Sparsity, or cardinality, as a tool for feature selection is extremely common in a vast number of current computer vision applications. The k-support norm is a recently proposed norm with the proven property of providing the tightest convex bound on cardinality over the Euclidean norm unit ball. In this paper we present a re-derivation of this norm, with the hope of shedding further light on this particular surrogate function. In addition, we also present a connection between the rank operator, the nuclear norm and the k-support norm. Finally, based on the results established in this re-derivation, we propose a novel algorithm with significantly improved computational efficiency, empirically validated on a number of different problems, using both synthetic and real world data.

This paper addresses the problem of semantic segmentation of 3D indoor scenes reconstructed from RGB-D images. Traditionally label prediction for 3D points is tackled by employing graphical models that capture scene features and complex relations between different class labels. However, the existing work is restricted to pairwise conditional random fields, which are insufficient when encoding rich scene context. In this work we propose models with higher-order potentials to describe complex relational information from the 3D scenes. Specifically, we relax the labelling problem to a regression, and generalize the higher-order associative P n Potts model to a new family of arbitrary higher-order models based on regression forests. We show that these models, like the robust P n models, can still be decomposed into the sum of pairwise terms by introducing auxiliary variables. Moreover, our proposed higher-order models also permit extension to hierarchical random fields, which allows for the integration of scene context and features computed at different scales. Our potential functions are constructed based on regression forests encoding Gaussian densities that admit efficient inference. The parameters of our model are learned from training data using a structured learning approach. Results on two datasets show clear improvements over current state-of-the-art methods.

Random hypothesis generation is central to robust geometric model fitting in computer vision. The predominant technique is to randomly sample minimal subsets of the data, and hypothesize the geometric models from the selected subsets. While taking minimal subsets increases the chance of successively "hitting" inliers in a sample, hypotheses fitted on minimal subsets may be severely biased due to the influence of measurement noise, even if the minimal subsets contain purely inliers. In this paper we propose Random Cluster Models, a technique used to simulate coupled spin systems, to conduct hypothesis generation using subsets larger than minimal. We show how large clusters of data from genuine instances of the model can be efficiently harvested to produce accurate hypotheses that are less affected by the vagaries of fitting on minimal subsets. A second aspect of the problem is the optimization of the set of structures that best fit the data. We show how our novel hypothesis sampler can be integrated seamlessly with graph cuts under a simple annealing framework to optimize the fitting efficiently. Unlike previous methods that conduct hypothesis sampling and fitting optimization in two disjoint stages, our algorithm performs the two subtasks alternatingly and in a mutually reinforcing manner. Experimental results show clear improvements in overall efficiency.

Recent works on multimodel fitting are often formulated as an energy minimization task, where the energy function includes fitting error and regularization terms, such as low-level spatial smoothness and model complexity. In this paper, we introduce a novel energy with high-level geometric priors that consider interactions between geometric models, such that certain preferred model configurations may be induced.We argue that in many applications, such prior geometric properties are available and should be fruitfully exploited. For example, in surface fitting to point clouds, the building walls are usually either orthogonal or parallel to each other. Our proposed energy function is useful in dealing with unknown distributions of data errors and outliers, which are often the factors leading to biased estimation. Furthermore, the energy can be efficiently minimized using the expansion move method. We evaluate the performance on several vision applications using real data sets. Experimental results show that our method outperforms the state-of-the-art methods without significant increase in computation.

Random hypothesis generation is central to robust geometric model fitting in computer vision. The predominant technique is to randomly sample minimal or elemental subsets of the data, and hypothesize the geometric model from the selected subsets. While taking minimal subsets increases the chance of simultaneously “hitting” inliers in a sample, it amplifies the noise of the underlying model, and hypotheses fitted on minimal subsets may be severely biased even if they contain purely inliers. In this paper we propose to use Random Cluster Models, a technique used to simulate coupled spin systems, to conduct hypothesis generation using subsets larger than minimal. We show how large clusters of data from genuine instances of the geometric model can be efficiently harvested to produce more accurate hypotheses. To take advantage of our hypothesis generator, we construct a simple annealing method based on graph cuts to fit multiple instances of the geometric model in the data. Experimental results show clear improvements in efficiency over other methods based on minimal subset samplers.

Multi-structure model fitting has traditionally taken a two-stage approach: First, sample a (large) number of model hypotheses, then select the subset of hypotheses that optimise a joint fitting and model selection criterion. This disjoint two-stage approach is arguably suboptimal and inefficient - if the random sampling did not retrieve a good set of hypotheses, the optimised outcome will not represent a good fit. To overcome this weakness we propose a new multi-structure fitting approach based on Reversible Jump MCMC. Instrumental in raising the effectiveness of our method is an adaptive hypothesis generator, whose proposal distribution is learned incrementally and online. We prove that this adaptive proposal satisfies the diminishing adaptation property crucial for ensuring ergodicity in MCMC. Our method effectively conducts hypothesis sampling and optimisation simultaneously, and gives superior computational efficiency over other methods.