аЯрЁБс>ўџ 9;ўџџџ8џџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџьЅС3 јПКjbjb^^ &h<h<l MџџџџџџlТТТТТТТђђђђ8*>ђ>h^LЊ(вввБББћ§§§§§§,І Цz)ТБ­БББ)IТТввл^IIIБ@ТвТвћIжфТТТТБћIЦI Њ[|ТТћR šТ5ИђђёXзћ>>щ@I@ћIУхCOMPARISON OF SQUARE-PIXEL AND HEXAGONAL-PIXEL RESOLUTION IN IMAGE PROCESSING Girish Tirunelveli1, Richard Gordon1,2 and Stephen Pistorius2,3 1 Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba, R3T2N2, Canada 2 Department of Radiology, University of Manitoba, Winnipeg, Manitoba, R3T2N2, Canada 3 Department of Medical Physics, CancerCare Manitoba, Winnipeg MB R3E 0V9, Canada E-mail:  HYPERLINK "mailto:tggirish@mts.net" tggirish@mts.net,  HYPERLINK "mailto:GordonR@ms.umanitoba.ca" GordonR@ms.umanitoba.ca,  HYPERLINK "mailto:Stephen@cancercare.mb.ca" Stephen@cancercare.mb.ca ABSTRACT In most applications of image processing, data is gathered and arranged in square pixels. Since hexagonal pixels offer the advantage of greater rotational symmetry, in addition to close packed structure and a nearly circular pixel, we studied image quality with square pixels compared to that of hexagonal pixels. The comparison was done using various images, each image considering a different aspect of geometry (i.e., lines at specific angles, curves, etc.). Each square pixel image was constructed using the average of a square area of smaller pixels. Hexagonal pixel images were constructed using two techniques. The first one is called the “two-template approach”, wherein two different templates are used to create a close packed hexagonal image from smaller square pixels. The second approach is called the “six-neighbor approach” which creates a rectangular template using the six neighbors of a hexagonal pixel. Euclidian distance and MTF are the two image quality measures we used to compare the square pixel and hexagonal pixel images. It was observed that not only does the hexagonal pixel image not offer a significant improvement in image quality, it is also more sensitive to angular variations, especially in the smaller ranges (0o to 5o). A brief explanation of the algorithm and the results are given. We conclude that, contrary to our intuition and their widespread use in nature (retinas and