A sequential algorithm is presented for computing the exact Euclidean distance transform (DT) of a k-dimensional binary image in time linear in the total number of voxels N. The algorithm, which is based on dimensionality reduction and partial Voronoi diagram construction, can be used for computing the DT for a wide class of distance functions, including the L/sub p/ and chamfer metrics. At each dimension level, the DT is computed by constructing the intersection of the Voronoi diagram whose sites are the feature voxels with each row of the image. This construction is performed efficiently by using the DT in the next lower dimension. The correctness and linear time complexity are demonstrated analytically and verified experimentally. The algorithm may be of practical value since it is relatively simple and easy to implement and it is relatively fast (not only does it run in O(N) time but the time constant is small). A simple modification of the algorithm computes the weighted Euclidean DT, which is useful for images with anisotropic voxel dimensions. A parallel version of the algorithm runs in O(N/p) time with p processors.

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A linear time algorithm for computing exact Euclidean distance transforms of binary images in arbitrary dimensions

The distance transform (DT) is a general operator forming the basis of many methods in computer vision and geometry, with great potential for practical applications. However, all the optimal algorithms for the computation of the exact Euclidean DT (EDT) were proposed only since the 1990s. In this work, state-of-the-art sequential 2D EDT algorithms are reviewed and compared, in an effort to reach more solid conclusions regarding their differences in speed and their exactness. Six of the best algorithms were fully implemented and compared in practice.

2D Euclidean distance transform algorithms: A comparative survey

The distance transform (DT) is a general operator forming the basis of many methods in computer vision and geometry, with great potential for practical applications. However, all the optimal algorithms for the computation of the exact Euclidean DT (EDT) were proposed only since the 1990s. In this work, state-of-theart sequential 2D EDT algorithms are reviewed and compared, in an effort to reach more solid conclusions regarding their differences in speed and their exactness. Six of the best algorithms were fully implemented and compared in practice.

2 2D Euclidean Distance Transform Algorithms: A Comparative Survey

In recent years, digital watermarking techniques have been proposed to protect the copyright of multimedia data. Different watermarking schemes have been suggested for images. The goal of this paper is to develop a watermarking algorithm based on the discrete cosine transform (DCT) and image segmentation. The image is first segmented in different portions based on the Voronoi diagram and features extraction points. Then, a pseudorandom sequence of real numbers is embedded in the DCT domain of each image segment. Different experiments are conducted to show the performance of the scheme under different types of attacks. The results show that our proposed watermark scheme is robust to common signal distortions, including geometric manipulations. The robustness against Joint Photographic Experts Group (JPEG) compression is achieved for a compression ratio of up to 45, and robustness against average, median, and Wiener filters is shown for the 3/spl times/3 up to 9/spl times/9 pixel neighborhood. It is observed that robustness against scaling was achieved when the watermarked image size is scaled down to 0.4% of its original size.

Digital watermarking-based DCT and JPEG model

We present an algorithm for fast computation of discretized 3D distance elds using graphics hardware. Given a set of primitives and a distance metric, our algorithm computes the distance eld for each slice of a uniform spatial grid by rasterizing the distance functions of the primitives. We compute bounds on the spatial extent of the Voronoi region of each primitive. These bounds are used to cull and clamp the distance functions rendered for each slice. Our algorithm is applicable to all geometric models and does not make any assumptions about connectivity or a manifold representation. We have used our algorithm to compute distance elds of large models composed of tens of thousands of primitives on high resolution grids. Moreover, we demonstrate its application to medial axis evaluation and proximity computations. As compared to earlier approaches, we are able to achieve an order of magnitude improvement in the running time.

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DiFi: Fast 3D Distance Field Computation Using Graphics Hardware

We propose an efficient Voronoi transform algorithm for constructing Voronoi diagrams using segment lists of rows. A significant feature of the algorithm is that it takes segments rather than pixels as the basic units to represent and propagate the nearest neighbor information. The segment lists are dynamically updated as they are scanned. A distance map can then be easily computed from the segment list representation of the Voronoi diagram. Experimental results have demonstrated its high efficiency. Extension of the algorithm to higher dimensions is also discussed.

Weiguang Guan

Papers

A list-processing approach to compute Voronoi diagrams and the Euclidean distance transform