# Generalized Design of Sampling Kernels for 2-D FRI Signals

## TL;DR

This paper introduces a comprehensive framework for designing compactly supported sampling kernels in 2-D signals, applicable to both separable and nonseparable cases, enabling efficient sampling and reconstruction of finite-rate-of-innovation signals.

## Contribution

It presents a novel frequency domain design method for 2-D sampling kernels, including sum-of-modulated-spline kernels and nonseparable kernels, expanding the tools for 2-D signal sampling.

## Key findings

- Design framework produces kernels with compact support.
- 2-D SMS kernels can reproduce polynomial-modulated exponentials.
- Simulation results validate the effectiveness of the proposed kernels.

## Abstract

One of the interesting problems in the finite-rate-of-innovation signal sampling framework is the design of compactly supported sampling kernels. In this paper, we present a generic framework for designing sampling kernels in 2-D. We consider both separable and nonseparable kernels. The design is carried out in the frequency domain, where a set of alias cancellation conditions are imposed on the kernel's frequency response. The Paley-Wiener theorem for 2-D signals is invoked to arrive at admissible kernels with a compact support. As a specific case, we show that a certain separable extension of the 1-D design framework results in 2-D sum-of-modulated-spline (SMS) kernels. Similar to their 1-D counterparts, the 2-D SMS kernels have the attractive feature of reproducing a class of 2-D polynomial-modulated exponentials of a desired order. Also, the support of the kernels is independent of the order. The design framework is generic and also allows one to design nonseparable sampling kernels. To this end, we demonstrate the design of a nonseparable kernel and present simulation results.