In this paper, we study almost minimizers for the parabolic thin obstacle (or Signorini) problem with zero obstacle. We establish their Hσ,σ/2-regularity for every 0<σ<1, as well as Hβ,β/2-regularity of their spatial gradients on the either side of the thin space for some 0<β<1. A similar result is also obtained for almost minimizers for the Signorini problem with variable Hölder coefficients.

Image segmentation has been explored for many years and still remains a crucial vision problem. Some efficient or accurate segmentation algorithms have been widely used in many vision applications. However, it is difficult to design a both efficient and accurate image segmenter. In this paper, we propose a novel method called DEL (deep embedding learning) which can efficiently transform superpixels into image segmentation. Starting with the SLIC superpixels, we train a fully convolutional network to learn the feature embedding space for each superpixel. The learned feature embedding corresponds to a similarity measure that measures the similarity between two adjacent superpixels. With the deep similarities, we can directly merge the superpixels into large segments. The evaluation results on BSDS500 and PASCAL Context demonstrate that our approach achieves a good tradeoff between efficiency and effectiveness. Specifically, our DEL algorithm can achieve comparable segments when compared with MCG but is much faster than it, i.e. 11.4fps vs. 0.07fps. 

Spectral Clustering is one of the most widely used cluster- ing algorithms. To find k clusters, it runs the K-means algorithm on the top k eigenvectors of a Laplacian matrix constructed from the data. As a consequence, it inherits the initialization issues of K-means. In this paper, we propose Viral Initialization (VI), a novel initialization procedure im- plemented in the Spectral Clustering algorithm before K-means is applied. VI is designed so that the resulting clusterings exhibit low normalized cut (Ncuts) values. This design principle is aligned with the recent observation that "good" clusterings have low Ncuts values. We show, through exten- sive numerical experiments, that the Spectral Clustering algorithm with VI consistently outperforms other state-of-the-art clustering techniques.

Cluster validation constitutes one of the most challenging problems in unsupervised cluster analysis. For example, identifying the true number of clusters present in a dataset has been investigated for decades, and is still puzzling researchers today. The difficulty stems from the high variety of the dataset characteristics. Some datasets exhibit a strong structure with a few well-separated and normally distributed clusters, but most often real-world datasets contain possibly many overlapping non-gaussian clusters with heterogeneous variances and shapes. This calls for the design of robust clustering algorithms that could adapt to the structure of the data and in particular accurately guess the true number of clusters. They have recently been interesting attempts to design such algorithms, e.g. based on involved non-parametric statistical inference techniques. In this paper, we develop Viral Clustering (VC), a simple algorithm that jointly estimates the number of clusters and outputs clusters. The VC algorithm relies on two antagonist and interacting components. The first component tends to regroup neighbouring samples together, while the second component tends to spread samples in various clusters. This spreading component is performed using an analogy with the way virus spread over networks. We present extensive numerical experiments illustrating the robustness of the VC algorithm, and its superiority compared to existing algorithms.