Max Operations Research Articles

Structured Possibilistic Planning Using Decision Diagrams

Qualitative Possibilistic Mixed-Observable MDPs (pi-MOMDPs), generalizing pi-MDPs and pi-POMDPs, are well-suited models to planning under uncertainty with mixed-observability when transition, observation and reward functions are not precisely known and can be qualitatively described. Functions defining the model as well as intermediate calculations are valued in a finite possibilistic scale L, which induces a finite belief state space under partial observability contrary to its probabilistic counterpart. In this paper, we propose the first study of factored pi-MOMDP models in order to solve large structured planning problems under qualitative uncertainty, or considered as qualitative approximations of probabilistic problems. Building upon the SPUDD algorithm for solving factored (probabilistic) MDPs, we conceived a symbolic algorithm named PPUDD for solving factored pi-MOMDPs. Whereas SPUDD's decision diagrams' leaves may be as large as the state space since their values are real numbers aggregated through additions and multiplications, PPUDD's ones always remain in the finite scale L via min and max operations only. Our experiments show that PPUDD's computation time is much lower than SPUDD, Symbolic-HSVI and APPL for possibilistic and probabilistic versions of the same benchmarks under either total or mixed observability, while still providing high-quality policies.

APCV 2014, 19–22 July 2014, Takamatsu, Japan Abstracts

How are the image statistics of global image contrast computed? We answered this by using a contrast-matching task for checkerboard configurations of ‘battenberg’ micro-patterns where the contrasts and spatial spreads of interdigitated pairs of micro-patterns were adjusted independently. Test stimuli were 20 × 20 arrays with various sized cluster widths, matched to standard patterns of uniform contrast. When one of the test patterns contained a pattern with much higher contrast than the other, that determined global pattern contrast, as in a max() operation. Crucially, however, the full matching functions had a curious intermediate region where low contrast additions for one pattern to intermediate contrasts of the other caused a paradoxical reduction in perceived global contrast. None of the following models predicted this: RMS, energy, linear sum, max, Legge and Foley. However, a gain control model incorporating wide-field integration and suppression of nonlinear contrast responses predicted the results with no free parameters. This model was derived from experiments on summation of contrast at threshold, and masking and summation effects in dipper functions. Those experiments were also inconsistent with the failed models above. Thus, we conclude that our contrast gain control model (Meese & Summers, 2007) describes a fundamental operation in human contrast vision.

Exploring biologically inspired shallow model for visual classification

Visual classification has long been a major challenge for computer vision. In recent years, biologically inspired visual models have raised great interests. However, most of the related studies mainly focus on learning features and representations from very large scale dataset relying on deep network architecture, which is doomed to fail with limited training samples due to its high complexity. In this paper, it is found that aside from the deep architecture, two other biologically inspired mechanisms, the pooling and nonlinear operations, also contribute to the improvement of classification performance. Based on this perspective, a new classifier of shallow architecture is proposed, in which the both mechanisms are implemented with max operation. Moreover, the architecture is derived in a probabilistic perspective to further explain the underlying rationale thereof. To train the classifier, a supervised learning algorithm is devised to minimize the hinge loss function under the new architecture. Based on the manifold assumption of continuously transforming features, an unsupervised learning algorithm is also presented to learn the features used by the classifier. Finally, the method is compared against other classifiers on several image classification benchmarks. The results demonstrate the strength of the proposed method when the training data source is limited.

Measuring the spatial extent of texture pooling using reverse correlation

The local image representation produced by early stages of visual analysis is uninformative regarding spatially extensive textures and surfaces. We know little about the cortical algorithm used to combine local information over space, and still less about the area over which it can operate. But such operations are vital to support perception of real-world objects and scenes. Here, we deploy a novel reverse-correlation technique to measure the extent of spatial pooling for target regions of different areas placed either in the central visual field, or more peripherally. Stimuli were large arrays of micropatterns, with their contrasts perturbed individually on an interval-by-interval basis. By comparing trial-by-trial observer responses with the predictions of computational models, we show that substantial regions (up to 13 carrier cycles) of a stimulus can be monitored in parallel by summing contrast over area. This summing strategy is very different from the more widely assumed signal selection strategy (a MAX operation), and suggests that neural mechanisms representing extensive visual textures can be recruited by attention. We also demonstrate that template resolution is much less precise in the parafovea than in the fovea, consistent with recent accounts of crowding.

