Minkowski Summation Research Articles

Coordinated scheduling of 5G base station energy storage for voltage regulation in distribution networks

With the rapid development of 5G base station construction, significant energy storage is installed to ensure stable communication. However, these storage resources often remain idle, leading to inefficiency. To enhance the utilization of base station energy storage (BSES), this paper proposes a co-regulation method for distribution network (DN) voltage control, enabling BSES participation in grid interactions. In this paper, firstly, an energy consumption prediction model based on long and short-term memory neural network (LSTM) is established to accurately predict the daily load changes of base stations. Secondly, a BSES aggregation model is constructed by using the power feasible domain maximal inner approximation method and Minkowski summation to evaluate the charging and discharging potential and adjustable capacity of BSES clusters. Subsequently, a BSES demand assessment and optimal scheduling model for low voltage regulation in DN is developed. This model optimizes the charging and discharging strategies of BSES to alleviate low voltage problems in DN. Finally, the simulation results effectively verify the feasibility of the proposed optimal scheduling method of BSES for voltage regulation in DN.

Sumset Estimates in Convex Geometry

Abstract Sumset estimates, which provide bounds on the cardinality of sumsets of finite sets in a group, form an essential part of the toolkit of additive combinatorics. In recent years, probabilistic or entropic forms of many of these inequalities were introduced. We study analogues of these sumset estimates in the context of convex geometry and the Lebesgue measure on ${\mathbb{R}}^{n}$. First, we observe that, with respect to Minkowski summation, volume is supermodular to arbitrary order on the space of convex bodies. Second, we explore sharp constants in the convex geometry analogues of variants of the Plünnecke-Ruzsa inequalities. In the last section of the paper, we provide connections of these inequalities to the classical Rogers-Shephard inequality.

Distributed energy management of electric vehicle charging stations based on hierarchical pricing mechanism and aggregate feasible regions

With the rapid development of electric vehicle charging stations, effective management of them has become challenging due to the high uncertainty of electric vehicles, the pricing mechanisms of charging stations, and their coupling with distribution networks. To address these challenges, this paper proposes a two-stage framework for energy management at charging stations. In the first stage, a resource allocation model considering the profits of distribution systems, charging stations, and electric vehicle users is established based on the aggregate feasible power regions of charging stations. The aggregate feasible region is obtained based on the combination of Minkowski summation and the data-driven method, which can preserve the privacy of electric vehicle data and reduce the computational burden. In the second stage, a novel hierarchical pricing mechanism is developed, which encompasses both the clearing price between charging stations and distribution networks and the retail electricity price between charging stations and electric vehicle users. Notably, charging stations participate in the power clearing of distributed networks based on the aggregate feasible power region, while a two-stage robust pricing strategy is established between electric vehicle users and charging stations. The model is finally optimized through a distributed coordination mechanism with a clear physical interpretation. The simulation results show that the proposed aggregation method enables charging stations to achieve a total economic profit at least 1.76 % higher than three competitive methods. The hierarchical pricing mechanism allows charging stations to achieve total economic profits 18.60 % and 2.94 % higher than those in the centralized dispatch and price-taker modes, respectively, while simultaneously reducing operating costs for the distributed network by 25.96 % and 27.99 %.

Multiple-layer energy management strategy for charging station optimal operation considering peak and valley shaving

Existing vehicle-to-grid (V2G) applications are aimed at the power grid and the government. It is difficult for charging stations (CSs) to execute the schedules in real time. To figure out the multiple-layer energy management from the perspective of CS, the dispatch potential assessment model is constructed based on the EV users’ charging demand and Minkowski summation. And the optimal energy management schedule model of CS with ESS is proposed considering peak shaving and valley filling under the time-in-use tariff. Besides, the real-time charging control model of EVs in CS is designed under the premise of meeting the charging needs. The simulation results show that the proposed strategy can promote CS operation revenues and track the scheduling plan of CS. The arbitrage of tariffs and peak shaving ancillary services are realized while the charging loads of CSs are smoothed by the charging/discharging of ESS. The proposed strategy is applicable for the CS aggregators and can help the grid operators for dispatch schedules.

