Adaptive Superresolution in Deconvolution of Sparse Peaks

TL;DR

The paper tackles sparse peak deconvolution under a smooth kernel, where the observed data $f$ is the convolution $f = G * \mu$ of a continuous sparse measure $\mu$ with a kernel $G$. It analyzes the discrete ${\ell_1}$-minimization (LASSO) formulation on a grid, showing that off-grid spikes produce multiple adjacent grid peaks and deriving an optimality-curve framework to connect grid solutions to true spike locations. The authors prove both exact recovery on-grid and structured recovery off-grid in 1D and extend to higher dimensions, then introduce a self-driven adaptive grid algorithm that refines the discretization around detected supports to achieve super-resolution in 1D and 2D. Extensive simulated experiments with Gaussian kernels and varying peak configurations demonstrate the method’s ability to localize amplitudes and positions, providing a practically applicable bridge between theory and fluorescence microscopy and spectral analysis. The work offers a principled approach to reduce discretization bias, quantify reconstruction error via a posteriori estimates, and iteratively sharpen resolution without manual parameter tuning.

Abstract

The aim of this paper is to investigate superresolution in deconvolution driven by sparsity priors. The observed signal is a convolution of an original signal with a continuous kernel.With the prior knowledge that the original signal can be considered as a sparse combination of Dirac delta peaks, we seek to estimate the positions and amplitudes of these peaks by solving a finite dimensional convex problem on a computational grid. Because, the support of the original signal may or may not be on this grid, by studying the discrete deconvolution of sparse peaks using L1-norm sparsity prior, we confirm recent observations that canonically the discrete reconstructions will result in multiple peaks at grid points adjacent to the location of the true peak. Owning to the complexity of this problem, we analyse carefully the de-convolution of single peaks on a grid and gain a strong insight about the dependence of the reconstructed magnitudes on the exact peak location. This in turn allows us to infer further information on recovering the location of the exact peaks i.e. to perform super-resolution. We analyze in detail the possible cases that can appear and based on our theoretical findings, we propose an self-driven adaptive grid approach that allows to perform superresolution in one-dimensional and multi-dimensional spaces. With the view that the current study can provide a further step in the development of more robust algorithms for the detection of single molecules in fluorescence microscopy or identification of characteristic frequencies in spectral analysis, we demonstrate how the proposed approach can recover sparse signals using simulated clusters of point sources (peaks) of low-resolution in one and two-dimensional spaces.

Figures (16)

• Figure 1: Supremum norm of $r^N$(residual) with respect to the size $N$ of the computational grid. Here, we considered $\mu = \delta_\xi$ and $\mu^N = \sum_{k=1}^N c_k\delta_{x_k}$. Thus, $r^N(x) = \frac{H(x-\xi) - \sum_{k=1}^N c_k H(x-x_k)}{\lambda}-q^N(x)$ based on \ref{['eq:subdiff_measures']} and $H(x)$ was a Gaussian kernel.
• Figure 2: If $\xi$ is in the blue interval, the reconstructed solution $\mu^N$ consists of only one peak. In the case that $\xi$ is located in the red interval, then one recovered peak is not sufficient.
• Figure 3: Function $p(x)$ around location $\xi$ for the cases where the reconstructed signal $\mu^N$ has (A) a single nonzero coefficient and (B) two nonzero coefficients between the location of the exact peak $\xi$.
• Figure 4: Function $p(x)$ around location $\xi$ when the reconstructed signal $\mu^N$ has four nonzero coefficients. (A) 3D plot of $p(x)$ and xy-plane with the computational grid (marked blue dots) and location of the nonzero coefficients (marked with blue circles) (B) Isocontours of $p(x)$, computational grid (marked with blue dots) and nonzero locations of $\mu^N$ (marked with blue circles). The exact location denoted by red $\mathrm{x}$.
• Figure 5: Numerical solution on a grid for decreasing value of the regularization parameter $\lambda$. The blue circles show the locations of the nonzero coefficients of the numerical solution. The blue dots are the grid points. The red circle is the smallest circle that encloses the largest convex hull formed by grid points that surround $\xi$ (points that can get nonzero entries). The maximum number of nonzero entries is depicted in (C).

Theorems & Definitions (6)

• Proposition 2.1
• proof
• Proposition 3.1
• proof
• Theorem 3.2
• proof