Probability, fuzziness and borderline cases

An integrated approach to truth-gaps and epistemic uncertainty is described, based on probability distributions defined over a set of three-valued truth models. This combines the explicit representation of borderline cases with both semantic and stochastic uncertainty, in order to define measures of subjective belief in vague propositions. Within this framework we investigate bridges between probability theory and fuzziness in a propositional logic setting. In particular, when the underlying truth model is from Kleene's three-valued logic then we provide a complete characterisation of compositional min–max fuzzy truth degrees. For classical and supervaluationist truth models we find partial bridges, with min and max combination rules only recoverable on a fragment of the language. Across all of these different types of truth valuations, min–max operators are resultant in those cases in which there is only uncertainty about the relative sharpness or vagueness of the interpretation of the language.

Central auditory neurons display flexible feature recombination functions

Recognition of natural stimuli requires a combination of selectivity and invariance. Classical neurobiological models achieve selectivity and invariance, respectively, by assigning to each cortical neuron either a computation equivalent to the logical "AND" or a computation equivalent to the logical "OR." One powerful OR-like operation is the MAX function, which computes the maximum over input activities. The MAX function is frequently employed in computer vision to achieve invariance and considered a key operation in visual cortex. Here we explore the computations for selectivity and invariance in the auditory system of a songbird, using natural stimuli. We ask two related questions: does the MAX operation exist in auditory system? Is it implemented by specialized "MAX" neurons, as assumed in vision? By analyzing responses of individual neurons to combinations of stimuli we systematically sample the space of implemented feature recombination functions. Although we frequently observe the MAX function, we show that the same neurons that implement it also readily implement other operations, including the AND-like response. We then show that sensory adaptation, a ubiquitous property of neural circuits, causes transitions between these operations in individual neurons, violating the fixed neuron-to-computation mapping posited in the state-of-the-art object-recognition models. These transitions, however, accord with predictions of neural-circuit models incorporating divisive normalization and variable polynomial nonlinearities at the spike threshold. Because these biophysical properties are not tied to a particular sensory modality but are generic, the flexible neuron-to-computation mapping demonstrated in this study in the auditory system is likely a general property.

Image focus volume enhancement in shape from focus systems

Three-dimensional object reconstructions is an active research area in digital imaging. In shape from focus approach, erroneous focus measurements result in inaccuracy in the depth map reconstruction of 2D object. Conventionally, to enhance the image focus volume, focus values are aggregated within a window, which is a linear filtering approach. Owing to the inherent limitation of linear process, optimal results may not be obtained. In order to overcome this limitation, a non-linear filtering approach is proposed to enhance the image focus volume for accurate depth estimation. The noisy focus values are restored in two steps. First, noisy focus values are detected using min–max operators. In order to increase the dynamic range between the minimum and the maximum focus values within the window, an appropriate power law function is designed. In second step, only the noisy measurements are replaced with the estimated ones. A refined depth map is obtained from the updated focus volume. This process continues until the difference between the previous and the current depth maps becomes very small. The performance of the proposed non-linear filtering approach is obtained for various synthetic and real objects. The results highlight the depth map estimates of the proposed approach more accurate while preserving object edges. Comparative analysis demonstrates the effectiveness of the proposed approach.

All-optical quaternary logic gates

Conventional binary logic based operations restrict the speed of operations as well as information handling capacity. A way to overcome these limitations is the implementation of multivalued logic operations in the optical domain. Multivalued logic operations not only enhance the data handling capacities but also increase the speed of processing. integrating enormous potential bandwidth of optical fiber as information carrying medium and faster optoelectronic/optical switches with no hardware complexity. A new method is proposed for the implementation of all-optical quaternary inversion, MAX, MIN, and equality operations using frequency-encoded data. Cross phase modulation-based frequency conversion, polarization switch (PSW) characteristics of a semiconductor optical amplifier (SOA), frequency routing by a wave division multiplexer (MUX), and a demultiplexer (DMUX) have been exploited to implement the desired quaternary logic operations. Simulation results support the feasibility of the proposed scheme.

An efficient similarity measure for intuitionistic fuzzy sets

We introduce a new methodology for measuring the degree of similarity between two intuitionistic fuzzy sets. The new method is developed on the basis of a distance defined on an interval by the use of convex combination of endpoints and also focusing on the property of min and max operators. It is shown that among the existing methods, the proposed method meets all the well-known properties of a similarity measure and has no counter-intuitive examples. The validity and applicability of the proposed similarity measure is illustrated with two examples known as pattern recognition and medical diagnosis.

