This article describes a general formalism for obtaining spatially localized ("sparse") solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems, such as the important case of Schrödinger's equation in quantum mechanics. Sparsity is achieved by adding an L 1 regularization term to the variational principle, which is shown to yield solutions with compact support ("compressed modes"). Linear combinations of these modes approximate the eigenvalue spectrum and eigenfunctions in a systematically improvable manner, and the localization properties of compressed modes make them an attractive choice for use with efficient numerical algorithms that scale linearly with the problem size. S ignificant progress has been recently achieved in a variety of fields of information science using ideas centered around sparsity. Examples include compressed sensing (1, 2), matrix rank minimization (3), phase retrieval (4), and robust principal component analysis (5), as well as many others. A key step in these examples is use of a variational formulation with a constraint or penalty term that is an ℓ 1 or related norm. A limited set of extensions of sparsity techniques to physical sciences and partial differential equations (PDEs) have also appeared recently, including numerical solution of PDEs with multiscale oscillatory solutions (6) and efficient materials models derived from quantum mechanics calculations (7). In all of these examples, sparsity is for the coefficients (i.e., only a small set of coefficients are nonzero) in a well-chosen set of modes (e.g., a basis or dictionary) for representation of the corresponding vectors or functions. In this article, we propose a use of sparsity-promoting techniques to produce "compressed modes" (CMs)-modes that are sparse and localized in space-for efficient solution of constrained variational problems in mathematics and physics.Our idea is motivated by the localized Wannier functions developed in solid state physics and quantum chemistry. We begin by reviewing the basic ideas for obtaining spatially localized solutions of the independent-particle Schrödinger's equation. For simplicity, we consider a finite system with N electrons and neglect the electron spin. The ground state energy is given by E 0 = P N j=1 λ j , where λ j are the eigenvalues of the Hamiltonian, H = − 1 2 Δ + V ðxÞ, arranged in increasing order and satisfyinĝ Hϕ j = λ j ϕ j , with ϕ j being the corresponding eigenfunctions. This can be recast as a variational problem requiring the minimization of the total energy subject to orthonormality conditions for wave functions:Here, Φ N = fϕ j g N j=1and hϕ j ;In most cases, the eigenfunctions ϕ j are spatially extended and have infinite support-that is, they are "dense." This presents challenges for computational efficiency [as the wave function orthogonalization requires OðN 3 Þ operations, dominating the computational effort for N ≈ 10 3 electrons and above] and is contrary to physical intuition, which suggests that the...

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In this paper we present a novel approach for the intrinsic mapping of anatomical surfaces and its application in brain mapping research. Using the Laplace-Beltrami eigen-system, we represent each surface with an isometry invariant embedding in a high dimensional space. The key idea in our system is that we realize surface deformation in the embedding space via the iterative optimization of a conformal metric without explicitly perturbing the surface or its embedding. By minimizing a distance measure in the embedding space with metric optimization, our method generates a conformal map directly between surfaces with highly uniform metric distortion and the ability of aligning salient geometric features. Besides pairwise surface maps, we also extend the metric optimization approach for group-wise atlas construction and multi-atlas cortical label fusion. In experimental results, we demonstrate the robustness and generality of our method by applying it to map both cortical and hippocampal surfaces in population studies. For cortical labeling, our method achieves excellent performance in a cross-validation experiment with 40 manually labeled surfaces, and successfully models localized brain development in a pediatric study of 80 subjects. For hippocampal mapping, our method produces much more significant results than two popular tools on a multiple sclerosis study of 109 subjects.

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This paper describes an L 1 regularized variational framework for developing a spatially localized basis, compressed plane waves, that spans the eigenspace of a differential operator, for instance, the Laplace operator. Our approach generalizes the concept of plane waves to an orthogonal real-space basis with multiresolution capabilities.G enerality, conceptual simplicity, and development of efficient numerical algorithms based on the fast Fourier transform (FFT) have facilitated the adoption of plane waves as canonical basis functions for countless applications in engineering, science, and mathematics (1, 2). Since the plane waves are continuously differentiable eigenfunctions of the Laplace operator, they are well suited for representing solutions to partial differential equations (PDEs) of mathematical physics, such as those arising in quantum mechanics and electrodynamics. One of the most attractive features of plane waves is the ability to systematically increase spatial (or temporal) resolution by including higher kinetic energy (or frequency) components. However, since the plane waves are global functions, resolution is increased uniformly throughout the entire space, while, in practice, high resolution may be required only in a small fraction of the problem domain. The need for functions that can represent multiple length scales has spurred the development of wavelets (3), which are localized basis functions with multiresolution capabilities. Wavelets have been tremendously successful in fields such as signal processing, image science, and data science, but adoption of wavelets as the basis for solving PDEs has been difficult because it is numerically complicated to evaluate the derivative of a wavelet in a wavelet expansion. Furthermore, canonical wavelet functions usually can only be defined on regular domains in R d by tensor products of wavelets in one dimension (1D), which makes them difficult to generalize to irregular domains.In this paper, we extend our earlier work in ref. 4 and propose a method for generating a localized orthonormal basis that is adapted to a given differential operator, in particular, the Laplace operator. In ref. 4, we showed that L 1 regularization of the variational formulation of the Schrödinger equation of quantum mechanics can be used to create compressed modes, a set of spatially localized functions fψ i g N i=1 in R d with compact support:is the Hamilton operator corresponding to potential V ðxÞ, Ψ N = fψ j g N j=1 are variational single-particle orbitals, and L 1 norm is defined as ψ j 1 = R Ω ψ j dx. This L 1 regularized variational approach describes a general formalism for obtaining localized (in fact, compactly supported) solutions to a class of mathematical physics PDEs, which can be recast as variational optimization problems. One of the main advantages of this variational approach is that one parameter μ controls both the physical accuracy and the spatial extent of the resulting solutions: The wave functions ψ j are nonzero only where required to achieve a give...

Recovering high quality surfaces from noisy triangulated surfaces is a fundamental important problem in geometry processing. Sharp features including edges and corners can not be well preserved in most existing denoising methods except the recent total variation (TV) and 0 regularization methods. However, these two methods have suffered producing staircase artifacts in smooth regions. In this paper, we first introduce a second order regularization method for restoring a surface normal vector field, and then propose a new vertex updating scheme to recover the desired surface according to the restored surface normal field. The proposed model can preserve sharp features and simultaneously suppress the staircase effects in smooth regions which overcomes the drawback of the first order models. In addition, the new vertex updating scheme can prevent ambiguities introduced in existing vertex updating methods. Numerically, the proposed high order model is solved by the augmented Lagrangian method with a dynamic weighting strategy. Intensive numerical experiments on a variety of surfaces demonstrate the superiority of our method by visually and quantitatively.