Celebratio Mathematica

R. Yin, T. Gao, Y. M. Lu, and I. Daubechies: “A tale of two bases: Local-nonlocal regularization on image patches with convolution framelets,” SIAM J. Imaging Sci. 10 : 2 (2017), pp. 711–750. MR 3650425 Zbl 06725650 ArXiv 1606.01377 article

We propose an image representation scheme combining the local and nonlocal characterization of patches in an image. Our representation scheme can be shown to be equivalent to a tight frame constructed from convolving local bases (e.g., wavelet frames, discrete cosine transforms, etc.) with nonlocal bases (e.g., spectral basis induced by nonlinear dimension reduction on patches), and we call the resulting frame elements convolution framelets. Insight gained from analyzing the1806 proposed representation leads to a novel interpretation of a recent high-performance patch-based image processing algorithm using the point integral method (PIM) and the low dimensional manifold model (LDMM) [S. Osher, Z. Shi, and W. Zhu, “Low Dimensional Manifold Model for Image Processing”, Tech. Rep., CAM report 16-04, UCLA, Los Angeles, CA, 2016]. In particular, we show that LDMM is a weighted \( \ell_2 \)-regularization on the coefficients obtained by decomposing images into linear combinations of convolution framelets; based on this understanding, we extend the original LDMM to a reweighted version that yields further improved results. In addition, we establish the energy concentration property of convolution framelet coefficients for the setting where the local basis is constructed from a given nonlocal basis via a linear reconstruction framework; a generalization of this framework to unions of local embeddings can provide a natural setting for interpreting BM3D, one of the state-of-the-art image denoising algorithms.

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Rujie (Rachel) Yin	Related
Yue M. Lu	Related

T. Gao, S. Z. Kovalsky, and I. Daubechies: “Gaussian process landmarking on manifolds,” SIAM J. Math. Data Sci. 1 : 1 (2019), pp. 208–236. MR 3949707 ArXiv 1802.03479 article

As a means of improving analysis of biological shapes, we propose an algorithm for sampling a Riemannian manifold by sequentially selecting points with maximum uncertainty under a Gaussian process model. This greedy strategy is known to be near-optimal in the experimental design literature, and appears to outperform the use of user-placed landmarks in representing the geometry of biological objects in our application. In the noiseless regime, we establish an upper bound for the mean squared prediction error (MSPE) in terms of the number of samples and geometric quantities of the manifold, demonstrating that the MSPE for our proposed sequential design decays at a rate comparable to the oracle rate achievable by any sequential or non-sequential optimal design; to our knowledge this is the first result of this type for sequential experimental design. The key is to link the greedy algorithm to reduced basis methods in the context of model reduction for partial differential equations. We expect this approach will find additional applications in other fields of research.

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Shahar Z. Kovalsky	Related

T. Gao, S. Z. Kovalsky, D. M. Boyer, and I. Daubechies: “Gaussian process landmarking for three-dimensional geometric morphometrics,” Siam J. Math. Data Sci. 1 : 1 (2019), pp. 237–267. MR 3949708 ArXiv 1807.11887 article

We demonstrate applications of the Gaussian process-based landmarking algorithm proposed in [T. Gao, S. Z. Kovalsky, and I. Daubechies, SIAM J. Math. Data Sci., 1 (2019), pp.208–236] to geometric morphometrics, a branch of evolutionary biology centered at the analysis and comparisons of anatomical shapes, and compares the automatically sampled landmarks with the “ground truth” landmarks manually placed by evolutionary anthropologists; the results suggest that Gaussian process landmarks perform equally well or better, in terms of both spatial coverage and downstream statistical analysis. We provide a detailed exposition of numerical procedures and feature filtering algorithms for computing high-quality and semantically meaningful diffeomorphisms between disk-type anatomical surfaces.

Tingran Gao	Related
Shahar Z. Kovalsky	Related
Douglas Martin Boyer	Related

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