Image restoration: Total variation, wavelet frames, and beyond
By Jian-Feng Cai, Bin Dong, Stanley J. Osher, Zuowei Shen. 2012
349 citations recorded by the source corpus.
From the beginning of science, visual observations have been playing important roles. Advances in computer technology have made it possible to apply some of the most sophisticated developments in mathematics and the sciences to the design and implementation of fast algorithms running on a large number of processors to process image data. As a result, image processing and analysis techniques are now applied to virtually all natural sciences and technical disciplines ranging from computer sciences and electronic engineering to biology and medical sciences; and digital images have come into everyone’s life. Image restoration, including image denoising, deblurring, inpainting, computed tomography, etc., is one of the most important areas in image processing and analysis. Its major purpose is to enhance the quality of a given image that is corrupted in various ways during the process of imaging, acquisition and communication, and enables us to see crucial but subtle objects reside in the image. Therefore, image restoration is an important step to take towards the accurate interpretations of the physical world and making the optimal decisions. Mathematics has been playing an important role in image and signal processing from the very beginning; for example, Fourier analysis is one of the main tools in signal and image analysis, processing, and restoration. In fact, mathematics has been one of the driving forces of the modern development of image analysis, processing and restorations. At the same time, the interesting and challenging problems in imaging science also gave birth to new mathematical theories, techniques and methods. The variational methods (e.g. total variation based methods) and wavelets and wavelet frame based methods developed in the last few decades for image and signal processing are two successful recent examples among many. This paper is designed to establish connections between these two major image restoration approaches: variational methods and wavelet frame based methods. Such connections provide new interpretations and understanding of both approaches, and hence, lead to new applications for both approaches. We start with an introduction of both the variational and wavelet frame based methods. The basic linear image restoration model used for variational methods is
Subfields: Computer Vision and Pattern Recognition
View Paper: https://www.semanticscholar.org/paper/c1d5f07184262d9efd84ed2c7c75ab3e4bc992d4
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In Computer Vision and Pattern Recognition
Citations in the corpus
- Data-driven Discovery of Partial Differential Equations for Multiple-Physics Electromagnetic Problem (2019, background): “Here, we demonstrate one simple example of their work [8] [10] to show how to express differentiation by convolution.”
- Deep Convolutional Framelets: A General Deep Learning Framework for Inverse Problems (2017, methodology): “More specifically, similar to the classical frame-based inpainting algorithms [11, 12, 10], we use the update algorithm in Eq.”
- PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network (2018, background): “A profound relationship between convolutions and differentiations was presented by [32, 33], where the authors discussed the connection between the order of sum rules of filters and the orders of differential operators.”
- PDE-Net: Learning PDEs from Data (2017, background): “The constraints on the filters are motivated by the earlier work of Cai et al. (2012); Dong et al. (2017) where general relations between wavelet frame transforms and differential operators were established.”