Performance Assessment and Construction of Compactly Supported Dual Windows for B-spline and Exponential B-spline Gabor Frames

TL;DR

This paper develops and evaluates compactly supported dual Gabor frames using B-spline and exponential B-spline generators, demonstrating their effectiveness for signal and image reconstruction with high accuracy and computational efficiency.

Contribution

It introduces new methods for constructing compactly supported dual Gabor frames with explicit formulas, improving reconstruction accuracy and efficiency for B-spline based systems.

Findings

Dual windows achieve low reconstruction error close to numerical precision.

Constructed duals are stable, computationally efficient, and suitable for practical signal processing.

Performance validated on one-dimensional signals and two-dimensional images using AMSE metrics.

Abstract

This manuscript focuses on the construction of compactly supported dual Gabor frames in $L^2(\mathbb{R})$. The performance of the constructed dual frames is analysed for Gabor systems generated by B-splines and exponential B-splines of orders 2 and 3. The reconstruction performance of these dual windows is evaluated using the average mean square error (AMSE) for standard one-dimensional benchmark signals. For two-dimensional data, image reconstruction experiments are carried out using tensor-product Gabor frames, and the reconstruction accuracy is also assessed using AMSE. Using the duality condition for Gabor systems \cite{jan}, several alternate dual windows with finite support are constructed under suitable assumptions, such as the partition of unity property. Additional dual windows can also be obtained from an existing dual window. The canonical dual window admits an explicit expression that avoids direct inversion of the frame operator and yields reconstruction errors close to numerical precision. The constructed non-canonical compactly supported duals also exhibit stable and competitive reconstruction performance. These findings indicate that compactly supported dual windows based on B-spline and exponential B-spline generators provide effective and practical alternatives for signal and image processing applications, particularly in situations where compact support and computational efficiency are important.