# Analysis Operator Learning and Its Application to Image Reconstruction Source: [Analysis Operator Learning and Its Application to Image Reconstruction]() Collection index: [llms.txt]() Data updated: 2025-10-25T20:09:28.486Z Authors: Simon Hawe, Martin Kleinsteuber, Klaus Diepold Published: 2012-04-24T08:56:42.000Z ## Abstract Exploiting a priori known structural information lies at the core of many image reconstruction methods that can be stated as inverse problems. The synthesis model, which assumes that images can be decomposed into a linear combination of very few atoms of some dictionary, is now a well established tool for the design of image reconstruction algorithms. An interesting alternative is the analysis model, where the signal is multiplied by an analysis operator and the outcome is assumed to be the sparse. This approach has only recently gained increasing interest. The quality of reconstruction methods based on an analysis model severely depends on the right choice of the suitable operator. In this work, we present an algorithm for learning an analysis operator from training images. Our method is based on an $\\ell\_p$-norm minimization on the set of full rank matrices with normalized columns. We carefully introduce the employed conjugate gradient method on manifolds, and explain the underlying geometry of the constraints. Moreover, we compare our approach to state-of-the-art methods for image denoising, inpainting, and single image super-resolution. Our numerical results show competitive performance of our general approach in all presented applications compared to the specialized state-of-the-art techniques. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Depth Map Super-Resolution on Middlebury Venus 8-bit (test)](): Depth Map Super-Resolution; Middlebury Venus 8-bit (test); RMSE: 0.105. - [Depth Map Super-Resolution on Middlebury Cones 8-bit (test)](): Depth Map Super-Resolution; Middlebury Cones 8-bit (test); RMSE: 0.939. - [Depth Map Super-Resolution on Middlebury Tsukuba v1 (test)](): Depth Map Super-Resolution; Middlebury Tsukuba v1 (test); Bad Pixel Rate (delta=1): 1.03. - [Depth Map Super-Resolution on Middlebury Venus v1 (test)](): Depth Map Super-Resolution; Middlebury Venus v1 (test); Bad Pixels (%) (delta=1): 22. - [Depth Map Super-Resolution on Middlebury Tsukuba 8-bit (test)](): Depth Map Super-Resolution; Middlebury Tsukuba 8-bit (test); RMSE: 0.278. - [Depth Map Super-Resolution on Middlebury Teddy 8-bit (test)](): Depth Map Super-Resolution; Middlebury Teddy 8-bit (test); RMSE: 0.996. - [Depth Map Super-Resolution on Middlebury Teddy v1 (test)](): Depth Map Super-Resolution; Middlebury Teddy v1 (test); Percentage Bad Pixels (delta=1): 2.95. - [Depth Map Super-Resolution on Middlebury Cones v1 (test)](): Depth Map Super-Resolution; Middlebury Cones v1 (test); Bad Pixel Rate (delta=1): 3.56. ## Original sources and implementation [Original paper]() [PDF]()