ommatidia), hexagonal pixels do not offer any advantage over conventional square pixels. Topic: Pattern Recognition and Image Analysis Keywords: hexagonal pixels, image processing, resolution Contact Person: Girish Tirunelveli; email:  HYPERLINK "mailto:tggirish@mts.net" tggirish@mts.net, Phone: (204) 227-6344 For Student Paper Competition CCECE’02 Winnipeg, MB; May 12-15, 2002 -  PAGE 1 of  NUMPAGES 1 -  )*/0=>CDNOabcdru‹ŽЉБВГийкъыэю3467def~€Š‹j k p q jpq—™ЂЃбв§§§§§§§§љїїїїѓёѓъёръмъёъёвъмъёъёШъмъёЦУЦРМРМРЦЕАЉЦёё 6B*ph B*ph 5B*phCJH*CJCJ 5jЊ6UjЧ6U0J6j6U j6U656H*5CJ CJ$BNOVЈЉ€Š‹ij™вNl”ЖЗИЙ§ћііхххзіћећггћћћЭЫЫЫћћ ЦрР! „а„„„Lџ]„а^„„`„Lџ$„а„„„Lџ]„а^„„`„Lџa$$a$NOVЈЉ€Š‹ij™вNl”ЖЗЙК§ћћћћјіієђђяэє вт§ў#$%56MNkl—˜žŸ ЁЅІАБВГЕЙК§ј№јјэъурулурурулуръ0JmH0J j0JUCJCJj‘U jU5ЙКљ ЦрР!$1hАа/ Ар=!А "А # $ %АААЧDаЩъyљКЮŒ‚ЊKЉ tggirish@mts.netрЩъyљКЮŒ‚ЊKЉ 0mailto:tggirish@mts.netуDаЩъyљКЮŒ‚ЊKЉ GordonR@ms.umanitoba.caрЩъyљКЮŒ‚ЊKЉ >mailto:GordonR@ms.umanitoba.caчDаЩъyљКЮŒ‚ЊKЉ Stephen@cancercare.mb.caрЩъyљКЮŒ‚ЊKЉ @mailto:Stephen@cancercare.mb.caЧDаЩъyљКЮŒ‚ЊKЉ tggirish@mts.netрЩъyљКЮŒ‚ЊKЉ 0mailto:tggirish@mts.net i(@ёџ( NormalCJmH :: Heading 1$$@&a$5CJ 4@4 Heading 2$@&5CJ00 Heading 3$@&588 Heading 4$„„@&^„„6<A@ђџЁ< Default Paragraph Font<Cђ< Body Text Indent „8^„8@R@ Body Text Indent 2 „а^„а,@, Header  ЦрР!, @", Footer  ЦрР!&)@Ђ1& Page NumberFSBF Body Text Indent 3$„а^„аa$,B@R, Body Text$a$8P@b8 Body Text 2$a$5CJ FYrF Document Map-D MЦ џ€OJQJ.Q@‚. Body Text 35(U@Ђ‘( Hyperlink>*B*К џџџџ џџ z™К LLLOвК ЙК К Вйъэ36e~§$ 5 К Xџ€Xџ€Xџ€Xџ„+249DFO!tџ•€tџ•€№8№@ёџџџ€€€ї№’№№0№( № №№B №S №ПЫџ ?№К :<OUwl Л  su  Чбl Л џџGirish Tirunelveli>C:\Girish\Education\Thesis\Thesis_Chapters\paper1_abstract.docGirish Tirunelveli>C:\Girish\Education\Thesis\Thesis_Chapters\paper1_abstract.docRichard GordonHDick 9:Dick's Library:Dick's T-Z Library:Tirunelveli,GordonPistorius2002Girish TirunelveliC:\Girish\THESIS.docGirish TirunelveliMC:\Girish\Education\Thesis\Thesis_Chapters\TirunelveliGordonPistorius2002.docGirish TirunelveliMC:\Girish\Education\Thesis\Thesis_Chapters\TirunelveliGordonPistorius2002.docGirish TirunelveliMC:\Girish\Education\Thesis\Thesis_Chapters\TirunelveliGordonPistorius2002.docGirish TirunelveliMC:\Girish\Education\Thesis\Thesis_Chapters\TirunelveliGordonPistorius2002.docGirish TirunelveliMC:\Girish\Education\Thesis\Thesis_Chapters\TirunelveliGordonPistorius2002.docRichard GordonGDick 9:Dick's Library:Dick's T-Z Library:TirunelveliGordonPistorius2002Уту`џџџџџџџџџжЫ~д!