Hexagonal Structure for Intelligent VisionXiangjian He and Wenjing JiaComputer Vision Research GroupFaculty of Information TechnologyUniversity of Technology, Sydney

Australia

Abstract-Using hexagonal grids to represent digitalimages have been studied for more than 40 years.Increased processing capabilities of graphic devicesand recent improvements in CCD technology havemade hexagonal sampling attractive for practicalapplications and brought new interests on thistopic. The hexagonal structure is considered to bepreferable to the rectangular structure due to itshigher sampling efficiency, consistent connectivityand higher angular resolution and is even provedto be superior to square structure in manyapplications. Since there is no mature hardware forhexagonal-based image capture and display,square to hexagonal image conversion has to bedone before hexagonal-based image processing.Although hexagonal image representation andstorage has not yet come to a standard,experiments based on existing hexagonalcoordinate systems have never ceased. In thispaper, we firstly introduced general reasons thathexagonally sampled images are chosen forresearch. Then, typical hexagonal coordinates andaddressing schemes, as well as hexagonal basedimage processing and applications, are fullyreviewed.

I. INTRODUCTION

Since Golay [1], the possibility of using ahexagonal structure to represent digital imagesand graphics has been studied by manyresearchers. Hexagonal grid is an alternativepixel tessellation scheme besides theconventional square grid for sampling andrepresenting discretized images. Sampling on ahexagonal lattice is a promising solution whichhas been proved to have better efficiency andless aliasing [2]. The importance of thehexagonal representation is that it possessesspecial computational features that are pertinentto the vision process. Its computational powerfor intelligent vision pushes forward the imageprocessing field. Dozens of reports describingthe advantages of using such a grid type arefound in the literature. Among these advantagesare higher degree of circular symmetry, uniformconnectivity, greater angular resolution, and a

reduced need of storage and computation inimage processing operations.

In spite of its numerous advantages, hexagonalgrid has so far not yet been widely used incomputer vision and graphics field. The mainproblem that limits the use of hexagonal imagestructure is believed due to lack of hardware forcapturing and displaying hexagonal-basedimages. In the past years, there have beenvarious attempts to simulate a hexagonal gridon a regular rectangular grid device. Thesimulation schemes include those usingrectangular pixels, pseudohexagonal pixels,mimic hexagonal pixels and virtual hexagonalpixels. Although none of these simulationschemes can represent the hexagonal structurewithout depressing the advantages that a realhexagonal structure possesses, the use of thesetechniques provides us the practical tools forimage processing on hexagonal grids and makesit possible to carry on theoretical study of usinghexagonal structure in existing computer visionand graphics systems.

The use of hexagonal grid is also fettered by itspixel arrangement. In the hexagonal structure,the pixels are no longer arranged in rows andcolumns. In order to take the advantages of thespecial structure of hexagonal grid, severaladdressing schemes and coordinate systems havebeen proposed. There exist a 2-axis obliquecoordinate system, a 3-axis oblique coordinatesystem, and a single dimensional addressingscheme.

This paper is organized as follows. In Section II,we list the major reasons to be based onhexagonal structure for intelligent vision system.In Section III, we introduce several typicalhexagonal simulation schemes. In Section IV,three addressing schemes on hexagonal structureare demonstrated. Image processing algorithmsusing hexagonal grid are discussed in Section V.

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II. WHY HEXAGONAL?

Since the introduction of computer graphics,one of the biggest problems that scientists haveto face is the fact that the physical screen is adiscrete set of points, i.e., a countable set ofisolated points, and the real world is in acontinuous environment. Moreover, in order tostore, process, display and transfer images bydigital devices, the image plane must bequantized into spatial elements of finitedimension, generally referred to as pixels.

Digitization, which is to convert real images intodiscrete sets of points, has been therefore one ofthe earliest subjects of study for computerscientists involved in vision and graphicsresearch. Each point which forms an image onthe screen must be properly addressed in order tobe indexed. The disposition of the points on theplane, called digitization scheme, however, cantake different choices. Considering technicalimplementation, these points must be placed asregularly as possible on the plane and they mustbe disposed so that the coverage of the plane isas efficient as possible.

A. Three Possible Regular Tessellation Schemes

There exist only three possible regulartessellation schemes to tile a plane withoutoverlapping among the samples and gapsbetween them, namely the tessellation withhexagons, with squares, and with regulartriangles [3, pages 61-64] (see Fig. 1). Any othertypes of spatial tessellation will result in eitherunequal distance between neighboring pixels, orintroduce gaps or overlaps among samples. Asimple explanation is given below. For moredetailed proof, please refer to [3, page 61-64].

