Set Of Local Maxima Research Articles

Minor Surfaces are Boundaries of Mode-Based Clusters

We show that mode-based cluster boundaries exhibit themselves as minor surfaces of the data probability density function. Based on this result, we provide a connectivity measure depending on minor surface search between sample pairs. Accordingly, we build a connectivity graph among data samples. The use of graph construction is particularly demonstrated for clustering, but applications in other machine learning areas are possible. On Gaussian mixture and kernel density estimate type probability density models, we illustrate the theoretical results with examples and demonstrate that cluster boundaries between sample pairs can be detected using a line integral. We also demonstrate an example where the data distribution has a continuous line segment as its set of local maxima (not strict), for which mean-shift like gradient flow and other mode-seeking algorithms fail to identify a single cluster, while the proposed approach successfully determines this fact.

Reconstructing the profile of a laser beam by measuring its power in a sequence of bands

A method is proposed for reconstructing noisy and ideal intensity profiles of a laser beam by using a sequence of integral measurements of the laser-radiation power that falls into a band of given width in a specified direction. For a spatially two-dimensional region, the proposed method makes it possible to obtain high-accuracy reconstruction of varied beam profiles with a set of local maxima of various amplitudes and different local distributions. It is shown that the number of measurements needed to achieve the required reconstruction accuracy can be reduced by increasing the width of the measurement bands and their partial spatial overlap. It is demonstrated that noise in the intensity distribution whose amplitude does not exceed 10% of the maximum beam intensity can be suppressed. The approach considered here can be applied to tomography and the pattern-recognition problem.

Properties of multilayer nonuniform holographic structures

Experimental results and analysis of properties of multilayer nonuniform holographic structures formed in photopolymer materials are presented. The theoretical hypotheses is proved that the characteristics of angular selectivity for the considered structures have a set of local maxima, whose number and width are determined by the thicknesses of intermediate layers and deep holograms and that the envelope of the maxima coincides with the selectivity contour of a single holographic array. It is also experimentally shown that hologram nonuniformities substantially distort shapes of selectivity characteristics: they become asymmetric, the local maxima differ in size and the depths of local minima reduce. The modelling results are made similar to experimental data by appropriately choosing the nonuniformity parameters.

Global Optimization: On Pathlengths in Min-Max Graphs

Let f be a smooth nondegenerate real valued function on a finite dimensional, compact and connected Riemannian manifold. The bipartite min-max graph Γ is defined as follows. Its nodes are formed by the set of local minima and the set of local maxima. Two nodes (a local minimum and a local maximum) are connected in Γ by means of an edge if some trajectory of the corresponding gradient flow connects them. Given a natural number k, we construct a function f such that the length of the shortest path in Γ between two specific local minima exceeds k. The latter construction is independent of the underlying Riemannian metric.

From Local Maxima to Connected Skeletons

In picture processing it is often convenient to deal with a stick-like version (skeleton) of binary digital images. Although skeleton connectedness is not necessary for storage and retrieval purposes, this property is desirable when a structural description of images is of interest. In this paper a parallel procedure is described which, applied to a connected image, originates a connected skeleton made by the union of simple digital arcs. The procedure involves a step by step propagation of the background over the image. At every step, contour elements either belonging to the significant convex regions of the current image or being local maxima of the original image are selected as skeleton elements. Since the final set so obtained is not ensured to be connected, the configurations in correspondence of which disconnections appear are investigated and the procedures to avoid this shortcoming are given. The presence of the whole set of local maxima among the skeleton elements ensures the possibility of recovering the original image by means of a reverse distance transform. The details of the program implementing the proposed algorithm on a parallel processor are finally included.