixel has a value of the minimum Euclidean distance from nonobject pixels. In this paper, the improved iterative erosion algorithm is proposed to avoid the redundant calculations in the iterative erosion algorithm. Furthermore, to avoid the iterative operations, the two-scan-based algorithm by a deriving approach is developed for achieving EDT correctly and efficiently in a constant time. Besides, we discover when obstacles appear in the image, many algorithms cannot achieve the correct EDT except our two-scan-based algorithm. Moreover, the two-scan-based algorithm does not require the additional cost of preprocessing or relative-coordinates recording.

Euclidean distance transform (EDT) using a gray-scale morphological erosion with a large distance structuring element is difficult to implement in real-time. Herein, a stack filtering method that permits the implementation of EDT in real-time using only binary logic gates is proposed. This letter reports a fast errorless method for EDT implementation.

Medial Axis (MA), also known as Centres of Maximal Disks, is a useful representation of a shape for image description and analysis. MA can be computed on a distance transform, where each point is labelled to its distance to the background. Recent algorithms allow to compute Squared Euclidean Distance Transform (SEDT) in linear time in any dimension. While these algorithms provide exact measures, the only known method to characterize MA on SEDT, using local tests and Look-Up Tables, is limited to 2D and small distance values [5]. We have proposed in [14] an algorithm which computes the look-up table and the neighbourhood to be tested in the case of chamfer distances. In this paper, we adapt our algorithm for SEDT in arbitrary dimension and show that results have completely different properties.

In this paper, we propose an efficient algorithm for computing the Euclidean distance transform of two-dimensional binary image, called PBEDT (Perpendicular Bisector Euclidean Distance Transform). PBEDT is a two-stage independent scan algorithm. In the first stage, PBEDT computes the distance from each point to its closest feature point in the same column using one time column-wise scan. In the second stage, PBEDT computes the distance transform for each point by row with intermediate results of the previous stage. By using the geometric properties of the perpendicular bisector, PBEDT directly computes the segmentation by feature points for each row and each segment corresponding to one feature point. Furthermore, by using integer arithmetic to avoid time consuming float operations, PBEDT still achieves exact results. All these methods reduce the computational complexity significantly. Consequently, an efficient and exact linear time Euclidean distance transform algorithm is implemented. Detailed comparison with state-of-the-art linear time Euclidean distance transform algorithms shows that PBEDT is the fastest on most cases, and also the most stable one with respect to image contents.

This paper proposes a new recognition approach for traffic light based on Euclidean distance transform (EDT) and local contour pattern (LCP). There are two main contributions of this paper. First, this paper combines principle component analysis (PCA) with EDT-based image to detect traffic light colors. The color space for specific colors is partitioned more precisely, which leads to a high recognition rate. Second, we incorporate the above color detection into the contour segmentation of traffic light holder based on the LCP to further improve the recognition rate of traffic light. The experimental results show that our approach is able to detect traffic light far away from camera about 50–80 m and the average recognition rate can reach up to 99.29 %.

The cortical thickness has been used as a biomarker to assess different cerebral conditions and to detect alterations in the cortical mantle. In this work, we compare methods from the FreeSurfer software, the Computational Anatomy Toolbox (CAT12), a Laplacian approach and a new method here proposed, based on the Euclidean Distance Transform (EDT), and its corresponding computational phantom designed to validate the calculation algorithm. At region of interest (ROI) level, within- and inter-method comparisons were carried out with a test-retest analysis, in a subset comprising 21 healthy subjects taken from the Multi-Modal MRI Reproducibility Resource (MMRR) dataset. From the Minimal Interval Resonance Imaging in Alzheimer's Disease (MIRIAD) data, classification methods were compared in their performance to detect cortical thickness differences between 23 healthy controls (HC) and 45 subjects with Alzheimer's disease (AD). The validation of the proposed EDT-based method showed a more accurate and precise distance measurement as voxel resolution increased. For the within-method comparisons, mean test-retest measures (percentages differences/intraclass correlation/Pearson correlation) were similar for FreeSurfer (1.80%/0.90/0.95), CAT12 (1.91%/0.83/0.91), Laplacian (1.27%/0.89/0.95) and EDT (2.20%/0.88/0.94). Inter-method correlations showed moderate to strong values (R > 0.77) and, in the AD comparison study, all methods were able to detect cortical alterations between groups. Surface- and voxel-based methods have advantages and drawbacks regarding computational demands and measurement precision, while thickness definition was mainly associated to the cortical thickness absolute differences among methods. However, for each method, measurements were reliable, followed similar trends along the cortex and allowed detection of cortical atrophies between HC and patients with AD.

