The B-splne Interpolaton n Vsualzaton of the Three-dmensonal Objects Zeljka Mhajlovc *, Alan Goluban **, Damr Kovacc ** * Department of Electroncs, Mcroelectroncs, Computer and Intellgent Systems, ** Department of Control and Computer Engneerng n Automaton Faculty of Electrcal Engneerng and Computng, Unversty of Zagreb, Unska 3, 0 000 Zagreb, Croata, zeljka.mhajlovc@fer.hr, alan.goluban@fer.hr, damr.kovacc@fer.hr Abstract The volume data s generally n the form of the large array of numbers. In order to render the object hdden n the volume, we need to reconstruct or nterpolate data values between the samples. The novelty presented n ths paper s B-splne nterpolaton n the volumetrc space. We show that ths approach s better then currently used methods. To enhance qualty durng the volume vsualzaton process t s mportant to enhance a qualty of the reconstructon. It s of crucal mportance to explore dfferent undesred effects. If better reconstructon s performed the more accurate result of volume vsualzaton process s acheved. Keywords: B-splne, volume renderng, volume reconstructon. Introducton The volume vsualzaton s based on the three-dmensonal scalar or vector feld. Object that should be vsualzed s represented by the array of dscrete samples. Durng renderng of the object t s necessary to reconstruct the contnuous three-dmensonal functon, defned by the samples, for any appled method. Classfcaton of the methods for the volume vsualzaton can be done regardng to the space where they bascally work. Development of the new methods extent the basc classfcaton proposed by Kaufman (Kaufman, 990). There are three groups of methods: the object space methods, the mage space methods and methods that are based on transformed object space. The object space methods manly creates polygons or classc geometrc prmtves and projects them n the projecton plane (Lorensen, 987). Methods that are based on the mage space start from the mage plane and cast the rays from each pcture element nto the scene (Levoy, 988). Methods that are based on the transformed object space work n transformed space, for example n the frequency doman (Totsuka, 993) or n the wavelet doman (Gross, 996). There are also some hybrd methods that employ coherency characterstcs from dfferent spaces. The object space s frst traversed to reorganze data to be prepared for traversng n the mage space. Durng traversng the mage space, rays are cast from each pcture element n the object space (Lacroute, 995). Organzaton of the volume elements s very mportant because sgnfcant performance benefts can be acheved f volume elements can be easly fetched along cast ray. Durng the volume renderng there are several layers where reconstructon s necessary, and the error caused by reconstructon may occur. Reconstructon s done n the three-dmensonal space based on the values of the volume elements. Numerc ntegraton along the ray path uses reconstructed values as sample ponts. To calculate value assgned to the ray, values n the sample ponts along the ray are accumulated. Fnal reconstructon s done based on each ray n order to produce the fnal mage (Fg. ).
Projecton plane Sample ponts on the ray Volume elements Fg.. Reconstructon n the volume renderng. It s mportant to be aware of lmtatons of the reconstructon because t can sgnfcantly nfluence the accuracy of the result. Investgaton of the reconstructon or the nterpolaton s requred to acheve compromse between dfferent undesred artfacts and achevng optmal result. 2 Reconstructon n the computer graphcs Development of new renderng algorthms for vsualzaton of the three-dmensonal scalar felds s recent area of research. Usually, related papers put the man accent on the proposal of new methods, whle to the problem of reconstructon s gven less attenton. Alasng s problem present n many areas of computer graphcs. Objects are usually defned procedurally and they are synthetc. Preflterng of such representaton s not practcal. Further more, transformaton between contnuous and dscrete representaton s often requred. Alasng may occur on every transformaton of representatons, and ths problem also appears when resamplng s requred. Multlayer resamplng s often requred and each layer may cause addtonal error. Ths problem s well recognzed n the computer graphcs, and nvestgated by many authors. Dsplay of the computer generated mage s nput object to our vsual system, and t s not completely understood how our vsual system works. Senstvty of the human eye s specfc, so mnmal devaton n mathematcs sense dffers from the most pleasant result for our perceptual system. Even a lttle dstorton n gray levels can cause unpleasant psychovsual result, especally n the areas wth smooth changes. In the analyss based on the perceptual approach, rather than mathematcs, some authors prefer lttle alasng n order to avod other vsual defects, that results from tryng to remove alas completely. The appearance of alasng s nvestgated when famly of pecewse cubc flters s appled to mage reconstructon (Mtchell, 988). Mtchell also presents (Mtchell, 996) how stratfed samplng reduces varance of the mean value of the mage pcture elements. The problem s to numercally express the result that depends on our vsual system. Marschner and Lobb (Marschner, 996) propose metrc that can be used to measure the flter characterstcs, n terms of smoothng and postalasng. On the three-dmensonal test sgnal they show the results when dfferent reconstructons are used. The proposed test functon s hghly senstve to the alasng, and dfferent undesred effects are vsble on the results. Dsadvantage s that the proposed test functon s contnuous, so drawback caused by dscontnuty usual n real data wll not appear. In volume renderng, gradent nformaton s used for shadng and classfcaton of the data set n combnaton wth the voxel ntenstes.
