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We formulate various variational problems in which the smoothness of functions is measured using Besov space semi-norms. Equivalent Besov space semi-norms can be defined in terms of moduli of smoothness or sequence norms of coefficients in appropriate wavelet expansions. Wavelet-based semi-norms have been used before in variational problems, but existing algorithms do not preserve edges, and many result in blocky artifacts. Here, we devise algorithms using moduli of smoothness for the B ∞ 1 ( L 1 ( I ) ) Besov space semi-norm. We choose that particular space because it is closely related both to the space of functions of bounded variation, BV ( I ) , that is used in Rudin–Osher–Fatemi image smoothing, and to the B 1 1 ( L 1 ( I ) ) Besov space, which is associated with wavelet shrinkage algorithms. It contains all functions in BV ( I ) , which include functions with discontinuities along smooth curves, as well as “fractal-like” rough regions; examples are given in an appendix. Furthermore, it prefers affine regions to staircases, potentially making it a desirable regularizer for recovering piecewise affine data. While our motivations and computational examples come from image processing, we make no claim that our methods “beat” the best current algorithms. The novelty in this work is a new algorithm that incorporates a translation-invariant Besov regularizer that does not depend on wavelets, thus improving on earlier results. Furthermore, the algorithm naturally exposes a range of scales that depends on the image data, noise level, and the smoothing parameter. We also analyze the norms of smooth, textured, and random Gaussian noise data in B ∞ 1 ( L 1 ( I ) ) , B 1 1 ( L 1 ( I ) ) , BV ( I ) and L 2 ( I ) and their dual spaces. Numerical results demonstrate properties of solutions obtained from this moduli of smoothness-based regularizer.

Image smoothing is a long-studied research area with tremendous approaches proposed. However, how to perform high-quality image smoothing with less computational cost still remains a challenging problem. In this paper, we try to solve this problem with a newly proposed global optimization based method named quasi weighted least squares. In our method, the 2D image is first re-ordered into a 1D vector via a newly proposed 2D-to-1D transformation. We then properly remove some original 2D neighborhood connections. The remaining neighboring pixels can simply form 1D neighborhood connections in the transformed 1D vector while they still contain the 2D neighborhood information in the original 2D image space. These together result in a quite compact linear system that can be easily and efficiently solved, which makes our method a fast global image smoothing approach. Our method is on par with the fastest approaches in terms of processing speed, however, it is able to yield comparable performance with the state-of-the-art ones in terms of smoothing quality. Our method can also work as a solver to approximate the weighted least squares problem in complex systems, and it can achieve similar results but runs much faster. The efficiency and effectiveness of our method are validated through comprehensive experiments in several tasks. Our code is publicly available at: https://github.com/wliusjtu/Q-WLS.

Edge-preserving image smoothing plays a vital role in the field of computational photography. In this paper, we propose a weighted and truncated L 1 -regularized optimization model for image smoothing. We show that the weighted and truncated scheme significantly promotes the edge-preserving property. Furthermore, we propose a deep unsupervised learning-based filter based on the loss function defined by the proposed optimization model. The proposed filter leverages a U-Net structure, which fully exploits the spatially varying smoothing scales of the edge-preserving filtering. We have conducted extensive experiments to evaluate the proposed filter. The results suggest that our filter outperforms the state-of-the-art filters in image quality on various tasks, such as image smoothing, detail enhancing, HDR tone mapping, and edge detection. Meanwhile, our filter is extremely efficient. It is able to process 720P images in real-time (more than 16 frames per second) on a modern desktop with an Intel i7-8700K CPU, an NVIDIA GTX 1080 GPU and 16GB memory.

The weighted least square (WLS) filter is a popular edge-preserving image smoother that is particularly useful for detail enhancing and HDR tone mapping. However, it suffers from limited edge-preserving capability and high computational cost. Existing deep learning-based filters under the WLS framework are mostly based on supervised learning. They improve the efficiency but not the quality. In this paper, we propose a novel edge-preserving filter under the weighted least square framework based on deep unsupervised learning. According to the spatial-varying smoothing property of the edge-preserving filter, we propose a lightweight fully convolution neural network based on dilated convolutions with varying expanding factors. The proposed filter fully makes use of the 2D neighborhood information, thus it is able to suppress various artifacts. Thanks to the highly optimized framework for deep learning, the proposed filter is highly efficient, enabling the processing of 720P images at interactive rates ( ≈ 12 fps) on a modern desktop. Experimental results indicate that the proposed filter achieves the state-of-the-art smoothing quality. Therefore, our filter benefits a variety of tasks in the field of image processing and computer graphics.