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We use the symbol {p, q} to denote thetessellation of regular p-gons, of which each hasq pixels (p-gond) surrounding each vertex. It iseasy to see that the {p,q} pair s are {4,4} , {6,3}

and {3,6} for the three tessellation schemes asillustrated in Fig. 1, where in each case thepolygon drawn in bold lines is the vertex figure,i.e., the q -gon whose vertices are the midpointsof the q edges connected to a vertex. Atessellation is said to be regular if it has regularfaces and a regular vertex figure at each vertex.

On the left is the square case {4,4}, which isfamiliar and usual because it is aligned with thestandard Cartesian axes, which helps to makeoperations simple and intuitive. The far rightillustrates the triangular case {3,6} , which yieldsa denser packing than the square case. Thismeans that more information is contained in thesame area of the image. The tessellation in themiddle figure, hexagonal case {6,3} , is oftenused for tiled floors and it can be seen in anybeehive. It is believed to be the most efficienttessellation scheme among them.

B. More Efficient Sampling Schemes

No matter which sampling scheme is chosen, aninsufficient sampling rate can always introduceunwanted effects in the reconstructed signal,referred as aliasing. Peterson and Middleton [5]investigated sampling and reconstructing wave-number-limited multi-dimensional functions andconcluded that the most efficient samplinglattice, i.e., which uses a minimum number ofsampling points to achieve exact reproduction ofa wave-number-limited function', is not ingeneral rectangular. Specially, when a two-dimensional isotropic function2 is considered, theoptimum sampling lattice is the 120q rhombic(hexagonal) with spacing of sample points equal

to t481 if the spectrum of a function is boundedby a circle of radius 2itB in the wave-numberplane (see Fig. 2). The sampling efficiency is90.8%, compared with 78.5% for the largestpossible square lattice.

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IWe call a function wave-number-limited if the Fouriertransform ofthe function lie within a bounded region ofspectrum space of the function [5].

An isotropic function is defined in the broad sense asdescribing a spectrum which cuts off at the same wave-number magnitude in all directions [5].

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A similar conclusion was obtained by Mersereau[2], who developed a hexagonal discrete Fouriertransform and hexagonal finite impulse responsefilters. Mersereau showed that for signals whichare band-limited over a circular region in Fourierspace, 13.4% fewer sampling points are requiredwith the hexagonal grid to maintain equal highfrequency image information with therectangular grid, thus less storage and lesscomputation time are required. An example isthat in image coding application, one may expectthat the coding efficiency can be increased byusing the hexagonal sampling scheme.

Recently, Vitulli, Armbruster and Santurri [6],after investigating the sampling efficiency ofhexagonal sampling, also concluded that usinghexagonal sampling, about 13% less number ofpixels are needed to obtain the same performanceas obtained using square sampling whensampling the same signal.

These conclusions are briefly illustrated below.Fig. 3((a) is a generic hexagonal sampling lattice.Goodman [6,7] showed that the Fouriertransform (FT) of a hexagonal lattice is still ahexagonal lattice. In [7, page 12], however, it issaid that the Fourier transform of a circularlysymmetric function is itself circularly symmetric,where the functionf can be said to be circularlysymmetric if it can be written as a function of ralone, that is, g(r, 0) g (r) R T [7, page 11]. Inhexagonal lattices, the inverse of the samplingsteps that corresponds with the distancesbetween two aligned rows and columns in FTdomain are twice the corresponding steps inspatial domain, as shown in Fig. 3(b). Similar tosquare case, the FT of the hexagonally sampledimage is also composed of infinite replicas of thespectrum G (,), the FT of the image g(x, y).These replicas are centered in the points of thehexagonal lattice, which is the FT of thehexagonal sampling lattice.

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Vitulli etc. compared the Nyquist constraints,i.e., the minimum sampling densities withoutaliasing, between rectangular and hexagonal

cases. The bigger the minimum sampling densityrequired is, -the better the sampling performancewill be. The maximum densities to tile spectra inspectrum space are illustrated in Fig. 4, forrectangular grid (left) and for hexagonal grid(right). The pixel density with hexagonalsampling that avoids aliasing is 3 2 and thus islower than rectangular one.