A distance transformation (DT) takes a binary image as input and generates a distance map image in which the value of each pixel is its distance to a given set of object pixels in the binary image. In this research, DT's for multi class data (MCDTs) are developed which generate both a distance map and a class map containing for each pixel the class of the closest object. Results indicate that the MCDT based on the Fast Exact Euclidean Distance (FEED) method is a factor 2 tot 4 faster than MCDTs based on exact or semi-exact euclidean distance (ED) transformations, and is only a factor 2 to 4 slower than the MCDT based on the crude city-block approximation of the ED. In the second part of this research, the MCDTs were adapted such that they could be used for the fast generation of distance and class maps for video sequences. The frames of the sequences contain a number of fixed objects and a moving object, where each object has a separate label. Results show that the FEED based version is a factor 2 to 3.5 faster than the fastest of all the other video-MCDTs which is based on the chamfer 3,4 distance measure. FEED is even a factor 3.5 to 10 faster than another fast exact ED transformation. With video, multi class FEED it will be possible to measure distances from a moving object to various identified stationary objects with nearly the frame rate of a webcam. This will be very useful when the risk exists that objects move outside surveillance limits.

Prediction method of dust pollutant diffusion range in building demolition based on Euclidean distance transformation

In this paper, we examine the use of implicit shape representations for nonrigid registration of serial CT liver examinations. Using ground truth in the form of corresponding landmarks manually labeled by a radiotherapist, we carry out an experiment to determine whether nonrigid registration performs better when applied to the original image data or to images constructed from implicit representations of the liver. We compare a variety of standard regularizers (elastic, diffusion, and curvature), similarity measures (sum of squared differences and mutual information), and weighting factors, using three different implicit shape representations: the Euclidean Distance Transform, the Poisson Transform (based on the expected hitting time of a random walk), and a new transform designed to highlight concavities in the shape.

SummaryResistivity measurements are a major input into hydrocarbon reserve estimation and are usually described by Archie’s laws. In this study, we use digital rock physics to analyze the mechanisms of non-Archie and Archie behavior of formation factor (FF) and resistivity index (RI) of low-porosity Fontainebleau (FB) sandstone for ambient conditions and under high confining pressure, respectively.FB sandstone was imaged by micro-X-ray computed tomography (micro-CT) at a resolution of 1 µm. Subresolution details of the grain contact width distribution along with their length were extracted from a set of scanning electron microscope (SEM) images. The nanoscale aperture of grain contacts, which is below tomogram resolution, is accounted for in micro-CT-based numerical calculations by assigning effective porosity and conductivity to individual voxels of the extracted grain contact network. A porosity reduction of grain contacts and open pore space as a function of applied confining pressure is introduced, capturing the pressure dependence. The concept was implemented by grain contact labeling, introducing an additional phase derived from a Euclidean distance transform (EDT). Subvoxel stress-strain effects were incorporated by attributing all compressibility effects to the pore space (open pore space and grain contacts), treating the solid phase as perfectly rigid. Voxel-scale input conductivities are assigned using Archie’s law followed by solving the Laplace equation for sample-scale rock resistivity and RI directly on the segmented image using the finite element method.For the numerical modeling of the FF and RI of low-porosity FB sandstone as a function of confining pressure, which depends on subresolution features, a set of hypotheses were tested. These are based on two segmentation scenarios incorporating the measured contact aperture distribution from SEM analysis—a homogeneous aperture-based segmentation by assuming all grain contacts as an average constant value and a heterogeneous aperture-based segmentation assigning two groups of grain contact apertures. The segmentation scenarios enable homogeneous and heterogeneous morphological change of grain contacts due to confining pressure effects. Furthermore, partial saturation of grain contacts is considered. In all cases, strong water-wetness was assumed, and discretization effects were analyzed carefully.The numerical results highlight the relative contribution of each of two conductive components of FB sandstone (open pores vs. grain contacts) over the full range of partial saturations. Of importance is the connectivity of the system, with discretization effects having a significant effect on FF, but a small effect on the RI. Grain contacts and confining pressure are found to have a significant impact on RI behavior of low-porosity FB sandstone. Both the grain contact network with homogeneous aperture and the heterogeneous grain contact network are able to describe experimental observations. However, it is not sufficient to assume a homogeneous change in contact area, and an inhomogeneous deformation of grain contact zones is required to match the experiment.

The Euclidean distance transform in arbitrary dimensions

Fast erosion and dilation by contour processing and thresholding of distance maps