Wavelet Analysis ofsolid Objects: Applications in Layered Manufacturing Mark D. Van Roosendaal, Peter Chamberlain, Charles Thomas University ofutah

Transcription

1 Wavele Analysis ofsolid Objecs: Applicaions in Layered Manufacuring Mark D. Van Roosendaal, Peer Chamberlain, Charles Thomas Universiy ofuah ABSTRACT In his paper, we inroduce wo-dimensional discree wavele basis funcions and heir applicaion in he analysis and modeling ofsurface opography in layered manufacuring objecs. In previous work, a one dimensional wavele ransform echnique was developed o generae variable hickness layers. [1] For verical edge layers Haar wavele decomposiion is used he slicing direcion bu is no useful in he slicing plane. For frequency analysis wihin he slicing plane, biorhogonal waveles provide he desired analysis abiliy. When analyzing layered manufacuring wih ruled edges a rue 2-D ransform is appropriae. Two-dimensional wavele analysis simulaneously conrols he layer hickness as well as he densiy ofconrol poins required he surface definiion ofeach layer edge. INTRODUCTION 1) Wha is frequency analysis in layered manufacuring? I is possible o analyze a 3-D solid objec in erms ofis spaial frequency conen. Using spaial frequency analysis, regions ofhigh complexiy appear as high frequencies and fla or genly curved regions appear as low frequencies. Wavele analysis allows spaially resolved frequency analysis, where he locaion ofhe high or low frequencies is idenified. By operaing on he wavele represenaion ofan objec wih cusom frequency filers, i is possible o simulae he resuls ofvarious layered manufacuring processes. [2] 2) Why do we wan a frequency analysis echnique? Simulae he resuls oflayered manufacuring - Haar wavele for verical edges - biorhogonal wavele for ruled Conrol variable layer hickness generaion Idenify appropriae slice direcion Poenial file size reducion WAVELETS A wavele ransform (WT) decomposes a funcion ino a summaion ofscaled and ranslaed basis funcions called waveles. [3] An inverse wavele ransform (IWT) sums he basis funcions "" aoo Figure 1. a) Haar wavele b) biorhogonal wavele 435

2 back ogeher o fonn he original signal. The waveles mus saisfy cerain crieria in order o be valid basis funcions. Among hese crieria, waveles mus be oscillaory and decay quickly o zero. [4] Two waveles ha have been found useful in layered manufacuring are he Haar wavele [5] and he biorhogonal wavele shown in Fig. 1. Discree wavele analysis (DWT, IDWT) assumes ha a signal f(x) has been sampled a equally spaced inervals. The sequence lengh N of he signal being analyzed deennines how many wavele levels can be represened. [6] When N=2 n here are n + 1 wavele levels. The levels represen differen frequencies in he signal. Fig. 2 demonsraes frequency decomposiion ofa signal ino wavele levels. Level al0 is he lowes frequency level and al is he highes frequency level. PROPOSED PROCESS OVERVIEW The proposed process ofanalyzing solid objecs for frequency conen is oulined in Fig. 4. From an STL file, in he caresian coordinae a. Signal and Approxlmalon(s) Figure 2. Frequency decomposiion ofhe objec. a) Using Haar, b) using biorhogonal waveles. a. Figure 3. Unwrapping ofhe objec. a) caresian represenaion, b) cylindrical presenaion. b. r 436

