Celebratio Mathematica

I. Daubechies, J. Lu, and H.-T. Wu: “Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool,” Appl. Comput. Harmon. Anal. 30 : 2 (2011), pp. 243–261. MR 2754779 Zbl 1213.42133 ArXiv 0912.2437 article

The EMD algorithm is a technique that aims to decompose into their building blocks functions that are the superposition of a (reasonably) small number of components, well separated in the time-frequency plane, each of which can be viewed as approximately harmonic locally, with slowly varying amplitudes and frequencies. The EMD has already shown its usefulness in a wide range of applications including meteorology, structural stability analysis, medical studies. On the other hand, the EMD algorithm contains heuristic and ad hoc elements that make it hard to analyze mathematically.

In this paper we describe a method that captures the flavor and philosophy of the EMD approach, albeit using a different approach in constructing the components. The proposed method is a combination of wavelet analysis and reallocation method. We introduce a precise mathematical definition for a class of functions that can be viewed as a superposition of a reasonably small number of approximately harmonic components, and we prove that our method does indeed succeed in decomposing arbitrary functions in this class. We provide several examples, for simulated as well as real data.

Jianfeng Lu	Related
Hau-Tieng Wu	Related

H. Yang, J. Lu, W. P. Brown, I. Daubechies, and L. Ying: “Quantitative canvas weave analysis using 2-D synchrosqueezed transforms: Application of time-frequency analysis to art investigation,” IEEE Signal Proc. Mag. 32 : 4 (July 2015), pp. 55–63. article

Quantitative canvas weave analysis has many applications in art investigations of paintings, including dating, forensics, and canvas rollmate identification. Traditionally, canvas analysis is based on X-radiographs. Prior to serving as a painting canvas, a piece of fabric is coated with a priming agent; smoothing its surface makes this layer thicker between and thinner right on top of weave threads. These variations affect the X-ray absorption, making the weave pattern stand out in X-ray images of the finished painting. To characterize this pattern, it is customary to visually inspect small areas within the X-radiograph and count the number of horizontal and vertical weave threads; averages of these then estimate the overall canvas weave density. The tedium of this process typically limits its practice to just a few sample regions of the canvas. In addition, it does not capture more subtle information beyond weave density, such as thread angles or variations in the weave pattern. Signal processing techniques applied to art investigation are now increasingly used to develop computer-assisted canvas weave analysis tools.

Jianfeng Lu	Related
Haizhao Yang	Related
Lexing Ying	Related
William P. Brown	Related

B. Cornelis, H. Yang, A. Goodfriend, N. Ocon, J. Lu, and I. Daubechies: “Removal of canvas patterns in digital acquisitions of paintings,” IEEE Trans. Image Process. 26 : 1 (2017), pp. 160–171. MR 3579347 article

We address the removal of canvas artifacts from high-resolution digital photographs and X-ray images of paintings on canvas. Both imaging modalities are common investigative tools in art history and art conservation. Canvas artifacts manifest themselves very differently according to the acquisition modality; they can hamper the visual reading of the painting by art experts, for instance, in preparing a restoration campaign. Computer-aided canvas removal is desirable for restorers when the painting on canvas they are preparing to restore has acquired over the years a much more salient texture. We propose a new algorithm that combines a cartoon-texture decomposition method with adaptive multiscale thresholding in the frequency domain to isolate and suppress the canvas components. To illustrate the strength of the proposed method, we provide various examples, for acquisitions in both imaging modalities, for paintings with different types of canvas and from different periods. The proposed algorithm outperforms previous methods proposed for visual photographs such as morphological component analysis and Wiener filtering and it also works for the digital removal of canvas artifacts in X-ray images.

Bruno Cornelis	Related
Jianfeng Lu	Related
Haizhao Yang	Related
Noelle Ocon	Related
Alex Goodfriend	Related

W. Zhu, Q. Qiu, B. Wang, J. Lu, G. Sapiro, and I. Daubechies: “Stop memorizing: A data-dependent regularization framework for intrinsic pattern learning,” Siam J. Math. Data Sci. 1 : 3 (2019), pp. 479–496. MR 3996914 ArXiv 1805.07291 article

Deep neural networks (DNNs) typically have enough capacity to fit random data by brute force even when conventional data-dependent regularizations focusing on the geometry of the features are imposed. We find out that the reason for this is the inconsistency between the enforced geometry and the standard softmax cross entropy loss. To resolve this, we propose a new framework for data-dependent DNN regularization, the Geometrically-Regularized-Self-Validating neural Networks (GRSVNet). During training, the geometry enforced on one batch of features is simultaneously validated on a separate batch using a validation loss consistent with the geometry. We study a particular case of GRSVNet, the Orthogonal-Low-rank Embedding (OLE)-GRSVNet, which is capable of producing highly discriminative features residing in orthogonal low-rank subspaces. Numerical experiments show that OLE-GRSVNet outperforms DNNs with conventional regularization when trained on real data. More importantly, unlike conventional DNNs, OLE-GRSVNet refuses to memorize random data or random labels, suggesting it only learns intrinsic patterns by reducing the memorizing capacity of the baseline DNN.

Guillermo R. Sapiro	Related
Jianfeng Lu	Related
Qiang Qiu	Related
Wei Zhu	Related
Bao Wang	Related

S. Z. Kovalsky, N. Aigerman, I. Daubechies, M. Kazhdan, J. Lu, and S. Steinerberger: Non-convex planar harmonic maps. Preprint, January 2020. ArXiv 2001.01322 techreport

We formulate a novel characterization of a family of invertible maps between two-dimensional domains. Our work follows two classic results: The Radó–Kneser–Choquet (RKC) theorem, which establishes the invertibility of harmonic maps into a convex planer domain; and Tutte’s embedding theorem for planar graphs–RKC’s discrete counterpart–which proves the invertibility of piecewise linear maps of triangulated domains satisfying a discrete-harmonic principle, into a convex planar polygon. In both theorems, the convexity of the target domain is essential for ensuring invertibility. We extend these characterizations, in both the continuous and discrete cases, by replacing convexity with a less restrictive condition. In the continuous case, Alessandrini and Nesi provide a characterization of invertible harmonic maps into non-convex domains with a smooth boundary by adding additional conditions on orientation preservation along the boundary. We extend their results by defining a condition on the normal derivatives along the boundary, which we call the cone condition; this condition is tractable and geometrically intuitive, encoding a weak notion of local invertibility. The cone condition enables us to extend Alessandrini and Nesi to the case of harmonic maps into non-convex domains with a piecewise-smooth boundary. In the discrete case, we use an analog of the cone condition to characterize invertible discrete-harmonic piecewise-linear maps of triangulations. This gives an analog of our continuous results and characterizes invertible discrete-harmonic maps in terms of the orientation of triangles incident on the boundary.

Jianfeng Lu	Related
Michael Kazhdan	Related
Shahar Z. Kovalsky	Related
Stefan Steinerberger	Related
Noam Aigerman	Related

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