Table Of ContentIntelligent Systems Reference Library 63
James F. Peters
Topology of Digital
Images
Visual Pattern Discovery in Proximity Spaces
Intelligent Systems Reference Library
Volume 63
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James F. Peters
Topology of Digital Images
Visual Pattern Discovery in Proximity Spaces
ABC
JamesF.Peters
DepartmentofElectricaland
ComputerEngineering
UniversityofManitoba
Winnipeg
Canada
ISSN1868-4394 ISSN1868-4408 (electronic)
ISBN978-3-642-53844-5 ISBN978-3-642-53845-2 (eBook)
DOI10.1007/978-3-642-53845-2
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013956647
(cid:2)c Springer-VerlagBerlinHeidelberg2014
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This book is dedicated to
Som Naimpally and
Anna, Amma, and Sheela
Preface
This book carries forward recent work on visual patterns and structures in
digital images and introduces a near set-based topology of digital images.
Visualpatternsarisenaturallyindigitalimagesviewedassetsofnon-abstract
points (also called picture points1 or places [140](see, also, [139])) endowed
withsome formofproximity(nearness)relation.Proximityrelationsmakeit
possibletoconstructuniformtopologiesonthesetsofpointsthatconstitutea
digitalimage.Auniformtopologyonadigitalimageisconstructedbyfinding
all sets of points that are near each given set in an image. By stepping back
and taking a look at any given set in an image, one finds that each selected
set of points serves as a visual pattern generator.
Thediscoveryofvisualpatternsindigitalimagesstartsbyviewing
each given set of image points as a pattern generator and then
finding all sets of points that are near the generator.
In keeping with an interest in gaining an understanding of digital images
themselvesasarichsourceofpatterns,thisbookintroducesthebasicsofdig-
ital images from a computer vision perspective. In parallel with a computer
vision perspective on digital images, this book also introduces the basics of
proximity spaces. Not only the traditional view of spatial proximity rela-
tions butalsothe morerecentdescriptiveproximityrelationsareconsidered.
Spatialproximityofsetsofpointsinimagesisimportanttoconsideringain-
ing an understanding of how visual set patterns fit together in a mosaic of
overlapping and adjacent geometric structures.
The spatial proximity approachto pattern recognitionis one of the oldest
and best understood forms of proximity, starting with the work of Poincar´e,
1 Unlike an abstract point in traditional topology, a picture point has both a
location and measurable content. The terms point and picture point are used
interchangeably in this topology of digital images.
VIII
Hadamard, Listing2 and others in the 19th century, leading to the work by
F. Riesz, F. Hausdorff, E. C˘ech, V. Efremovi˘c, J.M. Smirnov, S. Leader and
his student M. Lodato, S.A. Naimpally (and his students), W.J. Thron, H.
Herrlich and others. Descriptive proximity of sets of points in images is one
of the new kids on the block. Sets of image points are descriptively near
each other, whenever one or more of the points in the sets have matching
descriptions. Thanks to the descriptive proximity approach, it is possible to
find visual patterns that reachacross the borders of geometric constructions
suchasconnectedsets,adjacentpoints,paralleledges,andspatiallyseparated
configurations of points with matching descriptions.
The beauty of the descriptive proximity approach is that it is possible to
discover visual set patterns among sets that are non-overlapping and non-
adjacent spatially. By combining the spatial proximity and descriptive prox-
imity approaches, the search for salient visual patterns in digital images is
enriched, deepened and broadened.
Thisstudyleadstothediscoveryofvariousdigitalimagestructuressuchas
neighbourhoods of points, neighbourhoods of sets, boundary points, interior
points,clustersandcoversaswellasmanyotherstructuresthatcanbefound
in an image. A byproduct to the structured set view of digital images is the
discovery of a number of different, useful forms of visual patterns in image
structures. The study of digital image patterns leads to a deeper perception
and understanding of the underlying meanings in images.
A generous provision of Matlab® and Mathematica® scripts are used in
this book to lay bare the fabric and essential features of digital images for
thosewhoareinterestedinfindingvisualpatternsinimages.Thecombination
ofcomputervisiontechniquesandtopologicalmethodsleadtoadeepunder-
standing of images. Some of the shorter Matlab and Mathematica scripts
are included in some of the chapters as illustrations of what can be done.
Otherwise, the remaining Matlab and Mathematica scripts used to produce
the chapter examples can be found in the book Appendix. The Appendix
sections correspond to the book chapters where the scripts are referenced.
