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By Geometry Symposium (1980 University of Utrecht), Eduard Looijenga, D. Siersma, Floris Takens

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Due to the ambiguity of factorisation, this problem is not trivial. Given a measurement matrix W , factorisation yields a 3D motion matrix and a 3D structure matrix: ˆ ˆ Q)(Q−1 S), W = (M (4) ˆ represents the ˆ Q) represents the 3D motion and S = (Q−1 S) where M = (M 3D structure. The factorisation is ambiguous: the formula described in section 2 provides a correct result, but this result is not unique. If the coordinate system of the structure is rotated by a matrix A, the motion vectors by AT , where A is an Euclidean transformation matrix (AAT = I), then W = (M AT )(AS) is also a correct factorisation.

Let M be the number of hypotheses obtained in the sampling stage hj ; j = 1 . . M . Instead of studying the distribution of N residuals per hypothesis as in [7] when trying to determine the threshold for inlier classification, we propose to study the distribution of M residuals for each data point xi . We will show that this distribution reveals the presence of multiple models and further demonstrate how to estimate their number and their parameters. Nonparametric Estimation of Multiple Structures with Outliers 63 The rationale behind this choice is the following: when many samples are drawn from data containing multiple models, for each model, there will be a subset of samples which consist of only points belonging to it (inliers).

This is equivalent to assigning to a window the motion model that gives the least residual with respect to the BCC for that window. By applying this procedure to all pixels in the image, {xj }N j=1 , we can segment the entire scene. One can then refine the motion model parameters by re-calculating the motion parameters for each segment. 3 Experimental Results In this section, we analyze the performance of our proposed algorithm for segmenting image measurements arising from multiple motion models.

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