An efficient framework for exploiting operational flexibility of load energy hubs in risk management of integrated electricity-gas systems

Faced with increasing operational uncertainties, e.g. wind power variation, the risk management of integrated electricity-gas systems (IEGSs) has become extremely vital, which puts forward a higher requirement for flexible resources. On the demand side of IEGSs, local energy hubs (LEHs) composed of different energy devices can provide flexible resources for system risk management through multi-energy substitution. However, it can be difficult to exploit the operational flexibility of massive LEHs in IEGS operation owing to two major issues, namely 1) privacy concerns of individuals; 2) computation efficiency requirements. To address this, an efficient framework based on flexible regions is innovatively proposed for exploiting LEH flexibility in the risk management of IEGSs. The flexible region is estimated to characterize the adjustable range of energy inputs to LEHs considering the operational requirements of internal energy devices. On this basis, essential flexibility-related information of LEHs is directly utilized for system risk management, which can preserve individual privacy and avoid time-consuming iteration. Firstly, a generalized method based on Minkowski Summation is proposed to efficiently determine the flexible region of LEH, which contains time-independent and temporal-related parts. The two regions are formulated separately considering the multi-energy substitution of LEH and the charging/discharging process of electric storage. The flexible region is then mathematically formulated as a set of linearized equations, which can be directly incorporated into the system scheduling model. On this basis, a risk-based two-stage optimization model is developed for the coordinate dispatch of IEGSs and LEHs considering wind power uncertainties. Case studies demonstrate that the exploitation of LEH flexibility can effectively mitigate operational risk levels of IEGSs and reduce system costs. Besides, the proposed model has the advantage of high computation efficiency compared to the existing iteration-based method.

V1-based modeling of discrimination between natural scenes within the luminance and isoluminant color planes.

We have been developing a computational visual difference predictor model that can predict how human observers rate the perceived magnitude of suprathreshold differences between pairs of full-color naturalistic scenes (To, Lovell, Troscianko, & Tolhurst, 2010). The model is based closely on V1 neurophysiology and has recently been updated to more realistically implement sequential application of nonlinear inhibitions (contrast normalization followed by surround suppression; To, Chirimuuta, & Tolhurst, 2017). The model is based originally on a reliable luminance model (Watson & Solomon, 1997) which we have extended to the red/green and blue/yellow opponent planes, assuming that the three planes (luminance, red/green, and blue/yellow) can be modeled similarly to each other with narrow-band oriented filters. This paper examines whether this may be a false assumption, by decomposing our original full-color stimulus images into monochromatic and isoluminant variants, which observers rate separately and which we model separately. The ratings for the original full-color scenes correlate better with the new ratings for the monochromatic variants than for the isoluminant ones, suggesting that luminance cues carry more weight in observers' ratings to full-color images. The ratings for the original full-color stimuli can be predicted from the new monochromatic and isoluminant rating data by combining them by Minkowski summation with power m = 2.71, consistent with other studies involving feature summation. The model performed well at predicting ratings for monochromatic stimuli, but was weaker for isoluminant stimuli, indicating that mirroring the monochromatic models is not sufficient to model the color planes. We discuss several alternative strategies to improve the color modeling.

The convexification effect of Minkowski summation

Let us define for a compact set A \subset \mathbb{R}^n the sequence A(k) = \left\{\frac{a_1+\cdots +a_k}{k}: a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots + A}}\Big). It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that A(k) approaches the convex hull of A in the Hausdorff distance induced by the Euclidean norm as k goes to \infty . We explore in this survey how exactly A(k) approaches the convex hull of A , and more generally, how a Minkowski sum of possibly different compact sets approaches convexity, as measured by various indices of non-convexity. The non-convexity indices considered include the Hausdorff distance induced by any norm on \mathbb R^n , the volume deficit (the difference of volumes), a non-convexity index introduced by Schneider (1975), and the effective standard deviation or inner radius. After first clarifying the interrelationships between these various indices of non-convexity, which were previously either unknown or scattered in the literature, we show that the volume deficit of A(k) does not monotonically decrease to 0 in dimension 12 or above, thus falsifying a conjecture of Bobkov et al. (2011), even though their conjecture is proved to be true in dimension 1 and for certain sets A with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence A(k) , and both the Hausdorff distance and effective standard deviation are eventually monotone (once k exceeds n ). Along the way, we obtain new inequalities for the volume of the Minkowski sum of compact sets (showing that this is fractionally superadditive but not supermodular in general, but is indeed supermodular when the sets are convex), demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.