Modal Trajectory Estimation Using Maximum Gaussian Mixture

This technical note deals with the estimation of the whole trajectory of a stochastic dynamic system with highest probability, conditionally upon the past observation process, using a maximum Gaussian mixture. We first recall the Gaussian sum technique applied to minimum variance filtering. It is then shown that the same concept of Gaussian mixture can be applied in that context, provided we replace the Sum operator by the Max operator.

2-Tuple linguistic harmonic operators and their applications in group decision making

Harmonic mean is reciprocal of arithmetic mean of reciprocal, which is a conservative average to be used to provide for aggregation lying between max and min operators. In this paper, we develop some new linguistic aggregation operators such as 2-tuple linguistic harmonic (2TLH) operator, 2-tuple linguistic weighted harmonic (2TLWH) operator, 2-tuple linguistic ordered weighted harmonic (2TLOWH) operator, and 2-tuple linguistic hybrid harmonic (2TLHH) operator, which can be utilized to aggregate preference information taking the form of linguistic variables, and then study some desirable properties of the operators. Based on the 2TLWH and 2TLHH operators, we present an approach to multiple attribute decision making with 2-tuple linguistic information. Finally, illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness by comparing with the existing approaches.

Methods of Evaluation on Manufacturing Information Quality

Information quality (IQ) is one of the basic factors limiting the development of enterprises. And now it becomes a determinable factor. It’s urgent to improve IQ, especially methods of IQ evaluation. After comparing the views of the scholars at home and abroad, a method of IQ evaluation was developed. The evaluation of information quality includes two parts, data quality and information quality evaluation.Subjective and objective evaluation of data quality were listed, including three basic forms,simple ratio, min or max operation and weighed average. It also introduces developing data quality measurement in engineering practice.The method of IQ evaluation was studied ,and the basis of IQ evaluation was established. It reflected the status of organizational IQ using the data collected by the investigation questionnaires. These analyses are applied to analyze the gap between an organization and the best practice. The results of these techniques can be applied to assurance of IQ improvement and improvement of data quality. It’s also an effective way to lower information costs.

On Computing Criticality in Refactored Timing Graphs

The maximum operator in statistical static timing analysis (SSTA) is a decent approximation for timing sign-off, but often causes significant error in SSTA applications. This paper presents a timing criticality computation method based on non-maximum analytic operators in a parameterized SSTA. After an SSTA run, the proposed method computes the criticality for all edges and nodes in a single graph traversal. Although we do not employ the max operator in the computation process, the error in the maximum operator still degrades the accuracy of the computed criticality because the criticality is a joint probability of expressions, including arrival times, which are computed by the maximum operator during SSTA. To address this issue, we employ the refactoring technique, which was recently proposed to reduce common path pessimism in combinational circuits. This paper shows that refactoring is also very useful in reducing the maximum-induced error in arrival times, and how existing graph-based algorithms can be geared toward refactoring. Our experimental results show that the proposed method reduces the error of the criticality significantly compared to the conventional cutset-based method.

The ordered multiplicative modular geometric operator

The aim of this paper is to investigate an ordered multiplicative modular geometric operator and its relevant properties. The ordered multiplicative modular geometric operator is a generalized form of the ordered weighted geometric operator which has been designed incorporating the advantages of the geometric mean to deal with ratio judgments and the advantages of the ordered weighted averaging (OWA) operator to represent the concept of fuzzy majority in the process of information aggregation. Besides, the ordered multiplicative modular geometric operator can be seen as a symmetrized multiplicative modular aggregation function, characterized by comonotone multiplicative modularity. It is worth pointing that lots of the existing operators (such as the ordered weighted geometric operator, the weighted geometric operator, the ordered weighted maximum, and the Max and Min operators) can be regarded as the special cases of the ordered multiplicative modular geometric operator, which is of value in developing the theory of geometric operators.

Theory and data for area summation of contrast with and without uncertainty: Evidence for a noisy energy model

Contrast sensitivity improves with the area of a sine-wave grating, but why? Here we assess this phenomenon against contemporary models involving spatial summation, probability summation, uncertainty, and stochastic noise. Using a two-interval forced-choice procedure we measured contrast sensitivity for circular patches of sine-wave gratings with various diameters that were blocked or interleaved across trials to produce low and high extrinsic uncertainty, respectively. Summation curves were steep initially, becoming shallower thereafter. For the smaller stimuli, sensitivity was slightly worse for the interleaved design than for the blocked design. Neither area nor blocking affected the slope of the psychometric function. We derived model predictions for noisy mechanisms and extrinsic uncertainty that was either low or high. The contrast transducer was either linear (c(1.0)) or nonlinear (c(2.0)), and pooling was either linear or a MAX operation. There was either no intrinsic uncertainty, or it was fixed or proportional to stimulus size. Of these 10 canonical models, only the nonlinear transducer with linear pooling (the noisy energy model) described the main forms of the data for both experimental designs. We also show how a cross-correlator can be modified to fit our results and provide a contemporary presentation of the relation between summation and the slope of the psychometric function.

