Three families of super-resolution (SR) wavelets ${\mathrm{\Psi }}_{v,n}^g(x)$, ${\mathrm{\Psi }}_{u,n,m}^s(x)$ and ${\mathrm{\Psi }}_{w,n,m}^{ds}(x)$, to be called Gaussian SR (GSR), spline SR (SSR) and dual-spline SR (DSSR) wavelets, respectively, are introduced in this paper for resolving the super-resolution problem of recovering any point-mass $h(y)={\sum }_{\ell =1}^L{c}_{\ell }\delta (y-{\sigma }_{\ell })$, with $|{\sigma }_{\ell }-{\sigma }_k|\ge \eta$ for $\ell \ne k$, ${\sigma }_{\ell }\ne 0$, and $|c_{\ell }|>{\eta }^{⁎}$ for all $\ell ,k=1,\dots ,L$, where $\eta >0$ and ${\eta }^{⁎}>0$ are allowed to be arbitrarily small. Let ${\mathrm{\Psi }}_{\alpha ,n}={\mathrm{\Psi }}_{v,n}^g$, ${\mathrm{\Psi }}_{u,n,m}^s$ or ${\mathrm{\Psi }}_{w,n,m}^{ds}$, with $\alpha =v,u$ or w, respectively, where $m=\frac{1}{2},1,2,\cdots$ is suppressed. The SR wavelets are designed to have the n-th order of vanishing moments, with Fourier transform of their complex conjugates ${\widehat{\overline{\mathrm{\Psi }}}}_{\alpha ,n}(x)$ to possess the following properties: (i) ${\widehat{\overline{\mathrm{\Psi }}}}_{\alpha ,n}(x)\ge 0$ for all $x\in \mathbb{R}$, (ii) ${\max }_x{\widehat{\overline{\mathrm{\Psi }}}}_{\alpha ,n}(x)={\widehat{\overline{\mathrm{\Psi }}}}_{\alpha ,n}(\kappa )={\xi }^n$, where $\kappa \doteq 2.331122371$ and $\xi \doteq 1.449222080$ are positive constants independent of $n,\alpha$ and m, and (iii) the widths (or standard deviations) of ${\widehat{\overline{\mathrm{\Psi }}}}_{\alpha ,n}(x)$, with center at κ, tends to 0 very fast for large values of α. While the most popular approach to resolve this super-resolution problem is to consider the Fourier transform $d(x)={\sum }_{\ell =1}^L{c}_{\ell }{e}^{-i{\sigma }_{\ell }x}$ of $h(y)$ as the “data function” for solving the inverse problem of recovering L, ${\sigma }_1,\cdots ,{\sigma }_L$ and $c_1,\cdots ,c_L$ of the point-mass $d(x)$, our proposed approach is to consider the “enhanced data function” $D(a;\alpha ,n):=F_{{\mathrm{\Psi }}_{\alpha ,n}}(a)={\sum }_{\ell =1}^L{c}_{\ell }{\widehat{\overline{\mathrm{\Psi }}}}_{\alpha ,n}(a{\sigma }_{\ell })$, where $F_{{\mathrm{\Psi }}_{\alpha ,n}}(a)$, to be called the search function in this paper, is obtained by taking the continuous wavelet transform (CWT): $\left(W_{{\mathrm{\Psi }}_{\alpha ,n}}d\right)(t,a)={\int }_{-\infty }^{\infty }\overline{{\mathrm{\Psi }}_{\alpha ,n}\left(\frac{y-t}{a}\right)}d(y)\frac{dy}{a}$ of the data function $d(x)$, with ${\mathrm{\Psi }}_{\alpha ,n}$ as the analysis wavelet, followed by applying wavelet thresholding to “de-noise” the data function $d(x)$, by choosing an appropriate thresholding parameter $\gamma >0$, with $\gamma <{\eta }^{⁎}\times {\xi }^n$, in order not to remove any of the coefficients $c_{\ell }$, where $\ell =1,\cdots ,L$; and finally by setting $t=0$. Hence, the enhanced data function $D(a;\alpha ,n)$ is at least cleaner than the data function $d(x)$.

