Complex Orthogonal Designs with Forbidden 2×22 \times 2 Submatrices

Complex orthogonal designs (CODs) are used to construct space-time block codes. COD $\mathcal{O}_z$ with parameter $[p, n, k]$ is a $p \times n$ matrix, where nonzero entries are filled by $\pm z_i$ or $\pm z^*_i$, $i = 1, 2,..., k$, such that $\mathcal{O}^H_z \mathcal{O}_z = (|z_1|^2+|z_2|^2+...+|z_k|^2)I_{n \times n}$. Define $\mathcal{O}_z$ a first type COD if and only if $\mathcal{O}_z$ does not contain submatrix {\pm z_j & 0; \ 0 & \pm z^*_j}$or${\pm z^*_j & 0; \ 0 & \pm z_j}$. It is already known that, all CODs with maximal rate, i.e., maximal$k/p$, are of the first type. In this paper, we determine all achievable parameters$[p, n, k]$of first type COD, as well as all their possible structures. The existence of parameters is proved by explicit-form constructions. New CODs with parameters$[p,n,k]=[\binom{n}{w-1}+\binom{n}{w+1}, n, \binom{n}{w}], $for$0 \le w \le n$, are constructed, which demonstrate the possibility of sacrificing code rate to reduce decoding delay. It's worth mentioning that all maximal rate, minimal delay CODs are contained in our constructions, and their uniqueness under equivalence operation is proved

On the nonexistence of [(2mm−1),2m,(2m−1m−1)][\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}], mm odd, complex orthogonal design

Complex orthogonal designs (CODs) are used to construct space-time block codes. COD $\mathcal{O}_z$ with parameter $[p, n, k]$ is a $p\times n$ matrix, where nonzero entries are filled by $\pm z_i$ or $\pm z^*_i$, $i = 1, 2,..., k$, such that $\mathcal{O}^H_z \mathcal{O}_z = (|z_1|^2+|z_2|^2+...+|z_k|^2)I_{n \times n}$. Adams et al. in "The final case of the decoding delay problem for maximum rate complex orthogonal designs," IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 103-122, Jan. 2010, first proved the nonexistence of $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$, $m$ odd, COD. Combining with the previous result that decoding delay should be an integer multiple of $\binom{2m}{m-1}$, they solved the final case $n \equiv 2 \pmod 4$ of the decoding delay problem for maximum rate complex orthogonal designs. In this paper, we give another proof of the nonexistence of COD with parameter $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$, $m$ odd. Our new proof is based on the uniqueness of $[\binom{2m}{m-1}, 2m-1, \binom{2m-1}{m-1}]$ under equivalence operation, where an explicit-form representation is proposed to help the proof. Then, by proving it's impossible to add an extra orthogonal column on COD $[\binom{2m}{m-1}, 2m-1, \binom{2m-1}{m-1}]$ when $m$ is odd, we complete the proof of the nonexistence of COD $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$

Robust Saliency Detection via Fusing Foreground and Background Priors

Automatic Salient object detection has received tremendous attention from research community and has been an increasingly important tool in many computer vision tasks. This paper proposes a novel bottom-up salient object detection framework which considers both foreground and background cues. First, A series of background and foreground seeds are selected from an image reliably, and then used for calculation of saliency map separately. Next, a combination of foreground and background saliency map is performed. Last, a refinement step based on geodesic distance is utilized to enhance salient regions, thus deriving the final saliency map. Particularly we provide a robust scheme for seeds selection which contributes a lot to accuracy improvement in saliency detection. Extensive experimental evaluations demonstrate the effectiveness of our proposed method against other outstanding methods.Comment: Project website: https://github.com/ChunbiaoZhu/FB

Estimating the Hausdorff dimensions of univoque sets for self-similar sets

An approach is given for estimating the Hausdorff dimension of the univoque set of a self-similar set. This sometimes allows us to get the exact Hausdorff dimensions of the univoque sets.Comment: 11 page

Visibility of Cartesian products of Cantor sets

Let $K_{\lambda}$ be the attractor of the following IFS \begin{equation*} \{f_1(x)=\lambda x, f_2(x)=\lambda x+1-\lambda\}, \;\;0<\lambda<1/2. \end{equation*} Given $\alpha \geq 0$, we say the line $y=\alpha x$ is visible through $K_{\lambda}\times K_{\lambda}$ if $$\{(x, \alpha x): x\in \mathbb R\setminus \{0\}\}\cap ((K_{\lambda}\times K_{\lambda}))=\emptyset.$$ Let V=\left \{\alpha \geq 0: y=\alpha x \mbox{ is visible through } K_{\lambda}\times K_{\lambda} \right \}. In this paper, we give a completed description of $V$, e.g., its Hausdoff dimension and its topological property. Moreover, we also discuss another type of visible problem which is related to the slicing problem.Comment: 13 page

