A fast and flexible algorithm for computing watersheds in digital gray-scale images is introduced. A review of watersheds and related motion is first presented, and the major methods to determine watersheds are discussed. The algorithm is based on an immersion process analogy, in which the flooding of the water in the picture is efficiently simulated using of queue of pixel. It is described in detail provided in a pseudo C language. The accuracy of this algorithm is proven to be superior to that of the existing implementations, and it is shown that its adaptation to any kind of digital grid and its generalization to n-dimensional images (and even to graphs) are straightforward. The algorithm is reported to be faster than any other watershed algorithm. Applications of this algorithm with regard to picture segmentation are presented for magnetic resonance (MR) imagery and for digital elevation models. An example of 3-D watershed is also provided. >

/pdf/watersheds-in-digital-spaces-an-efficient-algorithm-based-on-320ti1t2p3.pdf

Watersheds in digital spaces: an efficient algorithm based on immersion simulations

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.

/pdf/a-linear-time-algorithm-for-computing-exact-euclidean-2e0cud5iyw.pdf

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

The image foresting transform (IFT) is a graph-based approach to the design of image processing operators based on connectivity. It naturally leads to correct and efficient implementations and to a better understanding of how different operators relate to each other. We give here a precise definition of the IFT, and a procedure to compute it-a generalization of Dijkstra's algorithm-with a proof of correctness. We also discuss implementation issues and illustrate the use of the IFT in a few applications.

/pdf/the-image-foresting-transform-theory-algorithms-and-4iwc6kdkzx.pdf

The image foresting transform: theory, algorithms, and applications

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

The uniform-cost algorithm is a special case of the A*-algorithm for finding the shortest paths in graphs. In the uniform-cost algorithm, nodes are expanded in order of increasing cost. An efficient version of this algorithm is developed for integer cost values. Nodes are sorted by storing them at predefined places (bucket sort), keeping the overhead low. The algorithm is applied to general distance transformation. A constrained distance transform is an operation which calculates at each pixel of an image the distance to the nearest pixel of a reference set, distance being defined as minimum path length. The uniform-cost algorithm, in the constrained case, proves to be the best solution for distance transformation. It is fast, the processing time is independent of the complexity of the image, and memory requirements are moderate. >

B.J.H. Verwer

Papers

An efficient uniform cost algorithm applied to distance transforms