As a result, using hexagonal grid, wider spectracan be sampled without aliasing with the samenumber of pixels, or less pixel than using squaregrid.

B. Smaller Quantization Error

As mentioned earlier, in order to process animage by a digital computer, the continuousimage in real world must be quantized intospatial elements of finite dimensions, generallyreferred as pixels. Due to the limited resolutioncapabilities of image sensors, this array isusually too small to adequately represent thescene in real world.

Quantization error, thus, is inevitable. Incomputer vision, quantization error is a veryimportant measurement to investigate the meritsof different types of sensory configurations inorder to find which spatial sampling wouldintroduce less quantization error intocomputations. Kamgar-Parsi [8-11] developedformal expressions for estimating quantizationerror in hexagonal spatial sampling and foundthat, for a given resolution capability of thesensor, hexagonal spatial sampling yields smallerquantization errors than square sampling. Fig 4.Spectral packaging for best rectangular andhexagonal sampling [6]

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C. Consistent Connectivity Definition

Connectivity between pixels is a fundamentalconcept that simplifies the definition ofnumerous digital image concepts, such asregions and boundaries. To decide if two pixelsare connected, it must be determined if they are

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neighbors and if they satisfy a specified criterionof similarity [12, pages 66-67].

On a square grid, there are two possible ways todefine neighbors of a pixel. We can either regardpixels as neighbors when they have a commonedge or when they have at least one commoncomer, so that four and eight neighbors exist(referred as a 4-neighborhood and an 8-neighborhood). Accordingly, on a square grid,object connectivity can be defined as 4-way toany of the four nearest neighbours, or 8-way ifconnectivity to diagonal neighbours is permitted.This is illustrated in Fig. 5.

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Correspondingly, background connectivity mustbe 8-way if object connectivity is four-way or 4-way if object connectivity is 8-way [13].Verification ofthis statement is presented below.

Consider the pattem shown in Fig. 6(a).Assuming 4-way connectivity for both the objectand the background, the number of vertices V inthe pattem is 16, the number of edges E is 16,and the number of faces F is 4.

Application of the Euler formula V= E+ F to thepattem should give its genus. Thus, by the aboveformula the genus isl6-16+4 = 4. However, thepattem has four objects and the background hastwo, so that the genus (the number of objectcomponents minus the number of backgroundcomponents +1) is 4 - 2 +1= 3. A similardisagreement in the value of the genus ariseswhen 8-way connectivity is assumed. For thenthe number of vertices V in Fig. 6(a) is 12, thenumber of edges E is 16, and the number of facesF is 4. Thus, by Euler's formula, the genus is 0,whereas in fact it should be 1.

(a) (b)FS. 6. Squme gnd and hexagonal rid

However, if 4-way connectivity is assumed forthe object and 8-way connectivity for thebackground then, according to Euler's formula,the genus is 4. Since the number of backgroundcomponents is now I (not 2), the value of thegenus obtained by counting the number ofcomponents is also 4. Similarly, when weassume the pattern to be 8-neighbor connectedand the background to be 4-neighbor connected,both methods of calculating the value of thegenus, which is 0, agree.

The hexagonal grid, however, offers noconnectivity choice. We can only define a 6-neighborhood. Neighboring pixels have alwaysone common edge and two common comers (seeFig. 6(b)). The absence of such choice inhexagonal grid results in easier and moreefficient algorithms, such as thinning algorithm[13][14][15], since fewer connectivity situationshave to be accounted for. Accordingly,connectivity in hexagonal objects is consistent asit is six-way to either of the nearest neighboursfor both the object and the background imagecomponents [16, 17].

Assuming 6-neighbor connectivity, the numberof vertices V in Fig. 6(b) is 24, the number ofedges E is 30 and the number of faces F is 6.Using the Euler formula, the genus is equal to 0.Since the number of components of the pattem isI and the number of components of thebackground is 2, the genus has the valuel-2+1=0. Thus, both values ofthe genus agree.

D. Equidistance

With the introduction of neighbourhood relation,distance function can be easily measured. Insquare grid we have two types of distances,where the distance between adjacent pixels in thediagonal direction is 42 times of that in thehorizontal (or vertical) direction (see Fig. 7(a)).

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While in hexagonal case, each hexagonal pixelhas and only has six neighboring pixels and eachpixel is equidistantly adjacent to their sixneighbors along the six sides of the pixels. Thecentroid of the middle pixel is at the samedistance from the centroids of the six adjacentpixels (see Fig. 7(b)).

