Soft Map Research Articles

New versions of maps and connected spaces via supra soft sd-operators.

In this manuscript we use novel types of soft operators to define new approaches of soft maps in the frame of supra soft topologies (or SSTSs), namely supra soft somewhere dens continuous (or SS-sd-continuous), SS-sd-open and SS-sd-closed maps. With the help of SS-closure (interior) operators and SS-sd-closure (interior) operators we succeed to introduce many equivalent conditions and several important properties to these notions. To name a few: We prove that there is an one to one between the SS-sd-open and SS-sd-closed maps under a bijective soft map, supported by counterexample to confirm the necessity of the bijectivity condition. Furthermore, we present the concept of SS-sd-separated sets with intersected characterizations, as a prelude to studying the connectedness in a supra soft topological space (or SSTS). Moreover, we show that, there is no priori relationship between supra soft-sd-connectedness in an SSTS and its parametric supra topological spaces in general, supported by concrete counterexamples. Finally, we prove that the image of an SS-sd-connected set under an SS-sd-irresolute map is an SS-sd-connected.

On b-open Sets via Infra Soft Topological Spaces

This work aims to present the concept of infra soft $b$-open sets(IS-b-open sets) as a generalized new class of infra open sets(IS-open sets). We first investigate their basic properties and study their behaviours under infra soft homeomorphism maps and finite product of soft spaces. Then, we establish some soft operators such as interior, closure, limit and boundary using IS-b-open sets and IS-$b$-closed sets. The relationships between them are illustrated and discussed. Finally, we display some soft maps(S-map) defined using IS-$b$-open and IS-$b$-closed sets and scrutinize their master properties.

New Techniques for Solutions of Fuzzy Volterra Integral Equations via Soft Set-Valued Maps

In this paper, new generalized concepts of set-valued maps under the name soft set-valued maps are established. The presented results herein are soft set extensions of various existing fuzzy set-valued and crisp multivalued fixed point theorems. Examples are provided to authenticate the hypotheses of our obtained results. Moreover, our main result is applied to establish conditions for existence of solutions to fuzzy Volterra integral equations.

Semantic Image Inpainting with Multi-Stage Feature Reasoning Generative Adversarial Network

Most existing image inpainting methods have achieved remarkable progress in small image defects. However, repairing large missing regions with insufficient context information is still an intractable problem. In this paper, a Multi-stage Feature Reasoning Generative Adversarial Network to gradually restore irregular holes is proposed. Specifically, dynamic partial convolution is used to adaptively adjust the restoration proportion during inpainting progress, which strengthens the correlation between valid and invalid pixels. In the decoding phase, the statistical natures of features in the masked areas differentiate from those of unmasked areas. To this end, a novel decoder is designed which not only dynamically assigns a scaling factor and bias on per feature point basis using point-wise normalization, but also utilizes skip connections to solve the problem of information loss between the codec network layers. Moreover, in order to eliminate gradient vanishing and increase the reasoning times, a hybrid weighted merging method consisting of a hard weight map and a soft weight map is proposed to ensemble the feature maps generated during the whole reconstruction process. Experiments on CelebA, Places2, and Paris StreetView show that the proposed model generates results with a PSNR improvement of 0.3 dB to 1.2 dB compared to other methods.

Infra Pre-open Sets and Their Applications to Generate New Types of Operators and Maps

Herein, we introduce the concepts of infra soft pre-open infra soft pre-closed sets which are respectively generalizations of infra soft open and infra soft closed sets.We characterize them and investigate their behaviours under infra soft homeomorphism maps and finite product of soft spaces. Then, we apply infra soft pre-open infra soft pre-closed sets to define the operators of infra pre-interior, infra pre-closure, infra pre-limit and infra pre-boundary. We discuss their main properties and show the interrelations between them. In the end, we introduce new types of soft maps using infra soft pre-open and infra soft pre-closed sets and explore their essential properties.

On soft set-valued maps and integral inclusions

As an improvement of fuzzy set theory, the notion of soft set was initiated as a general mathematical tool for handling phenomena with nonstatistical uncertainties. Recently, a novel idea of set-valued maps whose range set lies in a family of soft sets was inaugurated as a significant refinement of fuzzy mappings and classical multifunctions as well as their corresponding fixed point theorems. Following this new development, in this paper, the concepts of e-continuity and E-continuity of soft set-valued maps and αe-admissibility for a pair of such maps are introduced. Thereafter, we present some generalized quasi-contractions and prove the existence of e-soft fixed points of a pair of the newly defined non-crisp multivalued maps. The hypotheses and usability of these results are supported by nontrivial examples and applications to a system of integral inclusions. The established concepts herein complement several fixed point theorems in the framework of point-to-set-valued maps in the comparable literature. A few of these special cases of our results are highlighted and discussed.

