A New Morphological 3D Shape Decomposition: Grayscale Interframe Interpolation Method

Transcription

1 A ew Morphological 3D Shape Decompositio: Grayscale Iterframe Iterpolatio Method D.. Vizireau Politehica Uiversity Bucharest, Romaia R. M. Udrea Politehica Uiversity Bucharest, Romaia Abstract Oe of the mai image represetatios i Mathematical Morphology is the 3D Shape Decompositio Represetatio, useful for Image Compressio, ad Patter Recogitio. The 3D Morphological Shape Decompositio represetatio ca be geeralized a umber of times, to eted the scope of its algebraic characteristics as much as possible. With these geeralizatios, the Morphological Shape Decompositio 's role to serve as a efficiet image decompositio tool was eteded to discrete images ad grayscale images. This work follows the above lie, ad further develops it. A ew evolutioary brach is added to the 3D Morphological Shape Decompositio's developmet, by the itroductio of a 3D Multi Structurig Elemet Morphological Shape Decompositio, which permits 3D Morphological Shape Decompositio of 3D biary images (grayscale iamges ito "multiparameter" families of elemets. At the begiig, 3D Morphological Shape Decompositio represetatios are based oly o "1 parameter" families of elemets for image decompositio. This paper addresses the gray scale iterframe iterpolatio by meas of mathematical morphology. The ew iterframe iterpolatio method, called 3D Shape Decompositio iterpolatio is based o morphological 3D Shape Decompositio. This article will preset the theoretical backgroud of the morphological iterframe iterpolatio, deduce the ew represetatio ad show some applicatio eamples. Computer simulatios could illustrate results. 1. Itroductio Image Represetatio is a key compoet i may tasks i Computer Visio ad Image Processig. It cosists geerally of presetig a image i a form, differet from the origial oe, i which desired characteristics of the image are emphasized ad more easily accessed. I this work, we cosider morphological methods for both biary ad grayscale image represetatio. They are based o Mathematical Morphology, which is a relatively ew, ad rapidly growig, oliear theory for Image Processig. Mathematical Morphology is part of Set Theory, ad it has a strog geometric orietatio. Its theory was developed by Mathero ad Serra, i the middle 6's, with the purpose of describig the shape decompositio of materials by image aalysis of their sectios. Beig origially developed for biary images, it was later (durig the 7's geeralized for grayscale images as well. For biary images (2D ad 3D, Mathematical Morphology provides a well fouded theory for aalysis ad processig. Therefore, Biary Morphological Represetatios ca be developed ad aalyzed. Grayscale Morphological Represetatios are a geeralizatio of the biary represetatios i 3D space, ad they emphasize geometrical characteristics of the image, which are ot easily accessed i a liear represetatio. The mai morphological represetatio for 3D biary images is the Shape decompositio. The 3D Shape decompositio ca be calculated etirely by the basic operatios of Mathematical Morphology, which makes the 3D Shape decompositio a morphological represetatio, suitable for aalysis by morphological tools. Based o 3D Shape decompositio a morphological represetatio, called 3D Morphological Shape Decompositio is preseted. It cosists of first calculatig the ball (or other predefied cove 3D shapes with the greatest size cotaied iside the shape, the takig the residue (set differece betwee the origial shape to the above greatest balls, ad fially reiteratig the above procedure o the residue util the whole shape is decomposed. The resultig decompositio elemets are, therefore, disjoit. Followig the geeralizatio of the whole morphological theory, from biary to grayscale images, the Morphological Shape Decompositio Represetatio could be geeralized as well to grayscale images [2,3]. The motivatio of this work is to ivestigate the use of Morphological Shape Decompositio's for biary ad grayscale Image Represetatio, ad their applicatios, with special iterest i iterframe iterpolatio. Iterframe iterpolatio problem is the process of creatig itermediary frames betwee two gives oe. Two kow frames iput frame ad output frame, as well as itermediary frames are of the same type. Iterpolated frame ( i depeds o the cotet of iput frame ad output frame ad the positio of the iterpolated

