Efficient Adjoint Computation for Wavelet and Convolution Operators

TL;DR

This paper introduces efficient methods for computing adjoint operators in wavelet and convolution problems, enabling faster gradient calculations crucial for large-scale first-order optimization algorithms.

Contribution

It presents novel techniques for rapid adjoint computation applicable to wavelet reconstruction and convolution, generalizing to various boundary conditions and leveraging existing software.

Findings

Efficient adjoint computation for wavelet operators in image deblurring.

Fast adjoint calculation for convolution in blind channel estimation.

Generalization to multiple boundary conditions.

Abstract

First-order optimization algorithms, often preferred for large problems, require the gradient of the differentiable terms in the objective function. These gradients often involve linear operators and their adjoints, which must be applied rapidly. We consider two example problems and derive methods for quickly evaluating the required adjoint operator. The first example is an image deblurring problem, where we must compute efficiently the adjoint of multi-stage wavelet reconstruction. Our formulation of the adjoint works for a variety of boundary conditions, which allows the formulation to generalize to a larger class of problems. The second example is a blind channel estimation problem taken from the optimization literature where we must compute the adjoint of the convolution of two signals. In each example, we show how the adjoint operator can be applied efficiently while leveraging existing software.