E. Greater Angular Resolution

Image processing on a hexagonal lattice isadvantageous is also believed due to its greaterangular resolution to represent curved objects. Ithas been noted that hexagons offer greaterangular resolution as the nearest neighbors of thesame type are separated by 60q instead of 90q[1]. An example showing a familiar curvedfigure and a representation on square andhexagonal lattices is shown in Fig. 8.

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Notice that the hexagonal case, on the left of Fig.8, appears to have smoother curves than thesquare case. There are several reasons for this.The first is due to the consistent connectivity inthe hexagonal lattice. This means that allneighbours are uniform distances away fromeach other and leads to the smoother curvature.Another reason is what is known as the obliqueeffect in human vision (see web linkhttp://www.ecs.soton.ac.uk/-ljm/hip.php). Thismeans that we have a visual preference for linesat oblique angles. This also helps to make thehexagonal curves look smoother.

As a matter of fact, the theory developed and thesimulation done on a physical screen in [4]showed that hexagonal grids represent a

reasonable alternative to conventional squaregrid display techniques not only for circledrawing, which was somehow predictable, butalso for straight lines.

On the hexagonal grid, digitizations display abetter connectivity and are perceived as beingapproximated by small polylines, whereas on thesquare grid, digitizations are still perceived asbeing approximated by pixels. Such a perceptionof single pixels disturbs the impression ofcontinuity of the discretized line. This is due tothe fact that in the square grid neighbors of apixel are not placed all at the same distance.Moreover, two diagonal neighbors in the squaregrid have only one point in common, whereastwo horizontal or vertical neighbors of the squaregrid, and all the neighbors of a pixel in thehexagonal grid, have one segment in commonwith their neighbor. This fact produces thicknessvariations in square digitizations, leading togreater edge busyness and to a thinner averagewidth in a line digitization.

F. Higher Symmetry

Serra [19] has developed many of morphologicaloperators that were currently used for imageprocessing. He prefers the hexagonal grid to therectangular because ofthe connectivity definitionand the higher symmetry, which lead to simplerprocessing algorithms. It can be seen in Fig. 6that the cluster of hexagonal pixels possesses thesame symmetry about the three different linesconnecting pairs of two pixels and the centralpixel. This symmetry degree is one higher thanthat of square grid. This symmetric featuremakes image processing more accurate. Forexample, when an image on a hexagonal grid isrotated, more image information will be retainedcompared to the same rotation is performed onsquare grid.

G. Other Features ofHexagonal Grid

The research done in the biological domain ofanimal vision clearly demonstrates that in animalvision systems the arrangement of rods andcones in the fovea more nearly approximate ahexagonal tessellation than a rectangular one.Specifically the research by Hubel [20] showsthat the fovea can nearly be described by aregular hexagonal tessellation. Anothercompelling reason to investigate othertessellations of the plane is the well knownparadox concerning the definition of the nearest

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neighbor network such that edges are continuousand that the inside of an object not be connectedto the outside of the same object. [21]

III. HEXAGONAL IMAGEREPRESENTATION

In spite of the many advantages ofhexagonal structure, the hexagonal based imageprocessing has not been used widely inintelligent vision area. The main reason is thatcurrently there is no hexagonal-based deviceavailable to capture and display digital images onhexagonal grids. So how to simulate hexagonallysampled images on common square displayequipments has once become a serious problemthat affects the advanced research on hexagonalarchitecture in the field of computer vision andgraphics.

Fortunately, there have been several ways tosimulate a hexagonal grid on a regularrectangular grid. We list three most commonsimulations as follows. The use of thesetechniques allows us to take the advantages ofhexagonal grids for computer vision andcomputer graphics.

A. Mimic Hexagonal Pixels Using Square Pixels

Horn [22] has described how a practicalhexagonal data may be captured by delayingsampling by half a pixel width on altemate TVscan lines in horizontal direction (see Fig. 9). Inhis scheme, the pixel shape is square. In otherwords, the sampling intervals in horizontal andvertical directions are identical. This schemesimplifies the hardware design by settingidentical sampling intervals in both horizontaldirection and vertical direction. However, theequidistance property of hexagonal pixels is notpreserved. A shown in Fig.9, if we denote thedistance between any two neighbors inhorizontal and vertical direction as I unit, thedistance between any two neighboring pixels indiagonal direction will be 43/2.

