Table Of ContentIntelligent Systems Reference Library 162
James F. Peters
Computational
Geometry, Topology
and Physics of
Digital Images with
Applications
Shape Complexes, Optical Vortex Nerves
and Proximities
Intelligent Systems Reference Library
Volume 162
Series Editors
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James F. Peters
Computational Geometry,
Topology and Physics
of Digital Images
with Applications
Shape Complexes, Optical Vortex Nerves
and Proximities
123
James F.Peters
Department ofElectrical andComputer
Engineering, Engineering andInformation
Technology Complex
University of Manitoba
Winnipeg, MB,Canada
ISSN 1868-4394 ISSN 1868-4408 (electronic)
Intelligent Systems Reference Library
ISBN978-3-030-22191-1 ISBN978-3-030-22192-8 (eBook)
https://doi.org/10.1007/978-3-030-22192-8
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This book is dedicated to
Somashekhar (Som) Naimpally, 1931–2014
and Anna Di Concilio, Amma, and sweet P
for offering many glimpses of the proximities
of surface shapes.
Preface
This book introduces the computational geometry, topology and physics of digital
images and video frame sequences. It is the geometry of mesh generation by
Edelsbrunner [1] and thegeometry of polytopes by Ziegler [2] thatprovidea solid
basis for the computational geometry approach to the study of geometric struc-
tures that infuse triangulated visual scenes explored in this monograph. A planar
polytope is a filled polygon defined by the intersection of closed half-planes
covering the interior of the polygon. In addition, an introduction to computational
geometry in this monograph can be found on the geometric foundations of com-
puter vision by Peters [3]. It is the topology of cellular complexes introduced by
Alexandroff[4,5],(beautifullyextendedandelaboratedbyCookeandFinney[6]),
Borsuk [7, 8, 9], a recent formulation of this topology by Edelsbrunner and Harer
[10] and the work on persistence homology by Munch [11] that provide a solid
basis for an introductory study of the computational topology of visual scenes.
This form of topology exploresthe fabric, shapes and structurestypically found in
visual scenes. Coupled with the inherent geometry and topology of visual scenes,
thereisthecomputationalphysicsarisingfromthestructuresandeventsrecordedin
videos and the concomitant sensitivity concerning the fine structure of light to
consider. The fine structure of light and light caustics that we have in mind are
introduced by Nye [12]. A consideration of light caustics brings into play catas-
trophetheoryandtheappearanceoflightcausticfoldsandcusps,whichleadstothe
introductionof optical vortexnerves intriangulateddigital images. In thiscontext,
computational physics is synonymous with the study of the structure of light
choreographedinvideoframes.Thischoreographyofthestructureoflightappears
asasequenceofsnapshotsoflightreflectedandrefractedfromsurfaceshapes that
providesasolidbasisforthestudyofthestructuresandshapesthatappearinvisual
scenes.
Thestudyofthepersistenceofimageobjectshapesinsequencesofvideoframes
as well as sequences of photographs that record surface shape changes in a visual
scene,isimportant.Surfaceshapestendtoappear,undergoachangeinthevarying
ix
x Preface
light and surface conditions, and eventually disappear. The familiar tendency to
lookforunusualappearancesobjects(bothnaturalandartificial)invisualscenesis
atacitrecognitionofcontinuouschangeandthemomentarypersistenceofobserved
componentsinvisualscenes.Inotherwords,itisimportanttotakeintoaccountthe
spacetime character of visual scene shapes. By this, I mean an understanding of
visual scenes includes not only a study of the geometry and topology of visual
scenes but also a consideration of the physics of light, the character and energy
of the photons colliding with curved surfaces in a visual scene.
Physics enters into the picture in cases where we take into account the
descriptionofsurfaceshapesandthelightreflectedfromsurfaceshapesrecordedin
photos and, especially, in video frames. Computer engineering also enters in the
picture here with the study of photonics and reflected light-capturing devices. In
terms of the physics of digital images, the shape-shifting character of energy is
important. For more about this view of energy, see Susskind [13, x7, p. 126].
