Fourier-Domain Optimization for Image Processing
This work addresses computational efficiency issues for researchers and practitioners in image processing, though it appears incremental as it builds on existing Fourier domain techniques.
The paper tackles the problem of efficiently solving convex image optimization and deconvolution tasks, such as deblurring and dehazing, by presenting a framework that transforms these problems into the Fourier domain, enabling faster computation compared to conventional gradient descent methods.
Image optimization problems encompass many applications such as spectral fusion, deblurring, deconvolution, dehazing, matting, reflection removal and image interpolation, among others. With current image sizes in the order of megabytes, it is extremely expensive to run conventional algorithms such as gradient descent, making them unfavorable especially when closed-form solutions can be derived and computed efficiently. This paper explains in detail the framework for solving convex image optimization and deconvolution in the Fourier domain. We begin by explaining the mathematical background and motivating why the presented setups can be transformed and solved very efficiently in the Fourier domain. We also show how to practically use these solutions, by providing the corresponding implementations. The explanations are aimed at a broad audience with minimal knowledge of convolution and image optimization. The eager reader can jump to Section 3 for a footprint of how to solve and implement a sample optimization function, and Section 5 for the more complex cases.