论文标题
通过单角样品确定偏移不变空间中紧凑的功能
Determination of compactly supported functions in shift-invariant space by single-angle Radon samples
论文作者
论文摘要
传统上，l^{2}中的功能$ f \的计算机断层扫描（\ mathbb {r}^{2}）$取决于其radon的样本多个角度变换，有时需要由其radon transform $ f $ radon变换的样品重建$ \ Mathcal {r} _ {\ emph {\ textbf {p}}} f $以单个角度$θ$，其中$ \ emph {\ textbf {p}} =（\cosθ，\sinθ）$是方向向量。这自然会导致一个问题，即识别可以由单个角度$θ$确定其ra样品确定的功能。 $φ$生成的换档空间$ v（φ，\ mathbb {z}^2）$是一种在包括小波分析和信号处理在内的许多字段中广泛考虑的功能空间。在本文中，我们检查了v（φ，\ mathbb {z}^2）$的紧凑型功能的单角重建问题$ f \。该问题的主要问题是确定合格的$ \ emph {\ textbf {p}} $和采样设置$ x _ {\ emph {\ textbf {p textbf {p}} \ subseteq \ subseteq \ subseteq \ mathbb {r} $ f $ f $ f $ f $可以由其单端radon（W.R.W.R.）确定。 $ x _ {\ emph {\ textbf {p}}} $ $ x _ {\ emph {\ textbf {p}} $）y。对于通用发电机$φ$，我们解决了合格的$ \ emph {\ textbf {p}} $对于两种情况：（1）$φ$是不变的（$ \ int _ {\ mathbb {r}^{2}}φ（\ emph {\ textbf {x}}）d \ emph {\ emph {\ textbf {x}}} \ neq0 $）和（2）正在消失（$ \ int _ {\ mathbb {r}^2}φ（\ emph {\ textbf {x}}）d \ emph {\ textbf {\ textbf {x}}} = 0 $）。我们证明，对于常规$φ$，符合条件$ x _ {\ emph {\ textbf {p}}} $。特别是，如果$ x _ {\ emph {\ textbf {p}}}} $可以明确构造，如果$ c^{1}（\ Mathbb {r}^{2}）$，则可以构造。单角问题与$φ$为正定确定的情况相对应，以使$ x _ {\ emph {\ textbf {p}}}} $很容易构造。
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 $θ$, where $\emph{\textbf{p}}=(\cosθ, \sinθ)$ 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 $θ$. The shift-invariant space $V(φ, \mathbb{Z}^2)$ generated by $φ$ 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 $f\in V(φ, \mathbb{Z}^2)$. 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 $φ$, we address the eligible $\emph{\textbf{p}}$ for the two cases: (1) $φ$ being nonvanishing ($\int_{\mathbb{R}^{2}}φ(\emph{\textbf{x}})d\emph{\textbf{x}}\neq0$) and (2) being vanishing ($\int_{\mathbb{R}^2}φ(\emph{\textbf{x}})d\emph{\textbf{x}}=0$). We prove that eligible $X_{\emph{\textbf{p}}}$ exists for general $φ$. In particular, $X_{\emph{\textbf{p}}}$ can be explicitly constructed if $φ\in C^{1}(\mathbb{R}^{2})$. The single-angle problem corresponding to the case that $φ$ being positive definite is addressed such that $X_{\emph{\textbf{p}}}$ can be constructed easily.
