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 he used to determine the degree of resemblance between two objects that are superimposed on one another. In this paper, we provide efkient algorithms for computing the Hausdorff distance between all possible relative positions of a binary image and a model. We focus primarily on the case in which the model is only allowed to translate with respect to the image. Then, we consider how to extend the techniques to rigid motion (translation and rotation). 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. Moreover, we show how the method extends naturally to the problem of comparing a portion of a model against an image. can all be viewed as techniques for determining the difference between shapes. We have recently been investigating functions for determining the degree to which two shapes differ from one another. The goal of these investigations has been to develop shape comparison methods that are efficient to compute, produce intuitively reasonable results, and have a firm underlying theoretical basis. In order to meet these goals, we argue that it is important for shape comparison functions to obey metric properties (see [3] for related arguments). In this paper, we present algorithms for efficiently computing the Hausdorff distance between all possible relative positions of a model and an image. (The Hausdorff distance is a max-min distance defined below.) We primarily focus on the case in which the model and image are allowed to translate with respect to one another and then briefly consider extensions to handle the more general case of rigid motion. Here, we provide provably good approximation algorithms that are highly efficient both in theory and in practice. These methods operate on binary rasters, making them particularly well suited to image processing and machine vision applications, where the data are generally in raster form. The three key advantages of the approach are as follows: 1) Relative insensitivity to small perturbations of the image 2) simplicity and speed of computation 3) natural allowance for portions of one shape to be comWe discuss three different methods of computing the Hausdorff distance as a function of the translation of a model with respect to an image. The first of these methods is similar in many ways to binary correlation and convolution, except that the Hausdorff distance is a nonlinear operator. The second method extends the definition of the distance function to enable the comparison of portions of a model to portions of an image. The third method improves on the first two by using certain properties of the Hausdorff distance to rule out many possible relative positions of the model and the image without having to explicitly consider them. This speeds up the computation by several orders of magnitude. All three of these methods can be further sped up using special-purpose graphics hardware (in particular, a z buffer). We present a number of examples using real images. These examples illustrate the application of the method to scenes in which a portion of the object to be identified is hidden from view. Then finally, we show how the methods can be adapted to comparing objects under rigid motion (translation and rotation).
Huttenlocher et al. REF tried to avoid that shortcoming by introducing a generalized Hausdorff distance which only computes the maximum of the kth largest distances between the image and the model edges and the lth largest distances between the model and the image edges.
Comparing images using the hausdorff distance
IEEE Transactions on Pattern Analysis and Machine Intelligence
Mathematics Computer Science