This bookmarksthe beginningofastudyofdigitalimagesthatarestruc-
turedsets.Thetopologicalapproachtodigitalimagesextendsrecentworkon
the applications of topology by S.A. Naimpally and J.F. Peters [220], [194],
by S.A. Naimpally, J.F. Peters, M. Wolski [196] and has benefitted from re-
cent work by S.A. Naimpally [171, 25], R.S.D Thomas [283, 286, 285, 284],
U. Grenander[91,92,93, 94],T.Pavlidis[201], M.Kov´ar[139,140],F.W.M.
Stentiford [273, 274, 275], E.H. Kronheimer [142], S. Tiwari [289], M. Khare
and S. Tiwari [133], S. Tiwari and J.F. Peters [290], C.J. Henry [110, 109],
I. Du¨ntsch and D. Vakarelov [65], A. Rosenfeld [249, 251, 252, 250], T.Y.
Kong and A. Rosenfeld [137], R. Klette and A. Rosenfeld [135], A. Skowron,
e.g., [1, 2, 261], M. Wolski [195, 312, 311, 310, 309], M. Wolski [313], S. Pal
andJ.F.Peters[199,§4],H.Fashandi[72, 73],C.J.Henry[109, 110, 243], M.
2 J.B. Listing is thought to be the first to use the term topology (see, e.g., [160].
For more about this, see [135, §1.2.6].
IX
Khare [131, 134, 133], S.Tiwari [289, 288, 290], A.H. Meghdadi [177], S. Ra-
mannaandD.Chitcharoen[243],D.Chitcharoen[229],M.Borkowski[35,34],
L. Wang et al. [300], A. DiConcilio [60, 58, 56, 57](see, also, [52]), E. I˙nan
and M.A. O¨zturk [119], J.F. Peters,A. Skowronand J. Stepaniuk [223], J.F.
Peters and P. Wasilewski [230], Z. Pawlak, e.g., [203, 207, 206, 205, 204], A.
SkowronandZ. Pawlak[210, 209, 208], E. Or(cid:3)lowska,e.g., [64], L. Polkowski,
e.g., [235] and others.
Manythankstothefollowingpersonsfortheircomments,suggestionsand
corrections concerning topics in this book: S.A. Naimpally, S. Ramanna, S.
Tiwari, R.S.D. Thomas, M. Borkowski, C.J. Henry, H. Fashandi, A. Megh-
dadi,C.Uchime,D.Vafabakhsh,R.Hettiarachchi,E.A-IyehandD.Lockery,
anonymousreviewersandmyundergraduateandgraduatestudentswhoused
the course notes for this book.
Contents
Preface .................................................... VII
1 Topology of Digital Images: Basic Ingredients............ 1
1.1 What Is a Digital Image? ................................ 2
1.2 Digital Visual Space..................................... 4
1.3 Creating Your Own Images .............................. 6
1.4 Randomly Generated Images............................. 9
1.5 Ways to Display Images ................................. 10
1.6 Digital Image Formats................................... 11
1.7 Image Data Types....................................... 12
1.8 Colour Images .......................................... 15
1.8.1 Colour Spaces .................................... 15
1.8.2 Colour Channels.................................. 16
1.9 Colour Lookup Table .................................... 18
1.10 Accessing Values of Pixels ............................... 20
1.11 RGB, Greyscale, and Binary (BW) Images ................ 21
1.12 Displaying Colour Channels.............................. 22
1.13 Metric Space............................................ 25
1.14 Neighbourhood of a Point................................ 25
1.15 Set Interior, Set Boundary and Set Complement ........... 27
1.16 Descriptive Neighbourhoods of Picture Points ............. 29
1.16.1 Unbounded Descriptive Neighbourhoods ............ 31
1.16.2 Bounded Descriptive Neighbourhoods .............. 32
1.16.3 Indistinguishable Bounded Descriptive
Neighbourhoods .................................. 33
1.16.4 Descriptive Neighbourhood Patterns................ 33
1.17 Open Sets and Closed Sets............................... 34
1.17.1 Set Interior and Open Neighbourhoods Revisited .... 37
1.18 Spatially Near Sets...................................... 39
1.18.1 Closure of a Set .................................. 42
1.19 Descriptively Near Sets .................................. 43
1.19.1 Descriptively Near, Spatially Far................... 45
1.20 Continuity.............................................. 47
1.20.1 Proximally Continuous ............................ 48
1.20.2 Adherent Points .................................. 48