џџџџџџџџџ]s>Њ†Оѓџџџџџџџџџќ>ўў–дџџџџџџџџџ5bШ„d2џџџџџџџџџŒoљkFEЊџџџџџџџџџh „„˜ўЦ^„`„˜ўOJQJo(З№h „и „˜ўЦи ^„и `„˜ўOJQJo(oh „Ј „˜ўЦЈ ^„Ј `„˜ўOJQJo(Ї№h „x„˜ўЦx^„x`„˜ўOJQJo(З№h „H„˜ўЦH^„H`„˜ўOJQJo(oh „„˜ўЦ^„`„˜ўOJQJo(Ї№h „ш„˜ўЦш^„ш`„˜ўOJQJo(З№h „И„˜ўЦИ^„И`„˜ўOJQJo(oh „ˆ„˜ўЦˆ^„ˆ`„˜ўOJQJo(Ї№„а„˜ўЦа^„а`„˜ўo(.€„ „˜ўЦ ^„ `„˜ў.‚„p„LџЦp^„p`„Lџ.€„@ „˜ўЦ@ ^„@ `„˜ў.€„„˜ўЦ^„`„˜ў.‚„р„LџЦр^„р`„Lџ.€„А„˜ўЦА^„А`„˜ў.€„€„˜ўЦ€^„€`„˜ў.‚„P„LџЦP^„P`„Lџ.„а„˜ўЦа^„а`„˜ўo(.„ „˜ўЦ ^„ `„˜ўo(.„p„LџЦp^„p`„Lџ.€„@ „˜ўЦ@ ^„@ `„˜ў.€„„˜ўЦ^„`„˜ў.‚„р„LџЦр^„р`„Lџ.€„А„˜ўЦА^„А`„˜ў.€„€„˜ўЦ€^„€`„˜ў.‚„P„LџЦP^„P`„Lџ. „а„˜ўЦа^„а`„˜ўo(.€„ „˜ўЦ ^„ `„˜ў.‚„p„LџЦp^„p`„Lџ.€„@ „˜ўЦ@ ^„@ `„˜ў.€„„˜ўЦ^„`„˜ў.‚„р„LџЦр^„р`„Lџ.€„А„˜ўЦА^„А`„˜ў.€„€„˜ўЦ€^„€`„˜ў.‚„P„LџЦP^„P`„Lџ.„8„˜ўЦ8^„8`„˜ўo()€„„˜ўЦ^„`„˜ў.‚„и „LџЦи ^„и `„Lџ.€„Ј „˜ўЦЈ ^„Ј `„˜ў.€„x„˜ўЦx^„x`„˜ў.‚„H„LџЦH^„H`„Lџ.€„„˜ўЦ^„`„˜ў.€„ш„˜ўЦш^„ш`„˜ў.‚„И„LџЦИ^„И`„Lџ.„а„˜ўЦа^„а`„˜ўo(.„ј„@ќЦј^„ј`„@ќo(.‚„p„LџЦp^„p`„Lџ.€„@ „˜ўЦ@ ^„@ `„˜ў.€„„˜ўЦ^„`„˜ў.‚„р„LџЦр^„р`„Lџ.€„А„˜ўЦА^„А`„˜ў.€„€„˜ўЦ€^„€`„˜ў.‚„P„LџЦP^„P`„Lџ.Œoљk]s>жЫ5bќ>Уџџџџџџџџџџџџџџџџџџџџџџџџџџџџџ@€::Єk€ ::x”К P @GTimes New Roman5€Symbol3 Arial5 Tahoma?еFOND _№Courier New;€Wingdings"qˆ№аhЋ3\†Б4\†Л”S [МБ№  ДД20d ^2ƒq№џџfkjfkjdfGirish TirunelveliRichard Gordonўџ р…ŸђљOhЋ‘+'Гй0”˜ ДРмшє ( D P \ ht|„Œ' fkjfkjdft kjfGirish TirunelvelidiriiriNormalTRichard Gordone4chMicrosoft Word 9.0d@vнA@ЂmШАР@тЫ—–~С@&qѕИ~С[Мўџ еЭеœ.“—+,љЎDеЭеœ.“—+,љЎD hpŒ”œЄ ЌДМФ Ь с'Mastech Corporation Я  fkjfkjdf Titleќ 8@ _PID_HLINKS'AДA mailto:tggirish@mts.net €$= mailto:Stephen@cancercare.mb.ca €s mailto:GordonR@ms.umanitoba.ca €Amailto:tggirish@mts.net € ўџџџўџџџ !"#$%&'ўџџџ)*+,-./ўџџџ1234567ўџџџ§џџџ:ўџџџўџџџўџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџRoot Entryџџџџџџџџ РFYИМ†~С<€Data џџџџџџџџџџџџ1Tableџџџџ@WordDocumentџџџџ&SummaryInformation(џџџџџџџџџџџџ(DocumentSummaryInformation8џџџџџџџџ0CompObjџџџџџџџџџџџџXџџџџџџџџџџџџўџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџўџџџџџ РFMicrosoft Word DocumentўџџџNB6WWord.Document.8