Limitations of Geometric Hashing in the Presence of Gaussian Noise

This paper presents a detailed error analysis of geometric hashing in the domain of 2D object recogition. Earlier analysis has shown that these methods are likely to produce false positive hypotheses when one allows for uniform bounded sensor error and moderate amounts of extraneous clutter points. These false positives must be removed by a subsequent veri cation step. Later work has incorporated an explicit 2D Gaussian instead of a bounded error model to improve performance of the hashing method. The contribution of this paper is to analytically derive the probability of false positives and negatives as a function of the number of model features, image features, and occlusion, under the assumption of 2D Gaussian noise and a particular method of evidence accumulation. A distinguishing feature of this work is that we make no assumptions about prior distributions on the model space, nor do we assume even the presence of the model. The results are presented in the form of ROC (receiver-operating characteristic) curves, from which several results can be extracted; rstly, they demonstrate that the 2D Gaussian error model performs better for high clutter levels and degrades more gracefully as compared to the uniform bounded error model for the same conditions. They also directly indicate the optimal performance that can be achieved for a given clutter and occlusion rate, and how to choose the thresholds to achieve the desired rates. Lastly, we verify these ROC curves in the domain of simulated images.