Summary: 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 \(\mathcal{R}_{\mathfrak{p}} f\) at a single angle \(\theta\), where \(\mathfrak{p} = (\cos \theta, \sin \theta)\) 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 \(\mathfrak{p}\) and sampling set \(X_{\mathfrak{p}} \subseteq \mathbb{R}\) such that \(f\) can be determined by its single-angle Radon (w.r.t. \(\mathfrak{p}\)) samples at \(X_{\mathfrak{p}}\). For the general generator \(\varphi\), we address the eligible \(\mathfrak{p}\) for the two cases: (1) \(\varphi\) being nonvanishing \((\int_{\mathbb{R}^2} \varphi(\mathfrak{x}) d \mathfrak{x} \neq 0)\) and (2) being vanishing \((\int_{\mathbb{R}^2} \varphi (\mathfrak{x}) d \mathfrak{x} = 0)\). We prove that eligible \(X_{\mathfrak{p}}\) exists for general \(\varphi\). In particular, \(X_{\mathfrak{p}}\) can be explicitly constructed if \(\varphi \in C^1(\mathbb{R}^2)\). Positive definite functions form an important class of functions that have been widely applied in scattered data interpolation. For the case that \(\varphi\) is positive definite, the corresponding single-angle problem in SIS \(V(\varphi, \mathbb{Z}^2)\) is addressed such that \(X_{\mathfrak{p}}\) can be constructed easily. Besides using the samples of the single-angle Radon transform, another common feature for our recovery results is that the number of the required samples is minimum.