Kd-tree Research Articles

Model indexing and object recognition using 3D viewpoint invariance

Object recognition and location of 3D objects from range data are described, based on invariant representations of characteristic views of single objects. Depth data is processed to remove outliers, then smoothed and segmented into surface patches, fitted by polynomial equations. Mathematical invariants are derived from groups of these surface patches and used for indexing a model database stored in the form of a kd tree. The pose of the known object may then be determined from the Euclidean centre and orientation of the higher order surfaces formed from the feature groups. The approach is complete, scalable and extendable. The robustness of the method is evaluated in terms of the stability and discriminatory power of the representation, and the accuracy of the determined pose.

Fast implementations of nearest neighbor classifiers

Standard implementations of non-parametric classifiers have large computational requirements. Parzen classifiers use the distances of an unknown vector to all N prototype samples, and consequently exhibit O( N) behavior in both memory and time. We describe four techniques for expediting the nearest neighbor methods: replacing the linear search with a new kd tree method, exhibiting approximately O(N 1 2 ) behavior; employing an L ∞ instead of L 2 distance metric; using variance-ordered features; and rejecting prototypes by evaluating distances in low dimensionality subspaces. We demonstrate that variance-ordered features yield significant efficiency gains over the same features linearly transformed to have uniform variance. We give results for a large OCR problem, but note that the techniques expedite recognition for arbitrary applications. Three of four techniques preserve recognition accuracy.

KD trees and Delaunay-based linear interpolation for function learning: a comparison to neural networks with error backpropagation

We illustrate how a KD tree data structure with Delaunay triangulation can be used for function learning. The example function is the inverse kinematics of a three degree-of-freedom (DOF) robot. The result can subsequently be used for kinematic control. The KD tree is used to efficiently extract a set number of nearest neighbors to a query point. Delaunay triangulation provides a good criterion for constructing a continuous linear approximation to the true function from neighborhood points of the query. For comparison purposes we solve the same problem with a neural network trained with error backpropagation. We conclude that the KD/Delaunay approach, in comparison to neural networks, can potentially yield a massive reduction in training time and significantly improve function estimate performance.

Efficient Recognition of Partially Visible Objects Using a Logarithmic Complexity Matching Technique

An important task in computer vision is the recognition of partially visible two-dimensional objects in a gray scale image. Recent works addressing this problem have attempted to match spatially local features from the image to features generated by models of the objects. However, many algo rithms are considerably less efficient than they might be, typ ically being O(IN) or worse, where I is the number offeatures in the image and N is the number of features in the model set. This is invariably due to the feature-matching portion of the algorithm. In this paper we discuss an algorithm that significantly improves the efficiency offeature matching. In addition, we show experimentally that our recognition algo rithm is accurate and robust. Our algorithm uses the local shape of contour segments near critical points, represented in slope angle-arclength space (θ-s space), as fundamental fea ture vectors. These feature vectors are further processed by projecting them onto a subspace in θ-s space that is obtained by applying the Karhunen-Loève expansion to all such fea tures in the set of models, yielding the final feature vectors. This allows the data needed to store the features to be re duced, while retaining nearly all information important for recognition. The heart of the algorithm is a technique for performing matching between the observed image features and the precomputed model features, which reduces the runtime complexity from O(IN) to O(I log I + I log N), where I and N are as above. The matching is performed using a tree data structure, called a kD tree, which enables multidi mensional searches to be performed in O(log) time.

Spatial query processing in an object-oriented database system

DBMSs must offer spatial query processing capabilities to meet the needs of applications such as cartography, geographic information processing and CAD. Many data structures and algorithms that process grid representations of spatial data have appeared in the literature. We unify much of this work by identifying common principles and distilling them into a small set of constructs. (Published data structures and algorithms can be derived as special cases.) We show how these constructs can be supported with only minor modifications to current DBMS implementations. The ideas are demonstrated in the context of the range query problem. Analytical and experimental evidence indicates that performance of the derived solution is very good (e.g., comparable to performance of the kd tree.)

Pattern Data representation and management in image database systems

AbstractData representation and management in the image database system are discussed. It is pointed out first that both efficiency of representation (amount of data) and the efficiency of processing and retrieval are important in the representation and management of data. From this viewpoint, various kinds of image and graphics representations are discussed, arriving at the viewpoint of “management of M‐dimensional data in N‐dimensional space.” The point data (0 in N data) and the N‐dimensional image data (N in N data) are considered. Various kinds of data structures such as KD tree, KDB tree, 2N partition tree and linear tree, are discussed and analyzed from the viewpoint of the preceding two characteristics. Especially, a tree‐type data structure (called BD tree) is proposed where the N‐dimensional rectangular region is used as the partition key. The structure and the performance are discussed indicating that the performance is better than in other structures. Examples of applications of the proposed data structure in the image database are discussed, as well as the possibilities of future applications.

Multikey retrieval from K-d trees and QUAD-trees

Multikey retrieval from K-d trees and QUAD-trees