Celebratio Mathematica

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

J. Xu, H. Yang, and I. Daubechies: “Recursive diffeomorphism-based regression for shape functions,” SIAM J. Math. Anal. 50 : 1 (2018), pp. 5–32. MR 3740323 Zbl 1381.65104 ArXiv 1610.03819 article

This paper proposes a recursive diffeomorphism based regression method for one-dimensional generalized mode decomposition problem that aims at extracting generalized modes \( \alpha_k(t) \) \( s_k(2\pi N_k\phi_k(t)) \) from their superposition \[ \sum_{k=1}^K \alpha_k(t)\,s_k(2\pi N_k\phi_k(t)) .\] First, a one-dimensional synchrosqueezed transform is applied to estimate instantaneous information, e.g., \( \alpha_k(t) \) and \( N_k\phi_k(t) \). Second, a novel approach based on diffeomorphisms and nonparametric regression is proposed to estimate wave shape functions \( s_k(t) \). These two methods lead to a framework for the generalized mode decomposition problem under a weak well-separation condition. Numerical examples of synthetic and real data are provided to demonstrate the fruitful applications of these methods.

Jieren Xu	Related
Haizhao Yang	Related

J. Xu, Y. Li, D. Dunson, I. Daubechies, and H. Yang: Non-oscillatory pattern learning for non-stationary signals. Preprint, May 2018. ArXiv 1805.08102 techreport

This paper proposes a novel non-oscillatory pattern (NOP) learning scheme for several oscillatory data analysis problems including signal decomposition, super-resolution, and signal sub-sampling. To the best of our knowledge, the proposed NOP is the first algorithm for these problems with fully non-stationary oscillatory data with close and crossover frequencies, and general oscillatory patterns. NOP is capable of handling complicated situations while existing algorithms fail; even in simple cases, e.g., stationary cases with trigonometric patterns, numerical examples show that NOP admits competitive or better performance in terms of accuracy and robustness than several state-of-the-art algorithms.

David Brian Dunson	Related
Jieren Xu	Related
Haizhao Yang	Related
Yitong Li	Related

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