This paper presents an improvement of the J-linkage algorithm for fitting multiple instances of a model to noisy data corrupted by outliers. The binary preference analysis implemented by J-linkage is replaced by a continuous (soft, or fuzzy) generalization that proves to perform better than J-linkage on simulated data, and compares favorably with state of the art methods on public domain real datasets.

Geometric multi-model fitting aims at extracting parametric models from unstructured data in order to organize and aggregate visual content in suitable higher-level geometric structures. This ubiquitous task can be encountered in many Computer Vision applications, for example in 3D reconstruction, in the processing of 3D point clouds, in face clustering, in body-pose estimation or video motion segmentation, just to name a few.In practice, it is necessary to overcome the "chicken-&-egg dilemma" inherent to this problem: in order to estimate models one needs to first segment the data, but in order to segment the data it is necessary to know the models associated with each data point. The presence of multiple structures hinders robust estimation, which has to cope with both gross outliers and pseudo-outliers. Two somehow orthogonal strategies have been proposed in the literature in order to adress this challenging problem: consensus analysis and preference analysis. Consensus based methods, building on the RANSAC paradigm, instantiate a pool of tentative models and extract the strucutures that have maximal consensus. Preference oriented algorithms [2,3] instead tackle this problem by the data point of view. Residuals between point and putative models are used in order to build a conceptual space in which points are portrayed by their preferences with respect to the instantiated strucures, The multi model fitting problem is then solved by clustering points in this preference space.The method we present reduces the multi-model fitting task to many easier single robust model estimation problems, by combining preference analysis and robust low rank approximation. Three main step can be single out in our appraoch. At first data points are shifted in a conceptual space, where they are framed as a preference matrix Φ as shown in Fig model points preferences, in this way a first protection against outlier is achieved. The preference space is then equipped with a kernel, based on the Tanimoto distance, in this way an affinity matrix K, which measures the agreement between the preferences of points, is derived. The second step is devoted to robustly segment points explointing the information encapsulated in K. This stage can be thought as a sort of "robust spectral clustering". It is well known that spectral clustering produces accurate segmentations in two steps: at first data are projected on the space of the first eigenvectors of the Laplacian matrix and then k-means is applied. The shortcoming of spectral clustering however is that it is not robust to outliers. We propose to follow the same scheme enforcing robustness exploiting the low rank nature of the problem. As pictorially illustrated in Fig. 2, we decompose the affinity matrix asThe matrix S models the sparse preferences expressed by outliers, and is obtained by appling Robust PCA, which replaces the eigen-decomposition Finally, models are extracted inspecting the product of the preference matrix with a thresholded U, mimicking the MSAC strategy. The use of robust...

This paper deals with the extraction of multiple models from noisy or outlier-contaminated data. We cast the multi-model fitting problem in terms of set coverage, deriving a simple and effective method that generalizes Ransac to multiple models and deals with intersecting structures and outliers in a straightforward and principled manner, while avoiding the typical shortcomings of sequential approaches and those of clustering. The method compares favorably against the state-of-the-art on simulated and publicly available real data-sets

This paper deals with the rotation synchronization problem, which arises in global registration of 3D point-sets and in structure from motion. The problem is formulated in an unprecedented way as a "low-rank and sparse" matrix decomposition that handles both outliers and missing data. A minimization strategy, dubbed R-GoDec, is also proposed and evaluated experimentally against state-of-the-art algorithms on simulated and real data. The results show that R-GoDec is the fastest among the robust algorithms. of structure from motion, or local coordinates where 3D points are represented, in which case we are dealing with a 3D point-set registration problem.More abstractly, the goal of the group synchronization problem [1, 2] is to recover elements of a group from noisy measures of their ratios. In our case, absolute angular attitudes R 1 , . . . , R n are elements of the Special Orthogonal Group SO (3), and relative attitudes R ij = R i R T j are their ratios. The same problem is analysed in depth in [3], under the name "multiple rotation averaging". Our solution to the rotation synchronization problem is inspired by recent advances in the fields of robust principal component analysis and matrix completion. The main and original contribution of this paper is the formulation of the rotation synchronization problem as a "low-rank and sparse matrix decomposition", by conceiving a novel cost function that naturally includes missing data and outliers in its definition. Secondly, we develop a minimization scheme for that cost function -called R-GoDec -that leverages on the GoDec algorithm [4]. The resulting method is robust, by construction, fast, thanks to the use of Bilateral Random Projections (BRP) in place of Singular Value Decomposition (SVD), and compact, as it consist of a single fixed point iteration that can be coded in a few lines of MATLAB. Most of all, the framework is modular, as -in principle -any low-rank and sparse decomposition method able to deal with outliers and missing data can replace R-GoDec.This paper is organized as follows. Applications of the rotation synchronization problem are presented in Section 2 while existing solutions are described in Section 3. Section 4 is an overview of the theoretical background required to define our algorithm, i.e. lowrank and sparse matrix decomposition. Section 5 defines the rotation synchronization problem. Section 6 provides a detailed description of our robust solution. The method proposed in this section is supported by experimental results on both synthetic and real data, shown in Section 7. The conclusions are presented in Section 8. This paper is an extended version of [5].

This paper addresses the problem of multiple model fitting in the general context where the sought structures can be described by a mixture of heterogeneous parametric models drawn from different classes. To this end, we conceive a multi-model selection framework that extends Tlinkage to cope with different nested classes of models. Our method, called MCT, compares favourably with the stateof-the-art on publicly available data-sets for various fitting problems: lines and conics, homographies and fundamental matrices, planes and cylinders.