On extension of soft maps in soft topological spaces

Molodtsov introduced soft sets to deal with uncertainty. Shabir introduced soft topological spaces. In this paper the soft locally finite family of soft sets in a soft topological space is introduced and its properties are investigated. Further the soft function and soft continuous functions defined on soft topological spaces are studied using soft locally finite family of soft sets.

Idempotent measures: absolute retracts and soft maps

We investigate under which conditions the space of idempotent measures is an absolute retract and the idempotent barycenter map is soft.

Common $e$-soft fixed points of soft set-valued maps

Common $e$-soft fixed points of soft set-valued maps

WSCNet: Weakly Supervised Coupled Networks for Visual Sentiment Classification and Detection

Automatic assessment of sentiment from visual content has gained considerable attention with the increasing tendency of expressing opinions online. In this paper, we solve the problem of visual sentiment analysis, which is challenging due to the high-level abstraction in the recognition process. Existing methods based on convolutional neural networks learn sentiment representations from the holistic image, despite the fact that different image regions can have different influence on the evoked sentiment. In this paper, we introduce a weakly supervised coupled convolutional network (WSCNet). Our method is dedicated to automatically selecting relevant soft proposals given weak annotations (e.g., global image labels), thereby significantly reducing the annotation burden, and encompasses the following contributions. First, the proposed WSCNet detects a sentiment-specific soft map by training a fully convolutional network with the cross spatial pooling strategy in the detection branch. Second, both the holistic and localized information are utilized by coupling the sentiment map with deep features as semantic vector in the classification branch. The sentiment detection and classification branches are integrated into a unified deep framework optimized in an end-to-end manner. Extensive experiments demonstrate that the proposed WSCNet outperforms the state-of-the-art results on seven benchmark datasets.

Soft maps via soft somewhere dense sets

The concept of soft sets was proposed as an effective tool to deal with uncertainty and vagueness. Topologists employed this concept to define and study soft topological spaces. In this paper, we introduce the concepts of soft SD-continuous, soft SD-open, soft SD-closed and soft SD-homeomorphism maps by using soft somewhere dense and soft cs-dense sets. We characterize them and discuss their main properties with the help of examples. In particular, we investigate under what conditions the restriction of soft SD-continuous, soft SD-open and soft SD-closed maps are respectively soft SD-continuous, soft SD-open and soft SD-closed maps. We logically explain the reasons of adding the null and absolute soft sets to the definitions of soft SD-continuous and soft SD-closed maps, respectively, and removing the null soft set from the definition of a soft SD-open map.

Functional Maps Representation On Product Manifolds

AbstractWe consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.

Some categorical aspects of coarse spaces and balleans

Coarse spaces [26] and balleans [23] are known to be equivalent constructions ([25]). The main subject of this paper is the category, Coarse, having as objects these structures, and its quotient category Coarse/∼. We prove that the category Coarse is topological and hence Coarse is complete and co-complete and one has a complete description of its epimorphisms and monomorphisms. In particular, Coarse has products and coproducts, quotients, etc., and Coarse is not balanced. A special attention is paid to investigate quotients in Coarse by introducing some particular classes of maps, i.e. (weakly) soft maps which allow one to explicitly describe when the quotient ball structure of a ballean is a ballean. A particular type of quotients, namely the adjunction spaces, is considered in detail in order to obtain a description of the epimorphisms in Coarse/∼, shown to be the bornologous maps with large image. The monomorphisms in Coarse/∼ are the coarse embeddings; consequently, the bimorphisms in Coarse/∼ are precisely the isomorphisms, i.e., Coarse/∼ is a balanced category.

L-FUZZY (K,E)-SOFT TOPOLOGIES AND L-FUZZY (K,E)-SOFT CLOSURE OPERATORS

In this paper, we investigate the properties of fuzzy soft sets and fuzzy soft maps in stsc-quantales. We define a L-fuzzy (K,E)-soft topology and a L-fuzzy (K,E)-soft closure spaces as a Hohle's sense. We study the relations between L-fuzzy (K,E)-soft topologies and L-fuzzy (K,E)-soft closure spaces. We give their examples.

