Block Diagonalization of Quaternion Circulant Matrices with Applications

It is well-known that a complex circulant matrix can be diagonalized by a discrete Fourier matrix with imaginary unit $\mathtt{i}$. The main aim of this paper is to demonstrate that a quaternion circulant matrix cannot be diagonalized by a discrete quaternion Fourier matrix with three imaginary units $\mathtt{i}$, $\mathtt{j}$ and $\mathtt{k}$. Instead, a quaternion circulant matrix can be block-diagonalized into 1-by-1 block and 2-by-2 block matrices by permuted discrete quaternion Fourier transform matrix. With such a block-diagonalized form, the inverse of a quaternion circulant matrix can be determined efficiently similar to the inverse of a complex circulant matrix. We make use of this block-diagonalized form to study quaternion tensor singular value decomposition of quaternion tensors where the entries are quaternion numbers. The applications including computing the inverse of a quaternion circulant matrix, and solving quaternion Toeplitz system arising from linear prediction of quaternion signals are employed to validate the efficiency of our proposed block diagonalized results. A numerical example of color video as third-order quaternion tensor is employed to validate the effectiveness of quaternion tensor singular value decomposition

Phase Retrieval of Quaternion Signal via Wirtinger Flow

The main aim of this paper is to study quaternion phase retrieval (QPR), i.e., the recovery of quaternion signal from the magnitude of quaternion linear measurements. We show that all $d$-dimensional quaternion signals can be reconstructed up to a global right quaternion phase factor from $O(d)$ phaseless measurements. We also develop the scalable algorithm quaternion Wirtinger flow (QWF) for solving QPR, and establish its linear convergence guarantee. Compared with the analysis of complex Wirtinger flow, a series of different treatments are employed to overcome the difficulties of the non-commutativity of quaternion multiplication. Moreover, we develop a variant of QWF that can effectively utilize a pure quaternion priori (e.g., for color images) by incorporating a quaternion phase factor estimate into QWF iterations. The estimate can be computed efficiently as it amounts to finding a singular vector of a $4\times 4$ real matrix. Motivated by the variants of Wirtinger flow in prior work, we further propose quaternion truncated Wirtinger flow (QTWF), quaternion truncated amplitude flow (QTAF) and their pure quaternion versions. Experimental results on synthetic data and color images are presented to validate our theoretical results. In particular, for pure quaternion signal recovery, our quaternion method often succeeds with measurements notably fewer than real methods based on monochromatic model or concatenation model.Comment: 21 pages (paper+supplemental), 6 figure

Uniform Exact Reconstruction of Sparse Signals and Low-rank Matrices from Phase-Only Measurements

In phase-only compressive sensing (PO-CS), our goal is to recover low-complexity signals (e.g., sparse signals, low-rank matrices) from the phase of complex linear measurements. While perfect recovery of signal direction in PO-CS was observed quite early, the exact reconstruction guarantee for a fixed, real signal was recently done by Jacques and Feuillen [IEEE Trans. Inf. Theory, 67 (2021), pp. 4150-4161]. However, two questions remain open: the uniform recovery guarantee and exact recovery of complex signal. In this paper, we almost completely address these two open questions. We prove that, all complex sparse signals or low-rank matrices can be uniformly, exactly recovered from a near optimal number of complex Gaussian measurement phases. By recasting PO-CS as a linear compressive sensing problem, the exact recovery follows from restricted isometry property (RIP). Our approach to uniform recovery guarantee is based on covering arguments that involve a delicate control of the (original linear) measurements with overly small magnitude. To work with complex signal, a different sign-product embedding property and a careful rescaling of the sensing matrix are employed. In addition, we show an extension that the uniform recovery is stable under moderate bounded noise. We also propose to add Gaussian dither before capturing the phases to achieve full reconstruction with norm information. Experimental results are reported to corroborate and demonstrate our theoretical results.Comment: 39 pages, 1 figur

A One-step Image Retargeing Algorithm Based on Conformal Energy

The image retargeting problem is to find a proper mapping to resize an image to one with a prescribed aspect ratio, which is quite popular these days. In this paper, we propose an efficient and orientation-preserving one-step image retargeting algorithm based on minimizing the harmonic energy, which can well preserve the regions of interest (ROIs) and line structures in the image. We also give some mathematical proofs in the paper to ensure the well-posedness and accuracy of our algorithm.Comment: 24 pages, 10 figure