3 ~ SOLID MODEL INTO STL FILE UHJECI KhJ:'Kh:"oO-< 'lt (UNWRAP) FREQUENCY LEVEL DECOMPOSITION IN BOTH THETA (HORIZONTAL) AND RADIUS (VERTICAL) DIRECTIONS FREQUENCY FILTER A. LOW PASS FILTER - consan layer hickness B. THRESHOLD FILTER - variable layer hickness INVERSE WAVELET RECONSTRUCTION OF SYNTHESIZED IMAGE... ERROR EVALUATION sysem, 3-D solid objecs are sampled a equally spaced inervals in deermined sequence lenghs N. The objec is unwrapped by convering sampled daa ino cylindrical coordinae represenaions. Changes in he par radius (r) can herefore be analyzed in wo dimensions: verical (z) and horizonal (8). Fig. 3 shows how he par is unwrapped. [1] The discree wavele ransform is exended o wo dimensions, which are decomposed and filered simulaneously. Decisions regarding he ype ofwavele basis funcion and he ype offiler are made. Afer filering, an inverse discree wavele ransform is hen used o reconsruc he objec. From he resuling reconsrucion, error is evaluaed. CUTTER PATH POINT REDUCTION AND GENERATION I Figure 4. An overview ofhe 2-D wavele analysis procedure. FILTERING Two ypes offilers, low pass and hreshold, have been examined for applicaion in layered manufacuring. Low Pass Filer Here, all wavele coefficiens above a specified frequency level are removed. Afer inverse ransforming, he model conains no frequency componens above he cuoff. The resuling model looks as ifi were produced by a consan layer hickness LM echnique. Fig. 5b. shows hese resuls. Threshold Filer Here, all wavele coefficiens below a specified hreshold value are removed. The resuling model looks as ifi were consruced from variable hickness layers. Fig. 5c. shows his resul. Because ofhe abiliy o independenly vary he filering ofeach frequency level, remove wavele coefficiens above a specified frequency level, or perform global filering ofall frequency levels, housands ofpossible filer combinaions can be consruced. 437

4 .." DfllIpolna b. O ~F -0.1 O.O~F a. o"--_..~ ~. o ~ ~ ~ Daa polm.. Figure 5. a) This figure shows he Haar wavele frequency coefficiens for a single slice ofhe head. All en wavele levels are shown. Level dl is he highes frequency. b) Resuls from low pass filering, yielding uniform slice hickness. c) Resuls from hreshold filering, yielding adapive slice hickness. Filering was inenionally se high o beer illusrae he resuls. c. POINT REDUCTION When he original model is pu ino cylindrical forma, he file consiss of regularly spaced poins as shown in Fig. 6a. Threshold filering forces he poins ino sraigh lines, bu he oal number of poins in he file remains consan. Deleing redundan poins along each line segmen reduces boh poin densiy and file SIze.,... Figure 6. a) Poin densiy prior o wavele filering. b) Afer wavele filering. c) Afer poin reducion. a.."" b. c. 438

5 VERTICAL DIRECTION (z) RESULTS The abiliy o analyze he objec in he verical direcion ena~les he?~era~r o ~enerae variable hickness layers. In Figures 7 and 8, below, a single conour m he shcmg direcion was analyzed r r rr'r7 '7'" 7"'7'" ~ r a. b. Figure 7. a) Verical slice (side view offron porion ofhe head) afer wavele filering. Filering was inenionally se high o beer illusrae he resuls. b) Close up view ofhe nose. Noice he variable layer hickness. Gray represens he original conour; black represens he conour afer processing. The Haar WT is ideal for verical edge layered manufacuring. In his example he head was reconsruced from 76 variable hickness layers r----,---,------,----, ~00;;:---5;;;50:----=600;:----;6=5O---=7oo::----c:!750 a. b. Figure 8. a) Verical slice (side view offron porion ofhe head) afer wavele filering. Filering was inenionally se high o beer illusrae he resuls. b) Close up view ofhe nose. Noice he variable layer hickness. Gray represens he original conour; black represens he conour afer processing. The biorhogonal WT is ideal for ruled edge layered manufacuring. In his example he head was reconsruced from 40 variable hickness layers. 439

6 440 HORIZONTAL (e) DIRECTION RESULTS The abiliy o analyze he objec in he edirecion enables he operaor o filer unwaned informaion (high frequency seps) in he curer pah a. b. Figure 9. Original slice, gray (1024 poins), slice afer processing using filered Haar waveles and poin reducion scheme, black. a) 784 poins. b) 1020 poins a. b. Figure 10. Original slice, gray (1024 poins), slice afer processing using hreshold biorhogonal waveles and poin reducion scheme, black. a) 560 poins. b) 630 poins. The above figures (Fig. 9,10) show how wavele filering provides a reducion in he number ofconrol poins defining he layer edge while preserving an accurae edge