On the Minimum Decoding Delay of Balanced Complex Orthogonal Design

Complex orthogonal design (COD) with parameter $[p, n, k]$ is a combinatorial design used in space-time block codes (STBCs). For STBC, $n$ is the number of antennas, $k/p$ is the rate, and $p$ is the decoding delay. A class of rate $1/2$ COD called balanced complex orthogonal design (BCOD) has been proposed by Adams et al., and they constructed BCODs with rate $k/p = 1/2$ and decoding delay $p = 2^m$ for $n=2m$. Furthermore, they prove that the constructions have optimal decoding delay when $m$ is congruent to $1$, $2$, or $3$ module $4$. They conjecture that for the case $m \equiv 0 \pmod 4$, $2^m$ is also a lower bound of $p$. In this paper, we prove this conjecture

Lipschitz equivalence of a class of self-similar sets

We consider a class of homogeneous self-similar sets with complete overlaps and give a sufficient condition for the Lipschitz equivalence between members in this class.Comment: A remark was added. To appear in Ann. Acad. Sci. Fenn. Mat

Automatic Salient Object Detection for Panoramic Images Using Region Growing and Fixation Prediction Model

Almost all previous works on saliency detection have been dedicated to conventional images, however, with the outbreak of panoramic images due to the rapid development of VR or AR technology, it is becoming more challenging, meanwhile valuable for extracting salient contents in panoramic images. In this paper, we propose a novel bottom-up salient object detection framework for panoramic images. First, we employ a spatial density estimation method to roughly extract object proposal regions, with the help of region growing algorithm. Meanwhile, an eye fixation model is utilized to predict visually attractive parts in the image from the perspective of the human visual search mechanism. Then, the previous results are combined by the maxima normalization to get the coarse saliency map. Finally, a refinement step based on geodesic distance is utilized for post-processing to derive the final saliency map. To fairly evaluate the performance of the proposed approach, we propose a high-quality dataset of panoramic images (SalPan). Extensive evaluations demonstrate the effectiveness of our proposed method on panoramic images and the superiority of the proposed method against other methods.Comment: Previous Project website: https://github.com/ChunbiaoZhu/DCC-201

Far-field Imaging beyond the Diffraction Limit Using a Single Radar

Far-field imaging beyond the diffraction limit is a long sought-after goal in various imaging applications, which requires usually an array of antennas or mechanical scanning. Here, we present an alternative and novel concept for this challenging problem: a single radar system consisting of a spatial-temporal resonant aperture antenna (referred to as the slavery antenna) and a broadband horn antenna (termed the master antenna). We theoretically demonstrate that such resonant aperture antenna is responsible for converting parts of the evanescent waves into propagating waves, and delivering them to the far-field. We also demonstrate that there are three basic requirements on the proposed subwavelength imaging strategy: the strong spatial-temporal dispersive aperture, the near-field coupling, and the temporal (or broadband) illumination. Such imaging concept of a single radar provides unique ability to produce real-time data when an object is illuminated by broadband electromagnetic waves, which lifts up the harsh requirements such as near-field scanning, mechanical scanning or antenna arrays remarkably. We expect that this imaging methodology will make breakthroughs in super-resolution imaging in the microwave, terahertz, optical, and ultrasound regimes

Relative Pairwise Relationship Constrained Non-negative Matrix Factorisation

Non-negative Matrix Factorisation (NMF) has been extensively used in machine learning and data analytics applications. Most existing variations of NMF only consider how each row/column vector of factorised matrices should be shaped, and ignore the relationship among pairwise rows or columns. In many cases, such pairwise relationship enables better factorisation, for example, image clustering and recommender systems. In this paper, we propose an algorithm named, Relative Pairwise Relationship constrained Non-negative Matrix Factorisation (RPR-NMF), which places constraints over relative pairwise distances amongst features by imposing penalties in a triplet form. Two distance measures, squared Euclidean distance and Symmetric divergence, are used, and exponential and hinge loss penalties are adopted for the two measures respectively. It is well known that the so-called "multiplicative update rules" result in a much faster convergence than gradient descend for matrix factorisation. However, applying such update rules to RPR-NMF and also proving its convergence is not straightforward. Thus, we use reasonable approximations to relax the complexity brought by the penalties, which are practically verified. Experiments on both synthetic datasets and real datasets demonstrate that our algorithms have advantages on gaining close approximation, satisfying a high proportion of expected constraints, and achieving superior performance compared with other algorithms.Comment: 13 pages, 10 figure