Fi 9. U ing hf-pixtl shifd esquat pzxelsepit hexagonalstuc

Later on, Staunton [23] described a hexagonaldata structure with a rectangular shape, wherethe sic neighboring pixels of a pixel all lie on acircle with the centre of the circle being at thesampling point of the central pixel, as illustratedin Fig. 10. The major advantages with thisstructure are that, all sampling points areequidistant from their nearest neighbors, theangle subtended by two nearest neighboringpoints is 60', and Ithe horizontal samplingdistance is 2/43. The pixel size is one by 2/43and thus for systems employing an equal numberof pixels horizontally and vertically, the imageaspect ratio would be 2/43 :1

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B. Pseudo Hexagonal Pixel

Wuthrich [4] proposed a pseudo hexagonal pel(see Fig. l 1) in order to evaluate the visual effectof hexagonal pixel and square pixel. Acomparative simulation of two screens based onthe 'quare and the hexagonal lattices has beenmade. A hexagonal pixel, called a hyperpel, issimulated using a set of many square pels andthe simulated square grid had to be adapted inorder to make its density comparable with thehexagonal grid. This results in a great loss in thescreen resolution and to an inexact simulation ofthe square grid, reducing it to a rectangular grid.In order to approximate the square and the

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hexagonal grids, two ideal lattices has been

selected, that is A6=A((N5:/2,142 o1i)) andA-- AtI2,P2,0100) as an approximation ofA4. A has been chosen such that the points ofthe two lattices have the same density, i.e., thesame number of points per unit of surface. Asthere is no way to display exactly the point

(03/2,V2)on the square grid, in the practicalsimulator the lattices hat have been actuallydrawn on the simulator are thusA\4 A((7/8,O.l0J))andAX = A(t7/8,1/2).(O1respectively for the square and the hexagonallattices. The resulting hyperpels, which areillustrated in Fig. 11 were displayed at aresolution of60 u 60 pels.

This idea has been adopted by Yabushita in [18]who designed a similar pseudo hexagonal picture(hexelement), which is also composed of smallsquare pixels and which aspect ratio is 12:14.

hexagonal pixel consists of four traditionalsquare pixels and its grey level value is theaverage of the involved four pixels (see Fig. 12).This mimic scheme preserves the importantproperty of hexagonal architecture that eachpixel has exactly six surrounding neighbours.However, because the grey-level value of themimic hexagonal pixel is taken from the averageof the four corresponding square pixels, thismimic scheme introduces loss of resolution. Inaddition, we know that according to hexagonalstructure theory the distance between each of thesix surrounding pixels and the central pixel is thesame. However, this property is lost in the mimicSpiral Architecture.

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Simulated hyper pixel Middleton andSivaswamy [24, 25] proposed a framework forpractical hexagonal-image processing, where aprocess known as image re-sampling isemployed to generate a hexagonally sampledimage from normal square image.

D. Virtual Hexagonal Structure

Later, Wu [27] constructed a virtual hexagonalstructure which is an important milestone for thetheoretical research and the practical applicationexploration of this architecture. Using virtualSpiral Architecture, images on rectangularstructure (or called square grid as indicated inFig. 13) can be smoothly converted to SpiralArchitecture. Such virtual Spiral Architectureonly exists during the procedure of imageprocessing. It builds up a virtual hexagonal gridsystem on memory space on computer. Then,processing algorithms can be implemented onsuch virtual spiral space. Finally, resulted datacan be mapped back to rectangular architecturefor display (see Fig. 13). Unlike the previouslyproposed mimicking methods, this mimickingoperation nearly does not introduce distortion orreduce image resolution, which is the mostremarkable advantage over other mimickingmethods, while keeping the isotropic property ofthe hexagonal architecture.

C. Mimic Hexagonal Structure

He [26] proposed a mimic hexagonal structure,called mimic Spiral Architecture, where one

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IV. HEXAGONAL STRUCTUREADDRESSING

Obviously, no matter which kind ofsimulating scheme is chosen, there exists anotherbig problem that the hexagonal pixels cannot belabeled in normal column-row order as inrectangular grid. In order to properly address andstore hexagonal images data, different coordinatesystems have been proposed. In this section,typical coordinate systems are reviewed.