7 represenaion. Figures 9 and 10 demonsrae he advana~e ofusing h~ biorhogonal wavele over he Haar he hea direcion. Using Haar, 1020 poins were requlfed o reproduce he original slice geomery while only 630 poins were required using he biorhogonal wavele. Based on he resuls shown, we anicipae his analysis win simplify and smooh he cuer pah definiion a. b. c. Figure 11. a) Original objec. b) Prediced head afer WT filering and IDWT (low resoluion hea direcion). c) Prediced head afer WT filering and IDWT (low resoluion in he sacking direcion). 11 shows he original head and he reconsruced head afer unwrapping, biorhogonal wavele ransform decomposiion, hreshold filering, inverse ransforming, and rewrapping. Fig. 11 (a) is he original STL file. I is assumed ha he apparen layering effec is an arifac produced when he head was scanned. In Fig. 11 (b) he filers were se for fme resoluion he layering direcion (z) and low resoluion around he circumference direcion (8). Noe ha here is no apparen change in resoluion in he z direcion. The nose is broadened and he defmiion around he ears is poor due o he more aggressive filer in he 8 direcion. In Fig. 11 (c) he filers were se for low resoluion in he layering direcion (z) and fine resoluion around he circumference direcion (8). Now hicker layers are apparen in he z direcion. As expeced, he low resoluion filering grealy affeced he accuracy ofhe par causing he lips o be compleely removed. LIMITATIONS are wo primary limiaions for he process oulined in his paper. 1) Currenly, his echnique works only for single-value funcions. Decomposing complex geomery ino simple sub-pars prior o processing can eliminae his limiaion. 2) The sequence lengh N is chosen so ha f(x) is a desirable represenaion ofhe original daa. For simple, low-resoluion pars, a lower value ofn (say, 128) is adequae. For higher resoluion pars, a larger N (say, 1024) is recommended. Alhough a larger value ofn gives a beer approximaion, he value ofhis ineger is limied by he physical consrains ofhe decomposiion sofware or by he desire o perform compuaions effecively and efficienly. [5] 441

8 CONCLUSIONS The 2-D wavele ransform is a useful ool for mahemaical analysis ofhe spaial frequency conen ofhree-dimensional objecs. This analysis enables he predicion appropriae layer hickness as well as reducing he cuer pah poin load for layered manufacuring processes. In order o perform wavele analysis, 3-D objecs are decomposed (unwrapped) ino a cylindrical coordinae represenaion. This single valued represenaion is hen wavele ransformed, filered, and inverse ransformed. The choice offiler (low-pass or hreshold) is moivaed by he applicaion. Uniform layer hickness par decomposiion is creaed using low-pass filering. Threshold filering produces adapively sliced represenaions. As expeced, here is a direc relaionship beween filering and error. Under-sampling and over-filering were he mos common sources oferror. The Haar wavele is useful for analyzing verical edge layered manufacuring while he biorhogonal wavele proved useful for ruled edges. addiion, he wavele represenaion ofhe par is quie compac may be useful for file compression. FUTURE WORK Several areas have been idenified for furher work. These areas include echniques for error quanificaion and filer opimizaion. These echniques will allow user inpu ofmaximum error desired in boh direcions and will adjus he frequency filering accordingly. Fuure work includes he developmen ofmore sophisicaed sofware ools for analyzing, decomposing, and filering sereolihography files and he creaion ofcuer pahs. Fuure sofware will auomae he process oulined his paper. REFERENCES 1. C. "New Analysis mehods for Three-Dimensional Objecs in Solid Freeform Manufacuring." Ph.D. disseraion, Universiy ofuah Dep. ofmechanical Engineering, Lee, C. H., Thomas, C.L., "Wavele ransform based analysis for layered manufacuring", Proceedings ofhe Sevenh inernaional Conference on RapidProoyping, 3/31-4/3, Akansu, A. N., Smih, MJ.T, Subband and Wavele Transforms, Design andapplicaions. Kluwer Academic Publishers, 1996, Young, R. K., Wavele Theory and is Applicaions. Kluwer Academic Publishers, 1993, Chui,C.K, Waveles: A Mahemaical Toolfor Signal Analysis, Sociey for Indusrial and Applied Mahemaics, 1997, 6. Newland, D. Inroducion o Random Vibraions, Specral and Wavele Analysis. Longman Scienific & echnical, England,