©2001 CRC Press LLC same procedure recursively to the sublist greater than the median; otherwise apply it to the sublist less than the median (Figure 3.5). Eventually either q will be found — it will be equal to the median of some sublist — or a sublist will turn out to be empty, at which point the procedure terminates and reports that q is not present in the list. The efficiency of this process can be analyzed as follows. At every step, half of the remaining elements in the list are eliminated from consideration. Thus, the total number of comparisons is equal to the number of halvings, which in turn is O (log n ). For example, if n is 1,000,000, then only 20 comparisons are needed to determine if a given number is in the list. Binary search can also be used to find all elements of the list that are within a specified range of values ( min , max ). Specifically, it can be applied to find the position in the list of the largest element less than min and the position of the smallest element greater than max . The elements between these two positions then represent the desired set. Finding the positions associated with min and max requires O (log n ) comparisons. Assuming that some operation will be carried out on each of the m elements of the solution set, the overall computation time for satisfying a range query scales as O (log n + m ). Extending binary search to multiple dimensions yields a kd -tree. 7 This data structure permits the fast retrieval of all 3-D points; for example, in a data set whose x coordinate is in the range ( x min , x max ), whose y coordinate is in the range ( y min , y max ) and whose z coordinate is in the range ( z min , z max ). The kd -tree for k = 3 is constructed as follows: The first step is to list the x coordinates of the points and choose the median value, then partition the volume by drawing a plane perpendicular to the x -axis through this point. The result is to create two subvolumes, one containing all the points whose x coordinates are less than the median and the other containing the points whose x coordinates are greater than the median. The same procedure is then applied recursively to the two subvolumes, except that now the partitioning planes are drawn perpendicular to the y -axis and they pass through points that have median values of the y coordinate. The next round uses the z coordinate, and then the procedure returns cyclically to the x coordinate. The recursion continues until the subvolumes are empty.* FIGURE 3.5 Each node in a binary search tree stores the median value of the elements in its subtree. Searching the tree requires a comparison at each node to determine whether the left or right subtree should be searched. * An alternative generalization of binary search to multiple dimensions is to partition the dataset at each stage according to its distance from a selected set of points; 8-14 those that are less than the median distance comprise one branch of the tree, and those that are greater comprise the other. These data structures are very flexible because they offer the freedom to use an appropriate application-specific metric to partition the dataset; however, they are also much more computationally intensive because of the number of distance calculations that must be performed. q < Median  Median < q _ ©2001 CRC Press LLC 4 The Principles and Practice of Image and Spatial Data Fusion* 4.1 Introduction 4.2 Motivations for Combining Image and Spatial Data 4.3 Defining Image and Spatial Data Fusion 4.4 Three Classic Levels of Combination for Multisensor Automatic Target Recognition Data Fusion Pixel-Level Fusion • Feature-Level Fusion • Decision-Level Fusion • Multiple-Level Fusion 4.5 Image Data Fusion for Enhancement of Imagery Data Multiresolution Imagery • Dynamic Imagery • Three- Dimensional Imagery 4.6 Spatial Data Fusion Applications Spatial Data Fusion: Combining Image and Non-Image Data to Create Spatial Information Systems • Mapping, Charting and Geodesy (MC&G) Applications 4.7 Summary References 4.1 Introduction The joint use of imagery and spatial data from different imaging, mapping, or other spatial sensors has the potential to provide significant performance improvements over single sensor detection, classification, and situation assessment functions. The terms imagery fusion and spatial data fusion have been applied to describe a variety of combining operations for a wide range of image enhancement and understanding applications. Surveillance, robotic machine vision, and automatic target cueing are among the application areas that have explored the potential benefits of multiple sensor imagery. This chapter provides a framework for defining and describing the functions of image data fusion in the context of the Joint Directors of Laboratories (JDL) data fusion model. The chapter also describes representative methods and applications. Sensor fusion and data fusion have become the de facto terms to describe the general abductive or deductive combination processes by which diverse sets of related data are joined or merged to produce *Adapted from the principles and practice of image and spatial data fusion, in Proceedings of the 8th National Data Fusion Conference, Dallas, Texas, March 15–17, 1995, pp. 257–278. Ed Waltz Veridian Systems ©2001 