In our proposed approach, instead of directly recovering ${\sigma }_1,\cdots ,{\sigma }_L$ as in the published literature, we propose a “divide and conquer” strategy: first by applying “bottom-up thresholding” of the search function $F_{{\mathrm{\Psi }}_{\alpha ,n}}(a)$, with thresholding parameter ${\gamma }^{⁎}>0$ close to but not exceeding ${\eta }^{⁎}\times {\xi }^n$, to separate the set of the local extrema locations $a_{\ell }:=\frac{\kappa }{{\sigma }_{\ell }}$ of the function $F_{{\mathrm{\Psi }}_{\alpha ,n}}(a)$ in $\{a\in \mathbb{R}:a\ne 0\}$ into disjoint intervals of clusters, with more and smaller intervals and less number of local extrema $a_{\ell }$ in each interval for larger values of α; and secondly, by applying “top-down thresholding” to extract, one-by-one, of all local maxima, followed by all local minima (after a sign change), for each and every cluster. A desired leeway $\mathrm{\Delta }>0$ and lower bounds of the choice of the width parameter α are derived for the iterative application of top-down thresholding. Extension to ${\mathbb{R}}^s$ for $s\ge 2$ is also studied in this paper. For $s=2$, we observe that the imagery of the enhanced data function for a single point-mass at $({\sigma }^1,{\sigma }^2)$ where ${\sigma }^1,{\sigma }^2\ne 0$, resembles that of an “Airy disk” with center at $(\kappa /{\sigma }^1,\kappa /{\sigma }^2)$ in light microscopy and celestial telescopy of point-masses.

三族超分辨率 (SR) 小波${\mathrm{\Psi }}_{v,n}^G(X)$,${\mathrm{\Psi }}_{你,n,米}^s(X)$和${\mathrm{\Psi }}_{w,n,米}^{ds}(X)$，分别称为高斯SR（GSR），样条SR（SSR）和双样条SR（DSSR）小波，用于解决恢复任何点质量的超分辨率问题$H(\mathrm{是的})={\sum }_{\ell =1}^{\mathrm{大号}}{C}_{\ell }\delta (\mathrm{是的}-{\sigma }_{\ell })$， 和$|{\sigma }_{\ell }-{\sigma }_{ķ}|\ge \eta$为了$\ell \ne ķ$,${\sigma }_{\ell }\ne 0$， 和$|C_{\ell }|>{\eta }^{⁎}$对所有人$\ell ,ķ=1,\dots ,\mathrm{大号}$， 在哪里$\eta >0$和${\eta }^{⁎}>0$允许任意小。让${\mathrm{\Psi }}_{\alpha ,n}={\mathrm{\Psi }}_{v,n}^G$,${\mathrm{\Psi }}_{你,n,米}^s$或者${\mathrm{\Psi }}_{w,n,米}^{ds}$， 和$\alpha =v,你$或w，分别在哪里$米=\frac{1}{2},1,2,\cdots$被压制。