Fourier Series Approximation for Max Operation in Non-Gaussian and Quadratic Statistical Static Timing Analysis

The most challenging problem in the current block-based statistical static timing analysis (SSTA) is how to handle the max operation efficiently and accurately. Existing SSTA techniques suffer from limited modeling capability by using a linear delay model with Gaussian distribution, or have scalability problems due to expensive operations involved to handle non-Gaussian variation sources or nonlinear delays. To overcome these limitations, we propose efficient algorithms to handle the max operation in SSTA with both quadratic delay dependency and non-Gaussian variation sources simultaneously. Based on such algorithms, we develop an SSTA flow with quadratic delay model and non-Gaussian variation sources. All the atomic operations, max and add, are calculated efficiently via either closed-form formulas or low dimension (at most 2-D) lookup tables. We prove that the complexity of our algorithm is linear in both variation sources and circuit sizes, hence our algorithm scales well for large designs. Compared to Monte Carlo simulation for non-Gaussian variation sources and nonlinear delay models, our approach predicts the mean, standard deviation and 95% percentile point with less than 2% error, and the skewness with less than 10% error.

Fuzzy-based Medical X-Ray Image Classification

In this paper, a novel fuzzy scheme for medical X-ray image classification is presented. In this method, each image is partitioned into 25 overlapping subimages and then, we extracted the shape-texture features from shape and directional information of each subimage. In the classification step, we apply a fuzzy membership to each subimage considering the Euclidean distance between feature vector of each subimage and average of feature vectors of training subimages. At last, a hard classification of the test image can be obtained by performing a max operation on the summation of fuzzy memberships. The proposed method is evaluated for image classification on 2655 radiographic images from IRMA dataset with 300 training samples and 2355 test samples. Classification accuracy rates obtained by fuzzy classifier are higher than that of obtained by multilayer perceptron or even SVM classifier.

Linguistic harmonic mean operators and their applications to group decision making

Harmonic mean is the reciprocal of arithmetic mean of the reciprocal, which is a conservative average to be used to provide for aggregation lying between max and min operators. In this paper, we develop some new aggregation operators such as linguistic harmonic mean (LHM) operator, linguistic weighted harmonic mean (LWHM) operator, linguistic ordered weighted harmonic mean operator, and linguistic hybrid harmonic mean (LHHM) operator, which can be utilized to aggregate preference information taking the form of linguistic variables and then study some desirable properties of the operators. Based on the LHM and the LHHM operators, we propose a practical method for group decision making with linguistic preference relations and also give an illustrative example. Furthermore, based on the LWHM and LHHM operators, we develop a multiple attribute decision making applied in the group decision making to improving advanced manufacturing technology selection process.

Logical composition of Lyapunov functions

This article introduces the use of R-functions to compose single Lyapunov functions (LFs) via classic Boolean operators, with the aim to obtain a rich family of non-conventional, generally non-convex functions. The main benefit of the proposed composition is the nice geometric interpretation, since it corresponds to intersection and union operations in the phase space region. The composition of LFs is parameterised through a variable γ and classic compositions of LFs through min and max operations are recovered as a special case for a particular value of γ. The proposed logical composition is applied to region of asymptotic stability (RAS) estimation problems, where the union of several LFs corresponds to the union of the RAS estimates obtained from the separate use of each LF. Likewise, the intersection of several LFs defined on independent subsets of the state space variables provides a single LF for the overall dynamical system. Sufficient conditions for the composition function to be an LF are provided and results are described through several examples of classic nonlinear dynamical systems.

Robust local feature extraction algorithm with visual cortex for object recognition

A robust local feature extraction algorithm comprised of two key steps is proposed. The first step is the extraction of feature point candidates from a multi-scale fast Hessian detector at 0° and 45° based on Gabor rotation angles and a multi-scale Harris corner detector based on the Gabor phase. The second step is a MAX operation for selecting points from the outputs of the three feature point detectors used in the previous step. Performance analysis shows that the proposed algorithm significantly enhances the recognition rate and is better than SIFT and SURF.