Bentum presents the analyss of gradent estmators n frequency doman, and proposes takng the dervatve of the nterpolaton functon tself (Bentum, 996). Machraju and Yagel characterze and measure error by applyng Taylor seres expanson. They characterze error as truncaton error and non-snc error. The methods for error measurements are based on the spatal doman analyses. The Taylor seres expanson of the convoluton sum (Moller, 997) lead to the quanttatve and qualtatve compresson of the reconstructon and dervatve flters. The analyss s based on the BC-splnes defned by Mtchell. It s mportant to dstngush approxmaton and nterpolaton approach. The approxmaton curves are used to approxmate control polygon, and nterpolaton curves must pass through the defned vertces. Torach used nterpolaton quadratc B-splne for mage reconstructon (Torach, 988), and Unser presented B-splne transforms for the mage nterpolaton (Unser, 99). 3 Prealasng and postalasng Volumetrc space conssts of volume elements. Each volume element may represent result from real world object sample, from numerc smulatons, or may represent some pure mathematcal value. The samples are taken from contnuous space, but object wth sharp edges n that space creates dscontnuty. Accordng to Shannon theory, sgnal can be reconstructed from ts samples f two condtons are vald. Frst, spectrum of the sgnal must be bandlmted, and samplng frequency must be twce hgher than the largest frequency present n the sgnal. The alas that occurs durng samplng stage s called prealasng and postalasng s caused by the reconstructon. Natural forms often contan dscontnuty, so ther spectrum s not bandlmted. Before samplng, lowpass flterng must be appled. If deal (box) lowpass flterng s performed, Gbss phenomena wll appear on each dscontnuty. So, dscontnuty creates unbandlmted spectrum. Therefore, deal lowpass flter, used to elmnate hgher frequences, cause rngng effect near dscontnuty. When dscontnuty exsts on pecewse lnear functon, Fourer seres of functon overshoots the functon value near that dscontnuty. Lmes ( f x ) lm S, the n-th partal sum Sn(f, x n ) of the Fourer seres on the frst local maxmum (mnmum) x n near dscontnuty converges to hgher (lower) value then the value of the functon. Wlbraham-Gbbs constant quantfes the degree of overshoot. On the each sde of dscontnuty the lmtng crest of hghest wave converges to 8,949% of the dscontnuty heght. Ths s nherted property that should be taken nto the further consderaton. In the two-dmensons rngng exhbts on every dscontnuty n gray levels of the mage. In the three-dmenson, volume elements escape over the edge of the object and create vsual artfacts that manfest as clouds around the object. Some volume elements dve nto the object creatng caves n the object surface. To avod rngng, contnuous mpulse response of the lowpass flter s requred. Instead of box lowpass flterng, flters that have smooth mpulse response should be used. For lowpass flterng n two or three dmensons Gaussan flter wll be used, although further detaled nvestgaton s requred. Data acquston can be acheved by dfferent scanners: CT (Computer Tomography) or MR (Magnetc Resonance), for example. Durng the samplng process some lowpass flterng s performed, but nformaton about t for sequences of slces avalable on Internet, s usually unknown. If the samplng s not done correctly, nformaton can be rrecoverably lost. The resoluton of scanned slces s usually hgh, but number of slces s often nsuffcent because of radaton rsks for patent. To enlarge the number of slces, nterpolaton between the slces s requred. Compresson of the volume data s also desred n n n of
because the sze of dataset s large. Thus the reconstructon of the compressed volume, nterpolaton between slces, or nterpolaton of the volume elements becomes mportant step. The reconstructon s term that s usually used n sgnal processng, and nterpolaton s term used n mathematcs or computer graphcs. In ths paper those two terms wll be used nterchangeably. Both approaches: one from the nterpolaton of curves and the other from sgnal reconstructon, wll be confronted n order to analyze the problem. 4 The B-splne nterpolaton When desgnng the curves and surfaces for CAD applcatons some characterstc demands on the behavor of the curves and surfaces are requred (Farn, 990, Gud 990). The B- splne was created to fulfll certan requrements that wll reflect very well n solvng of our problem. 