Contents
　
　　Abstract ----------------------------------------------------------------- i
　　Acknowledgments ---------------------------------------------------------- v
　　Contents ---------------------------------------------------------------- vi
　　List of Figures ---------------------------------------------------------- x
　　List of Tables --------------------------------------------------------- xii
　　List of Symbols --------------------------------------------------------- xv
　
1　Introduction -------------------------------------------------------------- 1
　　1.1　Purpose of this research -------------------------------------------- 1
　　1.2　Background of this research ----------------------------------------- 3
　　　　1.2.1　History of Hilbert curve -------------------------------------- 3
　　　　1.2.2　Applications of the HH-expression to image processing and 
　　　　　　　 data compression for binary data ------------------------------ 5
　　　　1.2.3　Application to multi-valued images coding using bit-plane 
　　　　　　　 decomposition ------------------------------------------------- 6
　　　　1.2.4　Application to classifying LANDSAT multi-spectral image data -- 8
　　1.3　Contents of the thesis ---------------------------------------------- 9
　
2　Expression of N-dimensional data using a Hilbert scan -------------------- 11
　　2.1　Definition of the HH-expression ------------------------------------ 11
　　2.2　Implementation of a Hilbert address ordering algorithm in R2-------- 17
　　　　2.2.1　Address alignment and a recursive algorithm in R2------------- 17
　　　　2.2.2　Fast algorithm for Hilbert address ordering in R2------------- 20
　　　　2.2.3　Hardware implementation for Hilbert address ordering in R2---- 22
　　2.3　Implementation of a Hilbert address ordering algorithm in R3-------- 25
　　　　2.3.1　Recursive algorithm in R3------------------------------------- 25
　　　　2.3.2　Fast algorithm for Hilbert address ordering in R3------------- 28
　　　　2.3.3　Hardware implementation for Hilbert address ordering in R3---- 31
　　2.4　Implementation of a Hilbert address ordering algorithm in RN-------- 31
　　　　2.4.1　Hilbert address order in RN----------------------------------- 31
　　　　2.4.2　Description of Hilbert address orders in RN------------------- 32
　　　　2.4.3　Method for constructing look-up tables of terminal rules ----- 33
　　　　2.4.4　Look-up tables of induction rules ---------------------------- 37
　　　　2.4.5　Fast algorithm for Hilbert address ordering in RN------------- 44
　　　　2.4.6　Discussions -------------------------------------------------- 47
3　Applications to image processing and data compression for binary images -- 50
　　3.1　HH-expressions in image processing --------------------------------- 50
　　　　3.1.1　DF-expressions ----------------------------------------------- 50
　　　　3.1.2　Difference between the Hilbert scan and the Morton scan on the 
　　　　　　　 quadtree ----------------------------------------------------- 51
　　　　3.1.3　HH-expression for binary images ------------------------------ 51
　　　　3.1.4　Displaying a picture from the HH-expression ------------------ 54
　　　　3.1.5　Traversal algorithm for the HH-expression -------------------- 55
　　　　3.1.6　Previewing images -------------------------------------------- 57
　　　　3.1.7　Circular shifting -------------------------------------------- 58
　　　　3.1.8　Rotation by multiples of 90°and mirror operations ----------- 60
　　　　3.1.9　Magnification and reduction ---------------------------------- 61
　　　　3.1.10 Spectrum of a picture ---------------------------------------- 61
　　　　3.1.11 Logical operations on the HH-expression ---------------------- 61
　　3.2　Compression technique for long term storage ------------------------ 62
　　　　3.2.1　Run-length coding from the HH-expression --------------------- 63
　　　　3.2.2　Experiments for two-dimensional data ------------------------- 64
　　　　3.2.3　Experiments for three-dimensional data ----------------------- 68
　
4　Application to the compression of gray image data ------------------------ 73
　　4.1　Problem of the bit-plane decomposition for data compression -------- 73
　　4.2　Property of bit-plane decomposition -------------------------------- 74
　　　　4.2.1　Property of bit-planes and natural digital images ------------ 74
　　　　4.2.2　Some properties of bit-planes and real digital images -------- 77
　　　　4.2.3　Where and how should we uniformalize? ------------------------ 81
　　4.3　Uniformalization process using the HH-expression on each bit-plane - 82
　　　　4.3.1　Complexity measures on the HH-expression --------------------- 83
　　　　4.3.2　Measure for the uniformalization process using 
　　　　　　　 the HH-expression -------------------------------------------- 83
　　　　4.3.3　Hierarchical uniformalization process ------------------------ 85
　　4.4　Algorithm for the hierarchical uniformalization process ------------ 87
　　　　4.4.1　Variables and a stack operation ------------------------------ 88
　　　　4.4.2　Components of the uniformalization process ------------------- 91
　　4.5　Experimental results ---------------------------------------------- 100
　　　　4.5.1　Simulation of other methods --------------------------------- 101
　　　　4.5.2　Results and discussions ------------------------------------- 102
　
5　Application to LANDSAT image classification ----------------------------- 107
　　5.1　N-dimensional image expressions ----------------------------------- 107
　　5.2　Hierarchical histogram expression using a Hilbert scan ------------ 110
　　　　5.2.1　Hierarchical histogram expression --------------------------- 110
　　　　5.2.2　Problem of the division ------------------------------------- 113
　　5.3　Data structure of the tree structure expression ------------------- 114
　　　　5.3.1　List-like data structure ------------------------------------ 115
　　　　5.3.2　Image processing on this data structure --------------------- 116
　　　　5.3.3　Data compression of this data structure --------------------- 117
　　5.4　Interactive analysis for multi-spectral image classification ------ 117
　　　　5.4.1　LANDSAT-5 TM image data ------------------------------------- 117
　　　　5.4.2　Classification problem for multi-spectral images ------------ 118
　　　　5.4.3　Overview of our method -------------------------------------- 119
　　　　5.4.4　Constructing data tables ------------------------------------ 119
　　　　5.4.5　Interactive system for LANDSAT-5 TM image classification ---- 121
　　5.5　Experiments ------------------------------------------------------- 125
　　　　5.5.1　Experimental system ----------------------------------------- 125
　　　　5.5.2　System evaluations ------------------------------------------ 125
　　　　5.5.3　Discussions ------------------------------------------------- 126
　
6　Conclusions ------------------------------------------------------------- 129
　　6.1　HH-expressions ---------------------------------------------------- 129
　　6.2　Implementation of the Hilbert address ordering algorithm ---------- 129
　　6.3　Application to image processing and data compression 
　　　　 for binary images ------------------------------------------------- 130
　　6.4　Application to data compression of gray images -------------------- 130
　　6.5　Application to LANDSAT image classification ----------------------- 131