Linear Time Algorithms for Exact Distance Transform

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.

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. >

Distance transformations in arbitrary dimensions

Clustering of Medical Images for Analysis: A Fuzzy Approach

We consider Steiner minimum trees (SMT) in the plane, where only orientations with angle\({\sigma }\) , 0 ≤ i ≤ σ − 1 and σ an integer, are allowed. The orientations define a metric, called the orientation metric, λσ, in a natural way. In particular, λ2 metric is the rectilinear metric and the Euclidean metric can beregarded as λ∞ metric. In this paper, we provide a method to find an optimal λσ SMT for 3 or 4 points by analyzing the topology of λσ SMT's in great details. Utilizing these results and based on the idea of loop detection first proposed in Chao and Hsu, IEEE Trans. CAD, vol. 13, no. 3, pp. 303–309, 1994, we further develop an O(n2) time heuristic for the general λσ SMT problem, including the Euclidean metric. Experiments performed on publicly available benchmark data for 12 different metrics, plus the Euclidean metric, demonstrate the efficiency of our algorithms and the quality of our results.

The processing of three-dimensional (3-D) objects from 3-D digital image data is an important task in the image processing and the computer vision fields. The distance transform (DT) is extensively applied in the image processing and computer vision areas as a key operation. In a two or three-dimensional image array, the computation of distance transform (DT) is an important task. With the increasing application of 3-D voxel images, it is useful to consider the distance transform of a 3-D digital image array. In order to provide the efficient transform computations, parallelism is employed. We develop parallel algorithms for the three-dimensional Euclidean distance transform (3D-EDT) on the SIMD hypercube computer. The time complexity of our parallel algorithm is O(log/sup 2/N) for an N/spl times/N/spl times/N image array using N/sup 3/ processors. A generalized parallel algorithm for the 3D-EDT is also proposed and it runs O((N/p)/sup 3/log(N)+(N/p)/sup 2/log/sup 2/p) time for an N/spl times/N/spl times/N binary image array on the SIMD hypercube computer using p/sup 3/ PE's, where 1/spl les/p/spl les/N.

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.

Mathematical Morphology (MM) is a general method for image processing based on set theory. The two basic morphological operators are dilation and erosion. From these, several non linear filters have been developed, usually with polynomial complexity and this because the two basic operators depend strongly on the definition of the structural element. Most efforts to improve the algorithm's speed for each operator are based on structural element decomposition and/or efficient codification. In this second part, the concepts developed in part I (see Diaz de Leon and Sossa Azuela, “Mathematical morphology based on linear combined metric spaces on Z^1 (part I): Fast distance transforms,” Journal of Mathematical Imaging and Vision, Vol. 12, No. 2, pp. 137–154, 2000) are used to prove that it is possible to reduce the complexity of the morphological operators to zero complexity (constant time algorithms) for any regular discrete metric space and to eliminate the use of the structural element. In particular, this is done for an infinite family of metric spaces further defined. The use of the distance transformation is proposed for it comprises the information concerning all the discs included in a region to obtain fast morphological operators: erosions, dilations, openings and closings (of zero complexity) for an infinite (countable) family of regular metric spaces. New constant time, in contrast with the polynomial time algorithms, for the computation of these basics operators for any structural element are next derived by using this background. Practical examples showing the efficiency of the proposed algorithms, final comments and present research are also given here.

New quantum algorithm for visual tracking

The concept of the minimum spanning tree (MST) plays an important role in topological network design, because it models a cheapest connected network. In a tree, however, the failure of a vertex can disconnect the network. In order to tolerate such a failure, we generalize the MST to the concept of a cheapest biconnected network. For a set of points in the Euclidean plane, we show that it is NP-hard to find a cheapest biconnected spanning graph, where edge costs are the Euclidean distances of the respective points. We propose a different type of subgraph, based on forbidding (due to failure) the use of a vertex. A minimum spanning multi-tree is a spanning graph that contains for each possible forbidden vertex a spanning tree that is minimum among the spanning trees that do not use the forbidden vertex. We propose a worst-case time optimal algorithm for computing a minimum spanning multi-tree for a planar Euclidean point set. A minimum spanning multi-tree is cheap, even though it embeds a linear number of MSTs: Its cost is more than the MST cost only by a constant factor. Furthermore, we propose a linear time algorithm for computing a cheap vertex failure tolerant graph, given the Delaunay triangulation. This graph bounds the cost of the minimum spanning multi-tree from above.

This paper proposes a variable neighborhood distance transformation algorithm for two‐ and three‐dimensional (2‐D and 3‐D) digital binary pictures which is based on a new concept called path extension. The properties and performance of the method are experimentally evaluated. An algorithm was constructed using path extension which calculates the variable neighborhood distance for all one pixels in a 2‐D picture, using a sequence of four and eight neighborhoods composed in an appropriate way.In previous distance transformation algorithms, the distance for the central pixel under consideration is calculated uniquely from the distances of several neighborhood pixels, while the distance transformation in the path extension algorithm is executed by determining the distances of several neighborhood pixels from the distance of the central pixel under consideration.This paper proposes a serial path extension algorithm and two parallel path extension algorithms for both 2‐D and 3‐D pictures. For comparison of performance, one serial and one parallel algorithm, which are modifications of the traditional fixed neighborhood algorithms, are further analyzed. Finally, the results of an experimental evaluation are presented for computation time in the five proposed algorithms.

A non-photorealistic rendering method has been proposed to generate cell-like images in which cell-like patterns are represented in photographic images. Cell-like patterns are automatically generated by the change of density of photographic images. However, cell-like patterns are irregularly arranged. In this paper, we propose a method to arrange cell-like patterns along the edges of photographic images. We improves the conventional method by using Euclidean distance from the edges. We show that appealing cell-like images can be generated by our method through experiments using various photographic images.

MRF–MAP–MFT visual object segmentation based on motion boundary field

Planarity, bipartiteness and (directed) acyclicity are basic graph properties with classic linear time recognition algorithms. However, the problems of testing whether a given (di)graph has k vertices whose deletion makes it planar, bipartite or a directed acyclic graph (DAG) are all fundamental NP-complete problems when k is part of the input. As a result, a significant amount of research has been devoted to understanding whether, for every fixed k, these problems admit a polynomial time algorithm (where the exponent in the polynomial is independent of k) and in particular, whether they admit linear time algorithms. While we now know that for any fixed k, we can test in linear time whether a graph is k vertices away from being planar [FOCS 2009, SODA 2014] or bipartite [SODA 2014, SICOMP 2016], the best known algorithms in the case of directed acyclicity are the algorithm of Garey and Tarjan [IPL 78] which runs in time O(nk–1m) and the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008] which runs in time O(k!4kk4nm). In other words, it has remained open whether it is possible to recognize in linear time, a graph which is 2 vertices away from being acyclic! In this paper, we settle this question by giving an algorithm that decides whether a given graph is k vertices away from being acyclic, in time O(k!4kk5(n + m)). That is, for every fixed k, our algorithm runs in time O(m + n), thus mirroring the case for planarity and bipartiteness. Our algorithm is designed via a general methodology that shaves off a factor of n from some algorithms that use the powerful technique of iterative compression. The two main features of our methodology are: (i) This is the first generic technique for designing linear time algorithms for directed cut-problems and (ii) it can be used in combination with future improvements in algorithms for the compression version of other well-studied cut-problems such as Multicut and Directed Subset Feedback Vertex Set.

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.