Dirichlet Energy for Analysis and Synthesis of Soft Maps

AbstractSoft maps taking points on one surface to probability distributions on another are attractive for representing surface mappings in the presence of symmetry, ambiguity, and combinatorial complexity. Few techniques, however, are available to measure their continuity and other properties. To this end, we introduce a novel Dirichlet energy for soft maps generalizing the classical map Dirichlet energy, which measures distortion by computing how soft maps transport probabilistic mass from one distribution to another. We formulate the computation of the Dirichlet energy in terms of a differential equation and provide a finite elements discretization that enables all of the quantities introduced to be computed. We demonstrate the effectiveness of our framework for understanding soft maps arising from various sources. Furthermore, we suggest how these energies can be applied to generate continuous soft or point‐to‐point maps.

Soft Maps Between Surfaces

AbstractThe problem of mapping between two non‐isometric surfaces admits ambiguities on both local and global scales. For instance, symmetries can make it possible for multiple maps to be equally acceptable, and stretching, slippage, and compression introduce difficulties deciding exactly where each point should go. Since most algorithms for point‐to‐point or even sparse mapping struggle to resolve these ambiguities, in this paper we introduce soft maps, a probabilistic relaxation of point‐to‐point correspondence that explicitly incorporates ambiguities in the mapping process. In addition to explaining a continuous theory of soft maps, we show how they can be represented using probability matrices and computed for given pairs of surfaces through a convex optimization explicitly trading off between continuity, conformity to geometric descriptors, and spread. Given that our correspondences are encoded in matrix form, we also illustrate how low‐rank approximation and other linear algebraic tools can be used to analyze, simplify, and represent both individual and collections of soft maps.

Fuzzy soft hypergroups

We propose the concept of fuzzy soft hypergroups and explore two particular classes of them, namely fuzzy soft closed hypergroups and fuzzy soft ultraclosed hypergroups. Moreover, we study the image and inverse image of a fuzzy soft (closed, ultraclosed) hypergroup under a fuzzy soft map.

Probability measures and Milyutin maps between metric spaces

We prove that the functor P ˆ of Radon probability measures transforms any open map between completely metrizable spaces into a soft map. This result is applied to establish some properties of Milyutin maps between completely metrizable spaces.

Iterative Channel Estimation and Decoding of Turbo Coded SFBC-OFDM Systems

We consider the design of turbo receiver structures for space-frequency block coded orthogonal frequency division multiplexing (SFBC-OFDM) systems in the presence of unknown frequency and time selective fading channels. The turbo receiver structures for SFBC-OFDM systems under consideration consists of an iterative MAP expectation/maximization (EM) channel estimation algorithm, soft MMSE-SFBC decoder and a soft MAP outer-channel-code decoder. MAP-EM employs iterative channel estimation and it improves receiver performance by re-estimating the channel after each decoder iteration. Moreover, the MAP-EM approach considers the channel variations as random processes and applies the Karhunen-Loeve (KL) orthogonal series expansion. The optimal truncation property of the KL expansion can reduce computational load on the iterative estimation approach. The performance of the proposed approaches are studied in terms of mean square error and bit-error rate. Through computer simulations, the effect of a pilot spacing on the channel estimator performance and sensitivity of turbo receiver structures on channel estimation error are studied. Simulation results illustrate that receivers with turbo coding are very sensitive to channel estimation errors compared to receivers with convolutional codes. Moreover, superiority of the turbo coded SFBC-OFDM systems over the turbo coded STBC-OFDM systems is observed especially for high Doppler frequencies.

A hybrid PAPR reduction scheme for coded OFDM

We consider schemes for reducing the peak-to-average power ratio (PAPR) in coded orthogonal frequency-division multiplexing (OFDM) systems. We develop a new PAPR reduction technique using the label-inserted encoder of a random-like code and the soft amplitude limiter (SAL). Using this hybrid scheme provides 5.5 dB PAPR reduction in an OFDM system with 128 subcarriers, 4-bit selection and 3 dB clipping. Besides the significant PAPR reduction, the scheme also enjoys other advantages such as small overhead, low complexity, no side information transmission, and little performance loss. Among various random-like codes, the irregular repeat accumulate (IRA) code is the best choice for its simple encoder and capacity achieving performance. The scheme can be directly applied to multiple-input multiple-output (MIMO) OFDM systems. The capacity of the clipped MIMO-OFDM systems is analyzed based on a Gaussian approximation of the clipping noise. We consider an iterative receiver with soft MAP MIMO-OFDM detector. For both single antenna and multiple antenna systems, the encoder part is independent in the hybrid scheme, thus no additional constraint is applied to the IRA code optimization. The IRA codes are designed for the ergodic MIMO-OFDM systems with different PAPR reduction settings, more specifically different clipping ratios, based on the extrinsic information transfer (EXIT) charts. Simulation results show that the hybrid scheme with 3 dB clipping can achieve as good PAPR reduction performance as the simple clipping with 0 dB ratio but incurs much less performance loss at the receiver