A. 2-Axes Oblique Coordinate AddressingScheme

Using two oblique axes (see Fig. 14) to addresshexagonal structure is firstly suggested byLuczak and Rosenfeld [28], also referred asskewed coordinate system in [4], and h2 systemin [29], where two basis vectors are notorthogonal. With such an oblique coordinatesystem, each hexagonal pixel can be addressedby an ordered pair of unit vectors, u and v, asillustrated in Fig. 14, which indicate a horizontaldeflection and an upright deflection respectively.The system has been shown to have thefollowing properties:1. Complete: Be sufficient to represent any pointin a 2-dimensional space;2. Unique U: Any ordered pair corresponds toexactly one point;3. Convertible t: It can be easily converted to andfrom Cartesian coordinate; and4. Efficient: It is a convenient and efficientrepresentation.

In [30,311, Her developed a symmetricalhexagonal coordinateframe, denoted as *R3, forhexagonal grid, which uses three coordinates x,y, z, instead oftwo, to represent each pixel on thegrid plane, as shown in Fig. 15. The threecoordinates at any pixel has a relationship amongthem:

x +y + z =0.

Here the distance between two neighboring gridpoints is defined as one unit.

FR. 15. Synmmi hexagonl hRm *R3

The major advantage of this coordinate system isthat there is a one-to-one mapping between *R3and the 3-dimensional Cartesian frame R3, asillustrated in Fig. 16, where, x, y and z are thethree orthogonal axes of R3. Due to this reason,many geometrical properties of R3 can be readilytransferable to *RJ. Moreover, since the x and ycoordinates of a point of this symmetricalhexagonal coordinate frame*R3 are actually thetwo coordinates used in the oblique coordinateframe (see Fig. 14), theories and equationspreviously developed for the oblique coordihateframe can directly be used in*R3. Moreover, in[32], the use of this symmetrical hexagonalcoordinate frame is demonstrated to derivevarious affme transformations.

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Due to the physical relationships between thesymmetrical hexagonal coordinate frame and the3-dimensional Cartesian frame R3, geometrictransformations on the hexagonal grid areconveniently simplified and the beautifulsymmetry property of the hexagonal grid issuccessfully preserved.

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FIg. 16. Relation between ftmns *R3 ana R3This three-axis coordinate system is used in [33]for mathematically handling the hexagonalstructure, for example, numerically calculatingthe distance of two objects. This three-axiscoordinate system reflects the geometricalsymmetry of the grid.

C. Single Indexing System

Sheridan [34] proposed a one-dimensionaladdressing system, as well as two operationsbased on this addressing system, for hexagonalstructure. This system is called SpiralArchitecture (see Fig. 17). Spiral Architecture(SA) is inspired from anatomical considerationofthe primate's vision system.

Fig, 17. Spiti akhessn

Sheridan [34] presented a one dimensionalindexing scheme, called Spiral Addressing, toaddress each hexagon on the image. This addressgrows from the centre of image in powers ofseven along a spiral like curve. This addressingscheme combined with two later proposed

mathematic operations, spiral addition and spiralmultiplication is the basic of Spiral Architecture[26,34]. The spiral addition and spiralmultiplication correspond to image translationand image rotation respectively.

Middleton and Sivaswamy [24,25] also proposeda similar single-index system for pixeladdressing by modifying the GeneralizedBalanced Ternary system, as shown in Fig. 18.

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Neighbourhood operations are often used inimage processing. Finding the neighbour in ahexagonal image makes use of the spiral additionoperation, of which details can be found in [34].In a seven-pixel cluster, the neighbourhoodrelation can be determined by spiral addition asfollows.

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Let the spiral address of the central pixel, asshown in Fig. 19(a), be denoted by a,. Then thespiral address of its neighbour pixel can bedescribed by spiral addition denoted by +, with acertain number of displacements, as shown inFig. 19 (a). An example is given in Fig. 19(b).

For the whole image, following the spiralrotation direction, as shown in Fig. 20, one canfind out the spiral address of any hexagonal pixelwith centre on a certain hexagonal pixel whosespiral address is known.

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The Spiral Architecture has some distinguishingfeatures compared to the square imageprocessing. First, the one dimensional addressingscheme leads to an efficient storage and theplacement of the origin at the centre of the imagesimplifies geometric transformations of a givenimage. Finally, the hexagonally sampled imageallows non-traditional neighbourhoods withconsistent boundary connectivity, which is usefulfor many computer vision applications.

V. HEXAGONAL IMAGE PROCESSING

Although hexagonal image representationand storage has never yet come to any standard,theoretical studies on hexagonal imageprocessing have never ceased.

