Determination of compactly supported functions in shift-invariant space by single-angle Radon samples

While traditionally the computerized tomography of a function $f\in L^{2}(\mathbb{R}^{2})$ depends on the samples of its Radon transform at multiple angles, the real-time imaging sometimes requires the reconstruction of $f$ by the samples of its Radon transform \mathcal{R}_{\emph{\textbf{p}}}f at a single angle $\theta$, where \emph{\textbf{p}}=(\cos\theta, \sin\theta) is the direction vector. This naturally leads to the question of identifying those functions that can be determined by their Radon samples at a single angle $\theta$. The shift-invariant space $V(\varphi, \mathbb{Z}^2)$ generated by $\varphi$ is a type of function space that has been widely considered in many fields including wavelet analysis and signal processing. In this paper we examine the single-angle reconstruction problem for compactly supported functions $f\in V(\varphi, \mathbb{Z}^2)$. The central issue for the problem is to identify the eligible \emph{\textbf{p}} and sampling set X_{\emph{\textbf{p}}}\subseteq \mathbb{R} such that $f$ can be determined by its single-angle Radon (w.r.t \emph{\textbf{p}}) samples at X_{\emph{\textbf{p}}}. For the general generator $\varphi$, we address the eligible \emph{\textbf{p}} for the two cases: (1) $\varphi$ being nonvanishing (\int_{\mathbb{R}^{2}}\varphi(\emph{\textbf{x}})d\emph{\textbf{x}}\neq0) and (2) being vanishing (\int_{\mathbb{R}^2}\varphi(\emph{\textbf{x}})d\emph{\textbf{x}}=0). We prove that eligible X_{\emph{\textbf{p}}} exists for general $\varphi$. In particular, X_{\emph{\textbf{p}}} can be explicitly constructed if $\varphi\in C^{1}(\mathbb{R}^{2})$. The single-angle problem corresponding to the case that $\varphi$ being positive definite is addressed such that X_{\emph{\textbf{p}}} can be constructed easily

Single-shot phase retrieval: a holography-driven problem in Sobolev space

The phase-shifting digital holography (PSDH) is a widely used approach for recovering signals by their interference (with reference waves) intensity measurements. Such measurements are traditionally from multiple shots (corresponding to multiple reference waves). However, the imaging of dynamic signals requires a single-shot PSDH approach, namely, such an approach depends only on the intensity measurements from the interference with a single reference wave. In this paper, based on the uniform admissibility of plane (or spherical) reference wave and the interference intensity-based approximation to quasi-interference intensity, the nonnegative refinable function is applied to establish the single-shot PSDH in Sobolev space. Our approach is conducted by the intensity measurements from the interference of the signal with a single reference wave. The main results imply that the approximation version from such a single-shot approach converges exponentially to the signal as the level increases. Moreover, like the transport of intensity equation (TIE), our results can be interpreted from the perspective of intensity difference.Comment: 37page

Phase retrieval of analytic signals from short-time Fourier transform measurements

Analytic signals constitute a class of signals that are widely applied in time-frequency analysis such as extracting instantaneous frequency (IF) or phase derivative in the characterization of ultrashort laser pulse. The purpose of this paper is to investigate the phase retrieval (PR) problem for analytic signals in $\mathbb{C}^{N}$ by short-time Fourier transform (STFT) measurements since they enjoy some very nice structures. Since generic analytic signals are generally not sparse in the time domain, the existing PR results for sparse (in time domain) signals do not apply to analytic signals. We will use bandlimited windows that usually have the full support length $N$ which allows us to get much better resolutions on low frequencies. More precisely, by exploiting the structure of the STFT for analytic signals, we prove that the STFT based phase retrieval (STFT-PR for short) of generic analytic signals can be achieved by their $(3\lfloor\frac{N}{2}\rfloor+1)$ measurements. Since the generic analytic signals are $(\lfloor \frac{N}{2}\rfloor+1)$-sparse in the Fourier domain, such a number of measurements is lower than $4N+\hbox{O}(1)$ and $\hbox{O}(k^{3})$ which are required in the literature for STFT-PR of all signals and of $k^{2}$-sparse (in the Fourier domain) signals in $\mathbb{C}^{N^{2}}$, respectively. Moreover, we also prove that if the length $N$ is even and the windows are also analytic, then the number of measurements can be reduced to $(\frac{3 N}{2}-1)$. As an application of this we get that the instantaneous frequency (IF) of a generic analytic signal can be exactly recovered from the STFT measurements