SR 小波被设计为具有n阶消失矩，并对其复共轭进行傅里叶变换${\widehat{\overline{\mathrm{\Psi }}}}_{\alpha ,n}(X)$具备以下性质：(i)${\widehat{\overline{\mathrm{\Psi }}}}_{\alpha ,n}(X)\ge 0$对所有人$X\in \mathbb{R}$, (ii)${\mathrm{最大限度}}_X{\widehat{\overline{\mathrm{\Psi }}}}_{\alpha ,n}(X)={\widehat{\overline{\mathrm{\Psi }}}}_{\alpha ,n}(\kappa )={\xi }^n$， 在哪里$\kappa \doteq 2.331122371$和$\xi \doteq 1.449222080$是独立于的正常数$n,\alpha$和m，以及 (iii) 的宽度（或标准偏差）${\widehat{\overline{\mathrm{\Psi }}}}_{\alpha ,n}(X)$，以κ为中心，对于较大的α值，非常快地趋于 0 。虽然解决这个超分辨率问题最流行的方法是考虑傅里叶变换$d(X)={\sum }_{\ell =1}^{\mathrm{大号}}{C}_{\ell }{e}^{-\mathrm{一世}{\sigma }_{\ell }X}$的$H(\mathrm{是的})$作为解决恢复L的逆问题的“数据函数” ，${\sigma }_1,\cdots ,{\sigma }_{\mathrm{大号}}$和$C_1,\cdots ,C_{\mathrm{大号}}$质点的$d(X)$，我们提出的方法是考虑“增强数据功能”$D(\mathrm{一个};\alpha ,n)：=F_{{\mathrm{\Psi }}_{\alpha ,n}}(\mathrm{一个})={\sum }_{\ell =1}^{\mathrm{大号}}{C}_{\ell }{\widehat{\overline{\mathrm{\Psi }}}}_{\alpha ,n}(\mathrm{一个}{\sigma }_{\ell })$， 在哪里$F_{{\mathrm{\Psi }}_{\alpha ,n}}(\mathrm{一个})$，在本文中称为搜索函数，通过连续小波变换（CWT）得到：$\left(W_{{\mathrm{\Psi }}_{\alpha ,n}}d\right)(吨,\mathrm{一个})={\int }_{-\infty }^{\infty }\overline{{\mathrm{\Psi }}_{\alpha ,n}\left(\frac{\mathrm{是的}-吨}{\mathrm{一个}}\right)}d(\mathrm{是的})\frac{d\mathrm{是的}}{\mathrm{一个}}$数据函数$d(X)$， 和${\mathrm{\Psi }}_{\alpha ,n}$作为分析小波，然后应用小波阈值对数据函数进行“去噪”$d(X)$，通过选择适当的阈值参数$\gamma >0$， 和$\gamma <{\eta }^{⁎}\times {\xi }^n$, 为了不删除任何系数$C_{\ell }$， 在哪里$\ell =1,\cdots ,\mathrm{大号}$; 最后通过设置$吨=0$. 因此，增强的数据功能$D(\mathrm{一个};\alpha ,n)$至少比数据函数更干净$d(X)$.

在我们提出的方法中，不是直接恢复${\sigma }_1,\cdots ,{\sigma }_{\mathrm{大号}}$与已发表的文献一样，我们提出了“分而治之”的策略：首先通过应用搜索​​功能的“自下而上阈值”$F_{{\mathrm{\Psi }}_{\alpha ,n}}(\mathrm{一个})$, 带有阈值参数${\gamma }^{⁎}>0$接近但不超过${\eta }^{⁎}\times {\xi }^n$, 以分离局部极值位置的集合${\mathrm{一个}}_{\ell }：=\frac{\kappa }{{\sigma }_{\ell }}$功能的$F_{{\mathrm{\Psi }}_{\alpha ,n}}(\mathrm{一个})$在$\{\mathrm{一个}\in \mathbb{R}：\mathrm{一个}\ne 0\}$成不相交的簇间隔，间隔越来越小，局部极值的数量越来越少${\mathrm{一个}}_{\ell }$在每个区间中，对于较大的α值；其次，通过应用“自上而下的阈值”来逐一提取所有局部最大值，然后是所有局部最小值（在符号更改后），对于每个集群。理想的余地$\mathrm{\Delta }>0$为自上而下阈值的迭代应用推导出宽度参数α选择的下界。扩展至${\mathbb{R}}^s$为了$s\ge 2$本文也进行了研究。为了$s=2$，我们观察到单个点质量的增强数据函数的图像在$({\sigma }^1,{\sigma }^2)$在哪里${\sigma }^1,{\sigma }^2\ne 0$, 类似于中心在的“艾里斑”$(\kappa /{\sigma }^1,\kappa /{\sigma }^2)$在光学显微镜和点质量的天体望远镜中。