A. Hexagonal Image Transformation

For the purpose of efficient and fast processingand analysing, digital images that are originallydefined in spatial domain usually need to betransformed into another domain with certaintransformation and take use of some uniquecharacters of the transformed domain to processthe transformed image in the domain. Imagetransformation is the basis of many imageprocessing and analysing techniques. Aftertransformation, processing in spatial domain canbe converted into the corresponding processingin transformed domain, which has manyadvantages. Among them, the most importantones are the computation will be greatly reducedand various image filtering techniques can beapplied for image processing. For example,convolution operation becomes morecomputationally efficient when computed infrequency domain. Further examples includetransform coding for image compression purposeand filter design.

Among various image transformations, one ofthe most widely used is Fourier transformation,which transforms images from spatial domain tospectrum domain. Standard fast Fouriertransform (FFT) algorithms, however, are notapplicable to non-rectangularly sampled data.

Mersereau [35] developed a two-dimensionalfast Fourier transform (2-D FFT) for use withhexagonally sampled data. Nel [21] followed hisderivation and corrected a number of algebraicerrors in his derivation and derived the 2-DWalsh transformation. They were all derived innon-orthogonal axes. Later, Ehrhardt [36]claimed that Mersereauts 2-D FFT algorithmwould require an additional interpolation step,which might introduce artifacts. He presented aseparable fast discrete Fourier transformalgorithm where the data space is sampled withhexagonal grids and transform space is sampledwith rectangular grids. In [37], Middletonderived a Fast Fourier Transform (FFT) for thehexagonal lattice based upon the Cooley-Tukeyapproach [38] and the radix-7 decimation inspace algorithm.

In [39], a hexagonal discrete cosine transformwhich can be used in the applications of imagecoding is described and showed that theproposed HDCT is more efficient in energycompaction than the HDFT.

B. Edge Detection on Hexagonal Structure

When a scene is observed by a human, thehuman visual system first segments the scene.Edge detection is an important approach forimage segmentation in computer vision systems.This approach measures the rate of change anddecides the existence of an edge at each point.The basic assumption used in most edgedetection algorithms is that the edges arecharacterised by large (step) changes in intensity(or color in color images case). Hence, at thelocation of an edge, the first derivative of theintensity function should be a maximum or thesecond derivative should have a zero-crossing.

Middleton [24] investigated the performance ofusing a hexagonally sampled structure forimplementing classical edge detectors, includingPrewitt, Laplacian of Gaussian (LoG) and theCanny edge detector. Images that contain curvesand straight lines along with a variation incontrast are used for test. Equivalent edge

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detection masks have been designed forhexagonal images, where the horizontal mask forthe hexagonal case is equivalent to the squarecase, the vertical direction gradient mask isapproximated by a combination of two masksoriented at 60- and 120- to the horizontal. Fig. 21and Fig. 22 give different masks used in thePrewitt edge detector implementation.

Cho [40] applied the edge relaxation to thehexagonal grids. His experiments showed thathexagonal edge relaxation can detect betteredges than conventional edge relaxation. Thiscomes from the advantages of hexagonalsampling and unambiguous classification of edgetypes.

Furthermore, if a closed boundary is reached,then it is unchanged permanently and the openboundary is weakened as the iteration proceedsfrom the tail of the boundary. Therefore theoverall results are reliable in finding the edges inthe given edges.

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Several papers on edge detection based on SpiralArchitecture have been proposed since 1996. In[41], an overview on edge detection withinSpiral Architecture was given. In [42], edgedetection using edge focusing technique wasproposed. The second edge detection methodproposed by Zhou et al [43] applied a bilateralfilter which combines a domain filter with arange filter to suppress image noise for edgedetection. Another method for edge detection onSpiral Architecture, as shown in [44], was basedon triple-diagonal gradient. The gradient of grey-level function was defined as a combination ofthree vectors in three diagonal directions of

hexagonal image structure. This method is amore accurate detection mechanism where thegradient is implemented in a more accurate wayin the discrete image space.

Results of edge detection on hexagonal imagesshow that an edge map with better fidelity forcurved objects is obtained than with squareimages. In the case of straight edged objects theedge-maps are of similar quality. This is duemainly to the connectivity of the individualhexagonal pixels generating more consistentcontours. Furthermore, using Spiral Architecturefor edge detection has computational advantagesin order to achieve similar detection results. Inparticular, convolution operations which areroutinely used in edge detection can beimplemented with great efficiency. The twotogether make a strong case for hexagonal basededge detection and seem to reinforce the pointthat hexagonal image processing can be a viablealternative to conventional square imageprocessing.

