Smoothing and Matching of 3-D Space Curves

International audienceWe present a new approach to the problem of matching 3-D curves. The approach has a low algorithmic complexity in the number of models, and can operate in the presence of noise and partial occlusions. Our method builds upon the seminal work of Kishon et al. (1990), where curves are first smoothed using B-splines, with matching based on hashing using curvature and torsion measures. However, we introduce two enhancements: -- We make use of nonuniform B-spline approximations, which permits us to better retain information at highcurvature locations. The spline approximations are controlled (i.e., regularized) by making use of normal vectors to the surface in 3-D on which the curves lie, and by an explicit minimization of a bending energy. These measures allow a more accurate estimation of position, curvature, torsion, and Frtnet frames along the curve. -- The computational complexity of the recognition process is relatively independent of the number of models and is considerably decreased with explicit use of the Frtnet frame for hypotheses generation. As opposed to previous approaches, the method better copes with partial occlusion. Moreover, following a statistical study of the curvature and torsion covariances, we optimize the hash table discretization and discover improved invariants for recognition, different than the torsion measure. Finally, knowledge of invariant uncertainties is used to compute an optimal global transformation using an extended Kalman filter. We present experimental results using synthetic data and also using characteristic curves extracted from 3-D medical images. An earlier version of this article was presented at the 2nd European Conference on Computer Vision in Italy

Smoothing under Diffeomorphic Constraints with Homeomorphic Splines

In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms. This diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time-dependent vector field. This solution has a closed form expression which can be computed using classical unconstrained spline smoothing techniques. This method does not require the use of quadratic or linear programming under inequality constraints and has therefore a low computational cost. In a one dimensional setting incorporating diffeomorphic constraints is equivalent to impose monotonicity. Thus, as an illustration, it is shown that such a monotone spline can be used to monotonize any unconstrained estimator of a regression function, and that this monotone smoother inherits the convergence properties of the unconstrained estimator. Some numerical experiments are proposed to illustrate its finite sample performances, and to compare them with another monotone estimator. We also provide a two-dimensional application on the computation of diffeomorphisms for landmark and image matching

Disconnected Skeleton: Shape at its Absolute Scale

We present a new skeletal representation along with a matching framework to address the deformable shape recognition problem. The disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale. Our motivation is to rely on the stable properties of the shape instead of inaccurately measured secondary details. The new representation does not suffer from the common instability problems of traditional connected skeletons, and the matching process gives quite successful results on a diverse database of 2D shapes. An important difference of our approach from the conventional use of the skeleton is that we replace the local coordinate frame with a global Euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations. As a result, we can produce descriptions that are sensitive to any combination of changes in scale, position, orientation and articulation, as well as invariant ones.Comment: The work excluding {\S}V and {\S}VI has first appeared in 2005 ICCV: Aslan, C., Tari, S.: An Axis-Based Representation for Recognition. In ICCV(2005) 1339- 1346.; Aslan, C., : Disconnected Skeletons for Shape Recognition. Masters thesis, Department of Computer Engineering, Middle East Technical University, May 200

Landmark-Based Registration of Curves via the Continuous Wavelet Transform

This paper is concerned with the problem of the alignment of multiple sets of curves. We analyze two real examples arising from the biomedical area for which we need to test whether there are any statistically significant differences between two subsets of subjects. To synchronize a set of curves, we propose a new nonparametric landmark-based registration method based on the alignment of the structural intensity of the zero-crossings of a wavelet transform. The structural intensity is a multiscale technique recently proposed by Bigot (2003, 2005) which highlights the main features of a signal observed with noise. We conduct a simulation study to compare our landmark-based registration approach with some existing methods for curve alignment. For the two real examples, we compare the registered curves with FANOVA techniques, and a detailed analysis of the warping functions is provided

Person re-identification by robust canonical correlation analysis

Person re-identification is the task to match people in surveillance cameras at different time and location. Due to significant view and pose change across non-overlapping cameras, directly matching data from different views is a challenging issue to solve. In this letter, we propose a robust canonical correlation analysis (ROCCA) to match people from different views in a coherent subspace. Given a small training set as in most re-identification problems, direct application of canonical correlation analysis (CCA) may lead to poor performance due to the inaccuracy in estimating the data covariance matrices. The proposed ROCCA with shrinkage estimation and smoothing technique is simple to implement and can robustly estimate the data covariance matrices with limited training samples. Experimental results on two publicly available datasets show that the proposed ROCCA outperforms regularized CCA (RCCA), and achieves state-of-the-art matching results for person re-identification as compared to the most recent methods

Topology of Luminous Red Galaxies from the Sloan Digital Sky Survey

We present measurements of the genus topology of luminous red galaxies (LRGs) from the Sloan Digital Sky Survey (SDSS) Data Release 7 catalog, with unprecedented statistical significance. To estimate the uncertainties in the measured genus, we construct 81 mock SDSS LRG surveys along the past light cone from the Horizon Run 3, one of the largest N-body simulations to date that evolved 7210^3 particles in a 10815 Mpc/h size box. After carefully modeling and removing all known systematic effects due to finite pixel size, survey boundary, radial and angular selection functions, shot noise and galaxy biasing, we find the observed genus amplitude to reach 272 at 22 Mpc/h smoothing scale with an uncertainty of 4.2%; the estimated error fully incorporates cosmic variance. This is the most accurate constraint of the genus amplitude to date, which significantly improves on our previous results. In particular, the shape of the genus curve agrees very well with the mean topology of the SDSS LRG mock surveys in the LCDM universe. However, comparison with simulations also shows small deviations of the observed genus curve from the theoretical expectation for Gaussian initial conditions. While these discrepancies are mainly driven by known systematic effects such as those of shot noise and redshift-space distortions, they do contain important cosmological information on the physical effects connected with galaxy formation, gravitational evolution and primordial non-Gaussianity. We address here the key role played by systematics on the genus curve, and show how to accurately correct for their effects to recover the topology of the underlying matter. In a forthcoming paper, we provide an interpretation of those deviations in the context of the local model of non-Gaussianity.Comment: 23 pages, 18 figures. APJ Supplement Series 201