New sequential exact Euclidean distance transform algorithms based on convex analysis

A distance transformation (DT) takes a binary image as input and generates a distance map image in which the value of each pixel is its distance to a given set of object pixels in the binary image. In this research, DT's for multi class data (MCDTs) are developed which generate both a distance map and a class map containing for each pixel the class of the closest object. Results indicate that the MCDT based on the Fast Exact Euclidean Distance (FEED) method is a factor 2 tot 4 faster than MCDTs based on exact or semi-exact euclidean distance (ED) transformations, and is only a factor 2 to 4 slower than the MCDT based on the crude city-block approximation of the ED. In the second part of this research, the MCDTs were adapted such that they could be used for the fast generation of distance and class maps for video sequences. The frames of the sequences contain a number of fixed objects and a moving object, where each object has a separate label. Results show that the FEED based version is a factor 2 to 3.5 faster than the fastest of all the other video-MCDTs which is based on the chamfer 3,4 distance measure. FEED is even a factor 3.5 to 10 faster than another fast exact ED transformation. With video, multi class FEED it will be possible to measure distances from a moving object to various identified stationary objects with nearly the frame rate of a webcam. This will be very useful when the risk exists that objects move outside surveillance limits.

Chamfer distances play an important role in the theory of distance transforms. Though the determination of the exact Euclidean distance transform is also a well investigated area, the classical chamfering method based upon "small" neighborhoods still outperforms it e.g. in terms of computation time. In this paper we determine the best possible maximum relative error of chamfer distances under various boundary conditions. In each case some best approximating sequences are explicitly given. Further, because of possible practical interest, we give all best approximating sequences in case of small (i.e. 5x5 and 7x7) neighborhoods.

A fast and exact Euclidean distance transformation using decomposed grayscale morphological operators is presented. Applied on a binary image, a distance transformation assigns each object pixel a value that corresponds to the shortest distance between the object pixel and the background pixels. It is shown that the large structuring element required for the Euclidean distance transformation can be easily decomposed into 3/spl times/3 windows. This is possible because the square of the Euclidean distance matrix changes uniformly both in the vertical and horizontal directions. A simple extension for a 3D Euclidean distance transformation is discussed. A fast distance transform for serial computers is also presented. Acting like thinning algorithms, the version for serial computers focuses operations only on the potential changing pixels and propagates from the boundary of objects, significantly reducing execution time. Nonsquare pixels can also be used in this algorithm. An example application, shape filtering using arbitrary sized circular dilation and erosion, is discussed. Rotation-invariant basic morphological operations can be done using this example application. >

We propose BinaryRelax, a simple two-phase algorithm, for training deep neural networks with quantized weights. The set constraint that characterizes the quantization of weights is not imposed until the late stage of training, and a sequence of pseudo quantized weights is maintained. Specifically, we relax the hard constraint into a continuous regularizer via a Moreau envelope, which turns out to be the squared Euclidean distance to the set of quantized weights. The pseudo quantized weights are obtained by linearly interpolating between the float weights and their quantizations. A continuation strategy is adopted to push the weights toward the quantized state by gradually increasing the regularization parameter. In the second phase, an exact quantization scheme with a small learning rate is invoked to guarantee fully quantized weights. We test BinaryRelax on the benchmark CIFAR and ImageNet color image datasets to demonstrate the superiority of the relaxed quantization approach and the improved accuracy over the state-of-the-art training methods. Finally, we prove the convergence of BinaryRelax under an approximate orthogonality condition.

An antinorm is a concave nonnegative homogeneous functional on a convex cone. It is shown that if the cone is polyhedral, then every antinorm has a unique continuous extension from the interior of the cone. The main facts of the duality theory in convex analysis, in particular, the Fenchel–Moreau theorem, are generalized to antinorms. However, it is shown that the duality relation for antinorms is discontinuous. In every dimension, there are infinitely many self-dual antinorms on the positive orthant and, in particular, infinitely many autopolar polyhedra. For the two-dimensional case, we characterize them all. The classification in higher dimensions is left as an open problem. Applications to linear dynamical systems, to the Lyapunov exponent of random matrix products, to the lower spectral radius of nonnegative matrices, and to convex trigonometry are considered.

This paper looks into relations between repre-sentational methods and conceived architectural space on Le Corbusier's axonometric drawings.Conclusions are followings. 1) Axonometric drawings are made on and after La-Chaux-Fonds period, but military projection is conciously used nearly after 1925.2) Origins of using axonometric drawings are O. Choisy's “Histoire d'Architecture”and Exposition of De Stijl. 3) Time of starting to use military projection consciously corresponds to time of establishing modules on plan and using ‘polychromie’ as a design method. 4) By using ‘polychromie’, geometrical distances and sensuous distances are introduced into architectural design.We can understand that military projection is used not only to show exact geometrical distances between walls, but also to show sensuous distances by coloring walls.