C. Hexagonal Thinning

Thinning is the process which is used to reducethe amount of data of an object to obtain itsskeleton, which contains single pixel wide linesand can represent the shape of the object.Thinning has been applied to a great variety ofpatterns in the field of machine recognition [451.Wide range of applications show the usefulnessof reducing patterns to thin-line representations,which can be attributed to the need to process areduced amount of data, as well as to the fact thatshape analysis can be more easily made on line-like patterns. The thin-line representation ofcertain elongated patterns, for example,characters, would be closer to the humanconception of these patters; therefore, theypermit a simpler structural analysis and moreintuitive design of recognition algorithms. Askeleton should have the following properties[14, 15]:1. It contains a number of single pixel lines;2. Each element is connected to at least one otherwith no gaps in its structure;3. Skeletal legs are preserved;4. It is accurately positioned;5. Noise induced perimeter pixels are ignoredand limbs are not formed towards them.

Thinning algorithms for use with rectangular,hexagonal, and triangular arrays has been

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investigated by Deutsch [13]. He used the sameapproach to develop each algorithm, whereunnecessary pixels were iteratively deleted untilno more pixels can be removed. Experimentalresults on handwritten character recognitionshowed that the algorithm operating in thehexagonal grid was the most computationallyefficient. The resulting thinned images which areobtained using the triangular array contain theleast number of points per image, since on thisarray the neighbors span the largest distance. Theratio of the maximum distances of any neighboron the rectangular, hexagonal, and triangulararrays is 1: i 2: 3 / J 2 . However, the increasedsize of the basic window renders the processingon a triangular array, and thus the resultingimage, very sensitive to edge irregularities and,more important, to noise. From this point ofview, the hexagonal array is preferential, sinceall its neighbors, theoretically at least, areequidistant. Moreover, if the thinned image is tobe chain encoded the number of directionvectors, in the triangular array is 12, whichmeans that the maximum number of bits requiredto represent a single direction vector is four; thiscompares with the three bits required for theother two arrays. So for storage or transmissionof a complete resulting line drawing, thetriangular array will only be useful if the quartersthe number of points in the image on thehexagonal array.

Staunton [14] presented an analysis of thethinning operation from hexagonally sampledimages and compared the algorithmexperimentally to a similar parallel algorithmdesigned for a rectangular grid. He defined a setof structuring masks in order to decide whether apixel could be deleted from the object border.The hexagonal thinning algorithm requires onlysix masks containing seven elements each, whilethe rectangular algorithm requires eight maskscontaining nine elements each. This greatlyreduced the processing time for 55% of thatrequired to process the rectangular schemeskeleton. Experimental results also showed thatthe hexagonal skeleton exhibited more accuratecorner representation, noise immunity.

D. Hexagonal Interpolation: Hex-splines

Van De Ville etc. [46,47] constructed a newfamily of hex-splines which are specificallydesigned for hexagonal lattices and make use ofthese splines to derive the leastsquarereconstruction function. Hex-splines are a new

type of bivariate splines that are especiallydesigned for hexagonal lattices. Inspired by theindicator function of the Voronoi cell, they areable to preserve the isotropy of the hexagonallattice (as opposed to their B-splinecounterparts). They can be constructed for anyorder and are piecewise-polynomial (on atriangular mesh). Analytical formula have beenworked out in both spatial and Fourier domains.For orthogonal lattices, the hexsplines revert tothe classical tensor-product B-splines. While thestandard approach to represent two-dimensionaldata uses orthogonal lattices, hexagonal latticesprovide several advantages, including a higherdegree of symmetry and a better packing density.They discussed how to advantageously applythem for image processing. We show examplesof interpolation and least-squares resampling.

Yabushita [18] investigated the performance ofimage reconstruction on hexagonal grid.Conventional image reconstruction methods areimplemented on square structure. However, on asquare grid, the distance between adjacent pixelsis different in the horizontal (or vertical)direction from that in the diagonal direction. Thisdifference introduces inconsistency whenneighboring pixel values are interpolated with aspherically symmetric weighting function whichweight depends on the distance between a givenposition and the central pixel. Yabushitacompared the accuracy of the reconstructedimages and compared the results with thoseobtained on square grid. His experimental resultson disc-shaped images showed a betterreconstruction quality on hexagonal grid thanthat on rectangular grid.

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