On Addition of Sets in Boolean Space

In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operation of sets (namely, Minkowski addition) in Boolean space Bn are presented. Also, sums and multisums of various “classical figures” as: sphere, layer, interval etc. are considered. The obtained results make possible to describe multisums by such characteristics of summands as: the sphere radius, weight of layer, dimension of interval etc. using the methods presented in [2], as well as possible solutions of the equation X+Y=A, where , are considered. In spite of simplicity of the statement of the problem, complexity of its solutions is obvious at once, when the connection of solutions with constructions of equidistant codes or existence the Hadamard matrices is apparent. The present paper submits certain results (statements) which are to be the ground for next investigations dealing with Minkowski summation operations of sets in Boolean space.

Perception of differences in naturalistic dynamic scenes, and a V1-based model.

We investigate whether a computational model of V1 can predict how observers rate perceptual differences between paired movie clips of natural scenes. Observers viewed 198 pairs of movies clips, rating how different the two clips appeared to them on a magnitude scale. Sixty-six of the movie pairs were naturalistic and those remaining were low-pass or high-pass spatially filtered versions of those originals. We examined three ways of comparing a movie pair. The Spatial Model compared corresponding frames between each movie pairwise, combining those differences using Minkowski summation. The Temporal Model compared successive frames within each movie, summed those differences for each movie, and then compared the overall differences between the paired movies. The Ordered-Temporal Model combined elements from both models, and yielded the single strongest predictions of observers' ratings. We modeled naturalistic sustained and transient impulse functions and compared frames directly with no temporal filtering. Overall, modeling naturalistic temporal filtering improved the models' performance; in particular, the predictions of the ratings for low-pass spatially filtered movies were much improved by employing a transient impulse function. The correlations between model predictions and observers' ratings rose from 0.507 without temporal filtering to 0.759 (p = 0.01%) when realistic impulses were included. The sustained impulse function and the Spatial Model carried more weight in ratings for normal and high-pass movies, whereas the transient impulse function with the Ordered-Temporal Model was most important for spatially low-pass movies. This is consistent with models in which high spatial frequency channels with sustained responses primarily code for spatial details in movies, while low spatial frequency channels with transient responses code for dynamic events.

On the additive properties of the fat-shattering dimension.

The properties of the VC-dimension under various compositions are well-understood, but this is much less the case for classes of continuous functions. In this brief, we show that a commonly used scale-sensitive dimension, Vγ, is much less well-behaved under Minkowski summation than its VC cousin, while the fat-shattering dimension retains some compositional similarity to the VC-dimension. As an application, we analyze the fat-shattering dimension of trigonometric functions and series.

Orientation tuning in human colour vision at detection threshold

We measure the orientation tuning of red-green colour and luminance vision at low (0.375 c/deg) and mid (1.5 c/deg) spatial frequencies using the low-contrast psychophysical method of subthreshold summation. Orientation bandwidths of the underlying neural detectors are found using a model involving Minkowski summation of the rectified outputs of a bank of oriented filters. At 1.5 c/deg, we find orientation-tuned detectors with similar bandwidths for chromatic and achromatic contrast. At 0.375 c/deg, orientation tuning is preserved with no change in bandwidth for achromatic stimuli, however, for chromatic stimuli orientation tuning becomes extremely broad, compatible with detection by non-oriented colour detectors. A non-oriented colour detector, previously reported in single cells in primate V1 but not psychophysically in humans, can transmit crucial information about the color of larger areas or surfaces whereas orientation-tuned detectors are required to detect the colour or luminance edges that delineate an object's shape.

Experimental investigation of discomfort combination: toward visual discomfort prediction for stereoscopic videos

Automatic prediction of visual discomfort for stereoscopic videos is important to address the viewing safety issues in stereoscopic displays. In literature, disparity and motion characteristics have been known as major factors of visual discomfort caused by stereoscopic content. For developing an algorithm to accurately predict visual discomfort in stereoscopic videos, this study investigates how to combine the individual prediction values of disparity- and motion-induced discomforts into an overall prediction value of visual discomfort. To that end, subjective experiments were performed with various stereoscopic videos, and four possible combination methods were compared based on the results of subjective experiments. The combination methods used in this study were the weighted summation, multiplication, Minkowski summation, and max combination methods. Experimental results showed that the Minkowski summation combination with a high exponent and max combination yielded the best accuracy in predicting the overall level of visual discomfort. The results indicated that the overall level of perceived discomfort could be dominantly affected by the most significant discomfort factor, i.e., the winner-takes-all mechanism. Our results could be useful for the future development of an accurate and reliable algorithm for visual discomfort prediction.