4. The approxmaton B-splne The approxmaton B-splne curve wth degree k of each polynomal segment s defned wth p( u) = r N, r n = 0 k ( u), () where are ponts of the control polygon, and N k, are called B-splne weght functons, or B-splnes. The control polygon has n+ control ponts. The N k, are defned based on knot sequence { U u, u,... u knot = 0 wth recurson formula f u u< u+ No, ( u) =, 0 else N k, m }, (2) ( u u ) N, k ( u) ( u+ k+ u) N+, k ( u) ( u) = +. (4) u u u u + k + k+ + When denomnator s equal to zero, fracton s assumed to have value of zero. In our consderaton we restrct on the unform case, where parametrc ntervals between successve knot values are equal to one, and wth no multple knot values. U knot = { 23,,,..., m}, (5) In ths unform case, when k=3 (cubc case), formulaton of the -th B-splne segment s: 3 2 [ ] p ( u)= u u u u [ 0, ) 3 3 r r 3 6 3 0, (6) 6 3 0 3 0 r + 4 0 r + 2 where. For ths unform cubc case derved from () and (4), four ponts control each segment. The curve p ( ) u wll approxmate the control polygon. Boundary condtons can be handled by usng closed curves or crcular repetton of the control ponts, by zero paddng, or by settng some end condtons. For the sake of the (3)
smplcty crcular repetton of the control ponts wll be appled. The derved form (6) s dentcal to the cubc BC-splne derved by Mtchell (Mtchell 988), by settng B=, C=0: 3 2 3x 6x + 4 3 2 k( x)= x + 6x 2x + 8 6 0 for x < for x < 2 (7) else Wth few arthmetc manpulatons and reparameterzaton we can prove that (6) and (7) represents the same reconstructon form. It s obvous that the reconstructon n two dmensons (e.g. mages) approxmaton splne causes blur, because smaller or greater values only approxmate gray levels of the mage. In spte of that fact, many authors use BC-splne defned by Mtchell (Lacroute 995, Bentum 996, Moller 997). In the three dmensons fne detals are lost, and surface s smooth. The nterpolaton, as opposte to approxmaton of the control ponts, wll sgnfcantly mprove the resultng mage. The propertes of the B-splne curves or surfaces extend n the mage or volume reconstructon very well. These propertes are contnuty, convex hull, local control, varaton dmnshng and representaton of the multple values. The convex hull property ensures that each pont n the curve les n the convex hull of no more than k+ nearby control ponts. Thus, sample ponts bound the space of the reconstructed curve, surface or volume, so reconstructed values wll not escape outsde the convex hull. The local control property makes far ponts less nfluental on the segment of consderaton. In the terms of sgnal processng the local control property mples narrow mpulse response of the reconstructon flter. The mpulse response of deal reconstructon flter s snc functon, whch s very wde. The ponts far from the pont of reconstructon can have undesred nfluence. Varaton dmnshng property prevents varatons of the curve, or varatons n the gray levels of the reconstructed mage. The curve s not ntersected by any straght lne (or plane) more often than the control polygon. For a cubc case, control polygon conssts of the four control ponts and there are at most three ntersectons between straght lne and curve. Ths property s very mportant n the mage reconstructon, because the human eye s very senstve on small changes of the ntensty, especally n the areas where gray levels are changng smoothly. 4.2 The nterpolaton B-splne To buld the nterpolaton B-splne t s crucal to fnd the control polygon of the approxmaton B-splne, such that the resultant curve passes through the requested ponts. For the cubc unform closed curve, the matrx form defnes ponts of the control polygon: p0 p... pn pn 4 0... 0 0 r0 4... 0 0 0 r =............ 6 0 0 0... 4 rn 2 0 0... 0 4 rn 2 where p are the known ponts that must be nterpolated, and r s the unknown pont of the control polygon. Evaluaton of the nverse trdagonal matrx or the LU-decomposton can be used to fnd r. (8)
5 Results Sx dfferent reconstructon flters n the two and three-dmensonal space are 25 used. Appled reconstructons are: wth snc functon, nearest neghbor, approxmaton 20 snc B-splne, three-lnear nterpolaton, near Mtchell reconstructon wth BC-splne B= 5 B-apr C= /3 and nterpolaton B-splne. bln Dfferences between the ntal object and 0 Mtch the resultng objects are made and varances B-nt are calculated (Fg. 2). Reconstructon wth 5 snc functon exhbts strong rngng artfact, although mean square error s 0 mnmal. The best result s acheved for the Varance nterpolaton B-splne. Fg. 2. Sx dfferent reconstructon methods. The two-dmensonal examples (Fg. 3) and the three-dmensonal examples (Fg. 4) of the sx reconstructon methods are presented. For the three-dmensonal example the test functon proposed by Marshner and Lobb (Marshner 994) s used. Fg 3. Reconstructon wth snc, nearest neghbor, approxmaton B-splne, blnear nterpolaton, Mtchell reconstructon and nterpolaton B-splne. Acknowledgment Ths work has been carred out wthn the project 03604 Problem-Solvng Envronments n Engneerng, funded by Mnstry of Scence and Technology of the Republc of Croata.
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