Table of contents for Hexagonal image processing : a practical approach / Lee Middleton and Jayanthi Sivaswamy.


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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



Library of Congress subject headings for this publication: Image processing Digital techniques, Hexagons