Popular clustering algorithms based on usual distance functions (e.g., the Euclidean distance) often suffer in high dimension, low sample size (HDLSS) situations, where concentration of pairwise distances and violation of neighborhood structure have adverse effects on their performance. In this article, we use a new data-driven dissimilarity measure, called MADD, which takes care of these problems. MADD uses the distance concentration phenomenon to its advantage, and as a result, clustering algorithms based on MADD usually perform well for high dimensional data. We establish it using theoretical as well as numerical studies. We also address the problem of estimating the number of clusters. This is a challenging problem in cluster analysis, and several algorithms are available for it. We show that many of these existing algorithms have superior performance in high dimensions when they are constructed using MADD. We also construct a new estimator based on a penalized version of the Dunn index and prove its consistency in the HDLSS asymptotic regime. Several simulated and real data sets are analyzed to demonstrate the usefulness of MADD for cluster analysis of high dimensional data.

The interest in exploiting big datasets with machine learning has led to adapting classic strategies in this new paradigm determined by volume, speed, and variety. Because data quality is a determining factor in constructing a classifier, it has also been necessary to adapt or develop new data preprocessing techniques. One of the challenges of most significant interest is the class imbalance problem, where the class of interest has a smaller number of examples concerning another class called the majority. To alleviate this problem, one of the most recognized techniques is SMOTE, which is characterized by generating instances of the minority class through a process that uses the nearest neighbor rule and the Euclidean distance. Various articles have shown that SMOTE is not appropriate for datasets with high dimensionality. However, in big data, datasets with high dimensionality have contained many zeros. Therefore, in this article, our objective is to analyze the SMOTE-BD behavior on imbalanced big datasets with sparse and dense dimensionality. Experimental results using two classifiers and big datasets with different dimensionalities suggest that sparsity is a predominant factor than the dimensionality in the behavior of SMOTE-BD.KeywordsBig dataSMOTEClass imbalanceHigh dimensionalityDense datasetSparse dataset

In this chapter, we present basic notations and properties for functions which attain values in \(\mathbb {R}\cup \{-\infty ,+\infty \}\), the extended set of real numbers. Of course, such functions may also be real-valued. In the first two sections, we give a short overview about the extended set of real numbers and explain the way in which extended real-valued functions are handled in convex analysis. We also discuss problems that arise by admitting values \(\pm \infty \). Afterwards, statements about the continuity and the semicontinuity of extended real-valued functions are summarized. We investigate convexity, generalized convexity, linearity, sublinearity and related properties of the functionals. One section is devoted to monotonicity. Furthermore, basic notions for the variational analysis of extended real-valued functions are collected. Since extended real-valued functions belong to the standard in convex analysis, most of the results in this chapter are well known. We add several new statements in order to point out that some properties of the functions which we focus on in this book turn out to be properties of each extended real-valued function.

How can we efficiently and scalably cluster high-dimensional data? Thek-means algorithm clusters data by iteratively reducing intra-cluster Euclidean distances until convergence. While it finds applications from recommendation engines to image segmentation, its application to high-dimensional data is hindered by the need to repeatedly compute Euclidean distances among points and centroids. In this paper, we propose Marigold (k-means for high-dimensional data), a scalable algorithm fork-means clustering in high dimensions. Marigold prunes distance calculations by means of (i) a tight distance-bounding scheme; (ii) a stepwise calculation over a multiresolution transform; and (iii) exploiting the triangle inequality. To our knowledge, such an arsenal of pruning techniques has not been hitherto applied tok-means. Our work is motivated by time-critical Angle-Resolved Photoemission Spectroscopy (ARPES) experiments, where it is vital to detect clusters among high-dimensional spectra in real time. In a thorough experimental study with real-world data sets we demonstrate that Marigold efficiently clusters high-dimensional data, achieving approximately one order of magnitude improvement over prior art.

