The minimal and next minimal volumes of normal KSBA stable surfaces with pg≥2p_g\ge 2

In this paper we investigate the minimal and the next minimal volumes of normal KSBA stable surfaces with $p_g\ge 2$. We show that in case of $|K_X|$ not composed with a pencil, the minimal and next minimal volumes are $2p_g-4$ and $2p_g-4+\frac{1}{3}$. In case of $|K_X|$ composed with a pencil, the minimal and next minimal volumes are $\frac{p_g-1}{p_g+1}(p_g-1)$ and $\mathrm{min}\{\frac{2p_g-2}{2p_g+1}(p_g-1), \frac{(3p_g-2)p_g-4}{3(p_g+2)}\}$. We also characterize the surfaces achieving the minimal volumes.Comment: 31 pages, 6 figure

On Revenue Maximization with Sharp Multi-Unit Demands

We consider markets consisting of a set of indivisible items, and buyers that have {\em sharp} multi-unit demand. This means that each buyer $i$ wants a specific number $d_i$ of items; a bundle of size less than $d_i$ has no value, while a bundle of size greater than $d_i$ is worth no more than the most valued $d_i$ items (valuations being additive). We consider the objective of setting prices and allocations in order to maximize the total revenue of the market maker. The pricing problem with sharp multi-unit demand buyers has a number of properties that the unit-demand model does not possess, and is an important question in algorithmic pricing. We consider the problem of computing a revenue maximizing solution for two solution concepts: competitive equilibrium and envy-free pricing. For unrestricted valuations, these problems are NP-complete; we focus on a realistic special case of "correlated values" where each buyer $i$ has a valuation v_i\qual_j for item $j$, where $v_i$ and \qual_j are positive quantities associated with buyer $i$ and item $j$ respectively. We present a polynomial time algorithm to solve the revenue-maximizing competitive equilibrium problem. For envy-free pricing, if the demand of each buyer is bounded by a constant, a revenue maximizing solution can be found efficiently; the general demand case is shown to be NP-hard.Comment: page2

Bidirectional Multi-Scale Implicit Neural Representations for Image Deraining

How to effectively explore multi-scale representations of rain streaks is important for image deraining. In contrast to existing Transformer-based methods that depend mostly on single-scale rain appearance, we develop an end-to-end multi-scale Transformer that leverages the potentially useful features in various scales to facilitate high-quality image reconstruction. To better explore the common degradation representations from spatially-varying rain streaks, we incorporate intra-scale implicit neural representations based on pixel coordinates with the degraded inputs in a closed-loop design, enabling the learned features to facilitate rain removal and improve the robustness of the model in complex scenarios. To ensure richer collaborative representation from different scales, we embed a simple yet effective inter-scale bidirectional feedback operation into our multi-scale Transformer by performing coarse-to-fine and fine-to-coarse information communication. Extensive experiments demonstrate that our approach, named as NeRD-Rain, performs favorably against the state-of-the-art ones on both synthetic and real-world benchmark datasets. The source code and trained models are available at https://github.com/cschenxiang/NeRD-Rain.Comment: Project website: https://github.com/cschenxiang/NeRD-Rai

Look-Around Before You Leap: High-Frequency Injected Transformer for Image Restoration

Transformer-based approaches have achieved superior performance in image restoration, since they can model long-term dependencies well. However, the limitation in capturing local information restricts their capacity to remove degradations. While existing approaches attempt to mitigate this issue by incorporating convolutional operations, the core component in Transformer, i.e., self-attention, which serves as a low-pass filter, could unintentionally dilute or even eliminate the acquired local patterns. In this paper, we propose HIT, a simple yet effective High-frequency Injected Transformer for image restoration. Specifically, we design a window-wise injection module (WIM), which incorporates abundant high-frequency details into the feature map, to provide reliable references for restoring high-quality images. We also develop a bidirectional interaction module (BIM) to aggregate features at different scales using a mutually reinforced paradigm, resulting in spatially and contextually improved representations. In addition, we introduce a spatial enhancement unit (SEU) to preserve essential spatial relationships that may be lost due to the computations carried out across channel dimensions in the BIM. Extensive experiments on 9 tasks (real noise, real rain streak, raindrop, motion blur, moir\'e, shadow, snow, haze, and low-light condition) demonstrate that HIT with linear computational complexity performs favorably against the state-of-the-art methods. The source code and pre-trained models will be available at https://github.com/joshyZhou/HIT.Comment: 19 pages, 7 figure

StrokeGAN: Reducing Mode Collapse in Chinese Font Generation via Stroke Encoding

The generation of stylish Chinese fonts is an important problem involved in many applications. Most of existing generation methods are based on the deep generative models, particularly, the generative adversarial networks (GAN) based models. However, these deep generative models may suffer from the mode collapse issue, which significantly degrades the diversity and quality of generated results. In this paper, we introduce a one-bit stroke encoding to capture the key mode information of Chinese characters and then incorporate it into CycleGAN, a popular deep generative model for Chinese font generation. As a result we propose an efficient method called StrokeGAN, mainly motivated by the observation that the stroke encoding contains amount of mode information of Chinese characters. In order to reconstruct the one-bit stroke encoding of the associated generated characters, we introduce a stroke-encoding reconstruction loss imposed on the discriminator. Equipped with such one-bit stroke encoding and stroke-encoding reconstruction loss, the mode collapse issue of CycleGAN can be significantly alleviated, with an improved preservation of strokes and diversity of generated characters. The effectiveness of StrokeGAN is demonstrated by a series of generation tasks over nine datasets with different fonts. The numerical results demonstrate that StrokeGAN generally outperforms the state-of-the-art methods in terms of content and recognition accuracies, as well as certain stroke error, and also generates more realistic characters.Comment: 10 pages, our codes and data are available at: https://github.com/JinshanZeng/StrokeGA

Seeing the Unseen: A Frequency Prompt Guided Transformer for Image Restoration

How to explore useful features from images as prompts to guide the deep image restoration models is an effective way to solve image restoration. In contrast to mining spatial relations within images as prompt, which leads to characteristics of different frequencies being neglected and further remaining subtle or undetectable artifacts in the restored image, we develop a Frequency Prompting image restoration method, dubbed FPro, which can effectively provide prompt components from a frequency perspective to guild the restoration model address these differences. Specifically, we first decompose input features into separate frequency parts via dynamically learned filters, where we introduce a gating mechanism for suppressing the less informative elements within the kernels. To propagate useful frequency information as prompt, we then propose a dual prompt block, consisting of a low-frequency prompt modulator (LPM) and a high-frequency prompt modulator (HPM), to handle signals from different bands respectively. Each modulator contains a generation process to incorporate prompting components into the extracted frequency maps, and a modulation part that modifies the prompt feature with the guidance of the decoder features. Experimental results on commonly used benchmarks have demonstrated the favorable performance of our pipeline against SOTA methods on 5 image restoration tasks, including deraining, deraindrop, demoir\'eing, deblurring, and dehazing. The source code and pre-trained models will be available at https://github.com/joshyZhou/FPro.Comment: 18 pages, 10 figrue