简介
Summary:
Publisher Summary 1
An alternative, novel and practical approach to image processing on hexagonal grids - informs about the scope for its potential as a practical alternative to processing images defined on a square grid.
Publisher Summary 2
This book provides an introduction to the processing of hexagonally sampled images, includes a survey of the work done in the field, and presents a novel framework for hexagonal image processing (HIP) based on hierarchical aggregates.The strengths offered by hexagonal lattices over square lattices to define digital images are considerable:higher packing densityuniform connectivity of points (pixels) in the latticebetter angular resolution by virtue of having more nearest neighbourssuperlative representation of curvesThe utility of the HIP framework is shown by implementing several basic image processing techniques (for the spatial and frequency domain) and some applications. Theory and algorithms are covered as well as details such as accommodating hardware that support only images sampled on a square lattice.This fresh approach offers insight and workable know-how to both researchers and postgraduates.
目录
1 Introduction ........................................... 1
1.1 Scope of the book .................................... 2
1.2 Book organisation ....................................... 3
2 Current approaches to vision .............................. 5
2.1 Biological vision ....................................... 5
2.1.1 The human sensor array .......................... 6
2.1.2 Hierarchy of visual processes ........................ 9
2.2 Hexagonal image processing in computer vision.............. 10
2.2.1 Acquisition of hexagonally sampled images ........... 11
2.2.2 Addressing on hexagonal lattices .................... 15
2.2.3 Processing of hexagonally sampled images ............ 18
2.2.4 Visualisation of hexagonally sampled images .......... 21
2.3 Concluding Remarks ................ .................. 24
3 The Proposed HIP Framework .......................... 27
3.1 Sampling as a tiling..................................... 27
3.2 Addressing on hexagonal lattices ........... .............. 35
3.2.1 Hexagonal addressing scheme ............... ... .. 35
3.2.2 Arithmetic ..................................... 43
3.2.3 Closed arithmetic ................................. 49
3.3 Conversion to other coordinate systems .................... 52
3.4 Processing..................................... 54
3.4.1 Boundary and external points ...................... 54
3.4.2 Distance measures. ............ ............... 56
3.4.3 HIP neighbourhood definitions ..................... 59
3.4.4 Convolution ........................ ........... 61
3.4.5 Frequency Domain processing ................ ...... 62
3.5 Concluding remarks .................................... 68
4 Image processing within the HIP framework .............. 71
4.1 Spatial domain processing ............................ .. 71
4.1.1 Edge detection ......................... ... . . . 71
4.1.2 Skeletonisation ............................... . . 79
4.2 Frequency Domain Processing.................... .... ... 82
4.2.1 Fast algorithm for the discrete Fourier transform ..... 83
4.2.2 Linear Filtering ............................... . . 91
4.3 Image pyramids ................... ................ ... 96
4.3.1 Subsampling .. . ........ ................... .. . . 97
4.3.2 Averaging .......................... ....... .. . 98
4.4 Morphological processing ............................ . ... 100
4.5 Concluding remarks ................................. 103
5 Applications of the HIP framework ................... ... .105
5.1 Saccadic search ........................... ........ ... 105
5.1.1 Saccadic exploration ........................... ... 106
5.1.2 Discussion ................ ................ .. . 109
5.2 Shape extraction .................................. ... 111
5.2.1 Shape extraction system ................... ... . . 111
5.2.2 Critical point extraction ................. ... ... . 111
5.2.3 Attention window ............................. . 112
5.2.4 Feature extraction ......................... . .. ... 113
5.2.5 Integration .................... ............ .. 113
5.2.6 Discussion .................. ................ . 115
5.3 Logo shape discrimination ............................ ... 117
5.3.1 Shape extraction system ...................... . ... 117
5.3.2 Image conversion .............................. . . 120
5.3.3 Local energy computation ................... .. ... 120
5.3.4 Feature vectors. ...................... ........... 123
5.3.5 LVQ classifier .............................. ...123
5.3.6 Discussion ............ ............. ...... .... . 124
5.4 Concluding remarks .............. ................ . .. .125
6 Practical aspects of hexagonal image processing .......... 127
6.1 Resampling ....................................... . . 128
6.1.1 True hexagonal lattice ........................ . . . 128
6.1.2 Irregular hexagonal lattice ................... .. ... 136
6.1.3 Hexagonal to square resampling ................. ... 138
6.2 Display of hexagonal images .......................... . . 141
6.2.1 Approximation with rectangular hyperpixels ...... ... 142
6.2.2 Approximation with hexagonal hyperpixels .......... 143
6.2.3 Approximation via polygon generation ........... ... 144
6.2.4 Displaying HIP images ......................... . . 145
6.3 Concluding remarks . ............................. . .. .148
7 Processing images on square and hexagonal grids - a
comparison ................ ........................... 151
7.1' Sampling density ..................................... .151
7.2 Comparison of line and curve representation ............... 155
7.2.1 Algorithmic comparison .......................... 156
7.2.2 Down-sampling comparison ................... .. ..165
7.2.3 Overview of line and curve experiments ............. 167
7.3 General computational requirement analysis .............. .168
7.4 Performance of image processing algorithms .............. .175
7.4.1 Edge detection ................................ . 175
7.4.2 Skeletonisation ................. ............. . . 180
7.4.3 Fast Fourier transform ............... .......... . 182
7.4.4 Linear Filtering ................... . .......... . 186
7.4.5 Image pyramids ................................ .192
7.5 Concluding Remarks ................................ . . 195
8 Conclusion ............ ...................... ........ . 197
A Mathematical derivations ................................ 201
A.1 Cardinality of AA ................ ................... 201
A.2 Locus of aggregate centres ............................ . 202
A.3 Conversion from HIP address to Her's 3-tuple .............. 206
A.4 Properties of the HDFT ......... . . . ......... .. 208
A.4.1 Linearity .................................... 208
A.4.2 Shift/translation .............................. . 209
A.4.3 Convolution theorem ........................... 210
B Derivation of HIP arithmetic tables ...................... 211
B.1 HIP addition..................................... 211
B.2 HIP m ultiplication...................................... 213
C Bresenham algorithms on hexagonal lattices .............. 215
C.1 HIP Bresenham line algorithm ........................... 215
C.2 Hexagonal Bresenham circle algorithm .................... 215
D Source code ........................................... 219
D.1 HIP addressing............ ........................ . 219
D .2 HIP data structure ...................................... 227
D.3 HIP resampling .......................... .............230
D.4 HIP visualisation .......................... ............236
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