Prevalence of depression among university students in low and middle income countries (LMICs): a systematic review and meta-analysis

Introduction Though university years are peak time for the onset of many mental health problems including depression, knowledge on the prevalence of depressive symptoms among university students in low and middle-income countries (LMICs) is limited. This study examined the prevalence of depressive symptoms among university students in LMICs. Methods We systematically searched PubMed, Web of Science, and WHO Global Health Library for studies published between January 1, 2009 and December 31, 2018. Cross-sectional and longitudinal studies conducted in LMICs (as defined by World Bank), having a sample size≥500 and published in English were identified. Data on study characteristics and prevalence of depressive symptoms was extracted by two investigators. Estimate of prevalence was pooled in meta-analysis using random-effects meta-analysis. Sub-group differences were estimated using mixed-effects meta-analysis and meta-regression. Results Depressive symptoms prevalence data were extracted from 37 studies involving 76,608 individuals in 20 countries. Studies on depression among university students in LMICs were limited, and most were based on non-representative and small study samples. The overall prevalence of depressive symptoms was 24.4% (95% CI, 19.2% -30.5%). Subgroup analysis revealed that the prevalence did not vary by study design, sampling technique, sample size, study major, educational level, economic regions, and screening instrument. Limitations Only English language studies were included. Included studies were diverse in design, screening tool, and sample size that introduced substantial heterogeneity. Conclusion Overall prevalence of depressive symptoms among university students in LMICs was 24.4%, however, finding should be interpreted with caution. Further research is needed to address the issue

Prevalence, Management, and Associated Factors of Obesity, Hypertension, and Diabetes in Tibetan Population Compared with China Overall

Tibetans’ life expectancy lags behind China’s average. Obesity and noncommunicable diseases (NCDs) contribute to health disparity, but NCD patterns among Tibetans are unknown. To examine the prevalence, management, and associated factors for obesity, hypertension, and diabetes among Tibetans, compared with China’s average, we systematically searched PubMed and China National Knowledge Infrastructure databases for studies between January 2010 and April 2021. Thirty-nine studies were included for systematic review, among thirty-seven that qualified for meta-analysis, with 115,403 participants. Pooled prevalence was 47.9% (95% CI 38.0–57.8) for overweight/obesity among adults (BMI ≥ 24 kg/m2) and 15.4% (13.7–17.2) among children using Chinese criteria, which are lower than the national rates of 51.2% and 19.0%, respectively. The estimate for hypertension (31.4% [27.1–35.7]) exceeded China’s average (27.5%), while diabetes (7.5% [5.2–9.8]) was lower than average (11.9%). Men had a higher prevalence of the three conditions than women. Residents in urban areas, rural areas, and Buddhist institutes had monotonically decreased prevalence in hypertension and diabetes. Awareness, treatment, and control rates for hypertension and diabetes were lower than China’s average. Urban residence and high altitude were consistent risk factors for hypertension. Limited studies investigated factors for diabetes, yet none exist for obesity. Tibetans have high burdens of obesity and hypertension. Representative and longitudinal studies are needed for tailored interventions. There are considerable variations in study design, study sample selection, and data-analysis methods, as well as estimates of reviewed studies.This work was supported by the National Key Research & Development Program of China [grant numbers 2017YFC0907200 & 2017YFC0907201]; the Natural Scientific Foundation of China [grant number 82103846]; and the Chinese Nutrition Society [grant number CNS-SCP-2020-40]