We consider the problem of online and real-time registration of partial point clouds obtained from an unseen real-world rigid object without knowing its 3D model. The point cloud is partial as it is obtained by a depth sensor capturing only the visible part of the object from a certain viewpoint. It introduces two main challenges: 1) two partial point clouds do not fully overlap and 2) keypoints tend to be less reliable when the visible part of the object does not have salient local structures. To address these issues, we propose DeepPRO, a keypoint-free and an end-to-end trainable deep neural network. Its core idea is inspired by how humans align two point clouds: we can imagine how two point clouds will look like after the registration based on their shape. To realize the idea, DeepPRO has inputs of two partial point clouds and directly predicts the point-wise location of the aligned point cloud. By preserving the ordering of points during the prediction, we enjoy dense correspondences between input and predicted point clouds when inferring rigid transform parameters. We conduct extensive experiments on the real-world Linemod and synthetic ModelNet40 datasets. In addition, we collect and evaluate on the PRO1k dataset, a large-scale version of Linemod meant to test generalization to real-world scans. Results show that DeepPRO achieves the best accuracy against thirteen strong baseline methods, e.g., 2.2mm ADD on the Linemod dataset, while running 50 fps on mobile devices.

Human preferences are usually measured using ordinal variables. A system whose goal is to estimate the preferences of humans and their underlying decision mechanisms requires to learn the ordering of any given sample set. We consider the solution of this ordinal regression problem using a support vector machine algorithm. Specifically, the goal is to learn a set of classifiers with common direction vectors and different biases correctly separating the ordered classes. Current algorithms are either required to solve a quadratic optimization problem, which is computationally expensive, or based on maximizing the minimum margin (i.e., a fixed-margin strategy) between a set of hyperplanes, which biases the solution to the closest margin. Another drawback of these strategies is that they are limited to order the classes using a single ranking variable (e.g., perceived length). In this paper, we define a multiple ordinal regression algorithm based on maximizing the sum of the margins between every consecutive class with respect to one or more rankings (e.g., perceived length and weight). We provide derivations of an efficient, easy-to-implement iterative solution using a sequential minimal optimization procedure. We demonstrate the accuracy of our solutions in several data sets. In addition, we provide a key application of our algorithms in estimating human subjects' ordinal classification of attribute associations to object categories. We show that these ordinal associations perform better than the binary one typically employed in the literature.

Learning models for object detection is a challenging problem due to the large intra-class variability of objects in appearance, viewpoints, and rigidity. We address this variability by a novel feature pooling method that is adaptive to segmented regions. The proposed detection algorithm automatically discovers a diverse set of exemplars and their distinctive parts which are used to encode the region structure by the proposed feature pooling method. Based on each exemplar and its parts, a regression model is learned with samples selected by a coarse region matching scheme. The proposed algorithm performs favorably on the PASCAL VOC 2007 dataset against existing algorithms. We demonstrate the benefits of our feature pooling method when compared to conventional spatial pyramid pooling features. We also show that object information can be transferred through exemplars for detected objects.

Kernel mapping is one of the most used approaches to intrinsically derive nonlinear classifiers. The idea is to use a kernel function which maps the original nonlinearly separable problem to a space of intrinsically larger dimensionality where the classes are linearly separable. A major problem in the design of kernel methods is to find the kernel parameters that make the problem linear in the mapped representation. This paper derives the first criterion that specifically aims to find a kernel representation where the Bayes classifier becomes linear. We illustrate how this result can be successfully applied in several kernel discriminant analysis algorithms. Experimental results, using a large number of databases and classifiers, demonstrate the utility of the proposed approach. The paper also shows (theoretically and experimentally) that a kernel version of Subclass Discriminant Analysis yields the highest recognition rates.

2D Active Appearance Models (AAM) and 3D Morphable Models (3DMM) are widely used techniques. AAM provide a fast fitting process, but may represent unwanted 3D transformations unless strictly constrained not to do so. The reverse is true for 3DMM. The two approaches also require of a pre-alignment of their 2D or 3D shapes before the modeling can be carried out which may lead to errors. Furthermore, current models are insufficient to represent nonlinear shape and texture variations. In this paper, we derive a new approach that can model nonlinear changes in examples without the need of a pre-alignment step. In addition, we show how the proposed approach carries the above mentioned advantages of AAM and 3DMM. To achieve this goal, we take advantage of the inherent properties of complex spherical distributions, which provide invariance to translation, scale and rotation. To reduce the complexity of parameter estimation we take advantage of a recent result that shows how to estimate spherical distributions using their Euclidean counterpart, e.g., the Gaussians. This leads to the definition of Rotation Invariant Kernels (RIK) for modeling nonlinear shape changes. We show the superiority of our algorithm to AAM in several face datasets. We also show how the derived algorithm can be used to model complex 3D facial expression changes observed in American Sign Language (ASL).