CRC Press LLC 5 Data Registration 5.1 Introduction 5.2 Registration Problem 5.3 Review of Existing Research 5.4 Registration Using Meta-Heuristics 5.5 Wavelet-Based Registration of Range Images 5.6 Registration Assistance/Preprocessing 5.7 Conclusion Acknowledgments References 5.1 Introduction Sensor fusion refers to the use of multiple sensor readings to infer a single piece of information. Inputs may be received from a single sensor over a period of time. They may be received from multiple sensors of the same or different types. Inputs may be raw data, extracted features, or higher-level decisions. This process provides increased robustness and accuracy in machine perception. This is conceptually similar to the use of repeated experiments to establish parameter values using statistics. 1 Several reference books have been published on sensor fusion. 2-4 One decomposition of the sensor fusion process is shown in Figure 5.1. Sensor readings are gathered, preprocessed, compared, and combined, and a final result is derived. An essential preprocessing step for comparing readings from independent physical sensors is transforming all input data into a common coordinate system. This is referred to as data registration. In this chapter, we describe data registration, provide a review of existing methods, and discuss some recent results. Data registration transformation is often assumed to be known a priori , partially because the problem is not trivial. Traditional methods are based on methods developed by cartographers. These methods have a number of drawbacks and often make invalid assumptions concerning the input data. Although data input includes raw sensor readings, features extracted from sensor data, and higher- level information, registration is a preprocessing stage and, therefore, is usually applied only to either raw data or extracted features. Sensor readings can have one to n dimensions. The number of dimensions will not necessarily be an integer. Most techniques deal with data of two or three dimensions; however, same approaches can be trivially applied to one-dimensional readings. Depending on the sensing modal- ities used, occlusion may be a problem with data in more than two dimensions, causing data in the environment to be obscured by the relative position of objects in the environment. The specific case studies presented in this chapter use image data in two dimensions and range data in 2 1 / 2 dimensions. This chapter is organized as follows. Section 5.2 gives a formal definition of image registration. Section 5.3 provides a brief survey of existing methods. Section 5.4 discusses meta-heuristic techniques that have been used for image registration. This includes objective functions for sensor readings with various types Richard R. Brooks The Pennsylvania State University Lynne Grewe California State University ©2001 CRC Press LLC 6 Data Fusion Automation: A Top-Down Perspective 6.1 Introduction Biological Fusion Metaphor • Command and Control Metaphor • Puzzle-Solving Metaphor • Evidence Combination • Information Requirements • Problem Dimensionality • Commensurate and Noncommensurate Data 6.2 Biologically Motivated Fusion Process Model 6.3 Fusion Process Model Extensions Short-, Medium-, and Long-Term Knowledge • Fusion Classes • Fusion Classes and Canonical Problem-Solving Forms 6.4 Observations Observation 1 • Observation 2 • Observation 3 • Observation 4 • Observation 5 Acknowledgment References 6.1 Introduction This chapter offers a conceptual-level view of the data fusion process and discusses key principles associated with both data analysis and information combination. The discussion begins with a high-level view of data fusion requirements and analysis options. Although the discussion focuses on tactical situation awareness development, a much wider range of applications exists for this technology. After motivating the concepts behind effective information combination and decision making through a series of easily understood metaphors, the chapter •Presents a top-down view of the data fusion process, •Discusses the inherent complexities of combining uncertain, erroneous, and fragmentary information, • Offers a taxonomic approach for distinguishing classes of fusion algorithms, and •Identifies key algorithm requirements for practical and effective machine-based reasoning. 6.1.1 Biological Fusion Metaphor Multiple sensory fusion in biological systems provides a natural metaphor for studying artificial data fusion systems. As with any good metaphor, consideration of a simpler or more familiar phenomenon can provide valuable insight into the study of a more complex or less familiar process. Richard Antony VGS Inc. . practice of image and spatial data fusion, in Proceedings of the 8th National Data Fusion Conference, Dallas, Texas, March 15–17, 1995, pp. 25 7 27 8. Ed Waltz Veridian Systems 20 01 CRC. Recognition Data Fusion Pixel-Level Fusion • Feature-Level Fusion • Decision-Level Fusion • Multiple-Level Fusion 4.5 Image Data Fusion for Enhancement of Imagery Data Multiresolution Imagery. benefits of multiple sensor imagery. This chapter provides a framework for defining and describing the functions of image data fusion in the context of the Joint Directors of Laboratories (JDL) data fusion