2 oe betwee them. It is a umber betwee ad M deoted i this paper as m, M. For m = iterpolated frame is equal with iput frame,, for m = M iterpolated frame is equal with output frame, M. I the followig pages we will cosider a morphological methods for grayscale frame iterpolatio. These methods are based o 3D biary mathematical morphology. Mathematical Morphology is a geeral theory based of simple operators: erosio, dilatio, opeig ad closig. The morphological operatios have bee successfully used i may applicatios icludig object recogitio, frame ehacemet, teture aalysis, ad idustrial ispectio. The ew iterpolatio frame method preseted i this article is based o 3D morphological Shape Decompositio. It cosists of chagig, step by step, the Shape Decompositio subsets of iput frame with the Shape Decompositio subsets of out frame [1,4,5]. 2. 3D Morphological Shape Decompositio The 3D Shape Decompositio ca be calculated etirely usig the basic morphological operators. Dilatio ad erosio are the fudametal operators of the Mathematical Morphology. The key process i the dilatio ad erosio operators is the local compariso of a shape, called structurig elemet, with the object to be trasformed. The structurig elemet is a predefied shape, which is used for morphological processig of the frames. The most commo shapes used as structurig elemets are horizotal ad vertical lies, squares, digital discs, crosses, etc. The fudametal morphological operators are based o the operatio of traslatio. Let B be a set cotaied i Z 3, ad let be a poit i Z 3. The traslatio of the set B by the poit, deoted B, is defied as follows: { } B = b+ b B (1 The dilatio of the 3D biary shape by the structurig elemet B, deoted B, is defied by: B = B (2 The erosio of 3D biary shape by the structurig elemet B, deoted B, is defied i the followig way: B = b (3 b B Based o those fudametal operators, two morphological operators are developed. These are the opeig ad closig operators. The opeig operator, deoted, ca be epressed as a compositio of erosio followed by dilatio, both by the same iput structurig elemet: B B B (4 The closig operator, deoted, ca be epressed as compositio of dilatio followed by erosio by the same iput structurig elemet: B = ( B B (5 The Shape Decompositio ST(,B of 3D biary shape i Z 3 ca be calculated by meas of morphological operatios. The Morphological Shape Decompositio is a compact error free represetatio of images Origial Morphological Shape Decompositio The 3D morphological shape decompositio is a compact error free represetatio of images, a property useful for image compressio. The Shape decompositio is a redudat represetatio, i.e., some of its poits may be discarded without affectig its error free characteristic. The morphological shape decompositio represetatio of a biary image, with a give biary 3D structurig elemet B, is a collectio of sets SD (, B called Shape decompositio subsets of order. Morphological shape decompositio is defied us followig: I Iitial coditio a is the maimum value defied by with ad b We defie, B B B = > (6 B B B (7 ( (8 (, SD B B B B (9

3 II Recursive relatios We ca defie iteratively, for =, 1,...,2,1 ( 1 (, SD B B (1 (, SD B B B B (11 From the collectio of morphological shape decompositio subsets SD (, B the origial shape ca be perfectly recostruct by (, (12 = SD B B A geeralizatio of the 3D shape decompositio framework should be sought, based either o a topological or a algebraic approach. I the topological approach, such a geeralizatio should aspire to solve problems like robustess, coectivity ad precisio as a shape descriptor; whereas, i the algebraic oe, framework fleibility, self-duality, ad represetatio efficiecy are the mai issues. The 3D morphological shape decompositio satisfies the above requiremets. The algebraic framework of the 3D shape decompositio will be eteded several times. The purpose was always to obtai decompositios accordig to richer families of elemets, ot failig to satisfy the above algebraic requiremets. The above evolutio will be described. The morphological shape decompositio is a error-free represetatio sice the origial 3D biary image (gray scale image ca be recostructed from SD (, B Modified Shape decompositio The family of 3D elemets B used i the discrete morphological shape decompositio is geerated by recursively dilatig the structurig elemet B by itself. B serves here as a geerator, beig costat at every step of the family geeratio. The family geerator ca have a variable size. The Modified Shape decompositio decomposes ito maimal elemets from the family { B,1 B,2 B,4 B,8 B,16 B,...}. This family ca be geerated, as before, by a series of dilatios, but ot with a costat geerator. The sizes of the geerator, at the various steps, are the differeces { 1,1, 2, 4, 8,16,... } betwee the sizes of the elemets of the family. We ca observe, by comparig the coditios for the modified shape decompositio with those for the morphological shape decompositio that the oly sigificat differece is i the family geeratio. The modified shape decompositio also fully represets the 3D biary origial image. Geerally, codig simulatios, the modified morphological shape decompositio show better results tha the previous morphological shape decompositio. For modified shape decompositio, is bouded, the family is ideed by, ad epoetially geerated: 1 A = 2 B, 1 (13 The "sigle-poit'' elemet belogs to the family: A = {(, }. B is topologically ope ad cotais the origi. Geeratio formulas are give by: I Iitial coditio a is the maimum value defied by with ad b We defie 2 B 2 B B =, > (14 2 B 2 B B (15 ( 2 (16 SD, B 2 B 2 B B (17 2 II Recursive relatios We ca defie iteratively, for =, 1,...,2,1 ( 2 1 ( 2 SD (, B 2 B 2 SD (, B 2 B 2 ( 2 B B... (18 ( 1 SD B B 1 (, ( SD B B B B 1, 2.3. Geeralized-Step Shape decompositio ot oly the size of the geerator ca vary at each step of the family geeratio, but also the geerator's shape. Let B be a series of structurig elemets, topologically ope all of them cotaiig the origi ad satisfyig B B = B (19

