Distance transform

Abstract: The Hausdorff distance measures the extent to which each point of a model set lies near some point of an image set and vice versa. Thus, this distance can be used to determine the degree of resemblance between two objects that are superimposed on one another. Efficient algorithms for computing the Hausdorff distance between all possible relative positions of a binary image and a model are presented. The focus is primarily on the case in which the model is only allowed to translate with respect to the image. The techniques are extended to rigid motion. The Hausdorff distance computation differs from many other shape comparison methods in that no correspondence between the model and the image is derived. The method is quite tolerant of small position errors such as those that occur with edge detectors and other feature extraction methods. It is shown that the method extends naturally to the problem of comparing a portion of a model against an image. >

Abstract: A distance transformation converts a binary digital image, consisting of feature and non-feature pixels, into an image where all non-feature pixels have a value corresponding to the distance to the nearest feature pixel. Computing these distances is in principle a global operation. However, global operations are prohibitively costly. Therefore algorithms that consider only small neighborhoods, but still give a reasonable approximation of the Euclidean distance, are necessary. In the first part of this paper optimal distance transformations are developed. Local neighborhoods of sizes up to 7×7 pixels are used. First real-valued distance transformations are considered, and then the best integer approximations of them are computed. A new distance transformation is presented, that is easily computed and has a maximal error of about 2%. In the second part of the paper six different distance transformations, both old and new, are used for a few different applications. These applications show both that the choice of distance transformation is important, and that any of the six transformations may be the right choice.

Abstract: We describe a system for representing moving images with sets of overlapping layers. Each layer contains an intensity map that defines the additive values of each pixel, along with an alpha map that serves as a mask indicating the transparency. The layers are ordered in depth and they occlude each other in accord with the rules of compositing. Velocity maps define how the layers are to be warped over time. The layered representation is more flexible than standard image transforms and can capture many important properties of natural image sequences. We describe some methods for decomposing image sequences into layers using motion analysis, and we discuss how the representation may be used for image coding and other applications. >

Abstract: In many applications of digital picture processing, distances from certain feature elements to the nonfeature elements must be computed. In two dimensions at least four different families of distance transformations have been suggested, the most popular one being the city block/chessboard distance family. The purpose of this paper is twofold: To generalize these transformations to higher dimensions and to compare the computed distances with the Euclidean distance. All of the four distance transformation families are presented in three dimensions, and the two fastest ones are presented in four and arbitrary dimensions. The comparison with Euclidean distance is given as upper limits for the difference between the Euclidean distance and the computed distances.

Abstract: We describe linear-time algorithms for solving a class of problems that involve transforming a cost function on a grid using spatial information. These problems can be viewed as a generalization of classical distance transforms of binary images, where the binary image is replaced by an arbitrary function on a grid. Alternatively they can be viewed in terms of the minimum convolution of two functions, which is an important operation in grayscale morphology. A consequence of our techniques is a simple and fast method for computing the Euclidean distance transform of a binary image. Our algorithms are also applicable to Viterbi decoding, belief propagation, and optimal control.