Graphics and Image Processing J. D. Foley Editor Random Sample Consensus: A Paradigm for Model Fitting with Apphcatlons to Image Analysis and Automated Cartography Martin A. Fischler and Robert C. Bolles SRI International A new paradigm, Random Sample Consensus (RANSAC), for fitting a model to experimental data is introduced. RANSAC is capable of interpreting/ smoothing data containing a significant percentage of gross errors, and is thus ideally suited for applications in automated image analysis where interpretation is based on the data provided by error-prone feature detectors. A major portion of this paper describes the application of RANSAC to the Location Determination Problem (LDP): Given an image depicting a set of landmarks with known locations, determine that point in space from which the image was obtained. In response to a RANSAC requirement, new results are derived on the minimum number of landmarks needed to obtain a solution, and algorithms are presented for computing these minimum-landmark solutions in closed form. These results provide the basis for an automatic system that can solve the LDP under difficult viewing Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. The work reported herein was supported by the Defense Advanced Research Projects Agency under Contract Nos. DAAG29-76-C-0057 and MDA903-79-C-0588. Authors' Present Address: Martin A. Fischler and Robert C. Bolles, Artificial Intelligence Center, SRI International, Menlo Park CA 94025. © 1981 ACM 0001-0782/81/0600-0381 $00.75 381 and analysis conditions. Implementation details and computational examples are also presented. Key Words and Phrases: model fitting, scene analysis, camera calibration, image matching, location determination, automated cartography. CR Categories: 3.60, 3.61, 3.71, 5.0, 8.1, 8.2 I. Introduction We introduce a new paradigm, Random Sample Consensus (RANSAC), for fitting a model to experimental data; and illustrate its use in scene analysis and auto- mated cartography. The application discussed, the loca- tion determination problem (LDP), is treated at a level beyond that of a mere example of the use of the RANSAC paradigm; new basic findings concerning the conditions under which the LDP can be solved are presented and a comprehensive approach to the solution of this problem that we anticipate will have near-term practical appli- cations is described. To a large extent, scene analysis (and, in fact, science in general) is concerned with the interpretation of sensed data in terms of a set of predefmed models. Conceptually, interpretation involves two distinct activities: First, there is the problem of finding the best match between the data and one of the available models (the classification problem); Second, there is the problem of computing the best values for the free parameters of the selected model (the parameter estimation problem). In practice, these two problems are not independent--a solution to the parameter estimation problem is often required to solve the classification problem. Classical techniques for parameter estimation, such as least squares, optimize (according to a specified ob- jective function) the fit of a functional description (model) to all of the presented data. These techniques have no internal mechanisms for detecting and rejecting gross errors. They are averaging techniques that rely on the assumption (the smoothing assumption) that the maximum expected deviation of any datum from the assumed model is a direct function of the size of the data set, and thus regardless of the size of the data set, there will always be enough good values to smooth out any gross deviations. In many practical parameter estimation problems the smoothing assumption does not hold; i.e., the data con- tain uncompensated gross errors. To deal with this situ- ation, several heuristics have been proposed. The tech- nique usually employed is some variation of first using all the data to derive the model parameters, then locating the datum that is farthest from agreement with the instantiated model, assuming that it is a gross error, deleting it, and iterating this process until either the maximum deviation is less then some preset threshold or until there is no longer sufficient data to proceed. It can easily be shown that a single gross error ("poisoned point"), mixed in with a set of good data, can Communications June 1981 of Volume 24 the ACM Number 6