Computational geometry facilitates the capture of fine-grained structures
embeddedinimageobjectshapes.Andcomputationaltopologyenablesthecapture
andanalysesoftheproximitiesfoundincellularcomplexes(collectionsofvertices,
linesegments,filledtriangles,cycles,vortexes,nerves)embeddedinthegeometryof
triangulated visual scenes (see, e.g., Peters [14, 15]). It is the homology of cell
complexes (an offspring of Alexandroff’s approach to topology [4]) that is an
importantcomponent,here.Homologyisamathematicalframeworkthatfocuseson
howspaceisconnected,utilizingalgebraicstructuressuchasgroupsandmapsthat
relate topologically meaningful subsetsof a space to each other [10, xIV.1, p. 79].
A group G is a nonempty set equipped with a binary operation (cid:2) that is asso-
ciativeandinwhichthereisanidentityelementeandeverymemberainGhasan
inverse b, i.e., a(cid:2)b¼e . A cyclic group H is a group in which every member of
Gcanbewrittenasapositiveintegralpowerofasingleelementcalledagenerator.
AcyclicgroupisAbelian,provideda(cid:2)b¼b(cid:2)a,foreverypairelementsinG.A
freeabeliangroupisanAbeliPangroupwithmultiplegenerators,i.e.,everyelement
ofthegroupcanbewrittenas gaforgeneratorsg inG.Foragoodintroduction
i i
i
tocyclic groups from a homologyperspective, seeGiblin [16,A.1, p.216].
Inpracticalterms,homologyisasourceofinsightsintohowthepiecesofavisual
scenecanbeconnectedtoeachother.Cyclicgroupsareusefulinrepresentingina
concise way how the pieces of a visual scene that are attached to each other and
connected together. Cyclic groups with multiple generators are also a source of an
important feature inearmarkingsurface shapes of interest, namely, Betti numbers
(counts of the number of generators in a free Abelian group). H. Poincaré named
such numbers in honour of Enrico Betti, based on the paper [17]. The focus in
computationalgeometry,topologyandphysicsofdigitalimagesisonfinitespaces.
Betti observes that a finite space has properties independent of the size of its
dimensions,andfromtheshape ofitselements.Thesepropertiesrefertoitonlyto
the way of connection of its parts… [17, x3, p. 143]. The properties of a finite
bounded spatial region tend to be revealed by the path-connected vertices in a cell
complex covering the space. (see, for example, the path-connected vortices
Preface xi
coveringbounded region occupied bythesurfaceshape inFig. 1).Formore about
this, see Tucker and Bailey [18], Salepci and Welshinger [19] and Pranav and
Edelsbrunner and van de Weygaert and Vegter [20].
A thorough study of a computational approach to homology is given by
Kaczynski,MischaikovandMrozek[21].Thefocushereisdiscerningandtracking,
analyzing and representing, and approximating the closeness of shifting surface
shapes. To grapple with continual shape shifting from one visual scene (one video
frame to another one), feature vectors on descriptive proximity spaces, provide us
with a means of representing shape changes that are either close or sometimes far
apart. For more about this, see Di Concilio, Guadagni, Peters and Ramanna [22].
Fig.1 Nesting,
non-overlappingvortexes
coveringashape
In the Euclidean plane, these geometric structures are vertices, line segments,
andfilledtriangles(3-sidedpolytopes).Apolytopeisanintersectionofclosedhalf
planes[2]. Anindividualpolytope isaspatialregionwithafilledinteriorbounded
onallsides.Inatopologicalsetting,thefocusisonthedecompositionofregionsof
visual scenes into very simple polytopes such as filled triangles that are easily
measured and analyzed. The basic ingredients of this topology are simplicial
complexes, shape theory and persistent homology.
The secret underlying this work is the decomposition of closed digital image
regionsintosetsofshapecomplexesthatprovideabasisforshapeanalysis.Ashape
complex is a covering of a shape with a collection of nesting, usually
non-overlapping vortexes (see, for example, Fig. 1). A sample decomposition as a
partialtriangulationofaNapoliflowerdisplay1inFig.2aisshowninFig.2b.The
1ManythankstoArturoTozziforthisphoto.