Edge-preserving image smoothing is a fundamental step for many computer vision problems, and so far, countless algorithms have been proposed. Among these algorithms, bilateral filtering and its extensions are widely used in image preprocessing. However, several difficulties are hindering its further development. First, the phenomenon of \"halo artifact\" occurs along the edges. Second, most of the existing algorithms work only with a fixed filtering kernel and cannot accurately distinguish the edges and textures which leads to inappropriate filtering. To address these issues, we present a novel edge-preserving image smoothing via adaptive side window joint bilateral filtering. As a local optimized-based algorithm, different from the traditional filtering, the position of the target pixel in the filtering kernel is changed from the center to the optimal edge and the filtering kernel size of each pixel is effectively estimated. Combined side window filtering with the joint bilateral filter, the capability of texture removal and edge preservation is improved and the halo artifacts are alleviated. Experimental results show that the proposed method outperforms existing state-of-the-arts in removing the texture information while preserving the main image content.

Image smoothing techniques that can retain the significant structural information of an image and filter out texture and noise are particularly important. Aiming at the problems that existing methods do not deal with texture variations adequately and fail to efficiently mine and utilize the structural information of image edges, this paper proposes the texture intensity adaptive edge consistency network model (TIAEC-Net). In order to enhance the texture filtering effect, this paper proposes the texture intensity adaptive module, which captures the changes in texture information more accurately by predicting the salient structural information and applying the attention mechanism to filter the features. In the smoothing process, the method of extracting edge information and feature fusion with other input features is used to ensure that significant edge structures are not over-smoothed. Meanwhile, an edge consistency module is proposed to enhance the edge protection capability of the model by analyzing the edge response consistency relationship between the input image and the output image. In addition, a weak structure reinforce module is used to enhance the edge details in the image. After experimental validation, the TIAEC-Net model performs excellently in preserving the significant edge structure of the image and outperforms most of the current methods on publicly available datasets.

Image smoothing is an important processing operation that highlights low-frequency structural parts of an image and suppresses the noise and high-frequency textures. In the paper, we post an intriguing question of how to combine the paired unsmoothed/smoothed images and meaningful edge information to improve the performance of image smoothing. To this end, we propose a structure-preserving image smoothing network, which consists of a main interpreter (MI) and an edge map extractor (EME). The network is trained via contrastive learning on the extended BSD500 dataset. In addition, an edge-aware total variation loss function is utilized to distinguish between non-edge regions and edge maps via a pre-trained EME module, therefore improving the capability of structure preservation. In order to maintain the consistency in structure and background brightness, the outputs from MI are used as anchors for a ternary loss in 1:1 paired positive and negative samples. Experiments on different datasets show that our network outperforms state-of-the-art image smoothing methods in terms of SSIM and PSNR.

Bilateral filtering is a widely used image smoothing filter that preserves image edges while also smoothing texture. In previous research, the focus of improving the bilateral filter has primarily been on constructing an adaptive range kernel. However, recent research has shown that even slight noise perturbations can prevent the bilateral filter from effectively preserving image edges. To address this issue, we employ a neural network to learn the kernel parameters that can effectively counteract noise perturbations. Additionally, to enhance the adaptability of the learned kernel parameters to the local edge features of the image, we utilize the edge-sensitive indexing method to construct kernel parameter lookup tables (LUTs). During testing, we determine the appropriate spatial kernel and range kernel parameters for each pixel using a lookup table and interpolation. This allows us to effectively smooth the image in the presence of noise perturbation. In this paper, we conducted comparative experiments on several datasets to verify that the proposed method outperforms existing bilateral filtering methods in preserving image structure, removing image texture, and resisting slight noise perturbations. The code is available at https://github.com/FightingSrain/AdaBFLUT .

Texture filtering attempts to retain salient structures and remove insignificant textures. In this paper, we propose a highly effective iterative unsupervised deep bilateral texture filtering neural network for texture smoothing. The bilateral texture loss function is introduced to train the model without the ground truth smoothing images for guidance. The proposed model inherits well-known advantages of the bilateral texture filter to capture the texture information effectively. The model is trained solely using the training data, then the predicted outputs are generated iteratively through multiple forward passes. Extensive experiments demonstrate that our proposed iterative unsupervised deep bilateral texture filtering neural network outperforms existing methods in effectively removing textures while preserving the main structures of the image. The results showcase the superior performance of our approach and its ability to achieve high-quality texture smoothing without sacrificing important image features.

In this paper, we present feature/detail preserving models for color image smoothing and segmentation using the Hamiltonian quaternion framework. First, we introduce a novel quaternionic Gabor filter (QGF) which can combine the color channels and the orientations in the image plane. We show that these filters are optimally localized both in the spatial and frequency domains and provide a good approximation to quaternionic quadrature filters. Using the QGFs, we extract the local orientation information in the color images. Second, in order to model this derived orientation information, we propose continuous mixtures of appropriate exponential basis functions and derive analytic expressions for these models. These analytic expressions take the form of spatially varying kernels which, when convolved with a color image or the signed distance function of an evolving contour (placed in the color image), yield a detail preserving smoothing and segmentation, respectively. Several examples on widely used image databases are shown to depict the performance of our algorithms.