Shape analysis requires invariance under translation, scale, and rotation. Translation and scale invariance can be realized by normalizing shape vectors with respect to their mean and norm. This maps the shape feature vectors onto the surface of a hypersphere. After normalization, the shape vectors can be made rotational invariant by modeling the resulting data using complex scalar-rotation invariant distributions defined on the complex hypersphere, e.g., using the complex Bingham distribution. However, the use of these distributions is hampered by the difficulty in estimating their parameters and the nonlinear nature of their formulation. In the present paper, we show how a set of kernel functions that we refer to as rotation invariant kernels can be used to convert the original nonlinear problem into a linear one. As their name implies, these kernels are defined to provide the much needed rotation invariance property allowing one to bypass the difficulty of working with complex spherical distributions. The resulting approach provides an easy, fast mechanism for 2D & 3D shape analysis. Extensive validation using a variety of shape modeling and classification problems demonstrates the accuracy of this proposed approach.

We present an algorithm which provides the one-dimensional subspace where the Bayes error is minimized for the C class problem with homoscedastic Gaussian distributions. Our main result shows that the set of possible one-dimensional spaces v, for which the order of the projected class means is identical, defines a convex region with associated convex Bayes error function g(v). This allows for the minimization of the error function using standard convex optimization algorithms. Our algorithm is then extended to the minimization of the Bayes error in the more general case of heteroscedastic distributions. This is done by means of an appropriate kernel mapping function. This result is further extended to obtain the d-dimensional solution for any given d, by iteratively applying our algorithm to the null space of the (d - 1)-dimensional solution. We also show how this result can be used to improve up on the outcomes provided by existing algorithms, and derive a low-computational cost, linear approximation. Extensive experimental validations are provided to demonstrate the use of these algorithms in classification, data analysis and visualization.

Discriminant Analysis (DA) methods have demonstrated their utility in countless applications in computer vision and other areas of research - especially in the C class classification problem. The most popular approach is linear DA (LDA), which provides the C - 1-dimensional Bayes optimal solution, but only when all the class covariance matrices are identical. This is rarely the case in practice. To alleviate this restriction, Kernel LDA (KLDA) has been proposed. In this approach, we first (intrinsically) map the original nonlinear problem to a linear one and then use LDA to find the C - 1-dimensional Bayes optimal subspace. However, the use of KLDA is hampered by its computational cost, given by the number of training samples available and by the limitedness of LDA in providing a C - 1-dimensional solution space. In this paper, we first extend the definition of LDA to provide subspace of q < C - 1 dimensions where the Bayes error is minimized. Then, to reduce the computational burden of the derived solution, we define a sparse kernel representation, which is able to automatically select the most appropriate sample feature vectors that represent the kernel. We demonstrate the superiority of the proposed approach on several standard datasets. Comparisons are drawn with a large number of known DA algorithms.

Shape analysis requires invariance under translation, scale and rotation. Translation and scale invariance can be realized by normalizing shape vectors with respect to their mean and norm. This maps the shape feature vectors onto the surface of a hypersphere. After normalization, the shape vectors can be made rotational invariant by modelling the resulting data using complex scalar rotation invariant distributions defined on the complex hypersphere, e.g., using the complex Bingham distribution. However, the use of these distributions is hampered by the difficulty in estimating their parameters, which is shown to be very costly or impossible in most cases. The purpose of this paper is twofold. First, we show under which conditions the classification results obtained with complex Binghams are identical to those obtained with the easy-to-estimate complex Normal distribution. Second, we derive a kernel function which (intrinsically) maps the data into a space where the above conditions are satisfied and, hence, where the normal model can be successfully used. This results in a simple, low-cost algorithm for representing and classifying shapes. We demonstrate the use of this technique in several experimental results for object and face recognition. Comparisons to other statistical shape representation/classification approaches demonstrate the superiority of the proposed algorithms in classification accuracy and computational time.