Let P be a set of n points in /spl Ropf//sup d/. For any 1/spl les/k/spl les/d, the outer k-radius of P, denoted by R/sub k/(P), is the minimum, over all (d-k) -dimensional fiats F, of max/sub p/spl isin/P/ d(p, F), where d(p, F) is the Euclidean distance between the point p and fiat F. We consider the scenario when the dimension d is not fixed and can be as large as n. Computing the various radii of point sets is a fundamental problem in computational convexity with many applications. The main result of this paper is a randomized polynomial time algorithm that approximates Rk (P) to within a factor of O/spl radic/(log n/spl middot/log d) for any 1/spl les/k/spl les/d. This algorithm is obtained using techniques from semidefinite programming and dimension reduction. Previously, good approximation algorithms were known only for the case k=1 and for the case when k=d-c for any constant c; there are polynomial time algorithms that approximate Rk(P) to within a factor of (1+/spl epsi/), for any /spl epsi/>0, when d-k is any fixed constant. On the other hand, some results from the mathematical programming community on approximating certain kinds of quadratic programs imply an O/spl radic/(log n) approximation for R/sub 1/ (P), the width of the point set P. We also prove an inapproximability result for computing Rk (P), which easily yields the conclusion that our approximation algorithm performs quite well for a large range of values of k. Our inapproximability result for Rk (P) improves the previous known hardness result of Brieden, and is proved by improving the parameters in Brieden's construction using basic ideas from PCP theory.

There are many distance-based methods for classification and clustering, and for data with a high number of dimensions and a lower number of observations, processing distances is computationally advantageous compared to the raw data matrix. Euclidean distances are used as a default for continuous multivariate data, but there are alternatives. Here the so-called Minkowski distances, \(L_1\) (city block)-, \(L_2\) (Euclidean)-, \(L_3\), \(L_4\)- and maximum distances are combined with different schemes of standardisation of the variables before aggregating them. Boxplot transformation is proposed, a new transformation method for a single variable that standardises the majority of observations but brings outliers closer to the main bulk of the data. Distances are compared in simulations for clustering by partitioning around medoids, complete and average linkage, and classification by nearest neighbours, of data with a low number of observations but high dimensionality. The \(L_1\)-distance and the boxplot transformation show good results.

In this paper, a novel apple Physalospora recognition approach based on the Gabor feature-based principal component analysis (GBPCA) is proposed. In this method, the principal component analysis (PCA) is a powerful technique for finding patterns in data of high dimensionality and can reduce the high dimensionality of the data space to the low dimensionality of feature space effectively. Gabor filter is an effective tool because of its accurate time-frequency localization and robustness against variations caused by illumination and rotation. Three main steps are taken in the proposed GBPCA: Firstly, Gabor features of different scales and orientations are extracted by convoluting the Gabor filter bank and the original gray images. Then eigenvectors in the direction of the largest variance of the training vectors is computed by PCA. An eigenspace is composed of these eigenvectors. Thirdly, we project the testing images into the constructed eigenspace and the Euclidean distance and nearest neighbor classifier are adopted for classification. Therefore, the proposed method is not only insensitive to illumination and rotation, but also efficient in feature matching. Experimental results demonstrate the effectiveness of the proposed GBPCA.

Time profiled association mining is one of the important and challenging research problems that is relatively less addressed. Time profiled association mining has two main challenges that must be addressed. These include addressing i) dissimilarity measure that also holds monotonicity property and can efficiently prune itemset associations ii) approaches for estimating prevalence values of itemset associations over time. The pioneering research that addressed time profiled association mining is by J.S. Yoo using Euclidean distance. It is widely known fact that this distance measure suffers from high dimensionality. Given a time stamped transaction database, time profiled association mining refers to the discovery of underlying and hidden time profiled itemset associations whose true prevalence variations are similar as the user query sequence under subset constraints that include i) allowable dissimilarity value ii) a reference query time sequence iii) dissimilarity function that can find degree of similarity between a temporal itemset and reference. In this paper, we propose a novel dissimilarity measure whose design is a function of product based gaussian membership function through extending the similarity function proposed in our earlier research (G-Spamine). Our approach, MASTER (Mining of Similar Temporal Associations) which is primarily inspired from SPAMINE uses the dissimilarity measure proposed in this paper and support bound estimation approach proposed in our earlier research. Expression for computation of distance bounds of temporal patterns are designed considering the proposed measure and support estimation approach. Experiments are performed by considering naive, sequential, Spamine and G-Spamine approaches under various test case considerations that study the scalability and computational performance of the proposed approach. Experimental results prove the scalability and efficiency of the proposed approach. The correctness and completeness of proposed approach is also proved analytically.

We investigate the average minimum cost of a bipartite matching between two samples of n independent random points uniformly distributed on a unit cube in d $\ge$ 3 dimensions, where the matching cost between two points is given by any power p $\ge$ 1 of their Euclidean distance. As n grows, we prove convergence, after a suitable renormalization, towards a finite and positive constant. We also consider the analogous problem of optimal transport between n points and the uniform measure. The proofs combine sub-additivity inequalities with a PDE ansatz similar to the one proposed in the context of the matching problem in two dimensions and later extended to obtain upper bounds in higher dimensions.