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This paper explores possibilities for new uses of the Radon transform for imaging and analysis of lightfields. We show that the previously reported Dimansionality Gap can be derived from an ultrahyperbolic PDE, first proposed by F. John, which is satisfied by lightfields. Based on inverse John transform we demonstrate rigorous Focal Stack rendering and viewing from arbitrary angles. Based on Asgeirsson theorems for the ultrahyperbolic PDE we derive new kernels for processing lightfields. Our kernels provide alternative methods for depth computation and other image processing in lightfields.

Recommended citation: T. Georgiev, H. Qin, and H. Li, John transform and ultrahyperbolic equation for lightfields, arXiv preprint arXiv:1907.01186, (2019).

Metrics of graph Laplacian eigenvectors

Published in SPIE conference: Wavelets and Sparsity XVIII, 2019

The application of graph Laplacian eigenvectors has been quite popular in the graph signal processing field: one can use them as ingredients to design smooth multiscale basis. Our long-term goal is to study and understand the dual geometry of graph Laplacian eigenvectors. In order to do that, it is necessary to define a certain metric to measure the behavioral differences between each pair of the eigenvectors. Saito (2018) considered the ramified optimal transportation (ROT) cost between the square of the eigenvectors as such a metric. Clonginger and Steinerberger (2018) proposed a way to measure the affinity (or ‘similarity’) between the eigenvectors based on their Hadamard (HAD) product. In this article, we propose a simplified ROT metric that is more computational efficient and introduce two more ways to define the distance between the eigenvectors, i.e., the time-stepping diffusion (TSD) metric and the difference of absolute gradient (DAG) pseudometric. The TSD metric measures the cost of “flattening” the initial graph signal via diffusion process up to certain time, hence it can be viewed as a time-dependent version of the ROT metric. The DAG pseudometric is the $l_2$ distance between the feature vectors derived from the eigenvectors, in particular, the absolute gradients of the eigenvectors. We then compare the performance of ROT, HAD and the two new “metrics” on different kinds of graphs. Finally, we investigate their relationship as well as their pros and cons.

Recommended citation: H. Li and N. Saito "Metrics of graph Laplacian eigenvectors", Proc. SPIE 11138, Wavelets and Sparsity XVIII, 111381K (9 September 2019).

Lightfield Coordinates Adapted to Asgeirsson Theorem

Published in arXiv, 2019

John differential equation and its canonical form, the ultrahyperbolic equation, plays important role in lightfield imaging. The equation describes a local constraint on the lightfield, that was first observed as a dimensionality gap in the frequency representation. Related to the ultrahyperbolic equation, Asgeirsson theorems describe global properties. These indicate new, global, constraints on the lightfield. In order to help validate those theorems on real captured images, we introduce a coordinate system for the lightfield, which suits better the Asgeirsson theorems, and analyze behaviour in terms of the new coordinates.

Recommended citation: H. Li, H. Qin, and T. Georgiev, Lightfield coordinates adapted to asgeirsson theorem, arXiv preprint arXiv:1909.07923, (2019).

Natural Graph Wavelet Packet Dictionaries

Published in Journal of Fourier Analysis and Applications, 2021

We introduce a set of novel multiscale basis transforms for signals on graphs that utilize their “dual” domains by incorporating the “natural” distances between graph Laplacian eigenvectors, rather than simply using the eigenvalue ordering. These basis dictionaries can be seen as generalizations of the classical Shannon wavelet packet dictionary to arbitrary graphs, and do not rely on the frequency interpretation of Laplacian eigenvalues. We describe the algorithms (involving either vector rotations or orthogonalizations) to construct these basis dictionaries, use them to efficiently approximate graph signals through the best basis search, and demonstrate the strengths of these basis dictionaries for graph signals measured on sunflower graphs and street networks.

Recommended citation: Cloninger, A., Li, H. & Saito, N. Natural Graph Wavelet Packet Dictionaries. J Fourier Anal Appl 27, 41 (2021). https://link.springer.com/article/10.1007/s00041-021-09832-3

talks

Design natural Graph Wavelet by measuring Distance between Graph Laplacian eigenvectors

Published: January 24, 2019

This is my qualify exam talk, which is my first research talk! It demonstrates what is the problem of using graph Laplacian eigenvectors to design graph wavelets simply based on the ordering of corresponding eigenvalues. Moreover, it proposes a remedy strategy of this issue, which eventually leads to design natural graph wavelet packets in the future.

Metrics of graph Laplacian eigenvectors

Published: August 15, 2019

In this talk, we summerize the previous behavioral metrics of graph Laplacian eigenvectors and propose two new ways for measuring their behaviors on graphs. Then, we will compare their pros and cons in two different scenarios.

Natural Graph Wavelet Dictionaries: Methods and Applications

Published: June 07, 2021

This is my PhD exit seminar! The graph Laplacian is widely used in the graph signal processing field. When attempting to design graph wavelet transforms, people have been using its eigenvalues and eigenvectors in place of the frequencies and complex exponentials that are the backbone of the Fourier theory on Euclidean domains. However, this viewpoint could be misleading since the Laplacian eigenvalues cannot be interpreted as the frequencies of the eigenvectors on a general graph. Instead, we introduce and review several “natural” metrics of graph Laplacian eigenvectors, and propose a new way to naturally organize the eigenvectors by incorporating these metrics into a “dual” graph. We then introduce a set of novel multiscale basis transforms for graph signals fully utilizing this dual graph, rather than simply using the eigenvalue ordering. These basis dictionaries can be seen as generalizations of the classical Shannon/Meyer wavelet packet dictionary to arbitrary graphs, and do not rely on the frequency interpretation of Laplacian eigenvalues. We describe the algorithms (involving vector rotations, or orthogonalizations, or lapped orthogonal projections) to efficiently approximate and compress signals through the best-basis algorithm, and demonstrate the strengths of these basis dictionaries for graph signals on sunflower graphs and road traffic networks. Lastly, we propose a way to modify the spectral filters in the spectral graph wavelet transform by utilizing the structure of the dual graph instead of using the eigenvalue-dependent smooth functions. By doing so, we generate a redundant wavelet frame, propose a way to reduce its redundancy, and discuss its potential for applications.

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Teaching assistant

Math courses, UC Davis, 2017

I worked as a TA for various of math courses at UC Davis from Winter 2017 to Winter 2020.

Haotian Li

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