Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper

MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems

CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length (pseudoinverse) solution. This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems. MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned systems (singular or not), MINRES-QLP can give more accurate solutions than MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm, and the condition number.Comment: 26 pages, 6 figure

Reproducible Research in Computational Harmonic Analysis

Scientific computation is emerging as absolutely central to the scientific method. Unfortunately, it's error-prone and currently immature—traditional scientific publication is incapable of finding and rooting out errors in scientific computation—which must be recognized as a crisis. An important recent development and a necessary response to the crisis is reproducible computational research in which researchers publish the article along with the full computational environment that produces the results. The authors have practiced reproducible computational research for 15 years and have integrated it with their scientific research and with doctoral and postdoctoral education. In this article, they review their approach and how it has evolved over time, discussing the arguments for and against working reproducibly

Curvelets and Ridgelets

International audienceDespite the fact that wavelets have had a wide impact in image processing, they fail to efficiently represent objects with highly anisotropic elements such as lines or curvilinear structures (e.g. edges). The reason is that wavelets are non-geometrical and do not exploit the regularity of the edge curve. The Ridgelet and the Curvelet [3, 4] transforms were developed as an answer to the weakness of the separable wavelet transform in sparsely representing what appears to be simple building atoms in an image, that is lines, curves and edges. Curvelets and ridgelets take the form of basis elements which exhibit high directional sensitivity and are highly anisotropic [5, 6, 7, 8]. These very recent geometric image representations are built upon ideas of multiscale analysis and geometry. They have had an important success in a wide range of image processing applications including denoising [8, 9, 10], deconvolution [11, 12], contrast enhancement [13], texture analysis [14, 15], detection [16], watermarking [17], component separation [18], inpainting [19, 20] or blind source separation[21, 22]. Curvelets have also proven useful in diverse fields beyond the traditional image processing application. Let’s cite for example seismic imaging [10, 23, 24], astronomical imaging [25, 26, 27], scientific computing and analysis of partial differential equations [28, 29]. Another reason for the success of ridgelets and curvelets is the availability of fast transform algorithms which are available in non-commercial software packages following the philosophy of reproducible research, see [30, 31]

Content-Based VLE Designs Improve Learning Efficiency in Constructivist Statistics Education

Background: We introduced a series of computer-supported workshops in our undergraduate statistics courses, in the hope that it would help students to gain a deeper understanding of statistical concepts. This raised questions about the appropriate design of the Virtual Learning Environment (VLE) in which such an approach had to be implemented. Therefore, we investigated two competing software design models for VLEs. In the first system, all learning features were a function of the classical VLE. The second system was designed from the perspective that learning features should be a function of the course's core content (statistical analyses), which required us to develop a specific-purpose Statistical Learning Environment (SLE) based on Reproducible Computing and newly developed Peer Review (PR) technology. Objectives: The main research question is whether the second VLE design improved learning efficiency as compared to the standard type of VLE design that is commonly used in education. As a secondary objective we provide empirical evidence about the usefulness of PR as a constructivist learning activity which supports non-rote learning. Finally, this paper illustrates that it is possible to introduce a constructivist learning approach in large student populations, based on adequately designed educational technology, without subsuming educational content to technological convenience. Methods: Both VLE systems were tested within a two-year quasi-experiment based on a Reliable Nonequivalent Group Design. This approach allowed us to draw valid conclusions about the treatment effect of the changed VLE design, even though the systems were implemented in successive years. The methodological aspects about the experiment's internal validity are explained extensively. Results: The effect of the design change is shown to have substantially increased the efficiency of constructivist, computer-assisted learning activities for all cohorts of the student population under investigation. The findings demonstrate that a content-based design outperforms the traditional VLE-based design

Quality Control of Statistical Learning Environments and Prediction of Learning Outcomes through Reproducible Computing

This article introduces a new approach to statistics education that allows us to accurately measure and control key aspects of the computations and communication processes that are involved in non-rote learning within the pedagogical paradigm of Constructivism. The solution that is presented relies on a newly developed technology (hosted at www.freestatistics.org) and computing framework (hosted at www.wessa.net) that supports reproducibility and reusability of statistical research results that are presented in a so-called Compendium. Reproducible computing leads to responsible learning behaviour, and a stream of high-quality communications that emerges when students are engaged in peer review activities. More importantly, the proposed solution provides a series of objective measurements of actual learning processes that are otherwise unobservable. A comparison between actual and reported data, demonstrates that reported learning process measurements are highly misleading in unexpected ways. However, reproducible computing and objective measurements of actual learning behaviour, reveal important guidelines that allow us to improve the effectiveness of learning and the e-learning system

The Multiplicative Zak Transform, Dimension Reduction, and Wavelet Analysis of LIDAR Data

This thesis broadly introduces several techniques within the context of timescale analysis. The representation, compression and reconstruction of DEM and LIDAR data types is studied with directional wavelet methods and the wedgelet decomposition. The optimality of the contourlet transform, and then the wedgelet transform is evaluated with a valuable new structural similarity index. Dimension reduction for material classification is conducted with a frame-based kernel pipeline and a spectral-spatial method using wavelet packets. It is shown that these techniques can improve on baseline material classification methods while significantly reducing the amount of data. Finally, the multiplicative Zak transform is modified to allow the study and partial characterization of wavelet frames