4 Let this series geerate a family of elemets the followig way: ad A i A+ 1 = A B, =,1, 2,... (2 A A(, The collectio subsets (, = (21 SD B is called i this A case Geeralized-Step Morphological Shape decompositio. The coditios for the bouded shape decompositio are: I Iitial coditio a is the maimum value defied by with ad b We defie (, A A B = > (22 ( A A B (23 ( (24 (, = ( SD B A A B (25 II Recursive relatios We ca defie iteratively, for =, 1,...,2,1 ( 1 (, SD B A (26 (, = ( SD B A A B (27 From the collectio of morphological shape decompositio subsets SD (, B( the origial shape ca be perfectly recostruct by (, (28 = SD B A 3. Iterpolatio Method The ew iterpolatio frame method preseted i this article is based o 3D biary morphological Shape Decompositio. It cosists of chagig, step by step, the Shape Decompositio subsets of iput frame with the Shape Decompositio subsets of out frame Iterpolatio usig origial shape decompositio The method: a Shape Decompositio subsets ST (, B, for =, 1,...,1,, of iput frame are computed. The origial shape ca be perfectly recostructed i the followig way: (, (29 = ST B B = b Shape Decompositio subsets (, ST M B M, for = M, M 1,...,1,, of output frame are computed. The output frame M ca be perfectly recostructed i the followig way: M (, (3 = ST B B M M = c Iterpolated frame ( i depeds o the cotet of iput frame ad output frame ad the positio of the iterpolated oe betwee them ad must be computed usig morphological formulas. Iterpolated frame, i, is defied, for = ma (, M ad M = + 1, i= i 1 (, i M (, = ST B B ST B B, i= 1,..., M 1 = = i M, i= M (31

5 3.2. Iterpolatio usig Geeralized-Step Shape decompositio The method: a Shape Decompositio subsets ST (, B (, for =, 1,...,1,, of iput frame are computed. The origial shape ca be perfectly recostructed i the followig way: (, (32 = ST B A = b Shape Decompositio subsets (, ST M B M, for = M, M 1,...,1,, of output frame are computed. The output frame M ca be perfectly recostructed i the followig way: M M M = (, = ST B A (33 2D ad 3D Shape Decompositio. This article presets the theoretical backgroud of the morphological frame iterpolatio ad deduce the ew represetatio. Refereces [1] J. Serra, Frame Aalisis ad Mathematical Morphology, Lodo Academic Press, 1982 [2] J. Serra, Shape Decompositio Decompositios, SPIE Vol. 1769, Frame Algebra ad Morphological Frame Processig III, pp , [3] D.. Vizireau, Digital Image Processig A Itroductio, TTC, 1998, Bucharest, Romaia. [4] D.. Vizireau, C. Pirog, oliear Digital Sigal Processig, Electroica 2, Bucharest, 21, Romaia. [5] Gh. Gavriloaia, D.. Vizireau, Low level digital image processig, Techical Military Academy, Bucharest, Romaia, 21. c Iterpolated frame ( i depeds o the cotet of iput frame ad output frame ad the positio of the iterpolated oe betwee them ad must be computed usig morphological formulas. Iterpolated frame, i, is defied, for = ma (, M ad M = + 1, i= i 1 (, i M (, = ST B A ST B A, i= 1,..., M 1 = = i M, i= M (34 4. Coclusios Iterframe iterpolatio problem is the process of creatig itermediary frames betwee two gives oe. Two kow frames iput frame ad output frame, as well as itermediary frames are of the same type. Iterpolated frame depeds o the cotet of iput frame ad output frame ad the positio of the iterpolated oe betwee them. It is a umber betwee ad M deoted i this paper as,. For = iterpolated frame is equal with iput frame,, for = iterpolated frame is equal with output frame,. This paper addresses the gray scale iterframe iterpolatio by meas of mathematical morphology. The ew frame iterpolatio method, called Shape Decompositio iterpolatio is based o morphological