Two important tasks in the field of topological data analysis (TDA) are building practical filtrations on objects and using TDA to detect the geometry and primarily topological structures. Motivated by these tasks, we have defined the difference between the two group equivariant non-expansive operators (GENEOs) by DGENEO and built multiparameter filtrations by operators on images named the multi-GENEO, multi-DGENEO, and mix-GENEO, and we proved the stability of both the interleaving distance and multiparameter persistence landscape of the multi-GENEO with respect to the pseudometric on images, modeled as bounded functions. We also gave an upper bound for the multi-DGENEO and mix-GENEO. In practical applications, we regarded the space of images on a discrete domain, and then we built multifiltrations on the discrete function space. Finally, we conducted a comparable experiment on the MNIST dataset to demonstrate that our bifiltrations are superior to 1-parameter filtrations. The experiment results demonstrate that our bifiltrations have the ability to detect geometric and topological differences of digital images.

Existing deep unfolding methods unroll an optimization algorithm with a fixed number of steps, and utilize convolutional neural networks (CNNs) to learn data-driven priors. However, their performance is limited for two main reasons. Firstly, priors learned in deep feature space need to be converted to the image space at each iteration step, which limits the depth of CNNs and prevents CNNs from exploiting contextual information. Secondly, existing methods only learn deep priors at the single full-resolution scale, so ignore the benefits of multi-scale context in dealing with high level noise. To address these issues, we explicitly consider the image denoising process in the deep feature space and propose the deep unfolding multi-scale regularizer network (DUMRN) for image denoising. The core of DUMRN is the feature-based denoising module (FDM) that directly removes noise in the deep feature space. In each FDM, we construct a multi-scale regularizer block to learn deep prior information from multi-resolution features. We build the DUMRN by stacking a sequence of FDMs and train it in an end-to-end manner. Experimental results on synthetic and real-world benchmarks demonstrate that DUMRN performs favorably compared to state-of-the-art methods.

Regional facial image synthesis conditioned on a semantic mask has achieved great attention in the field of computational visual media. However, the appearances of different regions may be inconsistent with each other after performing regional editing. In this paper, we focus on harmonized regional style transfer for facial images. A multi-scale encoder is proposed for accurate style code extraction. The key part of our work is a multi-region style attention module. It adapts multiple regional style embeddings from a reference image to a target image, to generate a harmonious result. We also propose style mapping networks for multi-modal style synthesis. We further employ an invertible flow model which can serve as mapping network to fine-tune the style code by inverting the code to latent space. Experiments on three widely used face datasets were used to evaluate our model by transferring regional facial appearance between datasets. The results show that our model can reliably perform style transfer and multi-modal manipulation, generating output comparable to the state of the art.

Image super-resolution is essential for a variety of applications such as medical imaging, surveillance imaging, and satellite imaging, among others. Traditionally, the most popular color image super-resolution is performed in each color channel independently. In this paper, we show that the super-resolution quality can be further enhanced by exploiting the cross-channel correlation. Inspired by the High-Quality Linear Interpolation (HQLI) demosaicking algorithm by Malvar et al., we design an image super-resolution scheme that integrates intra-channel interpolation with cross-channel details by isotropic linear combinations. Despite its simplicity, our super-resolution method achieves the accuracy comparable with the existing fastest state-of-the-art super-resolution algorithm at 20 times faster speed. It is well applicable to applications that adopt traditional interpolations, for improved visual quality at trivial computation cost. Our comparative study verifies the effectiveness and efficiency of the proposed super-resolution algorithm.