SVD-Based Quality Metric for Image and Video Using Machine Learning

We study the use of machine learning for visual quality evaluation with comprehensive singular value decomposition (SVD)-based visual features. In this paper, the two-stage process and the relevant work in the existing visual quality metrics are first introduced followed by an in-depth analysis of SVD for visual quality assessment. Singular values and vectors form the selected features for visual quality assessment. Machine learning is then used for the feature pooling process and demonstrated to be effective. This is to address the limitations of the existing pooling techniques, like simple summation, averaging, Minkowski summation, etc., which tend to be ad hoc. We advocate machine learning for feature pooling because it is more systematic and data driven. The experiments show that the proposed method outperforms the eight existing relevant schemes. Extensive analysis and cross validation are performed with ten publicly available databases (eight for images with a total of 4042 test images and two for video with a total of 228 videos). We use all publicly accessible software and databases in this study, as well as making our own software public, to facilitate comparison in future research.

A general rule for sensory cue summation: evidence from photographic, musical, phonetic and cross-modal stimuli

The Euclidean and MAX metrics have been widely used to model cue summation psychophysically and computationally. Both rules happen to be special cases of a more general Minkowski summation rule , where m = 2 and ∞, respectively. In vision research, Minkowski summation with power m = 3-4 has been shown to be a superior model of how subthreshold components sum to give an overall detection threshold. Recently, we have previously reported that Minkowski summation with power m = 2.84 accurately models summation of suprathreshold visual cues in photographs. In four suprathreshold discrimination experiments, we confirm the previous findings with new visual stimuli and extend the applicability of this rule to cue combination in auditory stimuli (musical sequences and phonetic utterances, where m = 2.95 and 2.54, respectively) and cross-modal stimuli (m = 2.56). In all cases, Minkowski summation with power m = 2.5-3 outperforms the Euclidean and MAX operator models. We propose that this reflects the summation of neuronal responses that are not entirely independent but which show some correlation in their magnitudes. Our findings are consistent with electrophysiological research that demonstrates signal correlations (r = 0.1-0.2) between sensory neurons when these are presented with natural stimuli.

Summation of perceptual cues in natural visual scenes

Natural visual scenes are rich in information, and any neural system analysing them must piece together the many messages from large arrays of diverse feature detectors. It is known how threshold detection of compound visual stimuli (sinusoidal gratings) is determined by their components' thresholds. We investigate whether similar combination rules apply to the perception of the complex and suprathreshold visual elements in naturalistic visual images. Observers gave magnitude estimations (ratings) of the perceived differences between pairs of images made from photographs of natural scenes. Images in some pairs differed along one stimulus dimension such as object colour, location, size or blur. But, for other image pairs, there were composite differences along two dimensions (e.g. both colour and object-location might change). We examined whether the ratings for such composite pairs could be predicted from the two ratings for the respective pairs in which only one stimulus dimension had changed. We found a pooling relationship similar to that proposed for simple stimuli: Minkowski summation with exponent 2.84 yielded the best predictive power (r=0.96), an exponent similar to that generally reported for compound grating detection. This suggests that theories based on detecting simple stimuli can encompass visual processing of complex, suprathreshold stimuli.

A standard model for foveal detection of spatial contrast

The ModelFest data set was created to provide a public source of data to test and calibrate models of foveal spatial contrast detection. It consists of contrast thresholds for 43 foveal achromatic contrast stimuli collected from each of 16 observers. We have fit these data with a variety of simple models that include one of several contrast sensitivity functions, an oblique effect, a spatial sensitivity aperture, spatial frequency channels, and nonlinear Minkowski summation. While we are able to identify one model, with particular parameters, as providing the lowest overall residual error, we also note that the differences among several good-fitting models are small. We find a strong reciprocity between the size of the spatial aperture and the value of the summation exponent: both are effective means of limiting the extent of spatial summation. The results demonstrate the power of simple models to account for the visibility of a wide variety of spatial stimuli and suggest that special mechanisms to deal with special classes of stimuli are not needed. But the results also illustrate the limited power of even this large data set to distinguish among similar competing models. We identify one model as a possible standard, suitable for simple theoretical and applied predictions.

An Example Concerning Exposed Points

In his paper [1] V. L. Klee gave an example of a smooth bounded convex body C, in En, with the property that extC s closed and extC - expC is dense in extC. As in [1] extC and expC denote the sets of extreme and exposed points of C respectively. It is the purpose of this note to exhibit a similar example in a general separable Banach space using a direct construction which involves Minkowski summation of convex sets.