In this paper, we analyze the minimization of seminorms ∥L · ∥ on R^n under the constraint of a bounded I-divergence D(b,H·) for rather general linear operators H and L. The I-divergence is also known as Kullback–Leibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data but also in the case of multiplicative Gamma noise. Often H represents, e.g., a linear blur operator and L is some discrete derivative or frame analysis operator. A central part of this paper consists in proving relations between the parameters of I-divergence constrained and penalized problems. To solve the I-divergence constrained problem, we consider various first-order primal–dual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. One of these proximation problems is an I-divergence constrained least-squares problem which can be solved based on Morozov’s discrepancy principle by a Newton method. We prove that these algorithms produce not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which converges to a regularization parameter so that the corresponding penalized problem has the same solution. Furthermore, we derive a rule for automatically setting the constraint parameter for data corrupted by multiplicative Gamma noise. The performance of the various algorithms is finally demonstrated for different image restoration tasks both for images corrupted by Poisson noise and multiplicative Gamma noise.

It is well acknowledged in image processing domain that the information can be decomposed into different frequency parts and each part has its own merits. However, existing neural networks always ignore the distinctions and straightforwardly feed all the information into neural networks together, treating them equally. In this paper, we propose a novel neural networks framework named J-Net that decomposes images into different frequency bands and then processes them sequentially. Concretely, the images have been decomposed by wavelet transformation and then the wavelet coefficients are fed into neural networks gradually in different depth according to their decomposition levels. An attention module is utilized to facilitate the fusion of neural network features and injected information, yielding significant performance gain. Furthermore, we show how does the information with different frequency impact the accuracy of neural networks. Experiments show that 5.91%, 5.32% and 2.00% accuracy improvements on Caltech 101, Caltech256 and ImageNet, respectively.

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A method for exploring the structure of populations of complex objects, such as images, is considered. The objects are summarized by feature vectors. The statistical backbone is Principal Component Analysis in the space of feature vectors. Visual insights come from representing the results in the original data space. In an ophthalmological example, endemic outliers motivate the development of a bounded influence approach to PCA.